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Shape Statistics via Skeletal Structures
by
Mohsen Taheri Shalmani
Thesis submitted in fulfilment of
the requirements for the degree of
PHILOSOPHIAE DOCTOR
(PhD)
Faculty of Science and Technology
Department of Mathematics and Physics
2024
University of Stavanger
NO-4036 Stavanger
NORWAY
www.uis.no
©2024 Mohsen Taheri Shalmani
ISBN: 978-82-8439-261-5
ISSN: 1890-1387
PhD: Thesis UiS No. 781
In memory of Prof. James Damon
iv
Acknowledgments
I wish to convey my deepest gratitude and heartfelt appreciation to my
supervisors, Drs. Jörn Schulz, Stephen M. Pizer, Jan Terje Kvaløy,
and Guido Alves, for their invaluable guidance, unwavering support,
and expertise that have been indispensable throughout my doctoral
journey. I am also grateful to our institute leader, Dr. Bjørn Henrik
Auestad, for his unwavering support throughout this endeavor. My
appreciation extends to Drs. James Damon, J. S. Marron, Sigbjørn
Hervik, and Tore Selland Kleppe for their guidance and enlightening
scientific discussions. I would like to express my profound thanks to
my family and friends for their care and support.
I would also like to thank the friendly people of Stavanger who were
beside me during challenging times, whether by studying alongside me
in libraries, exercising with me at workout centers, or by accompanying
me in exploring the beautiful nature of Stavanger.
Mohsen Taheri Shalmani
Stavanger, April 2024
v
Abstract
Statistical shape analysis has emerged as a crucial tool for medical
researchers and clinicians to study medical objects such as brain
subcortical structures. The insights gained from such analyses hold
immense potential for diagnoses and enhancing our understanding of
various diseases, particularly neurological disorders.
This thesis explores three important areas of statistical shape
analysis, which are detailed in three separate papers: “Statistical
Analysis of Locally Parameterized Shapes,” “Fitting Discrete Swept
Skeletal Structures to Slabular Objects,” and “The Mean Shape under
the Relative Curvature Condition.” The innovative approaches
discussed in these papers offer a fresh perspective for representing
complex shapes, enabling more nuanced analysis and interpretation.
Central to this work is the discussion surrounding the introduction of
robust skeletal representations for establishing correspondences for a
class of swept regions called slabular objects and providing proper
mathematical methodologies supporting the statistical objectives such
as hypothesis testing and classification. The proposed skeletal models
are alignment-independent and invariant to the act of Euclidean
similarity transformations of translation, rotations, and scaling.
Damon’s criterion of the relative curvature condition (RCC) is an
essential factor for valid swept skeletal structures. This work extensively
discusses fitting skeletal models, defining shape space, and calculating
the mean shape for such models following the RCC.
The efficacy of the proposed methodology is underscored through
rigorous examinations, both visually and statistically. These
methodologies are specifically applied to medical contexts, focusing
on analyzing subcortical structures. Synthetic and actual datasets
serve for validation, facilitating a comprehensive comparison with
existing skeletal representations. This work highlights the resilience
and adaptability of innovative approaches, paving the way for further
medical research and diagnostic endeavors.
vii
List of papers
Paper I
Taheri, Mohsen, and Jörn Schulz (2022). “Statistical Analysis of
Locally Parameterized Shapes." Journal of Computational and
Graphical Statistics 32.2 (2023): 658-670.
Paper II
Taheri, Mohsen, Stephen M. Pizer, and Jörn Schulz (2023). “Fitting
the Discrete Swept Skeletal Representation to Slabular Object."
Submitted for publication in Journal of Mathematical Imaging and
Vision.
Paper III
Taheri, Mohsen, Stephen M. Pizer, and Jörn Schulz (2024). “The
Mean Shape under the Relative Curvature Condition." Submitted for
publication in Journal of Computational and Graphical Statistics
The three papers mentioned above utilize data from The Norwegian
ParkWest Study (Alves et al.,2009), a prospective population-based,
longitudinal cohort study of patients with incident Parkinson’s disease
in Western and Southern Norway in collaboration with The Stavanger
University Hospital.
ix
Abbreviations
Abbreviation Meaning
CMS Central Medial Skeleton
DSRep Discrete Skeletal Representation
DSSRep Discrete Swept Skeletal Representation
EDM Euclidean Distance Matrix
ETRep Elliptical Tube Representation
ETube Elliptical Tube
FDR False Discovery Rate
GC Generalized Cylinder
GOP Geometric Object Property
IDSREP Intermediate Discrete Skeletal Representation
LPDSRep Locally Parameterized Discrete Skeletal
Representation
LPDSSRep Locally Parameterized Discrete Swept Skeletal
Representation
MRep Medial Representation
MRI Magnetic Resonance Imaging
PD Parkinson’s Disease
PDM Point Distribution Model
PCA Principal Component Analysis
PGA Principal Geodesic Analysis
PNS Principal Nested Sphere
RCC Relative Curvature Condition
SlO Slabular Object
SPHARM-PDM Spherical Harmonic Point Distribution Model
SRep Skeletal Representation
xi
xii
Table of Contents
Acknowledgments .................................................... v
Abstract ............................................................... vi
List of papers ......................................................... viii
Abbreviations ......................................................... x
1 Introduction ....................................................... 1
1.1 Background and motivation ............................... 1
1.2 Skeletal structures.......................................... 5
2 Shape statistics of skeletal structures ........................... 11
2.1 Shape and shape space .................................... 11
2.2 Shape analysis .............................................. 13
3 Summary of the papers .......................................... 15
3.1 Paper I ...................................................... 15
3.2 Paper II ..................................................... 16
3.3 Paper III .................................................... 17
4 Discussion ........................................................ 19
4.1 Validity of the deformation-based DSRep fitting ........ 19
4.2 Intermediate DSRep ....................................... 19
4.3 CMS of a surface .......................................... 21
4.4 DSRep of objects with branching structures ............. 26
4.5 ETRep space as (S5)𝑛+1................................... 27
xiii
References ............................................................ 29
Papers
Paper I - Statistical Analysis of Locally Parameterized Shapes . . 36
Paper II - Fitting the Discrete Swept Skeletal Representation to
Slabular Objects .................................................. 63
Paper III - The Mean Shape under the Relative Curvature Con-
dition .............................................................. 103
Introduction
1 Introduction
1.1 Background and motivation
Over the years, scientists have endeavored to investigate the shapes and
formations of biological organisms. Among the early pioneers in this
field were the renowned Italian scholars Leonardo da Vinci (1452 -
1519) and Galileo Galilei (1564 - 1642). Leonardo documented the
anatomy of the human body and recorded his observations of diseases,
with a particular focus on conditions such as cardiovascular and mus-
culoskeletal diseases. He also tried to articulate the ideal proportions
of the human body reflected in his well-known artwork, the “Vitruvian
Man” (Oksanish,2019). Additionally, Galileo made a significant dis-
covery regarding the animals’ skeletons. He realized the skeletons of
large animals not only exhibit size variations compared to smaller ones
but also distinct shape differences. Bones in larger animals become
proportionally thicker to support their heavier weight, while bones in
smaller animals remain thinner (Dryden and Mardia,1998).
Despite the efforts of Leonardo and Galileo, we can attribute the
foundation of shape analysis to the Scottish biologist and mathematician
D’Arcy Wentworth Thompson (1860 - 1948). Thompson’s expertise in
biology led him to focus on morphogenesis, which refers to the de-
velopment of tissues and the comparative analysis of shapes across
various cells, organs, plants, and animals. His objective was to identify
similarities and differences in the appearance of living organisms via
mathematical approaches. In 1917, Thompson published the book “On
Growth and Form” (Thompson,1992), an influential work that gained
significant attention for its mathematical analysis of biological struc-
tures. For instance, the book explored a potential similarity between
two distinct fish species, Argyropelecus Olfersi versus Sternoptyx Di-
aphana. This suggests that Sternoptyx Diaphana can be regarded as
a moderately altered version of Argyropelecus Olfersi (or vice versa)
through a linear transformation.
1
Introduction
Coinciding with the emergence of shape analysis, mathematicians
such as Francis Galton (1822 - 1911) and Karl Pearson (1857 - 1936) pi-
oneered modern statistics and statistical methodologies. These method-
ologies were developed to rigorously study the information of a set of
entities. The primary goals of statistical analysis include collecting,
organizing, and summarizing data, which refers to a set of values of
qualitative or quantitative variables about one or more objects (OECD,
2008). Within this framework, statistical shape analysis refers to em-
ploying statistical methods to study the shapes of a group of objects1,
where the shape of an object encompasses all geometric information
that remains unchanged under the act of Euclidean similarity transfor-
mations of scaling, translation, and rotation (Kendall,1977;Dryden
and Mardia,1998;Lele and Richtsmeier,2001).
The emergence of computer science has revolutionized statistical
shape analysis, making it an essential tool for medical researchers and
clinicians. By leveraging advanced image analysis techniques, they can
gain valuable insights into patient data and improve medical outcomes
(Pennec et al.,2019;Marron and Dryden,2021). In fact, statistical
shape analysis offers valuable insights by facilitating the detection of
differences between samples of human organs. For instance, it enables
the comparison of the subcortical structures (like the hippocampi) of
patients with neurodegenerative disorders with those of a healthy control
group (Apostolova et al.,2012;Schulz et al.,2016).
A variety of human body parts like the kidney, mandible, and most
brain subcortical structures such as the caudate nucleus and hippocam-
pus are slab-shaped elliptical objects, namely slabular objects (SlOs)
(Pizer et al.,2022). Given the significant value of SlO analysis, partic-
ularly in the diagnosis and prognosis of neurological disorders, shape
statistics of SlOs have remained a focal point of research for decades.
Such analysis, alongside clinical assessments and genetic testing, is vital
in diagnosing disorders at early stages. Early diagnosis and prediction
1This work considers a set in R2,3as an object if it is homeomorphic to a closed ball
(Siddiqi and Pizer,2008).
2
Introduction
of the disorders’ behaviors leads to initiation of treatment at the outset,
reducing healthcare costs and improving patient quality of life.
Figure 1.1: Similarity between a caudate nucleus as an SlO (left) and an eccentric
ellipsoid (right). Grey disks represent the sweep of slicing planes along the objects’
center curves.
An SlO can be seen as a swept region with a center curve and
a sweep of slicing planes along the center curve such that the slicing
planes do not intersect within the object (Pizer et al.,2022;Taheri
and Schulz,2022). An SlO is analogous to an eccentric ellipsoid in
the sense that it has a crest corresponding to the crest of the ellipsoid
(i.e., the intersection of the ellipsoid’s first principal plane with its
boundary) and two vertices corresponding to the ellipsoid’s vertices
(i.e., the endpoints of the ellipsoid’s major axis) (Pizer et al.,2022;
Taheri et al.,2023). Figure 1.1 illustrates the mentioned similarities
between a caudate nucleus as an SlO and an eccentric ellipsoid.
Numerous techniques and approaches are available for SlO analy-
sis. For example, landmark-based analysis (Dryden and Mardia,2016),
the analysis of point distribution models (PDMs) based on spherical
harmonic PDM (SPHARM-PDM) (Styner et al.,2006) or elastic regis-
tration (Jermyn et al.,2017), persistent homology methods (Gamble and
Heo,2010;Turner et al.,2014), volumetric analysis of 3D objects (Ma-
lykhin et al.,2007), radial distance analysis (Apostolova et al.,2012),
etc. However, these techniques are basically alignment-dependent as
they aim to represent objects within a global coordinate system. Thus,
alignment (or superimposing) serves as a prerequisite for them. They
often follow the idea of Kendall (1984) to translate, rotate, and scale the
models to minimize their global differences prior to the analysis.
For instance, in PDM analysis, we present an SlO as a set of points
3
Introduction
distributed on the object’s boundary. Thus, an SlO can be seen as a
vector with 3𝑛elements as 𝒗=vec(𝒑1, ..., 𝒑𝑛), where 𝒑1, ..., 𝒑𝑛∈R3
are the distributed points on the boundary and vec is the vectorization
operator. To remove rigid transformations, we can translate and scale
the object’s representation so that the centroid of the points is located
at the origin of the coordinate system, and the size of the represen-
tation becomes equal to 1.0. Thus, by considering 1
𝑛Í𝑛
𝑖=1𝒑𝑖=0the
normalized vector 𝒖=𝒗
∥𝒗∥represents the object as a point on the unit
hypersphere S3𝑛−1(known as Kendall’s pre-shape space). Thus, having
a population of PDMs (associated with a population of SlOs), we have
a distribution of points on the hypersphere, where the distance between
the spherical points is the geodesic distance 𝑑𝑔(𝒙,𝒚)=cos−1(𝒚𝑇𝒙)
(Jung et al.,2012). To remove the effects of rotation, we can rotate
the PDMs around their centroids. This process can be carried out by
Procrustes analysis in a way that optimizes the sum of squared geodesic
distances between the corresponding points on the hypersphere, as com-
prehensively discussed by Dryden and Mardia (2016). Hence, we align
the PDMs using the Procrustes analysis and assume that alignment
removes the effect of Euclidean similarity transformations from the
PDMs.
Let 𝒖1, ..., 𝒖𝑁∈S3𝑛−1representing a set of aligned PDMs. The
shape statistics, including the mean shape and shape variation, can be
computed based on the distribution of 𝒖1, ..., 𝒖𝑁. In particular, the
mean shape can be considered as the Fréchet mean of the distribution
(Fréchet,1948;Pennec et al.,2019) as
¯
𝒖=argmin
𝒖∈S𝑛−1
𝑁
𝑗=1
𝑑2
𝑔(𝒖,𝒖𝑗).(1.1)
Nevertheless, such alignment-dependent analyses are misleading and bi-
ased, as explored by Lele and Richtsmeier (2001). Paper I of this thesis
(Taheri and Schulz,2022) offers examples to illustrate that alignment-
dependent methods usually struggle to detect differences, even between
very simple objects. In some sense, alignment is like the elephant
4
Introduction
in the room in the realm of shape analysis. It is a significant issue
that researchers frequently prefer to overlook. This tendency for imple-
menting alignment-dependent approaches may stem from the perceived
simplicity of their implementation. Further, SlOs as swept regions pos-
sess unique properties that are often ignored by many of the mentioned
approaches. For example, it can be shown that the mean shape of a set
of swept regions calculated via Procrustes analysis may not necessarily
represent a swept region or even an object homeomorphic to a closed
ball (see Supplementary Materials of Paper III). This topic is thoroughly
discussed in Paper III of this work (Taheri et al.,2024).
This work follows the principle of invariance of Berger (1985)
to provide SlO analysis in accordance with the definition of shape
considering the shape or structure of an organism is invariant to rigid
transformations. Such shape analysis methods can be established based
on the skeletal structure of objects (Blum,1973;Pizer et al.,1999;
Damon,2003;Fletcher et al.,2004;Siddiqi and Pizer,2008).
Skeletal structures provide an appropriate foundation to define ro-
bust invariant shape models, which are highly effective in identifying
local dissimilarities based on diverse deformations such as protrusions,
bending, shrinkage, elongation, twisting, and more (Pizer et al.,2022;
Taheri and Schulz,2022). Therefore, the objective of this work is to
define and employ robust SlO analysis based on skeletal structures. The
following section offers a simple and casual overview of the skeletal
structures of SlO and statistical concepts pertinent to the three papers
of this thesis.
1.2 Skeletal structures
This section presents an overview of skeletal models and definitions
studied in the three papers.
We realize the skeletal structure of an object Ωas a radial vector
field 𝑈defined on the object’s skeleton 𝑀denoted by (𝑀 , 𝑈), where
5
Introduction
𝑀in the generic case2is a Whitney stratified set as a union of disjoint
smooth strata (Damon,2003). Intuitively, 𝑀can be seen as a disjoint
union of a set of locally centered smooth manifolds obtained by the
process of continuous contraction of the boundary components (Siddiqi
and Pizer,2008;Bærentzen and Rotenberg,2021). The radial vector
field defines a flow from the skeleton to the boundary, which is similar
to the inverse of Blum’s grassfire flow (Blum et al.,1967). Thus, given
the skeletal structure (𝑀 , 𝑈)the boundary of Ωcan be defined as
𝜕Ω = {𝒑+𝑈(𝒑)| 𝒑∈𝑀}.
Therefore, Blum’s skeletal structure can be defined as (𝑀⊙, 𝑈⊙),
where 𝑀⊙is the medial skeleton (or medial axis), and 𝑈⊙is the Blum
radial vector field such that at each point of 𝑀⊙there exist at least two
radial vectors with equal lengths tangent to the boundary. Since each
radial vector defines a straight path from the skeleton to the boundary,
the medial skeleton 𝑀⊙can be seen as the locus of centers of inscribed
spheres bi-tangent or multi-tangent to the boundary as
𝑀⊙={𝒑∈Ω𝑖𝑛 |
{𝒒∈𝜕Ω| ∥ 𝒑−𝒒∥=𝑑𝑚 𝑖𝑛 (𝒑, 𝜕Ω) }
𝑐≥2},(1.2)
where Ω𝑖𝑛 denotes the interior of Ω,𝑑𝑚𝑖 𝑛 (𝒑, 𝜕Ω)is the minimum
Euclidean distance between point 𝒑and the object boundary 𝜕Ω,∥.∥
is the Euclidean norm, and |.|𝑐represents the cardinality of a set.
The primary key for statistical shape analysis is establishing a
meaningful correspondence (Van Kaick et al.,2011) across a popula-
tion of objects based on similar geometrical properties (Laga et al.,
2019). The medial skeleton is usually bushy and highly sensitive to
boundary noise. That is, even a small protrusion or intrusion can signif-
icantly alter the medial skeleton. As a result, it is difficult to establish a
meaningful correspondence using Blum’s skeletal structure. Addition-
ally, skeletal models based on the medial skeleton can be vastly different
even for very similar objects, which may lead to a large number of false
positives and make the analysis less informative.
2In simple terms, a property of a geometric object is considered generic, if it persists
under “almost all” small changes of the entity (Siddiqi and Pizer,2008)
6
Introduction
As discussed by (Giblin and Kimia,2003,2004) and (Damon,
2003), based on Blum radial vector field, at each smooth point of the
medial skeleton 𝒑∈𝑀⊙, there exist two radial vectors originating from
𝒑as 𝑈1
⊙(𝒑)and 𝑈2
⊙(𝒑), where ∥𝑈1
⊙(𝒑)∥ =∥𝑈2
⊙(𝒑)∥ and the vector
𝑈1
⊙(𝒑) − 𝑈2
⊙(𝒑)is orthogonal to the tangent plane 𝑇𝒑(𝑀⊙). Thus,
𝑈1
⊙(𝒑)and 𝑈2
⊙(𝒑)are perfectly symmetric relative to 𝑇𝒑(𝑀⊙).
Pizer et al. (1999) showed by slightly relaxing the medial skeleton,
it is possible to define a relaxed skeletal structure with a fixed branch-
ing topology suitable for correspondence establishment. He called such
quasi-medial skeletal models as medial representations (MRep) (Siddiqi
and Pizer,2008). An MRep is a field of non-crossing vectors called
medial spokes such that spokes with common tail positions have equal
lengths and the envelope of all spokes’ tips called the implied boundary
approximates the object boundary (Joshi et al.,2001,2002). The ro-
bustness of MRep in statistical shape analysis of subcortical structures
is explored in various studies such as the principal geodesic analysis
(PGA) of Fletcher et al. (2003,2004).
Pizer’s efforts inspired mathematician James Damon (1945-2022)
to propose an explicit mathematical approach for relaxing Blum’s cri-
teria. Eventually, Damon (2003) achieved success in relaxing Blum’s
criteria by defining his three conditions: 1. Radial curvature condition
(𝑟 < min{1
𝜅𝑟𝑖 }for all positive principal radial curvatures 𝜅𝑟𝑖 , where 𝑟is
the magnitude of the radial vector), 2. Edge condition (𝑟 < min{1
𝜅𝐸𝑖 }
for all positive principal edge curvatures 𝜅𝐸𝑖 ), and 3. Compatibility
condition (1-form 𝜂𝑈≡0for all singular points of 𝑀including its
edge points). The first two conditions control the local behavior of
the radial flow, ensuring the singularities do not develop from smooth
points. Thus, the level sets of the flow project diffeomorphically from
the skeleton to the boundary (and vice versa). The third condition
ensures that only when the flow reaches the boundary, all singularities
simultaneously disappear. Intuitively, the flow preserves the singulari-
ties until it reaches the boundary.
Based on Damon’s three conditions, the radial vectors originating
7
Introduction
from the same point do not need to be symmetric with equal lengths.
Later Damon (2008) combined his three conditions and defined the
relative curvature condition (RCC) for swept regions (as 𝑟 < 1
𝜅𝑟𝑒 𝑙 where
𝜅𝑟𝑒𝑙 is the principal relative curvature) (Ma et al.,2018). The RCC
regulates the behavior of the slicing planes established along the center
curve of a swept region, preventing them from intersecting with each
other within the object. Damon (2008) also defined the swept skeletal
structures of swept regions based on the RCC.
Assume Γas the center curve of a swept region Ω. Let Γ(𝑡)be the
curve length parameterization of Γsuch that 𝑡∈ [0,1], where Γ(0,1)
denotes the curves’ endpoints. Further, let Π(𝑡)be the slicing plane
crossing Γ(𝑡), and let Ω(𝑡)= Π(𝑡)∩Ωbe the cross-section at Γ(𝑡). The
skeletal structure (𝑀 , 𝑈)is a swept skeletal structure if it satisfies the
RCC and for each 𝒑∈Ω(𝑡)∩𝑀, the 𝑈(𝒑)∈Ω(𝑡). In this sense, a 3D
swept region (like a generalized cylinder or an SlO) is an object with a
swept skeletal structure defined based on a smooth sequence of affine
slicing planes along its center curve such that cross-sections do not
intersect within the object, and each cross-section is a two-dimensional
object with a skeletal structure. The union of the cross-sections’ skeletal
structures forms the swept skeletal structure of the object (Taheri and
Schulz,2022;Taheri et al.,2023).
Damon’s contributions facilitated the development of shape models
suitable for establishing correspondence. Pizer et al. (2013) defined a
skeletal structure called the skeletal representation (SRep) for an SlO.
Analogous to an MRep, an SRep defines the skeletal structure for an SlO
such that the SlO’s skeleton is a smooth two-dimensional topological
disc called skeletal sheet and the radial vector field is represented by a
field on non-intersecting vectors called skeletal spokes emanating from
the skeletal sheet. In discrete form, we call an SRep a discrete SRep
(DSRep) (Tu et al.,2016). The SRep can be seen as a penalized version
of an MRep because the skeletal spokes with common tail positions may
have non-identical lengths. The DSRep is an appropriate representation
of SlO for establishing correspondence and conducting statistical shape
8
Introduction
analysis due to its non-branching structure (Pennec et al.,2019).
Figure 1.2: Illustration of a DSSRep of a hippocampus (left). The vectors are skeletal
spokes representing the radial vectors 𝑈, while the blue surface denotes the skeletal
sheet representing the object’s skeleton 𝑀. In the right figure, slicing planes have been
incorporated into the model to demonstrate the coplanarity of the skeletal spokes
associated with each slicing plane.
An SlO can be conceptualized as a swept region where each cross-
section resembles a 2D generalized cylinder (GC), defined as a two-
dimensional swept region with a smooth center curve. In this sense,
an SlO is a swept region with a swept skeletal structure formed by
the swept skeletal structures of the cross-sections. Paper II of this
work discusses the discrete swept skeletal representation (DSSRep) for
SlOs. The DSSRep is defined similarly to a DSRep, with the difference
that skeletal spokes associated with each slicing plane are coplanar,
as defined by Damon (2008). Also, the edge of the skeletal sheet
coincides with the SlO’s crest by considering the center curve of each
cross-section as a smooth curve that connects the cross-section’s two
vertices (analogous to the chordal locus of Brady and Asada (1984)
for 2D GCs). Figure 1.2 illustrates a DSSRep of a hippocampus where
orange vectors are skeletal spokes representing the radial vectors and the
blue surface is the skeletal sheet representing the object’s skeleton. The
slicing planes sweep the object’s boundary and skeleton. Notice that the
skeletal spokes associated with each slicing plane are coplanar.
Within a certain category of SlOs, the cross-sections can be ap-
proximated by elliptical disks. Consequently, it becomes feasible to
streamline the shape representation and portray the object as a 3D gen-
eralized cylinder with elliptical cross-sections. Thus, the coplanar radial
9
Introduction
Figure 1.3: An ETRep of a hippocampus where the cross-sections are elliptical disks.
vectors can be seen as the chordal structure of an ellipse, as defined
by Brady and Asada (1984). This simplification aids in reducing the
dimensionality of the shape representation, thereby addressing the is-
sue of the curse of dimensionality. Additionally, it assists in mitigating
the problem of self-intersection when calculating the mean shape and
defining the shape space. In this regard, Paper III of this work intro-
duces elliptical tube representation (ETRep) as a simplified version of
the DSSRep for a class of SlOs. Figure 1.3 illustrates an ETRep of a
hippocampus.
10
Shape statistics of skeletal structures
2 Shape statistics of skeletal structures
Once a skeletal model is established, defining shape and shape space
concepts is essential for calculating the mean shape and shape distribu-
tions, which are crucial for hypothesis testing and classifications. The
following sections Section 2.1 and Section 2.2 discuss the shape, shape
space, and shape statistics of the introduced skeletal models.
2.1 Shape and shape space
This section first explains the shape and shape space of DSReps and
then discusses the shape and shape space of ETReps. All discussions
regarding DSReps are also valid for DSSReps.
As discussed in Section 1.2, a DSRep is a finite subset of an
SRep, where the SRep is a field of skeletal spokes emanating from the
skeletal sheet defined by the local geometry of the skeletal structure.
Therefore, in a simple format, a DSrep can be seen as an 𝑛-tuple of
spokes like 𝑠=(𝑠𝑖)𝑛
𝑖=1where 𝑠𝑖=(𝒑𝑖,𝒖𝑖,𝒓𝑖)is the 𝑖th spoke, and 𝒑𝑖,
𝒖𝑖and 𝑟𝑖are the tail position, unit direction and the length of the 𝑠𝑖,
respectively. Thus, a DSRep is living on a Cartesian product space of
(R3)𝑛× (S2)𝑛× ( R+)𝑛, where (R3)𝑛is the space of spokes’ tail positions,
the (S2)𝑛is the space of spokes’ directions based on the unit sphere
S2, and (R+)𝑛is the space of spokes’ lengths (Pizer et al.,2020).
However, such shape representation is alignment-dependent be-
cause the positional components and directional components of the
spokes are in the global coordinate system. Therefore, such a param-
eterization is not ideal for statistical analysis. As proposed in Paper I
and Paper II of this study, we can reparameterize the conventional rep-
resentation of the DSRep and define the locally parameterized DSRep
(LPDSRep). As depicted in Figure 2.1, an LPDSRep defines a tree-like
structure for the skeletal sheet equipped with a set of local frames.
The tree-like structure establishes a hierarchical connection among the
frames through a collection of vectors called connection vectors. In
11
Shape statistics of skeletal structures
Figure 2.1: Hierarchical structure of a skeletal sheet equipped with the local frames.
The blue vectors are connection vectors and the yellow vectors are the elements of
local frames.
this arrangement, each frame resides at the endpoint of a connection
vector, with a parent frame situated at the origin of the same vector.
Also, at each frame, we have two skeletal spokes, namely an up spoke
and a down spoke with opposite directions. Let 𝑛𝑓and 𝑛𝑐be the num-
ber of local frames and connection vectors, respectively, and assume
𝑖=1, . .., 𝑛 𝑓and 𝑗=1, ..., 𝑛𝑐. An LPDSRep can be expressed as
𝑠=(𝐹∗
1, ..., 𝐹∗
𝑛𝑓,𝒗∗
1, ..., 𝒗∗
𝑛𝑐,𝒖±∗
1, ..., 𝒖±∗
𝑛𝑓, 𝑣1, ..., 𝑣𝑛𝑐, 𝑟 ±
1, ..., 𝑟 ±
𝑛𝑓),(2.1)
where 𝐹∗
𝑖∈𝑆𝑂 (3)is the 𝑖th frame orientation based on its parent
coordinate system, 𝑆𝑂 (3)is the orthogonal rotation group, 𝒗∗
𝑗∈S2is
the direction of 𝑗th connection vector based on its local frame (i.e., the
frame that tail of the connection vector is located on), 𝑣𝑗∈R+is the
length of the 𝑗th connection vector, 𝒖±∗
𝑖∈S2are the directions of the
𝑖th up and down spokes based on their local frames, 𝑟±
𝑖∈R+are the
lengths of the up and down spokes at the 𝑖th frame. Thus, an LPDSRep
lives in the product space of
S=(𝑆𝑂 (3))𝑛𝑓× (S2)𝑛𝑐+2𝑛𝑓× (R+)𝑛𝑐+2𝑛𝑓.(2.2)
The LPDSRep constitutes an invariant shape representation as its com-
ponents are defined relative to the coordinate system of the local frames.
12
Shape statistics of skeletal structures
In essence, rigid transformations do not alter the representation. The
generalization of this approach is proposed by Pizer et al. (2022) based
on the level set of the radial flow. That is, the fitted frames are located
on the level sets of the flow corresponding to the configuration of the
skeletal sheet.
Let’s consider representing an SlO using an ETRep based on a
series of elliptical disks, as depicted in Figure 1.3. The combination of
the centroids of these disks delineates the object’s central curve, while
the union of their major axes constructs a developable skeletal sheet.
By considering a moving frame on the skeletal sheet along the center
curve, an ETRep can be defined as an invariant shape representation
based on 𝑖=1, ..., 𝑛 elliptical cross-sections as
𝑠𝑒=(𝐹∗
1, ..., 𝐹∗
𝑛, 𝑥1, ..., 𝑥𝑛, 𝑎1, ..., 𝑎𝑛, 𝑏1, ..., 𝑏𝑛),(2.3)
where 𝐹∗
𝑖is the orientation of the 𝑖th frame based on its previous frame
(i.e., (𝑖−1)th frame), 𝑥𝑖is the distance between the 𝑖th frame and its
previous frame, and 𝑎𝑖and 𝑏𝑖are the lengths of the semi-major and
semi-minor axes of the 𝑖th cross-section. Therefore, the ETRep space
is the product space (𝑆𝑂 (3))𝑛× (R3
+)3𝑛. The details of ETReps are
discussed in paper II.
2.2 Shape analysis
Section 2.1 discussed possible invariant shape representations for SlOs.
This section explains the statistical shape analysis based on the intro-
duced shape representations.
Assume two SlOs are represented by two LPDSReps as 𝑠1and
𝑠2with the same number of corresponding elements. We can define
the distance between 𝑠1and 𝑠2as defined by Equation 2 of Paper
I. Consequently, the mean shape can be considered as (the Fréchet
mean, i.e.,) a shape with minimum squared distance to 𝑠1and 𝑠2as
argmin𝑠∈SÍ2
𝑚=1𝑑2
𝑠(𝑠, 𝑠𝑚). Similarly, having a sample of LPDSReps as
13
Shape statistics of skeletal structures
𝑠1, ..., 𝑠𝑁, the sample mean shape can be defined as
¯𝑠=argmin
𝑠∈S
𝑁
𝑚=1
𝑑2
𝑠(𝑠, 𝑠𝑚).(2.4)
Note that we do not need to align the objects to calculate the mean
shape as the LPDSReps are alignment-independent.
Each element of an LPDSRep can be seen as a geometrical object
property (GOP) of the associated SlO (e.g., a spoke’s length represents
the object’s local width). Calculating the mean shape enables us to
compare two (or more) groups of LPDSReps to detect local dissimi-
larities by comparing their corresponding GOPs. This can be done by
designing partial hypothesis tests.
Let 𝐴={𝑠𝐴𝑚}𝑁1
𝑚=1and 𝐵={𝑠𝐵𝑚}𝑁2
𝑚=1be two groups of Eu-
clideanized LPDSReps of sizes 𝑁1and 𝑁2such that all the LPDSReps
have the same number of GOPs. To compare corresponding GOPs, we
can design partial tests as 𝐻0𝑖:¯𝑠𝐴(𝑖)=¯𝑠𝐵(𝑖)versus 𝐻1𝑖:¯𝑠𝐴(𝑖)≠¯𝑠𝐵(𝑖),
where ¯𝑠𝐴(𝑖)and ¯𝑠𝐵(𝑖)are the 𝑖th GOP of the observed sample mean
of 𝐴and 𝐵, respectively.
Further, since local frames can be seen as unit quaternion vectors
belonging to S3, we can Euclideanize and vectorize LPDSReps such
that each LPDSRep can be seen as a vector in a feature space, which is
a high-dimensional Euclidean space. The Euclideanization process can
be executed using principal nested spheres analysis (PNS) of Jung et al.
(2012). Consequently, besides hypothesis testing, diverse classification
techniques can be employed to classify LPDSReps within the high
dimensional feature space, as discussed in Paper I & II.
However, determining the mean shape and shape space by con-
sidering Damon’s criteria presents a fundamental challenge. In fact,
without considering the RCC for the underlying shape space, the ob-
tained mean shape by Equation (2.4) is not necessarily an SlO, as it
may have self-intersections. Paper III of this work devotes exclusive
attention to discussing the mean shape and shape space of SlOs, taking
into account the RCC.
14
Summary of the papers
3 Summary of the papers
3.1 Paper I
Taheri, Mohsen, and Jörn Schulz (2022). “Statistical Analysis of Lo-
cally Parameterized Shapes." Journal of Computational and Graph-
ical Statistics 32.2 (2023): 658-670.
The focus of the paper is to introduce an invariant skeletal shape
representation for SlO analysis capable of explicitly explaining types
of dissimilarities. The paper is divided into five sections. In the first
section, the authors comprehensively discuss critical issues related to
alignment-dependent methods such as point distribution analysis like
SPHARM-PDM (Styner et al.,2006) and explain why available invari-
ant methods, such as the Euclidean distance matrix (EDM) analysis
(Lele and Richtsmeier,2001), cannot reflect the type of dissimilari-
ties. The core section of the paper is the second section. The section
reviews the basic terms regarding DSReps and discusses the conven-
tional noninvariant parameterization of DSReps. It also introduces the
LPDSRep as a novel parameterization of a DSRep. The LPDSRep
represents an SlO based on a hierarchical tree-like structure of the
skeletal sheet equipped with the local frames, as defined at the begin-
ning of Section 2.1. The section also explains the Euclideanization
of spherical data, shape distance, mean shape, object deformation, and
boundary reconstruction utilizing the LPDSRep. In the third section,
the authors discuss hypothesis testing and controlling false positives
regarding LPDSRep analysis. The hypothesis pipeline is demonstrated
in the paper’s supplementary materials. The fourth section evaluates
the LPDSRep and proposed methodologies based on simulation and
the analysis of real data. For the real data analysis, the section studies
hippocampal differences between a group of patients with Parkinson’s
disease (PD) versus a control group (CG). Finally, the fifth section
summarizes and concludes the paper.
15
Summary of the papers
3.2 Paper II
Taheri, Mohsen, Stephen M. Pizer, and Jörn Schulz (2023). “Fit-
ting the Discrete Swept Skeletal Representation to Slabular Object."
Submitted for publication in Journal of Mathematical Imaging and
Vision.
The objective of the paper is to introduce a model fitting procedure
for DSSReps based on the swept skeletal structure of SlOs. The paper
consists of nine sections. The first section discusses the paper’s objec-
tive and explains the relationship between the swept skeletal structures
of an eccentric ellipsoid and SlOs. It also highlights the importance of
the RCC in the model fitting and serious issues in applying the curve
skeleton (Dey and Sun,2006) to define the center curve of a swept
region. Section two provides explicit definitions of swept regions,
SlO, skeletal structures, and swept skeletal structures and reviews basic
terms. In section three, the central medial skeleton (CMS) is intro-
duced. The CMS can be seen as a unique subset of the medial skeleton
that has no holes, discontinuity, or branches. The CMS of an SlO can
be calculated by dividing the Voronoi diagram (Attali and Montanvert,
1997) of the SlO based on the crest of the object. Section four pro-
poses a DSSRep model fitting based on the flattened CMS. The section
discusses CMS flattening based on dimensionality reduction methods,
such as principal component analysis and t-distributed stochastic neigh-
bor embedding (t-SNE) (Van Der Maaten,2014). Sections five and six
parameterize the DSSRep to introduce the LPDSSRep as an invariant
shape representation. They also discuss the goodness of fit based on
skeletal symmetry, tidiness, and the volume of the implied boundary
of a fitted model. In section seven, LPDSSRep hypothesis testing and
classification are demonstrated. Section eight compares the proposed
LPDSSRep and LPDSRep models, where the LPDSRep models are
based on the DSRep model fitting of Liu et al. (2021). The section
discusses that Liu’s method has critical issues in defining correspon-
dence, as it suffers from skeletal perturbation and asymmetricity. The
final section summarizes and concludes the paper.
16
Summary of the papers
3.3 Paper III
Taheri, Mohsen, Stephen M. Pizer, and Jörn Schulz (2024). “The
Mean Shape under the Relative Curvature Condition." Submitted
for publication in Journal of Computational and Graphical Statis-
tics.
The paper focuses on calculating the mean shape of SlO under
the relative curvature condition (RCC) across six sections. The first
section highlights the importance of RCC in defining the shape space
of swept regions and how it ensures that adjacent slicing planes do not
intersect within the object. The second section defines basic terms and
definitions regarding the swept skeletal structures of SlOs. The third
section explains the discrete material frames and Frenet frames along
the center curve of an SlO, which are crucial for defining the RCC. The
fourth section is the main core of the paper. It introduces the ETRep
and demonstrates the ETRep model fitting. An ETRep represents an
SlO as a combination of elliptical cross-sections (see Figure 1.3). The
section then discusses the intrinsic and semi-intrinsic approaches to
define the ETRep space by considering the RCC for ETReps. The
intrinsic approach defines the ETRep space as a high-dimensional space
with a hyperbolic boundary and a swept skeletal coordinate system.
The fifth section discusses statistical inferences derived from ETRep
hypothesis testing. Finally, the sixth section summarizes and concludes
the work.
In addition, the paper’s supplementary materials offer an algo-
rithm to address non-local self-intersection of swept regions. This is
important for future research since the RCC only addresses local self-
intersection.
17
Summary of the papers
18
Discussion
4 Discussion
The following sections investigate the strengths and weaknesses of the
methods suggested in this work’s three papers and explore the potential
directions for future research.
4.1 Validity of the deformation-based DSRep fitting
The initial paper in this study introduced LPDSRep, which is founded
upon the DSRep model as proposed by Liu et al. (2021). Damon
(2021) elucidated that a generic medial skeletal structure exhibits a
level of rigidity conducive to maintaining correspondence through mi-
nor boundary deformations. Consequently, if we can achieve an SIO by
slightly deforming an eccentric ellipsoid, the skeletal structure of the
deformed ellipsoid can be considered as the skeletal structure of the
SlO. While the subsequent paper highlighted critical issues with Liu’s
approach, it can be shown that the theoretical discussions of Liu’s ap-
proach hold validity. Thus, the theoretical frameworks delineated in the
first paper, along with their extension posited by Pizer et al. (2022), re-
main theoretically valid. In fact, the primary drawback of Liu’s method
stems from overlooking the crest and vertices of an SIO. Consequently,
after the deformation, we lose the correspondence between the crest
and vertices of the deformed ellipsoid and those of the original ellip-
soid. Therefore, addressing this concern could lead to the acquisition
of a DSRep characterized by elevated symmetricity and tidiness, which
could be the subject of future research.
4.2 Intermediate DSRep
The establishment of skeletal structures between two or more SlOs
presents an intriguing prospect, particularly because the intermediate
area between subcortical objects mirrors the geometric properties of the
brain’s white matter. As defined by Damon and Gasparovic (2017), such
intermediate skeletal structure can be defined based on the extension
19
Discussion
of the objects’ radial flows. That is, the intersection of the radial flows
outside the objects defines an intermediate skeleton. Liu et al. (2023)
proposed a model fitting of such intermediate skeletal structure based
on the extension of the skeletal spokes for well-posed SlOs, namely
intermediate DSREP (IDSRep). Well-posed SlOs are objects where
their skeletal sheets can be regarded as approximately parallel sheets.
The critical issue with Liu’s approach is that the intermediate area
between objects is not well-defined, as it is not bounded, leading to an
unbounded IDSRep where the skeleton extends infinitely. Additionally,
the approach fails to explicitly address the self-linking issue. The self-
linking arises when an object’s radial flow intersects with itself before
reaching other objects’ flows. Furthermore, well-posed SlOs are rare
occurrences.
Figure 4.1: Left: The IDSRep of two ellipsoids. Right: The IDSRep of a caudate and
putamen obtained by deforming the IDSRep of their corresponding ellipsoids.
Nonetheless, defining a well-defined bounded intermediate area
between any two ellipsoids is usually possible based on their convex
hull region as suggested by Damon and Gasparovic (2017). Using
this well-defined bounded region, we can define bounded IDSRep for
any arbitrary ellipsoids. Leveraging the rigidity of skeletal structures
discussed by Damon (2021), it appears feasible to define IDSReps for
ellipsoids and then slightly deform the space using a diffeomorphic
mapping (such as thin-plate splines) to fit the IDSRep between the
SlOs. Figure 4.1 illustrates a prototype IDSRep between a caudate and
20
Discussion
putamen (which are not well-posed objects) obtained by deforming the
intermediate space of their corresponding ellipsoids. This approach
avoids the self-linking issue as it is based on a diffeomorphic mapping.
Further, it is possible to parameterize the IDSRep locally (analogous to
the LPDSRep) so that it becomes alignment-independent.
4.3 CMS of a surface
Paper II used the DSSRep model to fit the skeletal sheet by flattening
it and defining its center curve (Taheri et al.,2023). The center curve
was then projected onto the skeletal sheet and considered as the center
curve of the SlO called spine. This method works well for regular SlOs
where the flattened sheet is similar to the original sheet. However, for
irregular SlOs, flattening the skeletal sheet is significantly different from
its flattened version. Thus, the spine obtained via surface flattening
might be an inappropriate representative of the center curve of the
irregular SlO. One alternative is to calculate the spine directly on the
skeletal sheet without flattening it.
Similar to the definition of the medial skeleton, a point can be
considered as a medial point on a surface if there are two shortest
straight geodesic paths that connect the point to the surface edge. This
concept is discussed by Chambers et al. (2013). We know the skeletal
sheet of an SIO is a surface that can be represented as a triangle surface.
Since straight paths on a triangle surface can be calculated (Polthier and
Schmies,1998), it seems possible to calculate the medial skeleton of the
skeletal sheet and consider such a skeleton as the spine of the object.
This section proposes a possible method that can be used to define the
spine without flattening the skeletal sheet based on the generalization
of the pruning methods of Foskey et al. (2003); Chazal and Lieutier
(2005). To achieve this, the section introduces the urchin.
As illustrated in Figure 4.2, an urchin with the core 𝒑∈Ω𝑖𝑛 is
a set of all spokes emanating from 𝒑as S( 𝒑)={𝒔(𝒑,𝒖)|𝒖∈S𝑑−1}.
We consider a discrete urchin as ˆ
S( 𝒑)={𝒔(𝒑,𝒖)|𝒖∈ˆ
S𝑑−1}, where
21
Discussion
ˆ
S𝑑−1represents the vertices of a discrete unit sphere as a symmetric
geodesic polyhedron (for example, an icosahedron sphere) in 3D or a
regular polygon in 2D.
Figure 4.2: Visualization of an urchin in a 3D object (left) and a 2D object (right).
Based on the Tomson (1904) problem, we can assume identical
angles between all adjacent spokes of an urchin. Let 𝑛ˆ
S=
ˆ
S( 𝒑)
𝑐be
the urchin’s number of spokes, and Δ𝜃∈ (0, 𝜋]be the angle between
two adjacent spokes of ˆ
S( 𝒑). Obviously, if Δ𝜃→0then 𝑛ˆ
S→ ∞ and
ˆ
S( 𝒑) → S( 𝒑), i.e., ˆ
S( 𝒑)includes all the possible straight paths (i.e.,
spokes) from 𝒑to 𝜕Ω.
Theoretically, 𝒑∈Ω𝑖𝑛 is a medial point if the shortest straight path
from 𝒑to 𝜕Ωis not unique. That is, at least two shortest straight
paths have equal lengths but different directions, i.e., ∃𝒖1≠𝒖2such
that ∥𝒔(𝒑,𝒖1)∥=∥𝒔(𝒑,𝒖2)∥=𝑑𝑚𝑖𝑛 (𝒑, 𝜕Ω). Thus, to check whether 𝒑is
a medial point or not, it is sufficient to check whether the two shortest
spokes of ˆ
S( 𝒑)have an equal length or not, given a sufficiently small
Δ𝜃. However, in practice, even if a point is indeed medial, almost
always all the urchin’s spokes have different lengths because of noise,
computation errors, and the limitation of choosing Δ𝜃small enough.
Further, the probability of picking a medial point by chance from Ωis
very low because the space that the medial skeleton occupies relative
to the object volume is close to zero. However, following the idea of
Foskey et al. (2003), it is possible to approximate the medial skeleton
based on the object angle (i.e., the angle between spokes with a common
tail position) Attali and Montanvert (1997).
Let the minimal angle 𝛿ˆ
S( 𝒑)∈ [0, 𝜋]be the angle between the
22
Discussion
two shortest spokes of ˆ
S( 𝒑). Assume 𝒑as a semi-medial point if
𝛿ˆ
S( 𝒑)>Δ𝜃, and otherwise, as is a nonsemi-medial point (i.e., 𝛿ˆ
S( 𝒑)=
Δ𝜃). Note that when 𝛿ˆ
S( 𝒑)is close to 𝜋, the point is located at the
middle of two approximately parallel surfaces as discussed by Giblin
and Kimia (2003). The semi-medial skeleton can be considered as the
union of all semi-medial points as 𝑀Δ𝜃
⊙={𝒑∈Ω𝑖𝑛 |𝛿ˆ
S( 𝒑)>Δ𝜃}.
Analogous to Theorem 4 of (Foskey et al.,2003), it can be shown
that the difference between the semi-medial axis and the medial axis
vanishes when Δ𝜃→0.
Theorem 1 For an object Ω⊂R𝑑=2,3with smooth boundary, the semi-
medial skeleton 𝑀Δ𝜃
⊙converges to the medial skeleton 𝑀⊙when Δ𝜃→0.
Proof: Recall the semi-medial skeleton of the object Ωis the collection
of all semi-medial points as 𝑀Δ𝜃
⊙={𝒑∈Ω𝑖𝑛 |𝛿ˆ
S( 𝒑)>Δ𝜃}, where
𝛿ˆ
S( 𝒑)∈ [0, 𝜋]is the angle between the two shortest spokes of the
discrete urchin ˆ
S( 𝒑). For any 𝒑∈𝑀Δ𝜃
⊙,∃𝒖1,𝒖2∈S𝑑−1such that
𝑑𝑔(𝒖1,𝒖2)>Δ𝜃, and 𝒔(𝒑,𝒖1)and 𝒔(𝒑,𝒖2)are the two shortest spokes.
If Δ𝜃→0, without loss of generality, we can assume ∥𝒔(𝒑,𝒖1)∥=
𝑑𝑚𝑖𝑛 (𝒑, 𝜕Ω)as ˆ
S( 𝒑)consists of all possible spokes with tail at 𝒑.
Thus, 𝒔(𝒑,𝒖1)is normal to 𝜕Ωbecause 𝒑is the center of an inscribed
sphere tangent to 𝜕Ω. Let 𝒒be the tip of 𝒔(𝒑,𝒖1). The 𝜕Ωhas positive,
negative, or zero curvature at 𝒒. Also, the positive curvature at 𝒒
cannot be greater than 1
∥𝒔(𝒑,𝒖1)∥, because the osculating sphere tangent
to 𝜕Ωat 𝒒with positive curvature includes the maximal inscribed
sphere with tangency point 𝒒(see Appendix of (Attali et al.,2007)
and (Siddiqi and Pizer,2008, Page. 41)). Thus, if we (infinitesimally)
move away from 𝒒to 𝒒′∈𝜕Ω, the length of the spoke with tail at
𝒑and tip at 𝒒′become greater or equal than ∥𝒔(𝒑,𝒖1)∥. Therefore,
∀𝒖∈ˆ
S𝑑−1:∥𝒔(𝒑,𝒖1)∥ ≤ ∥𝒔(𝒑,𝒖2)∥ ≤ ∥ 𝒔(𝒑,𝒖)∥. Let 𝒔(𝒑,𝒖)be an adjacent
spoke to 𝒔(𝒑,𝒖1)(e.g., 𝒖=(0,sin (Δ𝜃),cos (Δ𝜃))𝑇when 𝒖1=(0,0,1)𝑇
is the north pole of the unit sphere). Thus, ∥𝒔(𝒑,𝒖)∥→∥𝒔(𝒑,𝒖1)∥when
Δ𝜃→0and consequently based on the squeeze theorem ∥𝒔(𝒑,𝒖1)∥ →
∥𝒔(𝒑,𝒖2)∥. So the two shortest spokes would have equal lengths with
different directions as 𝑑𝑔(𝒖1,𝒖2)>0. Therefore, 𝒑∈𝑀⊙. On the
23
Discussion
other hand, if 𝒑∉𝑀Δ𝜃
⊙, then 𝑑𝑔(𝒖1,𝒖2)= Δ𝜃, i.e., the two shortest
spokes are adjacent. So when Δ𝜃→0, the two shortest spokes literally
coincide. Thus, the shortest path to the boundary is unique and 𝒑∉𝑀⊙.
□
Based on Theorem 1, by choosing a small enough Δ𝜃the semi-
medial skeleton approximates the medial skeleton. The resolution of
the semi-medial skeleton depends on two factors the number of urchins
and the value of Δ𝜃, i.e., more urchins and smaller Δ𝜃result in
a semi-medial skeleton closer to the actual medial skeleton. Recall
∀𝒑∈Ω𝑖𝑛 :𝛿ˆ
S( 𝒑)∈ [Δ𝜃, 𝜋]. Let 𝒔(𝒑,𝒖1)and 𝒔(𝒑,𝒖2)be the two shortest
spokes of ˆ
S( 𝒑), and consider Δ𝜃=𝜋
𝑛+1such that 𝑛∈N. There-
fore, we can categorize all the internal points into 𝑛semi-medial levels
𝐿𝑖={𝒑∈Ω𝑖𝑛 |𝛿ˆ
S( 𝒑)≥ (𝑖+1)Δ𝜃}, where 𝑖=0, ..., 𝑛 −1. Therefore,
𝐿1represents the semi-medial skeleton (i.e., the union of all interior
points that the angle between their two shortest spokes is greater than
Δ𝜃). Figure 4.3 depicts highest to lowest semi-medial levels of a 2D
object by Δ𝜃=𝜋
8.
Figure 4.3: Semi-medial skeleton levels of a 2D object.
Analogously, we can determine a non-linear urchin on a surface by
assuming its non-linear spokes as straight geodesic paths with starting
directions on the surface. Further, as Chambers et al. (2013) discussed,
for a medial point on a surface, there exist two shortest straight geodesic
paths connecting the point to the edge of the surface with different
starting directions. Thus, as depicted in Figure 4.4, we can define the
semi-medial skeleton for surfaces based on the non-linear urchins. In
this sense, the core of a non-linear urchin is a medial point if the angle
between the two shortest non-linear spokes of the urchin is greater than
24
Discussion
Δ𝜃, where Δ𝜃∈ (0, 𝜋]is the angle between the starting directions of
two adjacent non-linear spokes based on the tangent space of the core.
The detailed discussion of this approach is out of the scope of this work
and could be the subject of future research. For example, the surface
must be well-behaved in the sense that all the spokes of any non-linear
urchin must have finite lengths and not cross each other.
Figure 4.4: A non-linear urchin on a surface (Left) with semi-medial levels (Right).
Let 𝒑be a semi-medial point, and let 𝒔(𝒑,𝒖1)and 𝒔(𝒑,𝒖2)be its two
shortest non-linear spokes. Analogous to the boundary of a 2D object,
the edge of a connected surface can be divided into two components
as two non-overlapping curves such that the union of the curves forms
the edge. We say 𝒑belongs to the CMS if the tip of the two shortest
non-linear spokes is at different edge components. Thus, the CMS is a
subset of the semi-medial skeleton based on the two edge components
(i.e., the tips of the two shortest non-linear spokes are not at the same
edge component). Figure 4.5 depicts the semi-medial skeleton of a
surface (middle) and the CMS based on two edge components (right).
Figure 4.5: Left: A nonlinear urchin on the surface. Middle: The semi-medial
skeleton of a surface. Right: The CMS of the surface based on two edge components
that are depicted as blue and red curves separated by two bold dots.
25
Discussion
4.4 DSRep of objects with branching structures
Fitting skeletal structures to complex slab-shaped objects such as hemi-
mandibles with the coronoid process that are not necessarily SlOs is
particularly relevant given the frequent focus on such objects in research
studies (AlHadidi et al.,2012). Although such slab-shaped objects are
not necessarily SlOs, they can be divided into several sub-objects such
that each sub-object can be seen as an SlO. In this sense, the skeletal
structure of the object can be seen as the union of the skeletal structures
of its sub-objects.
Figure 4.6: Left: The skeletal structure of the flattened skeletal sheet of a
hemimandible. The six vertices with the highest local curvatures are depicted as bold
dots. The skeletal structure of each sub-object is depicted by the color of its associated
boundary components. Right: The DSRep of the hemimandible.
There are available approaches for obtaining 3D sub-objects such
as local separators of Bærentzen and Rotenberg (2021); Bærentzen et al.
(2023). However, such approaches usually ignore important boundary
properties such as the crest, leading to inappropriate boundary division
(see Supplementary Materials of Paper II). Paper II determined the
skeletal structure of an SlO using the skeletal structure of its flattened
skeletal sheet, where the skeletal sheet is the relaxed CMS delineated
by the object’s crest. It appears that for a class of slab-shaped objects,
such as a hemimandible, the flattened skeletal sheet has multiple distinct
vertices with the highest local curvature that can be used to divide it
into several 2D sub-objects. The combination of the skeletal structures
26
Discussion
of these sub-objects can be considered as the skeletal structure of the
flattened sheet. As a result, the projection of this skeletal structure
onto the skeletal sheet creates a tree-like structure that can be utilized
to determine the skeletal structure of the object. This approach has
shown promising results in initial implementations and can be a subject
of future research. Figure 4.6 (left) shows the skeletal structure of
the flattened skeletal sheet of a hemimandible as a union of three
skeletal structures of three 2D sub-objects depicted in different colors.
Figure 4.6 (right) illustrates the DSRep of the hemimandible obtained
by projecting the skeletal structure of the flattened sheet to the relaxed
CMS of the object. Note that it is also conceivable that we apply the
methodology discussed in Section 4.3 to calculate the skeletal structure
of the relaxed CMS without flattening it.
4.5 ETRep space as (S5)𝑛+1
Paper III introduces A†𝑛+1=(S3× (R+)3)𝑛+1as a convex space of
ETReps based on the ETRep representation as a sequence of ellipti-
cal cross-sections as 𝑠†=(𝜔𝑖)𝑛
𝑖=0, where 𝜔𝑖=(𝒇𝑖, 𝑥𝑖, 𝜏𝑖, 𝜌𝑖)𝑖represents
the 𝑖th cross-section, the unit quaternion 𝒇𝑖∈S3is the orientation
of 𝜔𝑖,𝑥𝑖is the distance between the centroid of 𝜔𝑖and 𝜔𝑖−1, and
𝜏𝑖, 𝜌𝑖∈ [0,1]reflect the scaled size of 𝜔𝑖. Considering the robust per-
formance demonstrated by the principal nested sphere analysis (PNS)
in classification and hypothesis testing across different studies, e.g.,
(Schulz et al.,2016;Liu et al.,2023), analyzing ETReps based on
PNS could be advantageous. This can be done by mapping the cross-
sectional data to a hypersphere. Thus, the ETRep space becomes the
product space of 𝑛+1hyperspheres.
Given a finite set of scalars 𝜙1, ...., 𝜙𝑚−1∈ [0, 𝜋], we can represent
these scalars as a vector 𝒙=(𝑥1, ..., 𝑥𝑚)′∈S𝑚−1using the generaliza-
27
Discussion
tion of the spherical coordinate system (Blumenson,1960)
𝑥1=cos 𝜙1
𝑥2=sin 𝜙1cos 𝜙2
𝑥3=sin 𝜙1sin 𝜙2cos 𝜙3
.
.
.
𝑥𝑚−1=sin 𝜙1... sin 𝜙𝑚−2cos 𝜙𝑚−1
𝑥𝑚=sin 𝜙1... sin 𝜙𝑚−2sin 𝜙𝑚−1.
(4.1)
Therefore, to apply PNS analysis, it is sufficient to map the elements of
𝜔𝑖to [0, 𝜋]. Since 𝒇𝑖∈S3is the unit quaternion representation of an
orthogonal frame, it can be represented as three Tait-Bryan angles (as
Euler angles) such as 𝜙1𝑖, 𝜙2𝑖, 𝜙3𝑖∈ [0, 𝜋](Hoffman et al.,1972;Murray
et al.,2017). Assume 𝜙4𝑖=2 tan−1(𝑥𝑖), and 𝜙5𝑖=𝜋𝜏𝑖and 𝜙6𝑖=𝜋𝜌𝑖.
Thus, we have 𝜙1𝑖, ..., 𝜙6𝑖∈ [0, 𝜋]. By considering Equation (4.1), the
𝜔𝑖can be represented as a vector in S5. Therefore, the ETRep space can
be seen as a poly-hypersphere (S5)𝑛+1. Consequently, with a population
of ETReps, the PNS can be applied 𝑛+1times on S5.
Moreover, Section 5 of Paper III explained how to map an ETRep
to R6(𝑛+1)based on ETRep vectorization. Let 𝒔be a vectorized ETRep.
Since 𝒔∈R6(𝑛+1), we have the normalized ETRep as 𝒔
∥𝒔∥∈S6𝑛+5. In
this sense, (analogous to Kendall’s pre-shape space) S6𝑛+5can be seen
as the shape space of normalized ETReps such that a normalized ETRep
is a point on the hypersphere. Thus, by having a population of ETReps,
one can directly use the PNS for statistical analysis on the hypersphere.
Although this method may result in strong statistical analysis, the main
challenge lies in defining an inverse mapping from S6𝑛+5to the ETRep
space A†𝑛+1, which can be explored in future studies.
28
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Paper I
Paper I
Statistical Analysis of Locally
Parameterized Shapes
36
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Statistical Analysis of Locally Parameterized
Shapes
Mohsen Taheri & Jörn Schulz
To cite this article: Mohsen Taheri & Jörn Schulz (2023) Statistical Analysis of Locally
Parameterized Shapes, Journal of Computational and Graphical Statistics, 32:2, 658-670, DOI:
10.1080/10618600.2022.2116445
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Paper I
37
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
2023, VOL. 32, NO. 2, 658–670
https://doi.org/10.1080/10618600.2022.2116445
Statistical Analysis of Locally Parameterized Shapes
Mohsen Taheriaand Jörn Schulzb
Department of Mathematics & Physics, University of Stavanger,Stavanger, Norway
ABSTRACT
In statistical shape analysis, the establishment of correspondence and dening shape representation are
crucial steps for hypothesis testing to detect and explain local dissimilarities between two groups of objects.
Most commonly used shape representations are based on object properties that are either extrinsic or
noninvariant to rigid transformation. Shape analysis based on noninvariant properties is biased because the
act of alignment is necessary, and shape analysis based on extrinsic properties could be misleading. Besides,
a mathematical explanation of the type of dissimilarity, for example, bending, twisting, stretching, etc., is
desirable. This work proposes a novel hierarchical shape representation based on invariant and intrinsic
properties to detect and explain locational dissimilarities by using local coordinate systems. The proposed
shape representation is also superior for shape deformation and simulation. The power of the method is
demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with
Parkinson’sdisease and controls. Supplementary materials for this article are available online.
ARTICLE HISTORY
Received August 2021
Accepted July 2022
KEYWORDS
Local coordinate system;
Local dissimilarity;
Parkinson’sdisease; Shape
alignment; Skeletal
representation; s-Rep
hypothesis testing
1. Introduction
In statistical shape analysis, detecting and characterizing loca-
tional dierences between two groups of objects is a matter of
special interest. For instance, in medical applications, analysis
of shape dissimilarities has the power to shed light on organ
deformations, supporting diagnosis and treatment.
Detecting locational dierences is a challenging task. For
decades, medical researchers have been trying to answer four
common questions w hen comparing a specic organ of a group
of patients versus a control group (CG). 1. Existence: Is there
any local dissimilarity? 2. Location: What is the location of the
dissimilarity? 3. Intensity: What is the size of the dissimilarity?
4. Type: What is the type or interpretation of the dissimilarity
(e.g., bending, twisting, or elongation)? Since a dissimilarity can
be seen as a distance between two entities, each shape analysis
method introduces distances between objects’ corresponding
parts (i.e., local dissimilarities) based on a specic shape repre-
sentation to answer these questions. The shape representation
could be invariant or noninvariant to object rigid transfor-
mation (i.e., translation and rotation). Therefore, as Lele and
Richtsmeier (2001)discussed,roughlywecancategorizeshape
analysis methods into alignment-independent and alignment-
dependent approaches that we call invariant and noninvariant
methods, respectively. Invariant methods use invariant shape
representations to follow the principle of invariance (Berger
1985) based on the fact that the true form of an organism does
not change if it translates or rotates. In contrast, noninvari-
ant methods follow the idea of Kendall (1977)tofactorout
translation, rotation,and (occasionally) scaling from noninvari-
ant shape representations by alignment. Usually, noninvariant
methods are more straightforward, faster, and provide a better
CONTACT Mohsen Taheri mohsen.taherishalmani@uis.no Department of Mathematics & Physics, University of Stavanger, Stavanger,4021, Norway.
Supplementary materials for this article are available online. Please go to www.tandfonline.com/r/JCGS.
intuition than invariantmetho ds,w hich explains their popular-
ity. In comparison, invariant methods are more reliable because
they are independent of choosing the alignment method or the
coordinate system.
In this work, we propose an invariant method equipped
with an invariant shape representation that benets from the
advantages of both types of methods. Further, it answers all
the four above questions in a single f ramework. For this, we
locally reparameterize a noninvariant skeletal representation (s-
rep) (Pizer et al. 2013) to an entirely invariant shape represen-
tation. To better understand and highlight the advantages of
our approach, rst we need to review ot her methods in more
detail.
Given two groups of objects, in the most common
approaches, whether invariant or noninvariant, researchers try
to answer the above questionsby hypothesis testing based on the
following steps. Step 1: Introduce shape representation as a tuple
of corresponding geometric object properties (GOPs) among
objects. Step 2: Dening a distance between the corresponding
GOPs of the two groups known as a test statistic representing
the local dissimilarity. Step 3: Measuring and analyzing the test
statistics to nd signicant GOPs. Step 4: Applying multiple
testing methods to control false positives.
A GOP can be a geometric or spatial fe ature (e.g., point’s
position, surface normal direction, Gaussian curvature, etc.), a
combination of features and their correlations (Tabia and Laga
2015), or more general a local descriptor as discussed in (Laga
et al. 2018,Ch.5).AGOPmayormaynotbeinvariantto
object translation and rotation. We call a shape re presentation
invariant if all of its GOPsare invariant, otherwise noninvariant.
© 2022 The Author(s). Published with license by Taylor& Francis Group, LLC.
This is an Open Access article distributed under the terms of the CreativeCommons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which
permits non-commercial re-use, distribution, and reproduction in any medium, providedthe original work is properly cited, and is not altered, transformed, or built upon in any way.
Paper I
38
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 659
Examples of noninvariant shape representations are the point
distribution model (PDM) and the discrete s-rep (ds-rep). A
PDM consists of an n-tuple of points (x1,...,xn),xi∈Rd
distributedonorinsidead-dimensional object where d=2,3
as comprehensively discussed in Srivastava and Klassen (2016),
Jermyn et al. (2017), Laga et al. (2018), and Dr yden and Mardia
(2016). Thus, the GOPs in a PDM are the points’ Cartesian
coordinates. A ds-rep (Pizer et al. 2013)consistsofatupleof
directions, tail positions and lengths of a set of internal vectors
and will be disc ussed in further detail in Sect ion2.1.Ads-repis
partly invariant as the vectors’lengths are invariant. An example
foraninvariantshaperepresentationistoconvertaPDMto
Euclidean distance matrix (EDM) representation as a tuple of
pairwise Euclidean distances of points (Lele and Richtsmeier
2001).
Having two groups of shape representations, we can dene
hypothesis tests based on the corresponding GOPs. In other
words, we simply test two groups of tuples element-wise. Note
that it is necessary to factor out translation and rotation from
noninvariant GOPs by alignment before the analysis. We say
the analysis is invariant if the shape representation is invari-
ant, otherwise it is noninvariant. For example, Styner et al.
(2006)andSchulz(2013) methods are noninvariant as Styner
et al. (2006) compared PDMs of brain objects of patients with
schizophrenia v.s.CG, and Schulz (2013)comparedtheobjects’
ds-reps. In contrast, Lele and Richtsmeier (1991) approach
is invariant as they used EDM representations to study skull
abnormality of patients with Crouzon and Apert syndromes. We
briey discuss both invariant and noninvariant met hods by an
example.
Figure 1(a) illustrates two ellipsoidal objectsasdenedin
Section 2.1.2, where the le one is an ellipse, and the right one
is a bent ellipse. The objects can be seen as an open arm (le
one) and a closed arm (right one), where each arm consists of
three parts, namely t he upper arm, elbow, and forearm. Since
the closed arm is a locally deformed version of the open arm,
we consider the main dierence at the elbow, which is com-
patible with our visual inspection. Both shapes are manually
registered with 20 corresponding boundary points depicted by
circles and crosses. By adding independent random noise to
each point, we simulated 15 PDMs for each object, as depicted
in Figure 1(b). Since a PDM is noninvariant, alignment is
necessary.
From several available alignment methods, we choose gener-
alized Procrustes analysis (GPA), weighted GPA (WGPA) (Dry-
den and Mardia 2016), and square root velocity framework
(SRVF) (Srivastava and Klassen 2016). Figure 1(c)–(e) illus-
trate alignments based on GPA, WGPA, and SRVF, respectively.
Apparently, there are two main issues. First, the outcomes of
dierent alignment methods are remarkably dierent as each
method tries to minimize a specic type of distance. Thus,
choosing the superior alignment method is challenging. S econd,
detecting locational dissimilarities could be extremely biased
because alignment aects the distributions of noninvariant
GOPs of points’ positions. As a result, PDM analysis introduces
false positives and false negatives. In Figure 1(c), WGPA (based
on a manually dened covariance matrix) reduces the variation
of forearm GOPs and increases the variation of upper arm
GOPs. Similarly, the right point at the elbow in Figure 1(e)
has a remarkably smaller GOP variation in comparison with
otherpoints.Basedonthesetypesofobservations,Leleand
Richtsmeier (2001) explained why noninvariant methods are
biased and why invariant methods are more reliable. However,
also for invariant methods a local dissimilarity can lead to
falsepositivesandfalsenegatives.Forinstance,ifweconvert
each PDM of our example to an invariant shape representa-
tion, where GOPs are invariant Euclidean distances between
points and the centroid of the PDM (i.e., center of gravity
¯
x=1
nn
i=1xidepicted by bold points in Figure 1(a)), then
almost all the GOPs in our example become signicantly dif-
ferent. Note, in this example, the GOPs are dened as extrin-
sic distances between the points and the extrinsic centroid.
If the centroid as well as the distances to the centroid are
dened intrinsic (e.g., by barycentroid (Rustamov, Lipman, and
Funkhouser 2009)), no dierences would be detected. To some
extent, the same discussion is valid for EDM analysis (EDMA)
as discussed in supplementary material (SUP). Besides, when
we consider only invariant GOPs, it is not always easy to map
various analysis results from the feature space to the object
space (Jermyn et al. 2017, 6). Consequently, some fundamental
aspects of shape analysis, such as mean shape, are unattainable.
For instance, it is easy to calculate the EDM of a point-based
model like a PDM, but it is dicult or sometimes impossible
toreconstructthemodelbasedonitsEDM.Wehavethesame
situation in persistent homology methods (Gamble and Heo
2010; Turner, Mukherjee, and Boyer 2014)wheretheinforma-
Figure 1. Problem of false positives due to alignment. (a) Twoellipsoidals are depicted by line and dashed line. Circles and crosses show corresponding boundary points.
Bold points are shapes’centroids. (b) Two populations of simulated PDMs. (c), (d), (e) Separation of correspondinglocal distributions.
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39
660 M. TAHERIAND J. SCHULZ
Figure 2. Skeletal structure of ellipsoidal objects. (a) 2D m-reps. sand sare corresponding spokes with unit directions uand u. (b) A ttedds-rep to a left hippocampus’s
mesh including up, down, crest spokes, and the skeletalsheet.
tion of the persistent diagram is not convertible to the object
space.
In summary, on the one hand, noninvariant methods are
biased due to alignment, and on the other hand, invariant
methods based on extrinsic properties could be misleading.
Thus, from our point of view, a suitable method should be
invariant, based on intrinsic object properties, ensure good
correspondence between the GOPs, and be able to answer the
fourth question, that is, to providea mathematical (and medical)
interpretation oft het ype ofdissimilarity such as bending, twist-
ing, stretching, protrusion, etc. For example, boundary PDMs
cannot explain the local bending in closed arms. In contrast,
askeletalmodel(seeFigure 2(a)) can explain the bending
mathematically, as we will discuss in Section 2.4.However,
the main obstacle in the skeletal analysis is the denition of
correspondence.
For a specic class of ellipsoidal objects, Pizer et al. (2013)
introduced s-rep and dened correspondence based on its dis-
crete version ds-rep (see Figure 2(b)). As pointed out above, ds-
reps are noninvariant and thus might bias t he analysis. Further,
ds-rep analysis is able to identify only a few types of dissim-
ilarities, for example, protrusion or bending. The identica-
tion of other types remains challenging. To overcome these
limitations, we propose a novel hierarchical ds-rep parame-
terization based on local coordinate systems known as local
frames. The proposed hierarchical local parameterization of
ds-rep, called LP-ds-rep, is an invariant shape representation
which supports sensitive hypothesis testing, that is, not biased
by alignment. Notethat the hierarchical structure equipped with
local frames can be modied and t to any kind of objects
(not only ellipsoidals) as long as a robust tree structure can be
established for the shape model. This is the subject of further
studies.
The article is structured as follows. In Section 2,werst
review basic notations and amenities of s-reps and discuss the
conventional noninvariant denition of ds-rep with the dis-
cussed challenges.Then, in Sections 2.1.3 and 2.1.4,wepropose
the novel LP-ds-rep parameterization. Further, we explain the
euclideaniz ation of spherical data, mean shape, t he transforma-
tion between two parameterizations, skeletal deformation, and
simulation. Section 3 introduces a hypothesis test method and
discusses controlling false positives. In Section 4 we study hip-
pocampal dierences between a group of patients with Parkin-
son’s disease and CG. Besides, we compare the results of both
parameterizations plus EDMA and show the advantages of our
method on simulated data. Finally, we summarize and conclude
the work in Section 5. A owchart depicting the framework of
the presented methods can be found in the SUP.
2. Skeletal Representation
To understand skeletal representation, we need to review some
fundamental denitions.
In this work, t he set ⊂Rdis a d-dimensional (or dD)
object if it is homeomorphic to the d-dimensional closed ball,
where d=2, 3. We denote the boundary and the interior of
,by∂ and in , respectively. Thus, =∂ ∪in.Also,
we consider only objects with smooth boundaries. Therefore,
∂ is a closed connected genus-zero smooth surface if d=3,
and it is a smooth closed c urve if d=2(Jermynetal.2017,
Ch.2). The medial locus of is a collection of entirely connected
curves or sheets in in forming the centers of all maximal
inscribed spheres bi-tangent or multi-tangent to ∂. We denote
the medial locus of by M.Theskeleton of is any curve or
sheet from which non-crossing spokes to ∂ emanate at each
point of it. Note that a spoke is a vector whose tail is on the
object’s skeleton, and its tip is on ∂. We consider a skeletal of
an object as a set of all non-crossing sp okes emanating from its
skeleton.Thus, the skeletal can be seenas a eld of spokes on the
skeleton. The medial locus is a form of skeleton where medial
spokes connecting the center of maximal inscribed spheres to
their tangenc y points. The union of the media l spokes forms the
medial skeletal (Siddiqi and Pizer 2008).
Medial re prese ntation (m-rep) and its properties have been
extensively studied in the literature (Pizer et al. 1999;Fletcher
et al. 2004; Siddiqi and Pizer 2008). Figure 2(a) illustratesm-reps
of two ellipsoidal objects. Briey, an m-rep is a discrete medial
skeletal (i.e., nite set of medial spokes). Thus, an m-rep reects
the interior object properties suchas lo cal widthsand directions.
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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 661
However, as pointed out in (Pizer et al. 2013), the m-rep is sensi-
tive to boundary noisebecause every protruding boundary kink
results in additional medial branches. This sensitivity aects m-
rep correspondence among a population as two versions of the
same objects can result in signicantly dierent m-reps. Thus,
Pizer et al. (2013) relaxed the medial conditions and dened s-
rep for a class of ellips oidal objects like hippoc ampus (discussed
in detail in Section 2.1.2) as a penal ized version of m-rep. As Liu
et al. (2021) described, the s-rep of 3D ellipsoidal object has
the form(M,S),whereskeletonM⊂in known as skeletal sheet
is a smooth 2-disk (i.e., an embedded, oriented two-dimensional
manifold of genus-ze ro with a single boundar y component), and
skeletal Sis the eld of noncrossing spokes on M.TheeldS
consists of three distinct elds of spokes: S0along M0where
M0is the boundary of M,S+(respectively, S−)denedonthe
relative interior of M, agreeing (respectively, disagreeing) with
the orientation of M.Thus,S+an d S−map M\M0to two sides
of ∂ considered as northern and southern part, and S0maps
M0to the crest of ∂.Wecallaspokesanup spoke ,down spoke,
or crest spoke ifit belongs to S+,S−,orS0, respectively.The same
denition is applicable for2D objects where Mis a smo oth open
curve. The relaxed conditions assure stability in the branching
structure and thus good case-to-case correspondence across a
population of s-reps. The ds-rep is a discrete form of s-rep (i.e.,
a nite set of spokes). The conventionalds-rep parameterization
is noninvariant as explained in more detail in Section 2.1.1.
Aerward, the propos ed invariant parameterization based on a
hierarchical structure of the local frames is introduced. Also, we
name the conventional parameterization as globally parameter-
ized ds-rep (GP-ds-rep), and the new parame terization as locally
parameterized ds-rep (LP-ds-rep). Further, sGP and sLP denote
GP-ds-rep and LP-ds-rep, respe ctively.
2.1. Parameterizations
2.1.1. GP-ds-rep
There are dierent ways to t and parameterize a GP-ds-rep.
Depending on the method of model tting, for example, (Liu
et al. 2021), some spokes may share a common tail position (see
Figure 2(b)). Let nsbe the numberof spokes, and npbe the num-
ber of tail positions s.t. np≤ns. A GP-ds-rep can be seen as a
tuple sGP =(pj,ui,ri)i,j=(p1,...,pnp,u1,...,uns,r1,...,rns)
where ∀j∈{1, ...,np}:pj∈R3is jth spoke’s tail position,
∀i∈{1, ...,ns}:ui∈S2,andri∈R+are ith spoke’s direction,
and length, respectively. Note Sd={x∈Rd+1|x=1}
is the unit d-sphere where d∈N. From now on we assume
i=1, ...,nsand j=1, ...,np.
The set {pj}np
j=1forms an np×3 conguration mat rix P
representing the skeletal PDM. Let Inpbe the np×npidentity
matrix and 1npbe the np×1 vector of ones. Location and scale
can be removed by centering and normalizing Pto obtain the
pre-shape ˜
P=CnpP
CnpP,whereCnp=Inp−1
np1np1T
npis the
centering matrix, and X=trace(XTX)is the Euclidean
norm. Since ˜
P=1, the pre-shape ˜
Plives on the hypersphere
S3np−1(Pizer et al. 2013). Therefore, a GP-ds-rep lives on a
manifold as a direct product of Riemannian symmetric spaces
S3np−1×(S2)ns×Rns+1
+where S3np−1indicates the pre-shape
space of the skeletal positions, (S2)nsis the space of nsspokes’
directions, and Rns+1
+is the space of spokes’ lengths and the
scaling factor. As we mentioned before, spoke positions and
directions are noninvariant as they are in a global coordinate
system (GCS). Thus, ds-rep analysis based on this representation
is biased.
For m-rep, a semi-local parameterization was proposed by
Fletcher, Lu,and Joshi (2003) based on local frames (n,b,b⊥)∈
SO(3),wherenis normal to the medial locus Mat p∈M,
b=u1+u2
u1+u2is the bisector direction of two equal-length
spokes with common position, b⊥=n×b,andSO(3)is
the 3D rotation group. Spokes’ directions are dened relative
to the local frames by the angle θ∈[0, π) between band
the spokes (see Figure 5(a)). Because the direction of band b⊥
depends on the spokes’ directions, if θ=π
2then btakes an
arbitrary direction that violates the uniqueness and consistency
of the tted frame. Besides, the spokes’ tail positions and frame
directions are noninvariantas they are in GCS.
Inspired by Cartan’s moving frames on space curves (Cartan
1937) and Fletcher’s semi-local parameterization, we propose a
fully local ds-rep parameterization. By uisng the inherent hier-
archical structure of ds-reps, we provide a consistent denition
of local frames independent of GCS that avoids arbitrary frame
rotation. This can be done by introducing a leaf-shaped skeletal
structure for ellipsoidal objects, that is, reected in (Liu et al.
2021)(seeFigure 6 on page 664). For this, we need to discuss
ellipsoidal objects.
2.1.2. Ellipsoidal Objects
Intuitively, an object is ellipsoidal if its skeletal structure corre-
sponds to the skeletal of an eccentric ellipsoid (i.e.,ellipsoid with
unequal principal radii).
Let E3⊂R3be a 3D eccentric ellipsoid. The medial locus
of E3is a 2D ellipsoid (i.e., an ellipse) E2⊂R2that we call
medial ellipse. The medial locus of E2is a 1D ellipsoid (i.e., a line
segment) E1⊂R1that we call medial line. The medial locus of
E1is a 0D ellipsoid (i.e., a point) E0⊂R0that we call medial
centroid.Thus,MEn=En−1is the medial locus of Enwhere
n=1, 2, 3. Analogous to backwardpr incipalcomponent analysis
(PCA) (Damon and Marron 2014), we consider E3,E2,E1,and
E0as four principal ellipsoids (see Figure 4(Le)).
We ca l l a 2 D o b j ect a perfect 2D-ellipsoidal if its medial locus
is a smooth open curve that we call medial curve (se e Figures 2a
and 3a). Let 2be a perfect 2D-ellipsoidal with medial locus
M2.SinceME2(i.e., medial line E1)isalsoasmoothopen
curve, we can dene correspondence between ME2and M2
basedon(SrivastavaandKlassen2016). We consider a point on
M2corresponding to E0as the medial centroid of 2.Letγ
represent the medial locus M2(or ME2) based on curve length
parameterization l. We know that for each medial point γ(l),
there are two media l spokes, one for each side of the medial
locus, with tail on γ(l)and tip γ±on the object boundary at
γ±=γ(l)−R(l)|d
dl R(l)|t±R(l)1−|d
dl R(l)|2n,(1)
where nand tare normal and tangent vectors of the medial
locus at γ(l),andRis the radius function such that R(l)is the
radius of the maximal inscribe sphere centered at γ(l)(Siddiqi
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41
662 M. TAHERIAND J. SCHULZ
Figure 3. (a) Illustration of a 2D-ellipsoidal (left) approximated by a perfect 2D-ellipsoidal (right). The solid curve and the bold dot (right) depict the medial curve and
medial centroid, respectively. (b) Left:A mandible (without coronoid processes) as an example of a 3D-ellipsoidal with slicing planes. The solid curve is the center curve.
Right: A cross-section as a 2D-ellipsoidal including its medial curve.
and Pizer 2008, Ch.2). Note that the two spokes at the edge (i.e.,
endpoints) of the medial locus coincide. Thus, in addition to
themediallocus,themedialskeletalof2corresponds to the
medial skeletal of E2.
We sa y a 2 D o b jec t ˆ
2is 2D-ellipsoidal if its boundary can
be precisely approximated1bytheboundaryofaperfect2D-
ellipsoidal 2. Following m-rep idea of Pizer et al. (1999), it is
reasonable to consider the skeletal of ˆ
2as the skeletal of 2
to have a better correspondence among a population. Thus, we
assume M2astheskeletonof ˆ
2as depicted in Figure 3(a).
In 3D, we dene 3D-ellipsoidal analogous to generalized oset
surface.
Damon (2008) dened generalized oset surfaces as 3D
objects similar to generalized tubes2basedonsequencesofane
slicing planes (not necessarily parallel) such that the cross-
sections of a generalized oset surface (i.e., the intersection of
the slicing planes with the object) do not intersect within the
object, and theboundary of the cross-sections forms the object’s
boundary. The skeleton of a generalized tube is a smooth curve,
and the skeleton of a generalized oset surface is a smooth
two-disk. In practice, we can represent a generalized tube or
an oset surface by a nite but large number of disj oint cross-
sections. Similarly, we say an object is 3D-ellipsoidal if it can
be represented by a large number ofdisjoint cross-sections such
that all the cross-sections are 2D-ellipsoidals, and the length of
acurvecalledthecenter curve connecting the medial centroids
of the successive cross-sections is remarkably larger than the
length of themedial curve of each cross-section. Sincethe union
of the medial curves can be seen as a discrete skeletal sheet,
Pizer et al. (2013) realized such 3D-ellipsoidals as slabular3
and introduced (slabular) ds-reps such that for a slabular, the
implied boundary of its ds-rep (i.e., envelope of the spokes’
tips) approximates the slabular’s boundary. Examples of 3D-
ellipsoidals are mandible (without considering the coronoid
processes), caudate nucleus, kidney, and hippocampus. Fig-
ure 3(b) illustrates the center curve and the slicing planes of
a mandible as a 3D-ellipsoidal (le) and a cross-section as a
2D-ellipsoidal (right).
Any eccentric ellipsoid E3canbeseenasa3D-ellipsoidal
such that parallel cross-sections are perpendicular to the center
1The required energy to deform one object to the other one is negligible, for
example, see Sorkine (2006).
2Tuberefers to a 3D object made by a sweeping disk such that its medial locus
is a smooth curve.
3Slab refers to a 3D object such that its medial locus is a sheet.
curve (i.e., the major axis of E2). In this sense, a meaningful
correspondence4between the skeletal of a 3D-ellipsoidal and
skeletal of E3is assumable as the skeleton of both of them
consists of a center curve, a set of medial curves emanating
from the center curves, and two spokes at each point of the
medial curves pointing toward two sides of the skeletal sheet
based on Equation (1). However, obtaining such correspon-
dence is dicult as it is challenging to dene corresponding
cross-sections for a population of c-shape objects, for example,
asetofhippocampi.
A possible approach for dening a skeletal sheet of a 3D-
ellipsoidal is to understand the object via a dieomorphism
from a reference 3D-ellipsoidal such as E3. Assume 3be a
3D-ellipsoidal. Since E3as a reference object is a 3D-ellipsoidal
and a meaningful correspondence between 3and E3is assum-
able, Liu et al. (2021) dened a (more or less) dieomorphic
transformation F:3→E3based on stratied mean
curvature ow (MCF). The transformation provides a boundary
registration b etween E3and 3.Then,theyappliedinverse
transformation F−1(based on the obtained registration and
inverse MCF) to deform E3and its interior (i.e., skeletal) to 3.
Aer deformation (i.e., F−1:E3→3), E2transforms to a
nonlinear surface Mthatcanbeseenasa2-disk.Consequently,
straight lines on E2(e.g., medial line and medial spokes) become
curves. Since we assumed a dieomorphic transformation, the
generated curves do not cross each other. We call the deformed
medial line F−1(E1)the spine, and deformed medial spokes
veins. Thus, veins are a set of noncrossing cur ves emanating
from the spine. Also, we assume the displaced medial centroid
F−1(E0)as an intrinsic centroid, and call it skeletal centroid
or s-centroid.Thus,Mhas curvilinear skeletal corresponding
to the medial skeletal of E2.Figure 4 provides an intuition
about the ellips oid’s media l locus deformation. Final ly, Liu e t al.
(2021) generated non-crossing spokes ont he skeletalsheet such
that the implied boundary approximates ∂3.Thegenerated
spokes represent a s-rep as a eld of noncrossing spokes on the
skeletal sheet.
Although weapply the method of Liu et al. (2021), web elieve
it is possible to improve the model tting in many aspects such
as a better boundary registration based on Jermyn et al. (2017)
that we leave for future studies.
4See VanKaick et al. (2011) for a comprehensive discussion about meaningful
correspondence.
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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 663
Figure 4. Skeletal sheet. Left: Ellipsoid’smedial locus. Right: s-rep skeletal sheet of a 3D- ellipsoidal.
Figure 5. Illustration of a local frame. nis normal to tangent planes Tp(M)and Tp(M).(a)s1and s2are equal-length spokes with unit directions u1and u2,andb=
u1+u2
u1+u2(b) cis a smooth curve on M.−p
1and p
2are the projection of p1and p2on Tp(M).ˆ
v
1=p−p
1
p−p
1,ˆ
v
2=p
2−p
p
2−p,andb=ˆ
v
2+ˆ
v
1
ˆ
v
2+ˆ
v
1.
2.1.3. Local Frames
Basedonthedenedcurvesonthes-repskeletalsheetofellip-
soidals, we can t local frames. Let c⊂Mbe a smooth curve in
R3. We consider b∈Tp(M)as the unit velocity vector tangent
to cwhere Tp(M)is the local tangent plane of Mat p∈cwith
normal n. The local frame can be dened as (n,b,b⊥)∈SO(3)
where b⊥=n×b(see Figure 5(b)). The unit vector bchooses
two opposite directions depending on the denitionof the cur ve
starting and ending p oints. Besides, the frame directions are
noninvariant. To have a consistent invariant frame denition,
we design a hierarchical structure. Then on the basis of the
structure, we dene consistent tted frames in a population of
GP-ds-reps.
Consider the principal ellipsoids. Similar to Blum’s grassre
ow (Blum 1967), we can say each point on ∂E2moves to
reach the medial line E1, and then moves to reach the medial
centroid E0. Thus, for each boundary point there is a path
from the point to the medial centroid. In discrete format, each
path can be represented by a nite set of consequent points
sorted based on the distance they travel to reach the medial
centroid. Imagine two consequent points on the same path. We
consider the point that takes the shorter route as the parent,
and the other one as the child.Therefore,likeaspanningtree,
each point (except medial centroid) has a parent but may have
multiple children (see Figure 6(Top)). Similarly, based on the
correspondence between the E2and the skeletal sheet M, each
point on the boundary (i.e., edge) of Mmoves on a vein to reach
the spine and then moves to reach the s-centroid. Therefore,
in discrete format, we dene parent and child relationship on
Mas we dened on E2. In addition, given a frame at each
skeletal point, we consider the same hierarchical structure for
the frames.
A vector that connects a frame to its parent frame is called
connection. The tip of a connection is at the frame’s origin, and
its tail is at the parent’s origin. Further, we assume that the s-
centroid frame is itsown parent without any connection to itself.
We approximate the dire ction of bat p oint p∈Mbased on
three consec utive frames. Frames on the spine are p arent of
multiple children. To have a consistent frame denition rst
we t frames on the spine. Except for the s-centroid frame and
two critical endpoints of the spine that we will explain later,
each spinal frame has a spinal parent frame and a spinal child
frame. Let p1and p2bethepositionoftheparentandthechild
frame of p.AsillustratedinFigure 5, assume v1=p−p1and
v2=p2−pas connections. Let p
1and p
2be the projectionof p1
and p2on Tp(M), respectively. We consider b=ˆ
v
2+ˆ
v
1
ˆ
v
2+ˆ
v
1where
ˆ
v
1=p−p
1
p−p
1,andˆ
v
2=p
2−p
p
2−p.Inthissense,bis a unit vector
tangent to a circle (or a line) crossing p−ˆ
v
1,p,andp+ˆ
v
2.
Theendpointsofthespinearecriticalbecausetheirframes
have no children on the spine. By construction, the medial line
is part of the major axis of the medial ellipse. Thus, there is a
curveontheskeletalsheetcorrespondtothemajoraxisthat
we call major curve. The major curve contains the spine and
two veins. We consider the closest skeletal point (in geodesic
sense) on these veins to the spine as the spine’s extension and
treat the critical points as any other spinal point. The s-centroid
frame has two spinal children. Let p1and p2be the position
of the children. We dene b=ˆ
v
2−ˆ
v
1
ˆ
v
2−ˆ
v
1,whereˆ
v
1=p−p
1
p−p
1,
and ˆ
v
2=p
2−p
p
2−p. Since a vein frame has a parent and a child
on the same vein, we consider the same denition for them as
discussedforspinalframes.Notethatwetreataveinframeat
the intersection of a vein and the spine as a spinalf rame.For the
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664 M. TAHERIAND J. SCHULZ
Figure 6. LP-ds-rep. Top: Hierarchical structure of the ellipsoid’s medial locus. Arrows are connections. The dot is the medial centroid. Bottom: A tted LP-ds-rep to a
hippocampus. Arrows indicate spokes,connec tions,and frames. The magnied image depicts a spinal frame. The dot is the s-centroid.
framesontheedgeoftheskeletal,weassumethetipofthecrest
spokes from Liu et al. (2021) as the p osition of the child frames.
The same procedure is applicable for the ellipsoid’s GP-ds-rep.
Figure 6 illustrates the hierarchical structure and a tted LP-
ds-rep to a le hippocampus as we discuss in the next section.
2.1.4. LP-ds-rep
Given the tted hierarchical frame structure introduced in the
previous section, we are now in the position todene LP-ds-rep.
In an LP-ds-rep, spokes and connections are me asured based on
their local frames, that is, their tails are located at the origin of a
frame. Assume ns,np,andncas the number of spokes, frames,
and connections, respectively. Note that nc=np−1. Let ui
and vkbe the ith spoke direction and kth connection direction
in GCS, respectively, where i=1,...,ns,j=1, ...,np,and
k=1, ...,nc. Consequently, we denote u∗
iand v∗
kas spoke
and connection d irections based on their lo cal frame, that is, we
reparameterize uiand vkto u∗
iand v∗
k, re sp e cti ve l y. Si mil arl y, if
Fj=(nj,bj,b⊥
j)be the frame Fjin GCS then F∗
j=(n∗
j,b∗
j,b∗⊥
j)
denotes Fjbasedonitsparentframe.
To calculate a vector direction according to a local frame, we
use the spherical rotation matrix R(x,y)=Id+(sin α)(ywT−
wyT)+(cos α−1)(yyT+wwT),wherex,y∈Sd−1,w=
x−y(yTx)
x−y(yTx),andα=cos−1(yTx). Therefore, R(x,y)transfers
xto yalong the shortest geodesic and we have R(x,y)x=y
(Amaral, Dryden, and Wood 2007).
For example, let frame F†=(n,b,b⊥)be the parent of
˜
F,bothinGCS.Lete1=(1, 0, 0)T,e2=(0, 1, 0)T,and
e3=(0, 0, 1)Tbe the axes unit vectors of GCS. We align
F†to ˜
I=(e3,e1,e2)such that R2R1F†=˜
I,whereR1=
R(n,e3),andR2=R(R1b,e1).Thus, ˜
F∗=R2R1˜
Frepresents
˜
Fin its parent coordinate system. In case we obtain R2R1F†=
(e3,e1,−e2),weadjusttheresultbyR2R1˜
F(13,13,−13)because
R2R1F†(13,13,−13)=˜
Iwhere 13=(1, 1, 1)T.Notethatframe
vectors are orthogonal, so aer rotating nto the north pole by
R1, the shortest geodesic between band e1wouldbeontheequa-
tor. This preserves the direction of R1nwhile R2rotates R1˜
F.
Wefollowthesameproceduretocalculatethespokes’and
connections’ directions based on their local frames F∗
j.Asa
result, we consider a LP-ds-rep as a tuple sLP =(u∗
i,ri,
F∗
j,v∗
k,vk)i,j,k=(u∗
1,...,u∗
ns,r1,...,rns,F∗
1,...,F∗
np,v∗
1,...,
v∗
nc,v1,...,vnc),suchthatu∗
i∈S2and v∗
k∈S2are ith and
kth spoke direction and connection direction relative to their
local frame with lengths ri∈R+and vk∈R+respectively, and
F∗
j∈SO(3)is the jthframeinitsparentcoordinatesystem.
Thus, by construction, the LP-ds-rep is invariant under the
act of rigid transformation. To remove the scale, we dene
LP-size as the geometric mean of the vectors’ lengths =
exp (1
ns+nc(ns
i=1ln(ri)+nc
k=1ln(vk))). Assume ρi=ri
,and
τk=vk
. A scaled LP-ds-rep can be expressed by sLP =
(u∗
i,ρi,F∗
j,v∗
k,τk)i,j,k.
Result 1. The LP-size of a scaled LP-ds-rep is equal to one
(see the proof in SUP).
Recall,foraGP-ds-rep,theGP-size is dened as the centroid
size of the skeletal PDM. As we discussed in the introduction,
the centroid is an extrinsic property. Thus, the centroid size
might be a poor measure for the size of an object. The same
discussion is also true for EDM-size where EDM-size is the
geometric mean of all pairwise distances (Lele and Richtsmeier
2001, Ch.4.7.3). Intuitively, by opening or closing an arm, the
arm’s volume remains the same despite its centroidsize or EDM-
size, that is, the closed arm has smaller centroid size and EDM-
size in comparison with the open arm (see Figure 1(a)).
As Section 2.1.1 discussed, the GP-ds-rep space is SGP =
S3np−1×(S2)ns×Rns+1
+.InLP-ds-rep,wedonothaveany
pre-shape space. The GOPs of an LP-ds-rep are directions
and lengths of spokes, directions and lengths of connections,
LP-size, and frames. Thus, the space is SLP =(S2)ns+nc×
(SO(3))np×Rns+nc+1
+,where(S2)ns+ncis the space of ve ctors’
directions, (SO(3))npis the space of the frames, and Rns+nc+1
+
is the space of vectors’ lengths plus LP-size. Further, we can
represent an LP-ds-rep as sLP =(u∗
i,ρi,q∗
j,v∗
k,τk)i,j,k,where
q∗
j∈S3is the unit quaternion representation of the frame F∗
j
(Huynh 2009). Thus, we have SLP =(S2)ns+nc×(S3)np×
Rns+nc+1
+,where(S3)npis the space of the framesbased on their
unit quaternion representations.
Paper I
44
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 665
2.2. Euclideanization and Mean Shape
Having a population of ds-reps, suitable methods to calculate
means are requiredin order to perform hypothesis tests on mean
dierences. The corresponding method should incorporate all
geometrical components of the model. Both shape spaces, the
GP-ds-rep space, and the LP-ds-rep space are composed of
several spheres and a real space. This section will rst discuss
an approach to analyze the spherical parts by principal nested
spheres (PNS). Aer ward, approaches to pro duce GP-ds-rep
means and LP-ds-rep means are discussed.
2.2.1. PNS
PNS (Jung, Dryden, and Marron 2012)estimatesthejointprob-
ability distribution of data on d-sphere Sdby a backward view,
that is, in decreasing dimension. Starting with Sd,PNSts
the best lower-dimensional subsphere in each dimension. A
subsphere is called great subsphere if its radius is equal to one;
otherwise, it is called small subsphere. To choose between the
great or small subsphere, we use the Kurtosis test from (Kim,
Schulz, and Jung 2020).
PNS is designed for spherical data (particularly for small
sphere distributions) to capture the curviness of circular dis-
tributions as discussed by Kim et al. (2019). PNS is similar
to PCA bec ause PCA provides obs ervations’ coordinates called
residuals as their distances from tted (hyper)planes, while PNS
residuals are the observations’ geodesic distances fromthe tted
subspheres. For example, the PNS residuals on S2consist of
the geodesic distances between the observations and the tted
circle and the mi nimal arc length between proj ected data on the
ttedcircletothePNSmean.Basically,PNSeuclideanizethe
data by dening a mapping from Sdto Rd.Inmanycases,the
distribution of the PNS residuals is similar to the mult ivariate
normal distribution (see an example in SUP).
Alternatively, a simpler but faster euclideanization is to map
the data on the tangent space. We transform observ ations to the
north pole e=(0, ...,0,1)T∈Sdby R(μF,e),whereμFis the
Fréchetmean.Then,wemapthetransformeddatatothetangent
space Te(Sd)by the Log map Loge(v)=θ
sin θ(v1,...,vd)T∈
Rd,wherev=(v1,...,vd+1)T∈Sd,andθ=cos−1(vd+1)
(Jung, Dryden, and Marron 2012;Kim,Schulz,andJung2020).
For concentrated von Mise s-Fisher distributi on, the distribution
of projected data to the tangent space is close to the distribution
of PNS residuals (see SUP).
2.2.2. Mean GP-ds-rep
A method to produce means and shape distributions of a pop-
ulation of GP-ds-reps is composite PNS (CPNS) introduced by
Pizeretal.(2013). The method consists of two steps. First,
thetwosphericalpartsoftheGP-ds-repshapespaceS3np−1×
(S2)ns×Rns+1
+areanalyzedbyPNS.Spokes’lengthsand
scaling factor can be mapped to Rns+1with the log .Aer-
ward, all Euclideanized variables are concatenated in addition
to some scaling factors that make all variables commensurate.
The covariance structure of the resulting matrix is investigated
by PCA. Consequently, the mean GP-ds-rep is dened as the
origin of the CPNS space. This method depends on a proper
pre-alignment and is computationally expensive because PNS
has to t sequential high dimensional sub-spheres to S3np−1.
2.2.3. Mean LP-ds-rep
To formalize the estimation of LP-ds-rep mean, rst we dene
a metric for the product space SLP. Assume metric spaces
(R+,dl),(S2,dg)and (SO(3),dR),wheredl(x,y)=|ln x−
ln y|is the Euclidean distance of log-scaled values, dg(x,y)=
cos−1(yTx)is the geodesic distance on the unit sphere (Jung,
Dryden, and Marron 2012), and dR(F1,F2)=1
√2log(FT
1F2)F
is the Riemannian distance on SO(3)where .Fis the
Frobenius nor m (Moakher 2002). The distance between two
scaled LP-ds-reps sLP
1=(u∗
1i,ρ1i,F∗
1j,v∗
1k,τ1k)i,j,kand sLP
2=
(u∗
2i,ρ2i,F∗
2j,v∗
2k,τ2k)i,j,kis given by
ds(sLP
1,sLP
2)=ns
i=1
d2
g(u∗
1i,u∗
2i)+
ns
i=1
d2
l(ρ1i,ρ2i)
+
np
j=1
d2
R(F∗
1j,F∗
2j)+
nc
k=1
d2
g(v∗
1k,v∗
2k)
+
nc
k=1
d2
l(τ1k,τ2k)1
2
.(2)
Remark 1. LP-ds-rep space SLP is a metric space equipped by
ds(.)(see the proof in SUP).
If sLP
1,...,sLP
Nbe a population of scaled LP-ds-repsthen mean
LP-ds-rep is
¯
sLP =argminsLP∈SLP
N
m=1
d2
s(sLP,sLP
m).(3)
Assume ¯
sLP =(¯
u∗
i,¯ρi,¯
F∗
j,¯
v∗
k,¯τk)i,j,kand ∀i,j,klet
¯
u∗
i=argminu∈S2
N
m=1
d2
g(u,u∗
im),
¯ρi=argminρ∈R+
N
m=1
d2
l(ρ,ρim ),
¯
F∗
j=argminF∈SO(3)
N
m=1
d2
R(F,F∗
jm),
¯
v∗
k=argminv∈S2
N
m=1
d2
g(v,v∗
km),
¯τk=argminτ∈R+
N
m=1
d2
l(τ ,τkm).(4)
By assuming the existence of unique solutions for optimization
problems (4), ¯
u∗
iand ¯
v∗
kcan be estimated as the Fréchet or PNS
mean of {u∗
im}N
m=1and {v∗
km}N
m=1,respectively.Obviously, ¯ρiand
¯τkrepresent the geometric means of {ρim}N
m=1and {τkm}N
m=1,
respectively. Further, we can calculate the mean frame ¯
F∗
jof
{F∗
jm}N
m=1as discussed by Moakher (2002).
Result 2. If ¯
sLP be the mean of a popu lation of scaled LP-ds-reps,
then LP-size of ¯
sLP is equal to one (see the proof in SUP).
Paper I
45
666 M. TAHERIAND J. SCHULZ
2.3. Converting LP-ds-rep to GP-ds-rep
Sections 2.1.3 and 2.1.4 discuss h ow to obtain an LP-ds-rep from
a GP-ds-rep. For several reasons, for example, for visualization,
we may need to reverse the procedure. For GP-ds-rep visual-
ization, it is sucient to draw spokes individually. To visualize
an LP-ds-rep, we convert it to a GP-ds-rep. We start from ˜
Ias
the s-centroid frame. Then, we reconstruct frames by nding
the position and orientation of the frame’s children based on
˜
I. Aerward, we nd the information of grandchildren frames
based on their parents and so on.
Let frame F∗be in the coordinate system of its parent F†.To
nd F∗basedonGCS,werotateF†by R2R1such that R2R1F†=
˜
I.Then[R2R1]−1F∗is the representation of F∗in GCS. Similarly,
wendthedirectionofconnectionsandspokesinGCS.
Finding the mean shape of a set of objects’ boundaries with-
out an alignment is almost impossible. But we can use LP-ds-
reps to estimate the mean boundary without alignment. First,
wecalculatethemeanLP-ds-rep.Then,weconvertthemeanLP-
ds-reptoaGP-ds-rep.Finally,wegeneratetheimpliedboundary
from the GP-ds-rep as demonstrated in (Liu et al. 2021). There-
fore, it is possible to approximate the mean boundary wit hout
alignment, which shows the power of LP-ds-reps.
2.4. Deformation
In statistical shape analysis generating random shapes is a matter
of interest. Designing simulations based on GP-ds-reps is chal-
lenging as we usually need to identify a local frame to bend or
twisttheobjectlocally.ItturnedoutthatLP-ds-repssupport
naturally skeletal deformations. We can stretch, shrink, bend,
and twist the skeletal by manipulating the frames’ orientations
and vectors’lengths. Then, we convert the LP-ds-rep to a GP-ds-
rep to generate the boundary. Consequently, we can add varia-
tion to a set of deformed LP-ds-reps’ GOPs to simulate random
ds-reps. Figure 7 shows a deformed hippocampus including
bending and twisting.The deformation is based on the rotation
of spinal frames.
3. Hypothesis Testing
For LP-ds-rep hypothesis testing, we consider frames as unit
quaternions (i.e., sLP =(u∗
i,ρi,q∗
j,v∗
k,τk)i,j,k). In this sense,
euclideanization of the frames b ased on their unit quaternion
representation is the same as other spherical data as we dis-
cussed in Section 2.2.
Let A={sAm}N1
m=1and B={sBm}N2
m=1be two groupsof either
GP-ds-reps or LP-ds-reps of sizes N1and N2.LetnGOP be the
total number of GOPs.To test GOPs’mean dierence, we design
nGOP partial tests. Let ¯
sA(n)and ¯
sB(n)be the observed sample
mean of the nth GOP from Aand Brespectively. The parti al test
is H0n:¯
sA(n)=¯
sB(n)versus H1n:¯
sA(n)= ¯
sB(n).Notethat
for GP-ds-rep, LP-ds-rep, and EDM of the skeletal PDM, nGOP
is (np+2ns+1),(2ns+np+2nc+1),and((np−1)np
2+1),
respectively.
To test mean dierences, we adapted a nonparametric per-
mutation test with minimal assumptions similar to Styner’s
approach (St yner et al. 2006). For the univariate data, that is,
vectors’ lengths and shapes’ sizes, the test statistic is t-statistic
T=¯
x−¯
y
Sp1
N1+1
N2
where Spis the pooled standard deviation.
For the multivariate data, that is, euclideanized directions and
GP-ds-rep skeletal positions, the test statistic is Hotelling’s T2
metric T2=(¯
x−¯
y)Tˆ
−1(¯
x−¯
y),where ˆ
is an unbiased
estimate of common covariancematrix (Martin and Maes 1979,
ch.3). Given the pooled group {A,B}, the permutation method
randomly partitions Btimes the pooled group into two paired
groups of sizes N1and N2without replacement, where usually
we consider B≥104. Aerward, it measures the test statis-
tic between the paired groups. The empirical p-value for the
nth GOP is ηn=1+B
h=1χE(|Tnh|≥Tno )
B+1,whereTno is the nth
observed test statistics, Tnh is the hth permutation test statistic,
and χEis the indicator function, that is, χE(ϕ) =1ifϕis
true, otherwise χE(ϕ) =0. Note that if we have normally
distributed data, it is reasonable to apply Hotelling’s T2test (with
normality assumption) instead of the p ermutation test as it is
much faster.
In order to account for the problem of multiple hypothesis
testing, one could use the metho d of Bonferroni (1936). Bon-
ferroni’s method tests each hypothesis at level α/nGOP and
guarantees the probability of at least one Type I error P(v≥1)
be less than the signicance level α. Since the method is highly
conservative we prefer to use Benjamini and Hochberg (1995)
(BH) method as a more moderate approach.
4. Evaluation
4.1. Data
To test our method, we study the hippocampal dierence
between earlyParkinson’sdisease (PD) and CG at baseline. Data
Figure 7. Skeletal deformation by LP-ds-rep. Left:A ds-rep with its implied boundar y in two angles. Middle: Shape bending by spinal frame rotationabout nand b⊥axes.
Right: Shape twisting by spinal frames rotation about baxis.
Paper I
46
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 667
areprovidedbyParkWest(http://parkvest.no), in cooperation
with Stavanger University Hospital (https://helse-stavanger.no).
At the baseline, we have 182 magnetic resonance images for
PD and 108 for CG with cor responding segmentation of hip-
pocampi. As described in Section 2,GP-ds-repsarettedto
le hippocampi by SlicerSALT toolkit (http://salt.slicer.org )and
reparameterize into LP-ds-reps. For the model tting, we used
GP-ds-reps with 122 spokes consisting of 51 up, 51 down, and
20crestspokes.Asupanddownspokessharethesametail
positions, we have in total 71 tail positions. The generated LP-
ds-reps have 122 spokes, 71 local frames, and 70 connections.
BeforeanalyzingtheParkWestdata,werststudyourmethod
based on simulations.
4.2. Simulation
For the simul ation study, we select a LP-ds-rep close to the
mean LP-ds-rep of CG as a template. Based on the template, we
generatetwogroupsofLP-ds-repseachofsize150withdier-
ent amount of tail bending, that is, bending in a local region.
Such bending was observed, for example, in (Pizer et al. 2003)
between schizophrenics and controls. Let Md(μ,κ) denotes
von Mises-Fisher distribution with mean μand concentration
parameter κon Sd−1(Dhillon and Sra 2003). For the special
case d=2weassumethedistributioninradian,thatis,θ,μ∈
[0, 2π) if θ∼M2(μ,κ). Given a random rotation angle of
bending θ∼M2(μ =0, κ=100)for the rst group and θ∼
M2(μ =−π
15 ,κ=100)for the second group, we simulate the
orientation of three spinal frames by successively rotating them
about their b⊥-axis with [R2R1]−1R(e3,(cos θ,0,sinθ)T)˜
I.This
means the tails in the second group are successively bent on
average 12◦downward for three consecutive spinal frames.
Chosen frames are the closest ones on the hippocampus tail to
thes-centroid.Thus,intotal,wehaveaslightdownwardbend-
ing about 36◦at the hippocampus tail. Finally, by preserving
frameorthogonality,weaddnoisetoalldirectionsbyM3(μ,κ),
where κfor frames’ vectors, spokes, and connections is equal
to 600, 250, and 5000, respectively. Further additional noise is
added to vectors’ lengths by the truncated normal distribution
ψ(μ,σ,a>0, b<∞)where μisthevectorlengthofthe
template, and parameters σ,a,andbare heuristically chosen.
As a result, we havetwo groups of random LP-ds-reps, which are
approximately similar in most of their GOPs but only dierent
in the orientation of three frames. Figure 8 illustrates twenty
samples of each group.Note that LP-ds-reps are not aligned, but
since we reconstruct them from the s-centroid frame, shapes
have Bookstein’s alignment (Dryden and Mardia 2016, Ch.2)
because the s-centroid frames are perfectly aligned.
As depicted in Figure 8(Right), hypothesis test on LP-ds-rep
from Section3 correctly detects signicant frame directions and
label almost all other GOPs as statistically nonsignicant given
a signicance level α=0.05. On the contrary, as depicted in
Figure 9, the test on GP-ds-repsindicates a large number of false
positives, that is, almost all of the positions and directions are
statistically signicant. Also, from EDMA on the skeletal PDM
we can see that about half of the distances are signicant. This
example conrms our observation from Figure 1 in Section 1,
and highlights the fact that noninvariant GP-ds-rep analysis is
biased and invariant EDMA could be misleading. The power
of LP-ds-rep is further highlighted by additional simulation
examples provided in SUP.
4.3. Real Data Analysis
The Parkinson dataset described in Section 4.1 was studied ear-
lier by (Apostolova et al. 2012) based on radial distance analysis
and parallel slicing and showed some regional atrophy. Since
shape correspondence in noninvariant parallel slicing method
is controversial, we attempt to reanalyze data by utilizing LP-ds-
reps.
Figure 8. Simulation. Left: Two groupsof simulated ds-reps. Middle: Overlaid mean LP-ds-reps. Right: Illustration of local frames. Bold frames are statistically signicant.
Figure 9. Sorted raw and adjusted p-values. The horizontal line indicates signicance level α=0.05.
Paper I
47
668 M. TAHERIAND J. SCHULZ
Tabl e 1. T-test on shape size.
MeanCG MeanPD SDCG SDPD p-value
Object volume (mm3) 3352.23 3271.44 563.39 616.68 0.26
LP-size 2.37 2.33 0.17 0.18 0.04
GP-size of spokes’tips 161.05 162.51 8.97 8.62 0.17
EDM-size of skeletal PDM 12.66 12.76 0.83 0.84 0.36
First let us compare the shape sizes fromTa ble 1 .Thevolume
measurement conrms the LP-size is more compatible with the
object volume because both, the mean object volume and the
LP-sizeofCG,aregreaterthanPD.InoppositethemeanGP-
sizeandEDM-sizeofCGaresmallerthanPD.Also,testson
shape size indicate signicant dierence in LP-size.
Figure 10 illustrates signicant LP-ds-rep and GP-ds-rep
GOPs before and aer BH adjustment. In LP-ds-rep, all the
spokes directions are insignicant. In contrast, about 40% of
GP-ds-rep spokes’ directions are signicant. Also, in LP-ds-rep,
there are a few signicant connection and frame directionsaer
the adjustment. Based on the LP-ds-rep analysis, it seems the
main dierence comes from connections’ lengths on the spine.
Figure 11 shows sorted p-values before and aer adjustment of
the applied methods. Based on Bonferroni adjustment, PD and
CG are similar because almost all adjusted p-values are greater
than 0.05. Based on BH adjustment, all GOPs in EDMA are
not signicant but about 30% of them are signicant before BH
adjustment. The reason is the sensitivity of BH to the number
of tests, that is, by increasing the number of tests, BH becomes
conservative. In GP-ds-rephalf of the GOPs are signicant even
aer the BH adjustment. In contrast, LP-ds-rep shows a small
portion of signicant GOPs before and aer the adjustment. In
addition, we analyzed the shapes without scaling to show the
sensitivity of GP-ds- rep to the scaling and the superi ority of LP-
ds-rep compared to GP-ds-rep and EDMA. Detailed results are
available in SUP.
5. Conclusion
Generally, it is common to detect locational dissimilarity
between two groups of objects based on the alignment. As
discussed, noninvariant (i.e., alignment-dependent) methods
such as GP-ds-rep analysiscould be hig hly biased,and invariant
methods based on extrinsic object properties like EDMA could
be misleading. Thus, we propose an invariant shape representa-
Figure 10. ds-rep signicant GOPs. Bold indicate signicant GOPs. FDR=0.05 for BH adjustment.
Figure 11. Test on real data. Sorted raw and adjusted p-values. Thehorizontal line indicates signicance level 0.05.
Paper I
48
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 669
tion called LP-ds-rep by putting a partial order on the skeletal
positions of a GP-ds-rep and constructing local frames at each
skeletal position. Such partial order exists on any tree struc-
ture, by considering the ow away from a chosen basepoint.
Therefore, the proposed idea is not limitedto ellipsoidal objects,
neither to skeletal models, as long as a tree structure can be
established for a shape model that ensuresgood correspondence
between objects. Further, we compared LP-ds-rep analysis with
GP-ds-rep analysis and EDMA to show the power and the
advantages of LP-ds-reps. For comparison, we applied simula-
tion and real data analysis. The simulation conrmed that even
if two populations of ds-reps dier only in a small local region,
thehypothesistestsbasedonGP-ds-repsandEDMsresultin
a large numbe r of signicant GOPs while the tests base d on
LP-ds-reps indeed detect the true underlying dierences. We
studied le hippocampi of PD versus CG for real data analysis.
Although hypothesis tests on GP-ds-reps and EDMs indicated
many signicant GOPs, tests on LP-ds-reps showed only a few,
which seems medically more reasonable. We concluded that PD
and CG groups are ver y similar, but the main dierence comes
from the spine length.
Acknowledgments
Special thanks to Profs. Stephen M. Pizer (UNC), Steve Maron (UNC),
James Damon (UNC), and Jan Terje Kvaløy (UiS) for insightful discussions
andinspirationforthiswork.WeareindebtedtoProf.GuidoAlves(UiS)for
providing ParkWest data. Weals o thank Zhiyuan Liu (UNC) for the model
tting toolbox.
Funding
This research is funded by the D epartment of Mathematics and Physics of
the University of Stavanger (UiS).
Supplementary Materials
Supplementary : SUP materials refe renced in this wor k are available as a
pdf. (pdf)
R-co de: In Supplementary.zip, simulationco des andles are placed. (zip)
ORCID
Mohsen Taher i http://orcid.org/0000-0003-4044-8507
Jörn Schul z http://orcid.org/0000-0002-6240-4794
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tions on Medical Imaging, 23, 995–1005. [660]
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for StatisticalAnalysis of Landmark-Based Shape Data,” Jour nal of Mul-
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[662]
Paper I
50
Supplementary materials of “Statistical
analysis of locally parameterized shapes”
Mohsen Taheri and J¨orn Schulz
Department of Mathematics & Physics, University of Stavanger
July 29, 2022
1 Proofs
Recall: The geometric mean of x1, ..., xn∈R+is exp( 1
nPn
i=1 ln xi)=(Qn
i=1 xi)1
n.
Proof of Result 1. Let ℓbe the LP-size of an LP-ds-rep. Thus the LP-size of the LP-ds-rep
after scaling is given by,
ns
Y
i=1
ri
ℓ·
nc
Y
k=1
vk
ℓ!1
ns+nc
=Qns
i=1 ri·Qnc
k=1 vk
ℓ(ns+nc)1
ns+nc=(Qns
i=1 ri·Qnc
k=1 vk)1
ns+nc
ℓ=ℓ
ℓ= 1
Proof of Remark 1. We know that if (O1, d1),(O2, d2), ..., (On, dn) are finite number of met-
ric spaces then (Pn
i=1 d2
i(xi, yi))1
2is a metric for the product space O1×O2×... ×Onwhere
xi, yi∈Oi, and i= 1, ..., n (O’Searcoid,2006, Section 1.6.1; Deza and Deza,2016, Section
4.2). Since the LP-ds-rep space SLP = (S2)ns+nc×(SO(3))np×Rns+nc+1
+is a product of
metric spaces (i.e., (S2, dg), (R+, dl) and (SO(3), dR)), the introduced distance function
ds:SLP ×SLP →R+is a metric for SLP.
Proof of Result 2. Let sLP
1, ..., sLP
Nbe a population of scaled LP-ds-reps. Then assume
ρm1, ..., ρmnsand τm1, ..., τmncbe the spokes’ lengths and connections’ lengths of the mth
scaled LP-ds-rep sLP
m, respectively. Also, let ¯ρ1, ..., ¯ρnsand ¯τ1, ..., ¯τncbe the spokes’ lengths
1
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51
and connections’ lengths of the mean LP-ds-rep ¯sLP , respectively. Based on Result 1, the
LP-size of sLP
mis equal to one. Thus the LP-size of ¯sLP is given by,
ns
Y
i=1
¯ρi·
nc
Y
k=1
¯τk!1
ns+nc
=
N
Y
m=1
ρm1!1
N
·... · N
Y
m=1
ρmns!1
N
· N
Y
m=1
τm1!1
N
·... · N
Y
m=1
τmnc!1
N
1
ns+nc
=
N
Y
m=1
ρm1·... ·
N
Y
m=1
ρmns·
N
Y
m=1
τm1·... ·
N
Y
m=1
τmnc!1
ns+nc
1
N
=
ns
Y
i=1
ρ1i·
nc
Y
k=1
τ1k!1
ns+nc
| {z }
=1
·... · ns
Y
i=1
ρNi ·
nc
Y
k=1
τNk !1
ns+nc
| {z }
=1
1
N
= 1
2 Examples
2.1 EDMA example
Figure 1a illustrates the pairwise distances of EDMA for the open and closed arm example
(see section 1 in the main manuscript). According to Figure 1b, EDMA for the open and
closed arm example shows that about half of the pairwise distances of the two groups are
significant. Based on landmark deletion approach (Lele and Richtsmeier,2001, Ch.4.10),
the EDMs of both groups become similar by removing the upper arm (or forearm) points.
Therefore, we can interpret that the upper arm (or forearm) points are significantly dif-
ferent. Although this interpretation makes sense, on the one hand, we cannot state which
part (i.e., upper arm or the forearm) is non-significant, and on the other hand, it is not
possible to explain the type of dissimilarity.
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(a) PDMs’ distances (b) EDM p-values
Figure 1: EDMA. (a) Visualization of pairwise distances between boundary points. (b) Raw and adjusted
p-values of the distances by BH and Bonferroni.
2.2 PNS example
Figure 2 illustrates a fitted circle to a cluster of 1000 observations on S2, and the PNS resid-
uals. Random points are generated from small sphere distribution X∼fS2(µ0, µ1, κ0, κ1)
(Kim et al.,2019) where µ0= (0,0,1)T,µ1= (cos π
3,0,sin π
3)T,κ0= 500, and κ1= 2.
Figure 2: PNS euclideanization. Left: Small circle distribution and the fitted circle on S2. Right:
Euclideanizated data.
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3 LP-ds-rep
3.1 Outline
Figure 3 shows LP-ds-rep hypothesis testing of two groups of 3D-ellipsoidal objects as the
outline of the main manuscript.
Figure 3: Outline of LP-ds-rep hypothesis testing for two groups of 3D-ellipsoidal objects.
3.2 Simulation
In the main manuscript, we explained how to simulate data with LP-ds-rep. This section
simulates ds-reps in various ways to show the superiority of LP-ds-rep analysis over GP-
ds-rep analysis. In Section 3.2.1 analogous to the main article, we discuss simulation and
results based on spinal frames’ orientations but with different parameters. In Section 3.2.2
we explain the obtained results from a simulation on vectors’ lengths.
3.2.1 Simulation based on frame orientation
In this section, we change the orientation of the same three spinal frames as selected in the
main paper. But we choose various bending rotation angles and different noise parameters.
Note that the orientation of only three spinal frames are significantly different. We simulate
4
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# significant GOPs # significant GOPs after BH
LP-ds-rep GP-ds-rep EDMA LP-ds-rep GP-ds-rep EDMA
Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
θ=−π
12 ,κ1= 100,
κ2= 600, κ3= 250
25.9 3.6 238.1 11.69 1606.4 153.2 3.3 0.67 232.6 12.8 1494.7 180.3
θ=−π
12 ,κ1= 10,
κ2= 60, κ3= 25
24.9 5.3 240.2 9.4 1609.2 100.7 3.3 0.4 234.6 10.1 1500.1 122.1
θ=−π
24 ,κ1= 100,
κ2= 600, κ3= 250
25.9 6.4 176.5 11.1 736.2 187.4 2.1 1.1 163.8 13.3 473.4 219.68
θ=−π
24 ,κ1= 10,
κ2= 60, κ3= 25
23.2 5.2 35.4 22.8 150.1 142.3 0.1 0.3 0.7 1.8 8.3 37.1
θ=−π
48 ,κ1= 100,
κ2= 600, κ3= 250
24.0 5.5 94.6 34.6 293.5 177.6 0.1 0.3 43.2 41.9 22.8 88.6
θ=−π
48 ,κ1= 10,
κ2= 60, κ3= 25
23.2 3.0 21.9 20.2 161.5 211.3 0.1 0.2 0.9 4.0 24.6 110.2
Table 1: Number of significant GOPs obtained from 50 simulations. Parameter θis the rotation angle of
the three spinal frames, while κ1,κ2, and κ3are the concentration factor of noise for θ, frames’ vectors,
and spokes, respectively.
two groups of LP-ds-rep of sizes 100 for 50 times. Then we estimate the average and
standard deviation of the number of significant GOPs of three methods (i.e., LP-ds-rep, GP-
ds-rep, and spokes’ tails EDMA) before and after p-value adjustment. For the simulation
we choose rotation angle θas 12◦, 6◦, and 3◦downward about b⊥-axis. Further, we add
noise to directional data by von Mises-Fisher distribution M3(µ, κ). Table 1 summarizes
the results where κ1,κ2, and κ3are the concentration factor of noise for the rotation
angle, frames’ vectors, and spokes, respectively. Obviously, by reducing the bending angle
or increasing the noise variations, we obtain less significant GOPs in all three methods.
Overall, LP-ds-rep analysis is more robust against noise and thus superior because of the
lower number of false positives. Theoretically, by considering significance level α= 0.05,
we expect to see nGOP /α false positives by chance where nGOP is the number of GOPs.
This fact is observable in the number of LP-ds-rep significant GOPs before the adjustment.
Note that in this study, nGOP is 456, 316, and 2485 for LP-ds-rep, GP-d-rep, and EDMA,
respectively.
3.2.2 Simulation based on vector length
In this section, we consider elongation simulation. To show the power of LP-ds-rep, we
slightly increased the length of only one connection. The chosen connection is between
the s-centroid frame and the children towards the hippocampus tail. We simulated 100
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ds-reps for two groups as discussed in Section 4.2 of the main manuscript. In one of the
groups, the length of the mentioned connection is approximately in average one unit larger
than the other group. Again as we can see in Figure 5b almost half of the GP-ds-rep
GOPs are statistically significant while LP-ds-rep according to Figure 5a, truly detects the
only difference as depicted by the bold solid line in Figure 4. Interestingly, as depicted in
Figure 5c, this slight change makes most of the pairwise distances significant in EDMA,
showing how EDMA is misleading when dealing with elongation.
Figure 4: Significant connection length based on LP-ds-rep analysis.
(a) LP-ds-rep p-values (b) GP-ds-rep p-values (c) EDMA p-values
Figure 5: Simulations’ p-values.
3.3 Real data analysis
In this section, we analyze the left hippocampi of PD vs. CG. First, we compare LP-ds-rep
analysis based on PNS vs. LP-ds-rep analysis based on tangent PCA. Then we analyze
PD vs. CG by LP-ds-rep, GP-ds-rep, and EDMA without scaling. As described in the
main article, the data are provided by ParkWest 2021 in cooperation with Helse Stavanger
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2021. Model fitting is implemented in SlicerSalt platform (Vicory et al.,2018) based on
(Liu et al.,2021).
3.3.1 LP-ds-rep analysis by PNS & Tangent PCA
We compared LP-ds-rep analysis based on PNS and tangent PCA. As depicted in Figure 6,
the result of both approaches is very similar. The reason is that we have concentrated
von Mises-Fisher distributions for the spherical data (i.e., spokes’ directions, connections’
directions, and frames based on their unit quaternion representations). Hence, as we men-
tioned in the main manuscript, the distributions of the projected data to the tangent space
is close to the distributions as PNS residuals.
(a) LP-ds-rep p-values based on PNS. (b) LP-ds-rep p-values based on tangent PCA.
Figure 6: Sorted p-values of PNS vs. Tangent PCA.
3.3.2 Analysis without scaling
In this section we analyze left hippocampi of PD vs. CG without scaling.
Figure 7 depicts the distribution of spokes’ tips and tails of PD GP-ds-reps and CG
GP-ds-reps after GPA alignment. Figure 8a and Figure 8b show the overlaid mean LP-
ds-reps and mean GP-ds-reps of PD and CG. Figure 10 and Figure 11 show the result of
hypothesis testing on LP-ds-rep and GP-ds-rep of left hippocampi of PD vs. CG without
scaling.
The general belief is scaling makes shapes more similar. But Figure 9 expresses the
percentage of significant GP-ds-rep GOPs increases dramatically after the scaling (from
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57
Figure 7: Distribution of spokes’ tips and tails of GP-ds-reps after GPA alignment.
(a) Mean LP-ds-rep (b) Mean GP-ds-rep
Figure 8: Mean shape. (a) Mean LP-ds-reps are overlaid by reconstruction based on s-centroid frame.
(b) Mean GP-ds-reps are overlaid by GPA alignment.
38% to 50%). In other words, scaling increases the number of raw p-values less than the
level of significance α= 0.05 and consequently increases the number of BH adjusted p-
values less than FDR=0.05. A possible explanation is that GPA tries to make shapes as
close as possible by reducing GOPs’ variation. By removing the scale, GPA reduces the
variation even more. Hotelling’s T2metric is proportional to the inverse common covariance
matrix. So by reducing the variation, the test statistic increases, and consequently, the p-
value decreases. On the contrary, the LP-ds-rep and EDMA are not sensitive to scaling as
they are invariant. Again, we conclude the GP-ds-rep analysis is biased
Since the mean shapes in both parameterizations are very similar, we conclude that LP-
ds-rep is superior because we see less significant GOPs based on raw p-values in comparison
with GP-ds-rep and EDMA.
8
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Figure 9: Scaling effect.
4 Conclusion
Simulation on bending and elongation confirms that noninvariant method like GP-ds-rep
analysis could be highly biased and invariant methods based on extrinsic GOPs like EDMA
can be missleading. Additional LP-ds-rep analysis on real data without scaling shows that
CG and early PD’s left hippocampi are similar. Since the mean shapes are very similar,
we consider most of the obtained significant p-values from GP-ds-rep as false positives.
Also, we see more significant p-values in EDMA by pairwise construction, making it more
difficult to detect location and the type of deformation.
References
Deza M, Deza E (2016) Encyclopedia of Distances. Springer Berlin Heidelberg, URL https:
//books.google.no/books?id=KQHdDAAAQBAJ
Helse Stavanger (2021) Stavanger university hospital. https://helse-stavanger.no/
Kim B, Huckemann S, Schulz J, Jung S (2019) Small-sphere distributions for directional
9
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data with application to medical imaging. Scandinavian Journal of Statistics 46(4):1047–
1071
Lele SR, Richtsmeier JT (2001) An invariant approach to statistical analysis of shapes.
Chapman and Hall/CRC
Liu Z, Hong J, Vicory J, Damon JN, Pizer SM (2021) Fitting unbranching skeletal struc-
tures to objects. Medical Image Analysis p 102020
O’Searcoid M (2006) Metric Spaces. Springer Undergraduate Mathematics Series, Springer
London, URL https://books.google.no/books?id=aP37I4QWFRcC
ParkWest (2021) Parkwest study. http://www.parkvest.no/
Vicory J, Pascal L, Hernandez P, Fishbaugh J, Prieto J, Mostapha M, Huang C, Shah
H, Hong J, Liu Z, et al. (2018) Slicersalt: Shape analysis toolbox. In: International
Workshop on Shape in Medical Imaging, Springer, pp 65–72, URL http://salt.slicer.
org/
10
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Significant GOPs without correction Benjamini-Hochberg correction
FDR=0.05
Spokes’ lengths
Connections’
lengths
Spokes’ directions
Connections’
directions
Frames’ directions
Figure 10: Test on LP-ds-rep. Bold indicate significant GOPs. The left and right columns illustrate
significant GOPs before and after BH adjustment with FDR=0.05, respectively.
11
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61
Spokes’ lengths
Spokes’ directions Spokes’ tail positions
Significant GOPs without correction BH correction FDR=0.05
Figure 11: Test on GP-ds-rep. Red indicates significant GOPs. The left and right columns illustrate
significant GOPs before and after BH adjustment with FDR=0.05, respectively.
12
Paper I
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Paper II
Paper II
Fitting the Discrete Swept
Skeletal Representation to
Slabular Objects
63
Paper II
64
Fitting the Discrete Swept Skeletal
Representation to Slabular Objects
Mohsen Taheri1*, Stephen M. Pizer2and J¨orn Schulz1
1*Department of Mathematics and Physics, University of
Stavanger, Norway.
2Department of Computer Science, University of North Carolina
at Chapel Hill, USA.
*Corresponding author(s). E-mail(s):
mohsen.taherishalmani@uis.no;
Contributing authors: pizer@cs.unc.edu;jorn.schulz@uis.no;
Abstract
Statistical shape analysis of slabular objects like groups of hippo campi
is highly useful for medical researchers as it can be useful for diag-
noses and understanding diseases. This work proposes a novel object
representation based on locally parameterized discrete swept skeletal
structures. Further, model fitting and analysis of such representations
are discussed. The model fitting procedure is based on boundary divi-
sion and surface flattening. The quality of the model fitting is evaluated
based on the symmetry and tidiness of the skeletal structure as well as
the volume of the implied boundary. The power of the method is demon-
strated by visual inspection and statistical analysis of a synthetic and an
actual data set in comparison with an available skeletal representation.
Keywords: Discrete skeletal representation, Medial axis, Medical image
analysis, Statistical shape analysis, Swept skeletal structure
1 Introduction
Statistical shape analysis of slabular objects (SlOs) (Pizer et al.,2022), such as
the hippocampus and caudate nucleus, is highly useful for medical researchers
1
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65
2Fitting the Discrete Swept Skeletal Representation to Slabular Objects
and clinicians. Such analysis offers valuable insights into detecting dissim-
ilarities between two sets of brain objects, for instance, by comparing the
hippocampi of patients with neurodegenerative disorders versus a healthy
group (Styner et al.,2006;Apostolova et al.,2012;Schulz,2013). The main
goal is to help physicians diagnose, predict, or understand disorders more
accurately and to start treatment at early stages.
The primary need for statistical shape analysis is to establish correspon-
dences among the objects in a population based on their geometric properties
(Laga et al.,2019). Thus, the target objective of this work is a shape rep-
resentation that supports statistical analysis, such as hypothesis testing and
classification. Our representation is designed to achieve correspondence for
SlOs by focusing on dividing the object through a sweep of slicing planes that
correspond accordingly.
Analogous to a generalized cylinder (GC)1, an SlO is a swept region with
a center curve called a spine and a smooth sequence of affine slicing planes
along the spine that do not cross within the object. The skeleton 2of an SlO
called the skeletal sheet is a smooth 2-dimensional topological disk (Damon,
2008;Pizer et al.,2022;Taheri and Schulz,2022). The slicing planes sweep the
SlO’s boundary and skeletal sheet, as illustrated in Figure 1.
Fig. 1 A hippocampus as an SlO. Grey disks are slicing planes along the spine. Left: The
black curve is the spine. Right: Two views of the SlO. The blue surface is the skeletal sheet.
Motivated by Damon (2003,2008), we understand the skeletal structure of
an SlO as a field of non-crossing internal vectors called skeletal spokes with tips
on the boundary and tails on the SlO’s skeletal sheet (see Figure 2). Skeletal
spokes provide geometric information such as locational width and direction.
One powerful SlO representation is the discrete skeletal representation (ds-
rep) as a finite subset of the SlO’s skeletal structure (Pizer et al.,2013;Liu
et al.,2021;Taheri and Schulz,2022). Assuming a sample of SlOs, their ds-reps
define a meaningful correspondence (Van Kaick et al.,2011) across the sample
based on the assumption that there is a correspondence between the SlOs
1A generalized cylinder is a swept region with a center curve and cross-sections that are star-
convex sets, as discussed by Ballard and Brown (1982); Damon (2008); Ma et al. (2018).
2The skeleton of an object is a curve or a sheet that can be understood as a locally centered shape
abstraction obtainable from a given shape by the process of continuous contraction (Bærentzen
and Rotenberg,2021;Siddiqi and Pizer,2008).
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66
Fitting the Discrete Swept Skeletal Representation to Slabular Objects 3
and an eccentric ellipsoid. That is, each SlO has a crest3corresponding to the
ellipsoid’s crest, two vertices corresponding to the ellipsoid’s vertices (i.e., the
endpoints of the ellipse’s major axis with maximum Gaussian curvature), and
the spine corresponding to the ellipsoid’s major axis (Pizer et al.,2013,2020,
2022) (see Figures 1 and 2). In this sense, a ds-rep is a tuple of skeletal spokes,
and a set of ds-reps is a set of tuples such that the tuples correspond to each
other element-wise. Therefore, we can compare and analyze the corresponding
ds-reps element-wise (Schulz et al.,2016;Pizer et al.,2020).
As Taheri and Schulz (2022) discussed, the advantage of the ds-rep over
most shape representations like the landmark-based model (Dryden and Mar-
dia,1998), point distribution model (PDM) (Laga et al.,2019;Styner et al.,
2006), Euclidean distance matrix (Lele and Richtsmeier,2001) and persis-
tent homology (Gamble and Heo,2010;Turner et al.,2014) is that a ds-rep
captures the interior curvature and width along the object. Moreover, we
can parameterize ds-rep so that it becomes invariant to rigid transformations
(i.e., alignment-independent), which enables us to detect local dissimilarities
between objects accurately. Further, such a skeletal model is able to explain
the types of dissimilarities like shrinkage, bending, and protrusion explicitly.
Although a ds-rep ensures good correspondence by construction, fitting a ds-
rep to an SlO remains challenging. Thus, the objective of this work is to define
a ds-rep such that fitting it to an object boundary defines a good correspon-
dence across a sample of SlOs (i.e., the geometric properties of the model
provide a meaningful relationship across a population, resulting in strong sta-
tistical performance). To define a suitable ds-rep, we need to have an explicit
definition of the skeletal structure.
A well-known skeletal structure is Blum’s medial skeletal structure. The
medial skeletal structure is a field of medial spokes on the medial skeleton (or
medial axis as defined in Section 2), where the skeleton is the locus of centers
of all bitangent inscribed spheres. The medial spokes are tangent to the bound-
ary, and medial spokes with common tail positions have equal lengths. The
medial skeletal structure is unique and defines a radial flow (i.e., the inverse
grassfire flow of Blum et al. (1967)) based on the portion of the spokes’ lengths
from the skeleton to the boundary. Therefore, the object’s boundary can be
reconstructed by having the medial skeleton and the radial flow. However,
Damon (2003) believed that Blum’s conditions were too strict for defining the
radial flow that leads to boundary formation. For example, spokes with com-
mon tail positions do not need (to be either symmetric relative to the skeleton
or) have equal lengths. Thus, he relaxed Blum’s conditions by defining three
conditions: 1. Radial curvature condition, 2. Edge condition, and 3. Compati-
bility condition. Based on the three conditions, the radial vector field does not
necessarily need to be Blum’s radial vector field. In other words, by having
the medial skeleton, we may define different radial vector fields satisfying the
3The crest of an SlO is a closed curve on the boundary such that at each crest point, the
curvature across the crest is convex, and the magnitude of the principal curvature has a relative
maximum. The crest of an ellipsoid is the intersection of its first principal plane with its boundary
(Siddiqi and Pizer,2008).
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4Fitting the Discrete Swept Skeletal Representation to Slabular Objects
three conditions and still construct the boundary. Moreover, as pointed out by
Damon (2003), as far as we preserve the three conditions, we may define dif-
ferent skeletal structures with different skeletons such that they produce the
same boundary (e.g., based on the chordal axis of Brady and Asada (1984)).
Thus, the skeletal structure of an object is not necessarily unique. Later Damon
(2008) used the three conditions and defined the swept skeletal structure for
swept regions by introducing the relative curvature condition (RCC).
The medial skeleton typically has a bushy structure. Thus, establishing
correspondence based on the medial skeleton is challenging. Pizer et al. (2013)
embraced Damon’s idea and introduced the skeletal representation (s-rep) for
SlOs. The s-rep is a quasi-medial representation designed to produce the cor-
respondences needed for statistical shape analysis. From Pizer’s point of view
based on the Jordan-Schoenflies Theorem (Mendelson,2012), the crest of an
SlO is a closed curve on the SlO’s boundary that divides the boundary into two
boundary components. Therefore, an SlO has a central medial skeleton (CMS)
as a unique connected manifold with no holes or branches corresponding to
the two boundary components (see Figure 6). The CMS can be seen as the
locus of all centers of inscribed spheres that are bitangent to both boundary
components, as we discuss in Section 3. Although the CMS is usually non-
smooth and bumpy, a relaxed version of the CMS can be considered as the
object’s skeleton. In this sense, an s-rep represents the SlO’s skeletal structure,
where the skeleton (i.e., the skeletal sheet) is a smooth sheet, and the enve-
lope of the non-crossing skeletal spokes represents the boundary (see Figures 1
and 2). We call the envelope of the spokes the implied boundary (Pizer et al.,
1999). The ds-rep is a finite subset of the s-rep. Thus, to fit a ds-rep, we need
to fit the skeletal sheet with non-crossing skeletal spokes on it such that the
implied boundary represents the actual boundary. Further, a SlO is a swept
region. Therefore, the skeletal structure of the fitted model should also reflect
the swept plane properties defining the SlO.
In the state-of-the-art model fitting, Liu et al. (2021) used boundary reg-
istration to deform the ds-rep of an ellipsoid as the reference object to fit the
model into a target SlO like a hippocampus. However, there are some con-
cerns with Liu’s method. Basically, it applies to any object homeomorphic
to an ellipsoid. Since almost all objects with no holes or handles are home-
omorphic to an ellipsoid (Jermyn et al.,2017), it is difficult to show that
the obtained ds-rep represents the skeletal structure of an SlO. Also, it is a
boundary deformation method. Thus, model fitting heavily relies on boundary
registration for deriving skeletal correspondence. Still, proper boundary reg-
istration is challenging and has been controversial for decades. Based on our
observations, various registration methods, including the spherical harmonic
PDM (SPHARM-PDM) (Styner et al.,2006), elastic registration (Srivastava
and Klassen,2016;Jermyn et al.,2017), or mean curvature flow registration
(Liu et al.,2021), fail to define an excellent correspondence between an SlO
and an ellipsoid, as discussed in the Supplementary Material (SUP). Besides,
such models are usually asymmetric and perturbed with an untidy structure.
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 5
The reason is that any boundary noise, protrusion, and intrusion significantly
affect the model. Analysis based on such models could be misleading because
they introduce false positives, as we discuss in Section 8.
As defined in Section 2, an SlO is a swept region such that each cross-
section (i.e., the intersection of a slicing plane with the object) is a symmetric
2D object with two vertices and a center curve. Thus, for each cross-section
there is a smooth sequence of line segments (i.e., 1D cross-sections) on its center
curve (see Figure 3). The line segments are coplanar, and each line segment
can be seen as two skeletal spokes with a common tail position in opposite
directions. Thus, an SlO has a swept skeletal structure (Damon,2008). The
union of the cross-sections’ center curves forms the skeletal sheet, and the union
of the skeletal spokes forms the radial vector field on the skeletal sheet that
defines the flow from the skeletal sheet to the boundary (Pizer et al.,2022).
The spine is a curve on the skeletal sheet that (approximately) transverses
the middle of cross-sections and connects two vertices of the SlO as depicted
in Figure 1. Therefore, we can define a meaningful correspondence across a
sample of SlOs by defining the correspondence between their spines based
on curve registration (Srivastava and Klassen,2016). Consequently, we have
corresponding spinal cross-sections associated with the corresponding spinal
points. In the same way that we define corresponding cross-sections on the
spine, we can define corresponding line segments on the center curves of the
spinal cross-sections (based on the curve registration in 2D). This approach
is thus designed to yield a finite set of corresponding skeletal spokes. We call
such a ds-rep a discrete swept skeletal representation (dss-rep). The dss-rep
represents the SlO’s swept skeletal structure, as depicted in Figure 2.
Fig. 2 A dss-rep of a hippocampus. Left: Discrete skeletal sheet. Right: Skeletal spokes in
grey with tails on the skeletal sheet.
The swept skeletal structure is also a form of skeletal structure that is not
unique. Therefore, the center curve of a swept region with a swept skeletal
structure is also not unique. In fact, as long as the center curve of a swept
region satisfies Damon’s criterion of the RCC, it can be bent such that the
cross-sections do not intersect within the object. The RCC defines a curvature
tolerance for the center curve to ensure the cross-sections do not intersect
within the object (Damon,2008;Ma et al.,2018) (see Figure 24 in SUP). For
example, for a 2D GC, the RCC can simply be determined at each point along
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6Fitting the Discrete Swept Skeletal Representation to Slabular Objects
the center curve based on its normal as r < 1
κ, where ris the object’s width
in the direction of the normal, and κis the curve’s curvature. Let us consider
an example in 2D to illustrate our approach.
Fig. 3 The skeletal structure of a 2D GC. (a) The approximation of the medial axis based
on the Voronoi diagram. The CMS is the blue curve connecting the two vertices (blue dots).
(b) A model based on a smooth curve very close to the CMS is invalid as it violates the RCC.
(c) A valid model based on a slightly relaxed CMS, which is tidy but not highly symmetric.
(d) A valid model with high symmetricity. The model is close to violating the RCC.
Assume a 2D GC as depicted in Figure 3 (a). The medial skeleton of the
object is calculated as shown in Figure 3 (a) based on the Voronoi diagram
(Attali et al.,2009;Dey and Zhao,2004). The object can be divided in two
parts, as discussed in Section 3, resulting in two vertices depicted by blue dots.
The CMS of the object is a unique curve (depicted in blue) with no branch
or discontinuity connecting the two vertices of the 2D GC. The CMS is a
pruned version of the medial skeleton and is unique. Thus, it seems reasonable
to construct the swept skeletal structure based on it. However, the CMS is
not smooth and has a bumpy structure (specifically at the intersection of
the branches). In this example, we cannot consider the CMS or a smooth
differentiable curve very close to the CMS as the center curve because they
violate the RCC (i.e., the cross-sections intersect within the object because
of its high local curvature), as shown in Figure 3 (b). Therefore, the dss-rep
cannot be established. However, following Pizer et al. (2013), by relaxing the
CMS, the center curve satisfies the RCC, as depicted in Figure 3 (c) and
Figure 3 (d). In Figure 3 (c), we slightly relaxed the CMS in the sense that
the relaxed version is very close to the CMS. Based on Damon’s discussions,
Figure 3 (c) is a valid model even though it is not perfectly symmetric (as
the spokes with a common tail position do not need to have equal length). In
Figure 3 (d), we increased the flexibility of the model so that it becomes more
symmetric, i.e., the center curve is closer to the middle of the cross-sections.
However, we observe that the center curve is very close to violating the RCC.
Note that defining a perfectly symmetric model with cross-sections normal to
the center curve usually is not feasible even if the object has a smooth non-
branching medial skeleton (Shani and Ballard,1984) (see Figure 23 in SUP).
Obviously, the structure of Figure 3 (c) is tidier than Figure 3 (d) as the center
curve has lower local curvature. In other words, the orientations of adjacent
cross-sections in Figure 3 (c) are not significantly different. There is a trade-off
between skeletal-symmetry and skeletal-tidiness, and both are crucial factors
in defining a model in addition to the volume-coverage. Thus, in Section 6, we
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 7
introduce an overall score based on these three criteria to measure the goodness
of fit within a set of model candidates for the swept skeletal structure.
One conceivable method for fitting a dss-rep into an SlO is to calculate the
spine as a curve skeleton (Dey and Sun,2006) in the first place and define
the cross-sections along the spine. However, there are concerns with the curve
skeleton. For example, even a smooth curve skeleton may still have branches
because there is no criterion that a curve skeleton must be a simple curve.
There are approaches for simplifying the curve skeleton to defining a simple
curve, for instance, the Laplacian contraction of Au et al. (2008), Mean curva-
ture skeleton of Tagliasacchi et al. (2012), L1-medial skeleton of Huang et al.
(2013), and skeletonization via local separators of Bærentzen and Rotenberg
(2021). However, according to our observations, they fail to offer a suitable
spine. In fact, the majority of these methods blindly search for the SlO’s skele-
ton without considering its important geometric properties, such as the crest
and vertices. Moreover, these methods commonly overlook the RCC entirely,
as it is not a prerequisite for defining the curve skeleton. Therefore, the spine
exhibits unpredictable behavior. For example, it may bend and swing freely
inside the object, as discussed in SUP.
By assuming a unique crest for an SlO, the CMS is unique. In this work,
we use the CMS to propose a novel dss-rep model fitting. We start by defining
the skeletal sheet by relaxing the SlO’s CMS. Then, we use the relaxed CMS
to define the spine as a curve located on the skeletal sheet connecting the SlO’s
two vertices, as we discuss in Section 4.2. The method we describe ensures that
it achieves good correspondence across the population samples based on the
uniqueness of the CMS. The method is independent of boundary registration
and complies with the definition of SlO. Further, the model fitting procedure
is flexible and can be tuned to obtain a tidy and symmetric model. We discuss
the tuning based on the essential properties of skeletal-symmetry, tidiness, and
the volume of the implied boundary.
To make the analysis alignment independent and to capture the type of
local dissimilarities (e.g., protrusion, elongation, etc.), we adapt the idea of
local ly parameterized ds-rep (LP-ds-rep) suggested by Taheri and Schulz (2022)
to introduce local ly parameterized dss-rep (LP-dss-rep) by parameterizing the
dss-rep based on a tree-like structure of its skeletal sheets equipped with local
frames as discussed in Section 5. The structure of this work is as follows.
Section 2 reviews basic terms and provides explicit definitions of a swept
region, SlO, skeletal structure, and swept skeletal structure. Section 3 intro-
duces the CMS. Section 4 uses the CMS to propose a dss-rep model fitting
procedure based on skeleton flattening using dimensionality reduction meth-
ods. Section 5 and Section 6 introduce the LP-dss-rep and discuss the goodness
of fit for a proper model based on essential skeletal symmetry and tidiness as
well as the volume of the implied boundary. Section 7 demonstrates LP-dss-
rep hypothesis testing and classification based on LP-dss-rep Euclideanization.
Section 8 compares the LP-ds-rep with the LP-dss-rep based on a set of toy
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8Fitting the Discrete Swept Skeletal Representation to Slabular Objects
examples and a real data set to discuss the pros and cons of our method.
Finally, Section 9 summarizes and concludes the work.
2 Basic terms and definitions
In this section, we review basic terms and definitions regarding skeletal
structures.
We consider the set Ω ⊂Rdas a d-dimensional object if Ω is homeomorphic
to the d-dimensional closed ball, where d∈N. We denote the boundary of Ω
by ∂Ω and its interior by Ωin, so Ω = Ωin ∪∂Ω. Assume point p∈Ωin and a
unit direction u∈Sd−1, where Sd−1={x∈Rd| ∥x∥= 1}is the unit (d−1)-
sphere. Assume Ω as a 2 or 3-dimensional object. If we start at pand move
straight forward based on u, we ultimately reach a boundary point. We call
such a straight interior path with starting point pand direction uaspoke. We
denote a spoke based on its positional and directional components by s(p,u).
We consider the skeletal spokes of Ω as a set of all non-crossing spokes
emanating from its skeleton M. The skeletal structure of Ω is a field of skeletal
spokes Uon Mdenoted by (M, U ) (see Figures 3 and 4). The medial skeletal
structure is one form of skeletal structure where the object’s skeleton is the
medial skeleton. The medial skeleton of object Ω is the set
M⊙={p∈Ωin | |{q∈∂Ω| ∥p−q∥=dmin(p, ∂ Ω)}|c≥2},(1)
where dmin(p, ∂ Ω) is the minimum Euclidean distance between pand ∂Ω,
and ∥.∥and |.|crepresent the Euclidean norm and cardinality, respectively. In
other words, M⊙is the center of all maximal (inscribed) spheres bi-tangent
(or multi-tangent) to ∂Ω. A medial spoke is a spoke connecting the center of
a maximal sphere to its tangency point. The collection of all medial spokes
U⊙={s(p,u)|p∈MΩand ∥s(p,u)∥=dmin(p, ∂ Ω)}on the medial skeleton
M⊙forms the medial skeletal structure (M⊙, U⊙) (Siddiqi and Pizer,2008).
Figure 4 illustrates the medial skeleton and a few medial spokes of a 2D object.
Fig. 4 Illustration of a medial skeleton and maximal spheres. The bold curve is the medial
skeleton. The dotted lines are medial spokes.
A swept region is a d-dimensional object with a smooth sequence of affine
slicing planes along a center curve (not necessarily normal to the center curve)
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 9
such that cross-sections do not intersect within the object, and each cross-
section is a (d−1)-dimensional object (Damon,2008). In other words, the
slicing planes sweep the object’s interior and boundary (see Figures 1 and 3).
Let Γ be the center curve of the swept region Ω, and let Mbe the skeleton
of Ω. Assume Γ(t)as the curve length parameterization of Ω such that t∈[0,1],
where Γ(0,1) denotes the curves’ endpoints (Srivastava and Klassen,2016).
Let Π(t)be the slicing plane crossing Γ(t), and let Ω(t)= Π(t)∩Ω be the
cross-section at Γ(t). Let U(p)denote the set of skeletal spokes with tails on
p∈M. The skeletal structure of Ω is a swept skeletal structure if, for each
p∈Ω(t)∩M, the U(p)∈Ω(t), i.e., all the skeletal spokes with tails on a cross-
section are coplanar. Thus, the slicing planes also sweep the object’s skeletal
structure (Damon,2008). We consider an object as a GC if its skeletal structure
is a swept skeletal structure and its skeleton Mis a smooth open curve. For
example, a cylinder is a 3D GC, and an ellipse is a 2D GC, where the skeleton
is the major axis and the slicing planes are perpendicular to the center curve
(Brady and Asada,1984;Giblin and Brassett,1985;Ma et al.,2018).
Following Pizer et al. (2022) and Taheri and Schulz (2022), in this work,
we consider an SlO as a swept region with a swept skeletal structure such that
each cross-section is a 2D GC, the length of the spine (i.e., the SlO’s center
curve) is notably larger than the length of the skeleton of each cross-section.
The intersection of the spine with each 2D GC is a point on and approximately
at the middle of the 2D GC’s skeleton. The union of the 2D GCs’ boundaries
forms the SlO’s boundary, and the union of the 2D GCs’ skeletons forms the
SlO’s skeleton, called the skeletal sheet. For instance, any eccentric ellipsoid
(i.e., an ellipsoid with unequal principal radii) is an SlO by considering the
ellipsoid’s major axis (i.e., the intersection of the first principal axis with the
ellipsoid) as the spine and (parallel) slicing planes perpendicular to the spine
with cross-sections as ellipses which are 2D GCs. Thus, the skeletal sheet of
the ellipsoid is the union of the ellipses’ skeletons, i.e., the intersection of the
first principal plane of the ellipsoid with itself.
3 Central medial skeleton
In this section, we discuss the CMS of 2D GCs and SlOs as a subset of their
medial skeleton. In Section 4, we use the CMS to fit the SlO’s dss-rep.
Fig. 5 Left: The Voronoi diagram of a 2D GC. Right: The top and bottom parts are
depicted as blue and red curves. The two sub-regions are two union polygons associated
with the top and bottom parts with the same color. The CMS is the black curve as part of
the shared boundary of the union polygons connecting the object vertices.
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10 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
We know the medial skeleton can be approximated by the Voronoi diagram
(Attali et al.,2009). A Voronoi diagram is a geometric structure that partitions
a space into regions based on the proximity to a specified set of points called
sites. In a Voronoi diagram, each region is a polygon consisting of all the points
in the space that are closer to a particular site than to any other site. By
assuming the sites as a large number of points uniformly distributed on the
object’s boundary, the polygons’ borders located inside the object approximate
the medial skeleton (Dey and Zhao,2004), as depicted in Figure 5 (Left).
Assume Ω as a 2D GC object with two vertices. The two vertices divide
the boundary ∂Ω into two non-overlapping components, namely top part and
bottom part. Each part is a simply connected manifold with no holes or dis-
continuity. These two parts cover the entire boundary without any gaps. In a
discrete sense, ∂Ω and its two parts can be represented by a set of adjacent
points. Thus, each boundary point belongs to only one part, and each part is a
set of adjacent points (De Berg,2000). Let ∂ˆ
Ω be the discrete form of ∂Ω, and
∂ˆ
Ω+and ∂ˆ
Ω−be the top and bottom parts, respectively. Thus, ∂ˆ
Ω = ∪∂ˆ
Ω±.
Assume a box that contains ∂ˆ
Ω. The Voronoi diagram of each part consists of
a set of adjacent polygons such that adjacent polygons share a common edge.
The union of these adjacent polygons is a union polygon as a connected subset
of the box. Therefore, the box (or the embedding space) is partitioned into two
sub-regions. Let Ω+and Ω−be the intersection of the two sub-regions with
Ω associated with ∂ˆ
Ω+and ∂ˆ
Ω−. Thus, Ω+and Ω−can be seen as two sub-
objects, such that Ω = ∪Ω±. The intersection of these sub-objects Ω+∩Ω−
defines their shared boundary that we consider as the central medial skeleton
(CMS) of Ω.
The CMS can be seen as the locus of the centers of all inscribed spheres
bitangent to both parts. In other words, the CMS is a subset of the medial
skeleton that is central relative to the ∂ˆ
Ω+and ∂ˆ
Ω−. This occurs because the
distances from any point on the CMS to both the top and bottom parts are
equal. Therefore, the CMS is a unique subset of the medial skeleton. Further,
the CMS has no branches because it is part of ∂Ω+(i.e., the boundary of the
object Ω+). Also, the CMS has no discontinuity because if it has a discontinu-
ity, it means Ω+and Ω−are adjacent, but there is a gap between them. Thus,
there is a polygon inside Ω that its interior does not belong to Ω+or Ω−,
which contradicts the definition of the Voronoi diagram. Figure 5 illustrates
the Voronoi diagram and the CMS of a 2D GC.
Similarly, by assuming a unique crest of an SlO as a closed curve (cor-
responding to the crest of an eccentric ellipsoid), the crest divides the SlO’s
boundary into two parts- the top and bottom parts. The Voronoi diagrams of
the two parts define two union polyhedrons (i.e., two 3D polygons) as two sub-
objects. The CMS of a SlO is the shared boundary of these two sub-objects.
Thus, the CMS is a unique subset of the medial skeleton. Analogous to the
CMS of a 2D GC, the CMS of an SlO is manifold with no holes or branches.
Figure 6 (a) illustrates the top part, the bottom part, and the crest of a cau-
date nucleus as an SlO. Figure 6 (b) shows the Voronoi diagram of the SlO
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 11
Fig. 6 (a) The top part, the bottom part, and the crest of a caudate are depicted in blue,
red, and yellow, respectively. (b) Voronoi diagram of the caudate in two views. (c) Top:
Illustration of the two sub-regions in blue and red. Bottom: The sub-regions are separated
to provide a better intuition. (d) The CMS is depicted as a black triangle surface.
located in a box. Figure 6 (c) depicts the two sub-regions associated with the
top and bottom parts. The sub-regions are also separated to provide a clear
intuition about the sub-regions. Figure 6 (d) illustrates the CMS.
4 Fitting swept skeletal structure
In this section, we discuss fitting swept skeletal structures for 2D GCs, and
then we generalize the method for SlOs.
We assume the skeleton or the center curve of a 2D GC is a smooth open
curve. If we move on the boundary of a 2D GC, the directions of the normals
do not change intensively except near the vertices associated with the endpoint
of the center curve Siddiqi and Pizer (2008, Ch.8.8). The same discussion is
valid for an SlO near the crest (Giblin and Kimia,2003,2004;Siddiqi and
Pizer,2008;Fletcher et al.,2004).
Thus, for an SlO, the normal directions of any two close points on each
boundary part are not significantly different unless at the area near the crest.
Abulnaga et al. (2021) used this property to calculate the crest and the two
boundary parts. They considered the discrete boundary and measured the
geodesic distance between any two boundary points (to find how far apart
they are) and their normal difference. Then, they applied spectral clustering
of Ng et al. (2001) (as a binary classification method) to classify the boundary
points into the top and bottom parts.
The main idea of our model fitting is to apply Abulnaga’s method to divide
the SlO’s boundary, obtain the CMS based on the two boundary parts, fit a
smooth manifold representing the skeletal sheet close to the CMS, and finally
find the spokes from the skeleton to the boundary such that the obtained
skeletal structure is a swept skeletal structure. For better intuition, we start
our discussion using a 2D GC, as depicted in Figure 7.
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12 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
4.1 Fitting a swept skeletal structure to a 2D GC
Assume a 2D GC with two vertices. The objective is to calculate the center
curve as an open curve connecting the two vertices and a sequence of slicing
planes along the center curve so that the slicing planes do not intersect within
the object. To do so, first, we find the top and bottom parts of the object.
Then, we calculate the CMS. The CMS is a curve connecting the two vertices.
Finally, we relax CMS to define the center curve and slicing planes.
Let Ω2be a 2D GC with the center curve ΓΩ2. Assume Γ0
Ω2and Γ1
Ω2as the
curve’s endpoints corresponding to the two vertices. Also, let ∂Ω+
2and ∂Ω−
2
be the top and bottom parts of ∂Ω2, respectively. Thus, Γ0,1
Ω2are at the edge
of the top and bottom parts. Therefore, to find the Γ0,1
Ω2and the CMS of the
object, it is sufficient to find the top and bottom parts.
For this purpose, we cluster boundary points into two groups as explained
by Abulnaga et al. (2021) with the assumption that the directions of the bound-
ary normals only change significantly around Γ±
Ω2, the same way they behave
at the vertices of an ellipse. Abulnaga et al. (2021) defined the affinity matrix
[W]i,j = exp{δ⟨ni,nj⟩dg(xi,xj)}, where niis the normal vector at the ith
boundary vertex xi,dgindicate the geodesics distance and δis a penalization
parameter to control the local effect of normals’ variation. Therefore, the ele-
ments of the affinity matrix reflect the distance between two boundary points
and how much their normals are different. Thus, based on spectral clustering
of Ng et al. (2001), the sign of the second smallest eigenvectors of the normal-
ized Laplacian L=I−D
−1
2W D
−1
2classifies the boundary points into two
classes as top and bottom parts that are consistent with the orientation of
the normals, where Iis the identity matrix, and [D]i,j = Σj[W]i,j. The same
discussion is valid for SlOs (see Figure 8). In this work, we choose δ= 0.5
as it usually results in a reasonable crest. In SUP, the boundary division of a
caudate is visualized based on different values of δ.
Fig. 7 Obtaining the center curve of a 2D GC. (a) Blue and red show the sub-regions
associated with the boundary’s top and bottom parts. (b) The CMS. (c) The relaxed CMS.
(d) The maximal inscribed circles are centered at the relaxed CMS. The implied boundary
is the envelope of the inscribed circles. (e) Up and down skeletal spokes of a large number of
points on the relaxed CMS in blue and red. (f) Chordal structure of the implied boundary.
(g) Semi-chordal structure of the object. Black dots show the semi-chordal skeleton. (h) The
dss-rep of the 2D GC based on the curve length registration.
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 13
As discussed in Section 3, by having the top and bottom parts, we can
calculate the CMS based on the Voronoi diagram, as shown in Figure 7 (a).
The two sub-regions are in blue and red. Figure 7 (b) depicts the CMS,
which is not smooth. We relax the CMS by fitting a smooth curve close to it
(e.g., by implicit polynomial (Unsalan and Er¸cil,1999), principal curve (Hastie
and Stuetzle,1989), nonlinear regression (Ritz and Streibig,2008), etc.), as
depicted in Figure 7 (c). Let ˆ
ΓΩ2be the relaxed CMS. We generate a large
number of maximal inscribed circles inside Ω2with a center on ˆ
ΓΩ2as illus-
trated in Figure 7 (d). Since ˆ
ΓΩ2is continuous, we can consider the envelope of
the maximal inscribed spheres as the boundary of an object ˆ
Ω2such that ˆ
ΓΩ2
is the medial skeleton of ˆ
Ω2. We know when the medial skeleton is a smooth
open curve, for each medial point p; there are two spokes at two sides of the
medial skeleton with the tail at pand tip at the object boundary as
b=p−R(l)|d
dl R(l)|t±R(l)r1− | d
dl R(l)|2n,(2)
where nand tare normal and tangent vectors of the medial skeleton at p, and
Ris the radius function based on curve length parameterization lsuch that
R(l) is the radius of the maximal inscribe sphere centered at p(Giblin and
Kimia,2003). Therefore, for a large number of medial points, we generate two
spokes pointing toward the top and bottom parts as depicted in Figure 7 (e).
Note that we can also make the relaxed CMS curvier as long as it produces
non-intersecting spokes (similar to Figure 3 (d)).
We consider two top and bottom parts for ∂ˆ
Ω±
2based on the endpoints
of ˆ
ΓΩ2. We call spokes connecting medial points to the ∂ˆ
Ω+
2and ∂ˆ
Ω−
2as up
and down spokes, respectively. The relaxed CMS equipped with the spokes as
shown in Figure 7 (e) can be seen as the medial representation (m-rep) of the
2D GC if the implied boundary of the spokes approximates the object bound-
ary (Pizer et al.,1999;Fletcher et al.,2004). In other words, ˆ
Ω2represents
Ω2if the Jaccard index J(Ω2,ˆ
Ω2)≈1, where J(Ω2,ˆ
Ω2) = ∥Ω2∩ˆ
Ω2∥A
∥Ω2∪ˆ
Ω2∥Aand ∥.∥A
measures the area. In Section 6.1, we discuss the Jaccard index based on the
volume coverage for our 3D models.
If we connect the tips of the up and down spokes, we obtain the chordal
structure as a set of non-crossing line segments connecting ∂ˆ
Ω+
2and ∂ˆ
Ω−
2as
discussed by Giblin and Brassett (1985). Thus, ˆ
Ω2is a 2D GC with cross-
sections defined as the chordal structure of ˆ
Ω2. Note that the chordal skeleton
depicted in black in Figure 7 (g) (i.e., the union of the middle of the chords) is
slightly different from the medial skeleton of Figure 7 (f) (see (Giblin and Bras-
sett,1985)). In the next step, we stretch the chords until they reach the actual
boundary ∂Ω, as shown in Figure 7 (g). We consider the stretched chords as
semi-chords. The semi-chords represent the cross-sections of the 2D GC Ω2,
and the curve connecting the middle of the semi-chords, called the semi-chordal
skeleton, represents the center curve of Ω2. The semi-chords may intersect
somewhere between the implied boundary ∂ˆ
Ω2and the actual boundary ∂Ω2.
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14 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
In this case, we trim the semi-chords based on the point of intersection. How-
ever, the trimming stage can be skipped if the relaxed CMS has low curvature
everywhere (because based on the RCC, r→ ∞ when κ→0). Thus, the semi-
chordal structure (i.e., semi-chords plus the semi-chordal skeleton) satisfies the
RCC. The implied boundary of the semi-chordal structure (i.e., the envelope
of the cross-sections) represents the actual boundary even though they are
not exactly the same. Similar to m-rep, a good semi-chordal structure should
approximate the actual boundary with the Jaccard index as ≈1.0. The semi-
chordal structure can also be used for straightening the 2D GC, as discussed
in SUP. Finally, based on the discussion of Srivastava and Klassen (2016) on
curve registration, we can choose corresponding semi-chords (e.g., based on
curve length registration) across a sample of 2D GCs. In this sense, we have
the dss-rep of the 2D GC as shown in Figure 7 (h).
4.2 Fitting a swept skeletal structure to an SlO
For fitting a dss-rep to an SlO, similar to fitting a dss-rep to a 2D GC, we
divide ∂Ω3into the top and bottom parts ∂Ω±
3.Figure 8 shows the top and
bottom parts of a mandible (without coronoid processes), a caudate nucleus,
and a hippocampus. We consider the border between ∂Ω+
3and ∂Ω−
3as the
Fig. 8 Visualization of SlOs. Blue and yellow indicate the top and bottom parts of a
mandible (in two angles), a caudate, a hippocampus, and an ellipsoid from left to right.
crest denoted by ∂Ω0
3as shown in Figure 6 (a). Thus, ∂Ω0
3corresponds to the
crest of an eccentric ellipsoid, which is an ellipse, i.e., the intersection of the
ellipsoid’s first principal plane with its boundary. Based on ∂Ω±
3we obtain the
CMS, as depicted in Figure 6 (d) and Figure 9 (Left). We relaxed the CMS by
fitting a smooth surface close to it (e.g., based on spline surface fitting (Lee
et al.,1997)). We consider the relaxed CMS as the skeletal sheet, as visualized
in Figure 9 (Right).
Fig. 9 Left: The CMS of a caudate. Right: Two views of the relaxed CMS.
Based on our observations, the skeletal sheet of most brain objects has low
curvature everywhere, and its flattened version can be seen as 2D GC with two
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 15
vertices. We call an SlO with such a skeletal sheet a regular SlO; otherwise,
an irregular SlO.
4.2.1 Regular SlO
For fitting the spine (and consequently slicing planes) of a regular SlO, the
idea is to flatten the skeletal sheet based on a suitable manifold dimensionality
reduction method to become a 2D GC. Then, we fit the center curve of the 2D
GC, map the fitted center curve to the skeletal sheet, and consider it as the
SlO’s spine. In this sense, the spine is a curve on the skeletal sheet connecting
two vertices of the SlO corresponding to the curve connecting the two vertices
of the flattened skeletal sheet.
Fig. 10 Left: (1) Skeletal sheet of a caudate in red. (2) The PCA projection of the skeletal
sheet as a 2D GC. (3) Relaxed CMS and the implied boundary of the 2D GC. (4) Semi-
chordal skeletal of the 2D GC. Middle: Visualization of the slicing planes along the spine
of a caudate (top) plus the skeletal sheet of the object (bottom). Right: cross-sections and
their center curves as the intersection of the slicing planes with the skeletal sheet.
An appropriate dimensionality reduction method preserves the structure
of the high-dimensional data properly in the mapped low-dimensional data
(Van der Maaten and Hinton,2008). For example, principal component analysis
(PCA) (Hotelling,1933) is a suitable method when the data has an approx-
imately flat structure relative to the first PCA principal plane (i.e., a plane
that is expanded by the first and second eigenvectors originated at the data
centroid). In other words, the flattened version of a 3D surface (flattened by
PCA) should not be significantly different from its original version. There are
various ways to quantify the irregularity or non-flatness of a surface (Bosch´e
and Guenet,2014;Haitjema,2017;Mik´o,2021). In this work, we quantify the
irregularity based on the maximum local curvature.
For an entirely flat 2D surface, the absolute value of the two principal curva-
tures (i.e., the eigenvalues of the second fundamental form) are zero everywhere
(Pressley,2013). Assume Mas a 2D surface. There is a point on Mwith abso-
lute average curvature κmax ∈[0,∞) such that ∀p∈M;κp≤κmax , where κp
is the average absolute value of the two principal curvatures at p. Therefore,
κmax can be assumed as the total curvature of the surface. Thus, the irregu-
larity of Mcan be strictly quantified as 2 arctan(κmax )/π ∈[0,1]. Obviously,
Mis entirely flat if it has zero irregularity.
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16 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
Let ˜
MΩ3be the skeletal sheet of Ω3. In this work, we say ˜
MΩ3is semi-flat if
its irregularity is less than 0.01. Let ˜
MΩ3be semi-flat and assume the mapping
F:˜
MΩ3→˜
M′
Ω3as the orthogonal projection that maps ˜
MΩ3to the PCA
first principal plane along the third principal axis. We say the semi-flat ˜
MΩ3
is PCA flatable if Fis a diffeomorphism (i.e., Fis a topological preservative
bijective mapping between ˜
MΩ3and ˜
M′
Ω3). We consider Ω3as a regular SlO
if ˜
MΩ3is semi-flat, PCA flatable, and ˜
M′
Ω3can be seen as a 2D GC.
Let Ω3be a regular SlO, and Γ′
2be the center curve of the embedded 2D
GC ˜
M′
Ω3. We consider F−1(Γ′
2) as the spine of Ω3. Also, if cis a semi-chord of
˜
M′
Ω3we consider F−1(c) as a non-linear semi-chord of ˜
MΩ3. Note that since
we apply orthogonal projection, both cand F−1(c) are located on a plane.
Therefore, we can consider these planes as the slicing planes of Ω3.Figure 10
shows the cross-sections of a caudate obtained based on the chordal structure
of PCA projection of the skeletal sheet.
As depicted in Figure 11, the semi-chordal skeleton (i.e., the union of mid-
dle points of the semi-chords) is usually wavier than the CMS, which is not
desirable as it violates the skeleton tidiness that we discuss in Section 6.3.
Therefore, we prefer to consider Γ′
2as the relaxed CMS of ˜
M′
Ω3. Further, if
the spine (i.e., F−1(Γ′
2)) has low curvature everywhere and is not very wavy,
we can consider cross-sections perpendicular to the spine based on the spine
moving Frenet frame as discussed by Ma et al. (2018). Figure 11 illustrates the
spine based on the semi-chordal skeleton (left column), relaxed CMS (middle
column), and relaxed CMS such that cross-sections are normal to the spine
(right column).
Fig. 11 Fitted tree-like structure of the skeletal sheet of a caudate. Left: Slicing planes
and the spine are based on chordal structure. Middle: Slicing planes are based on chordal
structure, but the spine is based on the relaxed CMS. Right: The spine is based on the
relaxed CMS, and the slicing planes are normal to the spine.
4.2.2 Irregular SlO
But what if the skeletal sheet is not semi-flat? Theoretically, the skeletal sheet
of an irregular SlO can be bent and twisted with an extremely curvy structure.
However, there are irregular SlOs, such as the mandible, whose skeletal sheet
has low local curvature. Thus, the skeletal sheet can be properly flattened. We
consider such irregular SlOs as treatable SlOs if the flattened skeletal sheet
can be seen as a 2D GC. Therefore, analogous to a regular SlO, we consider
the inverse map of the center curve of the flatted skeletal sheet, i.e., F−1(Γ′
2)
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 17
of a treatable SlO as the spine of the object, where Fis a proper embedding.
In this case, we define cross-sections normal to the spine.
From initial studies, we found the t-distributed stochastic neighbor embed-
ding (t-SNE) method (Van Der Maaten,2014) suitable for flattening the
skeletal sheet of treatable SlOs. The t-SNE maps high-dimensional data to a
lower-dimensional space in such a way that preserves the local structure and
relationships between data points as much as possible. It is similar to the SNE
of Hinton and Roweis (2002), but instead of using a Gaussian distribution to
model these relationships, it employs the t-distribution (Bunte et al.,2012).
Figure 12 illustrates the skeletal sheet of a mandible (Left), its t-SNE flattened
version as a 2D GC plus the center curve in 2D (Middle), and the mandible
with the slicing planes along its spine (Right).
Fig. 12 Left: Skeletal sheet of a mandible (without coronoid processes). Middle: Flattened
skeletal sheet by t-SNE (left), and center curve of the flattened skeletal sheet (right). Right:
Slicing planes of the mandible along the spine.
Often, the complexity of the model fitting problem can be reduced by
partitioning an irregular SlO into several regular SlOs. For example, skeletal
analysis of a mandible can be based on skeletal models of two segmented
hemimandibles (AlHadidi et al.,2012) as regular SlOs (see Figure 14 (d)).
However, since the focus of this article is on regular SlOs, we leave further
discussions about the generation of slicing planes along the spine of irregular
SlOs to our further studies.
5 Parameterization
As Lele and Richtsmeier (2001) discussed, it is crucial for statistical shape
analysis that the shape representation is invariant to rigid transformations as
shape analysis based on noninvariant shape representation is alignment depen-
dent, while the act of alignment makes the analysis biased and misleading. For
example, hypothesis testing to detect local dissimilarity between two groups of
objects based on a noninvariant shape representation (e.g., SPHARM-PDM)
introduces a large number of false positives and false negatives (Taheri and
Schulz,2022). On the other hand, explaining the type of dissimilarity, such as
protrusion, bending, and twisting, is essential for medical researchers. Fortu-
nately, the ds-rep can be parameterized such that the shape representation is
invariant and is able to explain the type of dissimilarity. In this regard, Taheri
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18 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
and Schulz (2022) discussed local frames for ds-reps and introduced the LP-ds-
rep as an invariant skeletal shape representation based on a tree-like structure
of the skeletal sheet. The tree-like structure of LP-dss-rep is established based
on the spine and a set of non-intersecting curves emanating from the spine
called veins, where the spine and veins are located on the skeletal sheet. The
veins connect the spine to the SlO’s crest (analogous to Figure 13). In this
sense, the spine and veins define paths for a moving frame on the skeletal sheet.
By considering veins as the center curves of the non-intersecting cross-sections,
we can define a tree-like structure for the skeletal sheet analogous to LP-ds-
rep. Therefore, following the idea of Taheri and Schulz (2022), we parameterize
the dss-rep to introduce an invariant shape representation called LP-dss-rep.
5.1 LP-dss-rep
Fig. 13 Tree-like structure of the skeletal sheet of a caudate, including local frames in
yellow and connection vectors in blue.
Let ˜
MΩ3be the skeletal sheet of the regular SlO Ω3. For each point p
inside ˜
MΩ3, there is a line segment based on the semi-chordal structure of the
cross-sections that contains pwith endpoints at ∂Ω±
3. We consider one up and
one down spoke along the line segment with a tail at pand tips on ∂Ω3+
and ∂Ω3−, respectively. Thus, the length of the up and down spokes reflect
the local width of Ω3, and their difference reflects the local symmetry relative
to ˜
MΩ3(i.e., higher differences indicate higher local asymmetry). Assume the
middle point of the spine is the skeletal-centroid, as depicted in Figure 13.
Thus, the spine can be seen as two curves with the starting point at the
skeletal-centroid. Also, the center curve of each cross-section consists of two
typically non-straight curves called veins emanating from the spine. Thus, we
can consider a vein’s starting point on the spine and its ending point on the
SlO’s crest. Since pis on the skeletal sheet, it belongs to a curve γ⊂˜
MΩ3,
where γis the spine or a vein. We define the orthogonal local frame at pas
Fγ
p= (nγ
p,bγ
p,bγ⊥
p)∈SO(3), where npis normal to ˜
MΩ3,bγ
pis the velocity
vector tangent to γ, and bγ⊥
p=nγ
p×bγ
p. For points on the intersections of
the spines and cross-sections’ center curves (i.e., veins’ starting points), we
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 19
consider γas the spine. We do not consider frames at the endpoints of the
spine and veins as they are at the edge of the skeletal sheet. Therefore, each
frame has only one parent frame but up to three children frames. A vector
connecting a frame to its parent is called a connection vector, with the tail at
the origin of the parent frame and the tip at the origin of the (child) frame. In
this sense, the union of the connection vectors has a tree-like structure along
the spine from the skeletal-centroid out and along the veins from the spine
out. Figure 13 illustrates the tree-like structure of a caudate’s skeletal sheet.
We define an LP-dss-rep as a tuple of geometric object properties (GOPs)
of the object as
s= (F∗
1, ..., F ∗
nf,v∗
1, ..., v∗
nc,u±∗
1, ..., u±∗
nf, v1, ..., vnc, r±
1, ..., r±
nf),(3)
or in the short form s= (F∗
i,v∗
j,u±∗
i, vj, r±
i)i,j , where i= 1, ..., nfand
j= 1, ..., nc,nfis the number of frames, ncis the number of connection vec-
tors, F∗
i∈SO(3) is the ith frame (orientation) based on its parent coordinate
system, v∗
j∈S2, and vj∈R+are jth connection vector’s direction and length,
where the direction is based on the local frame (i.e., a frame that tail of the
vector is located on), u±∗
i∈S2are the directions of the ith up and down
spokes based on their local frames, r±
i∈R+are the lengths of the up and
down spokes at the ith frame. Thus, an LP-dss-rep lives in a Cartesian product
of Euclidean symmetric spaces as
S= (SO(3))nf×(S2)nc+2nf×(R+)nc+2nf.(4)
Based on our model fitting u+∗
i=−u−∗
i, we can ignore down spokes’ direc-
tions. Thus, we rewrite s= (F∗
i,v∗
j,u+∗
i, vj, r±
i)i,j . Further, if the skeletal
sheet of a regular SlO has low curvature everywhere, we can consider up and
down spokes normal to the skeletal sheet. Therefore, the first frame element
at each point represents the direction of its up and down spokes, and we have
s= (F∗
i,v∗
j, vj, r±
i)i,j and consequently S= (SO(3))nf×(S2)nc×(R+)nc+2nf.
However, to avoid ambiguity, we stick to the definition of Equation (3) and
Equation (4) as they cover both the LP-ds-rep and the LP-dss-rep. Figure 14
depicts the fitted LP-dss-rep to a caudate (a), a hippocampus (b), a mandible
(c), and a hemimandible (d).
Fig. 14 Fitted LP-dss-rep to a caudate (a), a hippocampus (b), a mandible (c), and a
hemimandible (d), including local frames and spokes.
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20 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
The LP-dss-rep can be used for both shape analysis (i.e., after removing
the size) and size-and-shape analysis (i.e., by preserving the size) (Dryden and
Mardia,2016). Since the LP-dss-rep is invariant to rigid transformation, for
shape analysis, we remove the scale by dividing the vectors’ lengths by the
size of the LP-dss-rep, which we call LP-size (Taheri and Schulz,2022). The
LP-size is the geometric mean of the vectors’ length as
∥s∥LP = exp ( 1
2nf+nc
(
nc
X
j=1
ln(vj) +
nf
X
i=1
ln(r+
i) +
nf
X
i=1
ln(r−
i))).
In Section 8, we analyze real data based on LP-dss-rep shape analysis according
to features given earlier.
6 Goodness of fit
In Section 4, we discussed the SlO’s relaxed skeletal structure by fitting the
skeletal sheet close to the CMS and defining the spine by fitting a smooth
curve close to the relaxed CMS of the flattened skeletal sheet. Thus, we need to
explain the level of relaxation. Also, we explored the possibility of choosing dif-
ferent spines based on the chordal skeleton or the relaxed CMS of the flattened
sheet (see Figure 11). In addition, we know by definition the swept skeletal
structure of an SlO is not unique. Therefore, we need to discuss which model
is superior for establishing correspondence and statistical analysis. Obviously,
we prefer a model that represents the object such that the implied bound-
ary of the skeletal model approximates the actual boundary. Also, the model
should be locally as symmetric as possible because symmetricity is the key
element of any skeletal structure. Further, it should be tidy to avoid false
positives in the analysis, as we discuss in Section 8. Thus, it is reasonable to
search for a superior model with a more tidy and symmetric structure. This
can be done by tuning the model fitting procedure based on the flexibility of
the curve or surface fitting methods (e.g., by modifying the number of basis
functions in B-spline or Fourier expansion (Kokoszka and Reimherr,2017) or
by changing the degree of the polynomial in polynomial regression (Jorgensen,
1993;Montgomery et al.,2015)). In this section, we discuss the goodness of fit
of an LP-dss-rep by considering its volume-coverage,skeletal-symmetry, and
skeletal-tidiness. Then, in Section 6.4, we discuss our fitting strategy by an
example.
6.1 Volume-coverage
Let sbe a fitted LP-dss-rep to SlO Ω, where its spokes’ tips are located on
∂Ω±, and the endpoints of its veins are located on ∂Ω0. Therefore, we have a
point cloud of the spokes’ tips and veins endpoints representing the implied
boundary of s. We can use this point cloud to generate the implied boundary
as a triangular mesh, e.g., by (Pateiro L´opez,2008), or simply by collapsing
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 21
the boundary mesh representing Ω on the point cloud (i.e., by displacing the
position of the boundary point p∈∂Ω to the nearest point of the point cloud).
Alternatively, we can use the quadrilateral patches of the skeletal sheet to
interpolate spokes and generate the implied boundary as discussed by Han
et al. (2006); Liu et al. (2021).
Let ˆ
Ω be the object generated by the implied boundary. One important fac-
tor for the goodness of fit is to see how well ˆ
Ω approximates Ω. In other words,
we prefer a model in which the implied boundary is as close as possible to the
actual boundary. Thus, we consider the volume-coverage as the Jaccard index
of the model as J(Ω,ˆ
Ω) ∈[0,1], where J(Ω,ˆ
Ω) = 1 reflects identical volume.
Obviously, we prefer models with higher volume-coverage (see Figure 15).
6.2 Skeletal-symmetry
According to the definition of SlOs, the spine is located on the center curve of
each cross-section, approximately at its middle point. The union of the cross-
sections’ center curves forms the object’s skeletal sheet. Thus, the skeletal
sheet represents the SlO’s skeleton, and the spine can be seen as a nonlinear
skeleton of the skeletal sheet (Taheri and Schulz,2022).
In LP-dss-rep, for each up spoke s(p,u+)we have a down spoke s(p,u−)
with lengths r+=∥s(p,u+)∥and r−=∥s(p,u−)∥, respectively. Also, the spine
divides the skeletal sheet into two parts, namely, the right side and the left
side. For each vein ν+
psat the right side, we have a coplanar vein on the left
side ν−
pswith a common starting point psp on the spine. Thus, in a symmetric
model, we expect r+∼r−, and ∥ν+
psp ∥g∼ ∥ν−
psp ∥g, where ∥.∥gmeasures the
curve length. Assume we have nvveins on each side of the spine. Assume vector
l+= (r+
1, ..., r+
nf,∥ν+
1∥g, ..., ∥ν+
nv∥g)T, where ν+
tis the tth vein on the right side
and t∈ {1, ..., nv}. Similarly, assume l−= (r−
1, ..., , r−
nf,∥ν−
1∥g, ..., ∥ν−
nv∥g)T.
For simplicity, we write l+= (l+
1, ..., l+
nf+nv)Tand l−= (l−
1, ..., l−
nf+nv)T.
We define skeletal length as ls=Pnf+nv
i=1 l+
i+Pnf+nv
i=1 l−
iand define the
weight vector w= (w1, ..., wnf+nv)Tsuch that wi=l+
i+l−
i
ls. Then, we define
skeletal-symmetry as Pnf+nv
i=1 wi·fi, where fi=l+
i
l−
i
if l−
i≥l+
iand otherwise
fi=l−
i
l+
i
. Thus, skeletal-symmetry is ∈[0,1]. If we have a perfect symmetry
(i.e., ∀i:l+
i=l−
i), then skeletal-symmetry is 1.0.
6.3 Skeletal-tidiness
A model with good volume-coverage and skeletal-symmetry may still be messy.
Obviously, we prefer skeletal models that are as tidy as possible for two main
reasons. First, untidy models increase the number of false positives and make
the analysis misleading because the analysis is a frame-by-frame analysis. Since
we consider frames on the skeletal sheet, even a small perturbation on the
skeletal sheet violates frame correspondence and hugely affects the results, as
we demonstrate by a simple example in Section 8.2.1. Thus, we prefer a model
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22 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
such that the spine and cross-sections’ center curves have the lowest possible
curvature. Second, as discussed in Section 1, the center curve at each point
must satisfy the RCC.
Let us assume a moving frame Fγon an open curve γ(representing the
spine or a vein) located on the skeletal sheet M. In discrete format we assume
a sequence of N+ 1 equidistant points p0, ..., pNon γand define average
perturbation of γas ζ(γ) = 1
NPN
i=1 dg(qi−1,qi)∈[0,π
2], where qi∈S3is the
unit quaternion representation of the frame Fγ
pi, and dg(x,y) = arccos(|xTy|)
is the first quadrant geodesic distance (Huynh,2009). Thus, ζ(γ) is the mean
integrated rotation of the moving frame along γwhen N→ ∞.
Assume a discrete skeletal sheet Mwith a spine γs, spinal points
p0, ..., pN+1, and Ncross-sections’ center curves {γi}N
i=1 such that γs∩γi=pi.
We know ∀i:nγs
pi=nγi
pi. We define the degree of rotation of the ith cross-
section relative to the spine by dr(γs, γi) = dg(b⊥γs
pi,bγi
pi). Finally, the average
perturbation of the skeletal sheet Mis defined as
ζ†(M) = 2
(2N+ 1)π ζ(γs) +
N
X
i=1
ζ(γi) +
N
X
i=1
dr(γs, γi)!∈[0,1),
and the average tidiness of Mas 1 −ζ†(M). Thus, the average tidiness of
the ellipsoid’s skeleton is 1.0 if the spine is the ellipsoid’s major axis, and
the cross-sections (with straight center curves) are normal to the spine, i.e.,
∀i:dr(γs, γi) = 0.
In practice, we need the skeletal sheet as tidy as possible. Thus, it is reason-
able to consider ζ(γ) = 2
πmax{dg(qi−1,qi)}N
i=1 and consequently define strict
tidiness as 1 −ζ‡(M) to judge a model based on its most disordered element,
where ζ‡(M) = 2
πmax{ζ(γs),max{ζ(γi)}N
i=1,max{dr(γs, γi)}N
i=1}. We define
the goodness of fit score as the multiplication of volume-coverage, skeletal-
symmetry, and (average or strict) tidiness. In this sense, a model is superior if
its score is closer to 1.0.
6.4 Example
The flexibility of our method comes from fitting the skeletal sheet ˜
MΩ3and the
center curve of the flattened skeletal sheet Γ′
2. Although it is possible to define
a standard approach to make the fitting procedure straightforward by, for
example, fitting ˜
MΩ3using the B-spline surface fitting of Lee et al. (1997) and
fitting Γ′
2using the principal curve fitting of Hastie and Stuetzle (1989), such
a standard approach might not be flexible enough to generate distinct LP-dss-
reps for an SlO. Therefore, in this work, we prefer to use polynomial regression
(PR). Thus, by changing the PR degree from 1 to n(where we consider n≤7),
we can fit n×ndifferent LP-dss-reps to an SlO. Then, we select the best model
based on the goodness of fit. We leave a detailed discussion about choosing the
best curve or surface fitting method (from various available methods such as
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 23
implicit polynomial (Unsalan and Er¸cil,1999), nonlinear regression (Ritz and
Streibig,2008), etc.) for our future studies.
Figure 15 depicts 5 (from 49) fitted LP-dss-reps to a hippocampus at the
top row and their corresponding implied boundary at the bottom row in blue.
The PR degrees for fitting the ˜
MΩ3and Γ′
2are shown in Table 1. Note that
the boundary division and, consequently, the CMS for all five samples are the
same. For the fit #1 we have a straight line segment as the spine on a perfectly
flat skeletal sheet. Although the model is perfectly tidy, its volume-coverage
and skeletal-symmetry are weak, resulting in a low model score; see Table 1,
where score 1 is based on average tidiness and score 2 is based on strict tidiness.
By adding more flexibility to the spine and the skeletal sheet, we obtained
models #2-5. The best skeletal-symmetry and score 1 value belong to model
#5. However, as it is obvious from Figure 15 (5), we have a few sudden changes
in spinal frames (i.e., cross-sections with high rotation degrees), which reduces
the strict tidiness. In contrast, based on score 2, model #3 is the winner as
it has good volume-coverage (≈90%), skeletal-symmetry (≈87%), and strict
tidiness (≈90%) as we do not observe any extreme changes in frame rotations.
Fig. 15 Top row: Fitted LP-dss-reps with parameters given in Table 1. Bottom: Implied
boundary in blue.
Table 1 Goodness of fit.
Fit PR degrees
˜
MΩ3, Γ′
2
Volume-coverage skeletal-symmetry Average tidiness ζ†Strict tidiness ζ‡Score 1 Score 2
#1 1, 1 0.812 0.724 1 1 0.588 0.588
#2 3, 4 0.877 0.883 0.969 0.861 0.750 0.667
#3 4, 4 0.890 0.873 0.973 0.890 0.756 0.692
#4 4, 5 0.876 0.861 0.972 0.870 0.733 0.656
#5 7, 7 0.878 0.894 0.971 0.735 0.762 0.577
7 Hypothesis testing
One of the main objectives of LP-dss-rep analysis is to detect dissimilarities
between two groups of objects. We explain hypothesis testing for LP-ds-reps
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24 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
similar to LP-dss-reps (Taheri and Schulz,2022). To do so, in this section,
we discuss global and partial LP-dss-rep hypothesis testing based on LP-
dss-rep Euclideanization. In addition to the hypothesis testing, LP-dss-rep
classification is explained in SUP.
Let q∗
i∈S3be the unit quaternion representation of the ith frame (Huynh,
2009). From Equation (3) and Equation (4) we have s= (q∗
i,v∗
j,u±∗
i, vj, r±
i)i,j
and S= (S3)nf×(S2)nc+2nf×(R+)nc+2nf, where S3is the space of local
frames based on their unit quaternion representations. Therefore, for a pop-
ulation of LP-dss-reps, we have nc+ 3nfsets of spherical data (on S2and
S3) and nc+ 2nfsets of positive real numbers. The spherical data can be
Euclideanized by principal nested spheres (PNS) (Jung et al.,2012) or for
reducing the computational cost by projecting data to the tangent space (that
we use in this work) as discussed by (Kim et al.,2020). Given the map-
ping F:Sd→Rd,se= (F(q∗
i),F(v∗
j),F(u±∗
i),log vj,log r±
i)i,j is the
Euclideanized version of sfrom Equation (3) that lives on the product space
(R3)nf×(R2)nc+2nf×(R+)nc+2nf.
Now, assume A={se
Am}N1
m=1 and B={se
Bm}N2
m=1 are two groups of
Euclideanized LP-dss-reps (or LP-ds-rep) of sizes N1and N2. We can consider
the vectorized version of LP-dss-reps in the feature space (see classification
in SUP) and design a global test to compare Aand Bas two multivariate
distributions. Or we can consider LP-dss-reps as tuples and can compare the
two populations of tuples element-wise to detect locational dissimilarities, as
discussed by Taheri and Schulz (2022).
7.1 Global test
Let µAand µBbe the means of the sets Aand B, respectively. For the global
test, we test H0:µA=µBversus H0:µA=µB. Since the feature space is
a high dimensional space, we apply direction projection permutation (DiProP-
erm) (Wei et al.,2016) with distance weighted discriminator (DWD) (Marron
et al.,2007).
7.2 Partial test
The main objective of LP-dss-rep is to detect local dissimilarities. Thus, we
design partial hypothesis tests on the GOPs (of Equation (3)). Basically, when
the result of the global test shows significant dissimilarities, we apply partial
tests to detect and explain local dissimilarities.
Let ntbe the total number of GOPs. To test GOPs’ mean difference,
we design ntpartial tests. Let µA(k) and µB(k) be the observed sam-
ple mean of the kth GOP from Aand B, respectively. The partial test is
H0k:µA(k) = µB(k) versus H1k:µA(k)=µB(k), where k∈ {1, ..., nt}. For
the partial test, we can apply Hotelling’s T2test with normality assumption or
permutation test. Finally, we need to control false positives by methods such
as the family wise error rate of Bonferroni (1936) or the false discovery rate
(FDR) of Benjamini and Hochberg (1995) (BH).
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 25
8 Evaluation
The proposed LP-dss-rep is evaluated in comparison to the available LP-ds-
rep that is based on the fitting method of Liu et al. (2021) based on visual
inspection and statistical analysis.
8.1 Visual inspection
Fig. 16 Toy example for comparing the LP-ds-rep (top row) and the LP-dss-rep (bottom
row) of SlOs. Bold blue points are the spines’ endpoints.
For the visual inspection, first, we start with a few toy examples. To design
the toy examples, we slightly deform an ellipsoid to make an SlO as a reference
object, as illustrated in column (b) of Figure 16. The reference object can
be seen as an arm. By bending the reference object with different degrees of
bending at the elbow, we make two other SlOs, as depicted in columns (c)
and (d). To obtain LP-ds-reps, we used the SlicerSALT toolkit (Vicory et al.,
2018). Apparently, by increasing the degree of bending, the LP-ds-rep (i.e.,
top row) fails to define a good correspondence as the spine does not show
enough flexibility to bend according to the degree of bending we impose on the
reference object. Also, the structure of the skeletal sheet becomes more and
more chaotic. A possible explanation could be the poor boundary registration
or the effect of boundary deformation on the skeletal sheet. In contrast to the
LP-ds-rep, we can see the structure of the skeletal sheet in the LP-dss-rep is
tidier as depicted in the bottom row of Figure 16. Also, the spine shows more
flexibility, such that the endpoints of the spines (depicted by bold blue points)
are located at the points corresponding to the vertices of the ellipsoid.
Fig. 17 Fitted LP-ds-rep (Left) and LP-dss-rep (right) in a hippocampus in two angles.
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26 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
Analogous to the toy examples, we can see the same issues of the LP-ds-
rep on real data. Figure 17 illustrates the LP-ds-reps of hippocampus in two
angles with the LP-ds-rep (left), and the LP-dss-rep (right). It seems the spine
provides a better description of the SlO center curve in LP-dss-rep.
8.2 Statistical analysis
For the statistical analysis, we start with a simulation of a toy example before
analyzing real data.
8.2.1 Toy example
We design a very simple example by simulating two groups of fifty ellipsoids
based on randomly generated principal radii such that the ellipsoids of the first
group are perfectly symmetric. In contrast, in the second group, there is a small
protrusion on the boundary, as illustrated in Figure 18 (a). Figure 18 (d) and
(e) show the results of the partial tests from Section 7.2 on the fitted LP-ds-
reps and LP-dss-reps, respectively. Although both methods point to the area
where we expect to see the difference, the LP-dss-rep reflects the dissimilarity
more accurately based on the length of the top spokes in the critical region.
In fact, the LP-ds-rep introduces more significant GOPs such that we may
conclude that the left part of the two groups is totally different. The reason
is that there is a perturbation of the skeletal sheet caused by the boundary
deformation, as depicted in Figure 18 (b). In comparison, LP-dss-reps show
more resistance against the protrusion and thus better correspondence among
the skeletal sheets of the two groups (see Figure 18 (c)). The reason is that
the relaxed CMS is less sensitive to boundary protrusion.
Fig. 18 (a) Two groups of simulated ellipsoids with and without boundary protrusion. (b)
and (c) LP-ds-rep and LP-dss-rep of two samples from each group. (d) and (e) results of the
partial tests on LP-ds-rep and LP-dss-rep, respectively. Red indicates significant GOPs.
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 27
8.2.2 Real data
For the statistical analysis, we compare the left hippocampus of 182 patients
with early Parkinson’s disease (PD) with the left hippocampus of 108 healthy
people as a control group (CG). The data is provided by the Stavanger Uni-
versity Hospital (https://helse-stavanger.no) and the ParkWest study (http:
//parkvest.no). For this, LP-dss-reps are fitted to the hippocampi of both pop-
ulations and compared with global and partial hypothesis tests as described
in Section 7.1 and Section 7.2 to evaluate if there are differences between the
two populations.
Global test
Figure 19 depicts the result of the global test based on DiProPerm. The p-
value is less than 0.001 for LP-dss-reps and 0.103 for LP-ds-reps. Thus, given
a significance level of α= 0.05, PD and CG are statistically significantly
different based on LP-dss-reps. Notably, LP-dss-reps have a better performance
compared to LP-ds-reps based on the higher Z-score (6.54 vs. 1.32), which
indicates that the data are more separated for LP-dss-reps. This might be the
effect of the above-described stiffness of LP-ds-reps. In other words, generated
LP-ds-reps are relatively similar to each other, which is also reflected in the
classification results as discussed in SUP.
Fig. 19 DiProPerm plots of LP-dss-rep (top) and LP-ds-rep (bottom).
Partial tests
The partial tests are illustrated in Figure 20. Both the LP-ds-rep and the LP-
dss-rep show significant differences in the spinal connection vectors. The LP-
dss-rep introduces fewer significant connection vectors’ directions and frames’
orientations. This behavior supports our conclusion about the skeletal sheet
from the toy example.
9 Conclusion
In this work, we introduced a novel model of discrete swept skeletal represen-
tation for SlOs called the LP-dss-rep. The LP-dss-rep is designed to support
good correspondence between a population of objects, which is important for
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28 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
Fig. 20 Partial tests. Red indicates significant GOPs. Results based on the LP-dss-rep
(Top) and the LP-ds-rep (Bottom) before and after BH p-value adjustment in the first and
second rows, respectively. Significant level α= 0.05 and FDR=0.1.
statistical shape analysis. The fitting is based on dividing the SlO’s surface into
two parts, obtaining a skeletal sheet based on the relaxed central medial skele-
ton, and finding a tree-like structure of the skeletal sheet by the act of mapping
and inverse mapping between R3and R2. The LP-dss-rep is more flexible
in comparison with the currently available skeletal model, namely LP-ds-rep.
To have a standard measure for choosing the best LP-dss-rep, we intro-
duced goodness of fit criteria based on volume-coverage, skeletal-symmetry,
and skeletal-tidiness. The superiority of LP-dss-rep in hypothesis testing over
the available LP-ds-rep was demonstrated by visual inspection and statisti-
cal analysis on two sets of toy examples and real data. This suggests that by
having a proper goodness of fit in a population, the LP-dss-rep provides fewer
significant GOPs and a better description of local dissimilarities.
Acknowledgments
This work is funded by the Department of Mathematics and Physics of the
University of Stavanger (UiS). We thank Profs. James Damon (late of UNC),
J.S. Marron (UNC), and Jan Terje Kvaløy (UiS) for insightful discussions and
inspiration for this work. We also thank Prof. Guido Alves (UiS) for providing
ParkWest data.
Supplementary Material
SPHARM-PDM correspondence
As depicted in Figure 21, SPHARM-PDM fails to define a good correspondence
between the boundary of an ellipsoid and the bent versions of it, as the vertices
of the ellipsoid do not correspond to the vertices of the objects.
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 29
Fig. 21 SPHARM-PDM correspondence. Points with the same color are in correspondence.
Choosing δfor the affinity matrix
Choosing the value of δthat defines the affinity matrix can slightly change the
boundary division. However, based on our observation for most of the objects
of our study, δ= 0.5 is a reasonable choice. Figure 22 depicts the boundary
division of a caudate based on δas 0.1, 0.2, 0.5, 0.8, and 0.9, from left to right,
respectively
Fig. 22 From left to right, the boundary divisions are based on δas 0.1, 0.2, 0.5, 0.8, and
0.9, respectively.
Normality versus symmetricity
Fig. 23 (a) Cross-sections are normal to the medial skeleton. The RCC is violated. (b)
Cross-sections are normal to the center curve, but the center curve is not medial. (c) Cross-
sections are not normal to the center curve, but the center curve as the chordal axis traverses
the cross-section’s middle points.
In this section, we use Figure 23 (which is similar to Figure 3 of (Shani
and Ballard,1984)) to show even if a 2D GC has a non-branching smooth
medial skeleton defining a perfectly symmetric model with normality condition
is not feasible. In Figure 23 (a), the cross-sections are normal to the medial
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30 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
skeleton, but the RCC is violated. In Figure 23 (b), cross-sections are normal
to the center curve, but the center curve does not meet the middle of the
cross-sections. Thus, the model is not symmetric. The model of Figure 23 (c)
is perfectly symmetric as the center curve is the chordal skeleton. However,
the cross-sections are not normal to the chordal skeleton.
Importance of the RCC
In this section, we provide an illustration to show the importance of the RCC
in defining the spine of a swept skeletal structure. Figure 24 depicts two swept
skeletal structures of a hippocampus. In Figure 24 (Right), the RCC is satisfied
by the spine while in Figure 24 (Left) the RCC is violated. While the spines
in both figures exhibit notable similarity, it becomes apparent that defining
the spine solely as a curve positioned in the middle of the object is insufficient
without considering the RCC.
Fig. 24 Left: The RCC is violated. A few cross-sections are intersected within the object.
Right: The RCC is satisfied.
Issues with the curve skeleton
There are available methods for calculating the spine of SlOs as a curve skele-
ton. The curve skeleton, as defined by Dey and Sun (2006), is unique but
typically has a branching structure. Based on our experiences, most of the
recent methods for pruning and smoothing the curve skeleton, such as Lapla-
cian contraction of Au et al. (2008), mean curvature skeleton of Tagliasacchi
et al. (2012), L1-medial skeleton of Huang et al. (2013), and skeletonization via
local separators of Bærentzen and Rotenberg (2021); Bærentzen et al. (2023))
are more or less suitable for tube-like objects with circular cross-sections. But,
they do not provide a suitable spine for slabular objects. The primary issue
is that almost all of these methods blindly search for the curve skeleton with-
out considering important boundary properties, such as the crest. We believe
a proper spine should be on the skeletal sheet, connecting the SlO’s vertices
together, and be relatively in the middle of the cross-sections’. Based on our
observations, the mentioned methods do not provide such a spine, even for
simple SlOs. In addition, they entirely ignore the RCC. The RCC is not a pre-
requisite for calculating the curve skeleton because the curve skeleton does not
necessarily represent the center curve of a swept region. However, the RCC
is vital for defining the center curve of a swept region (see Figure 24). Based
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 31
on our observations, the curve skeleton exhibits unpredictable behavior inside
an SlO. For example, it bends and swings freely inside the object. Therefore,
defining corresponding skeletal structures based on the curve skeleton is highly
questionable.
For a better intuition, we provide a very simple example to discuss the
issue based on skeletonization via local separators of Bærentzen and Roten-
berg (2021); Bærentzen et al. (2023). In simple words, the method searches
for closed rings called local separators on the boundary. The center points
of local separators define the skeleton. Figure 25 shows the spine of two
SlOs with smooth boundaries. The obtained spines are based on the PyGEL
library (https://www2.compute.dtu.dk/projects/GEL/PyGEL). The spine of
each object is not satisfactory as it is non-smooth and perturbed. By consid-
ering a smooth version of the spine as depicted by a black curve, the spine still
suffers from critical issues. It seems the spine swings and bends freely towards
the left parts of the objects. Probably the reason is that the algorithm detects
the local separator as the blue curve rather than the yellow curve. On the
other hand, the spine violates the RCC.
Fig. 25 Visualisation of the spine of two SlOs based on the local separator skeleton. The
black curve represents the smooth version of the spine. The spines of both objects are bent
and swung toward the left side of the objects.
2D GC straightening
As depicted in Figure 26, we can use the chordal structure of a 2D GC to
straighten an object. Let {ci}n
i=1 be nconsecutive chords (i.e., cross-sections)
with points {pi}n
i=1, such that ∀i;piis the middle of ci. Let ∥ci∥be the length
of the ith chord. To accomplish the ob ject straightening we use a sequence of
points {p′
i}n
i=1 corresponding to {pi}n
i=1 on a straight line ℓsuch that ∀i, j,
∥p′
i−p′
j∥=dg(pi,pj), and line segments {c′
i}n
i=1 corresponding to {ci}n
i=1
such that ∀i,c′
iis perpendicular to ℓ, the middle of c′
iis on ℓ, and ∥c′
i∥=∥ci∥.
Classification
For classification, we follow the idea of composite PNS (CPNS) Pizer et al.
(2013). We normalize and vectorize the Euclideanized LP-dss-reps (or LP-ds-
reps). Thus, the feature space becomes R2nc+9nf(or R2nc+9nf+1 in case we
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32 Fitting the Discrete Swept Skeletal Representation to Slabular Objects
Fig. 26 Illustration of 2D GC straightening based on its chordal structure.
consider scaled LP-dss-reps and include the LP-size). In this sense, the problem
becomes a binary classification problem of two groups of vectors that can be
solved by, e.g., support vector machine (SVM)(Noble,2006).
For the classification of the real data from Section 7, we applied different
standard binary classification methods including k nearest neighbor (KNN),
SVM with linear and radial kernels, and naive Bayesian (NB) classification.
Further, we applied 10-fold cross-validation to evaluate the accuracy, speci-
ficity, and sensitivity of the outcomes based on Cohen’s kappa (Cohen,1960)
as shown in Table 2. As we expected, the classification based on both LP-ds-
rep and LP-dss-rep is not promising because, basically, we do not expect to
observe separable local distributions at the early stages of PD. However, over-
all, it seems that the LP-dss-rep has a better performance compared to the
LP-ds-rep according to both accuracy and Cohen’s kappa.
Table 2 LP-ds-rep classification based on 10-fold cross-validation.
Classes Model fitting Assessment
Classification methods
KNN SVM-linear SVM-radial NB
PD & CG
LP-ds-rep Accuracy 0.60 0.55 0.63 0.57
Kappa 0.10 0.05 0.06 0.14
LP-dss-rep Accuracy 0.63 0.63 0.60 0.61
Kappa 0.18 0.17 0.10 0.11
Declarations
Ethical Approval
Not applicable.
Competing interests
The authors declare no competing interests.
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Fitting the Discrete Swept Skeletal Representation to Slabular Objects 33
Authors’ contributions
Mohsen Taheri is the first author and writer of the manuscript. J¨orn Schulz
and Stephen Pizer have supervised the project. All authors read and approved
the final manuscript.
Funding
Not applicable
Availability of data and materials
Implementation of the manuscript’s methodology is available as R scripts on
https://github.com/MohsenTaheriShalmani/LP-dss-rep. Due to limited per-
mission to share the Parkwest data, we only provide R code for producing toy
examples and synthetic data in the repository.
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The Mean Shape under the
Relative Curvature Condition
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The Mean Shape under the Relative
Curvature Condition
Mohsen Taheri∗, Stephen M. Pizer†and J¨orn Schulz‡
May 28, 2024
Abstract
The relative curvature condition (RCC) serves as a crucial constraint, ensuring the
avoidance of self-intersection problems in calculating the mean shape over a sample
of swept regions. By considering the RCC, this work discusses estimating the mean
shape for a class of swept regions called elliptical slabular objects based on a novel
shape representation, namely elliptical tube representation (ETRep). The ETRep
shape space equipped with extrinsic and intrinsic distances in accordance with object
transformation is explained. The intrinsic distance is determined based on the intrin-
sic skeletal coordinate system of the shape space. Further, calculating the intrinsic
mean shape based on the intrinsic distance over a set of ETReps is demonstrated. The
proposed intrinsic methodology is applied for the statistical shape analysis to design
global and partial hypothesis testing methods to study the hippocampal structure in
early Parkinson’s disease.
Keywords: Elliptical Tube, Generalized Cylinder, Swept Skeletal Structure, Statistical
Shape Analysis, Slabular Objects, Relative Curvature Condition.
∗Mohsen Taheri, Department of Mathematics and Physics, University of Stavanger (UiS), Email:
mohsen.taherishalmani@uis.no
†Prof. Stephen M. Pizer, Department of Computer Science, University of North Carolina at Chapel
Hill (UNC), Email: pizer@cs.unc.edu
‡Assoc. Prof. J¨orn Schulz, Department of Mathematics and Physics, University of Stavanger (UiS),
Email: jorn.schulz@uis.no
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1 Introduction
Object shapes in 3D or 2D are important for statistical analysis, for example, in the char-
acterization of diseases of the human body. Such analysis is commonly based on moments,
especially means and covariances, on abstract metric spaces. For these spaces to support
these moments, paths on these spaces are best if they pass through only geometrically valid
shapes; distances of these paths then form the basis for calculating means and covariances.
Here we create such a shape space from an object representation provided by elliptical
tubes, which form a useful approximation of so-called slabular objects, and we focus on the
calculation of means.
Considering that a variety of human body parts like the kidney, mandible, and hip-
pocampus are slab-shaped objects known as slabular objects (SlOs) (Pizer et al.,2022;
Taheri and Schulz,2022), calculating the (sample) mean shape of such objects is crucial for
hypothesis testing and statistical inferences to reveal underlying patterns and differences
within SlO groups (e.g., patients vs. controls) (Fletcher et al.,2004;Styner et al.,2006;
Schulz et al.,2016;Taheri et al.,2023). However, calculating the SlO mean shape is not
straightforward as the mean shape may violate SlO conditions or might be an inappropriate
representative of the sample. In this work, we explore the problem and propose a solution
for a class of SlOs called elliptical SlOs (E-SlOs) by discussing their shape, shape space,
and shape distance.
An SlO is a swept region based on a smooth sequence of slicing planes along the SlO’s
center curve called the spine such that the slicing planes do not intersect within the object.
Each cross-section (i.e., the intersection of a slicing plane with the object) can be seen as a
2-dimensional (2D) generalized tube1with a (planar) center curve, and the spine of the SlO
intersects each cross-section roughly at the midpoint of the cross-section’s center curve.
For a 2D generalized tube, the center curve can be considered as the skeleton2, and
the cross-sections are line segments divided by the skeleton. Thus, each line segment can
be seen as two vectors with opposite directions and a common tail position (see Figure 1
1A generalized tube or a generalized cylinder is a swept region defined by a center curve where each
cross-section is a star-convex set such that the center curve intersects a star center of each cross-section
(Ballard and Brown,1982;Damon,2008;Ma et al.,2018b).
2The skeleton of an object is a curve or a sheet (or a combination of both) that can be understood as
a locally centered manifold obtainable by the process of continuous contraction (Siddiqi and Pizer,2008;
Bærentzen and Rotenberg,2021).
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(right)). In this sense, a 2D generalized tube has a skeletal structure as a set of non-
intersecting vectors called skeletal spokes emanating from the skeleton with tails on the
skeleton and tips on the boundary. The skeletal structure can be seen as a radial vector
field that defines an outward flow from the skeleton to the boundary (Damon,2003).
Since an SlO is the union of its cross-sections that are 2D generalized tubes, we consider
aswept skeletal structure for the SlO as the union of the cross-sections’ skeletal structures
(Damon,2008;Taheri et al.,2023). In this sense, SlO’s swept skeletal structure consists of
the SlO’s skeleton called the skeletal sheet as the union of the cross-sections’ center curves
and the vector field on the skeletal sheet as the union of the skeletal spokes.
Theoretically, SlO analysis could be highly challenging because an SlO as a swept region
can be tangled like a knot with an extremely complex structure. However, a fraction of SlOs
in a human’s body, including most of the brain’s subcortical structures (e.g., hippocampus
and caudate nucleus), can be seen as E-SlOs. We define an E-SlO as an SlO that can be
properly inscribed inside an optimal elliptical tube (E-tube), as we discuss in Section 2,
where an E-tube is a 3D generalized tube such that all its cross-sections are elliptical discs
with non-zero eccentricity centered at the tube’s center curve. Figure 1 illustrates a swept
skeletal structure of the hippocampus as an E-SlO inscribed inside an optimal E-tube.
Figure 1: Visualization of a left hippocampus as an E-SlO inscribed inside an optimal E-tube. The right
figure illustrates the skeletal structure of a cross-section with its center curve. The skeletal spokes are
vectors with tails on the center curve. The cross-section is inscribed inside its corresponding elliptical
cross-section of the E-tube.
A suitable shape or shape representation should establish locational correspondence
among a population of objects based on their common geometric characteristics (Laga
et al.,2019). For example, assume a sample of mE-SlOs as {Pj}m
j=1. In a naive ap-
proach, each E-SlO can be represented by a distribution of npoints called the point dis-
tribution model (PDM) on the object’s boundary as the n-tuple Pj= (pj1, ..., pjn ), where
∀i∈ {1, ..., n};pji ∈R3(Styner et al.,2006). That is, the E-SlOs are in a point-wise
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correspondence. By assuming that the centroids of the PDMs are located at the origin
of the global coordinate system (i.e., shapes are aligned such that ∀j;1
nPn
i=1 pji =0),
we can scale and represent each PDM as a unit vector as pj=vec(Pj)
∥vec(Pj)∥∈S3n−1, where
S3n−1={x∈R3n|∥x∥= 1}is the (n−1)-sphere, vec() is the vectorization operator, and
∥.∥is the Euclidean norm. Thus, each PDM becomes a point on the hypersphere (namely
Kendall’s pre-shape space) (Kendall,1984;Dryden and Mardia,2016), which is a manifold
equipped with the geodesic distance dg(x,y) = cos−1(x·y) as the shape distance, where
(·) is the dot product (Jung et al.,2012). Therefore, {Pj}m
j=1 can be seen as a distribution
of points {pj}m
j=1 on the manifold, and the mean shape (as the Fr´echet mean) is a point
on the manifold with the minimum sum of squared geodesic distances to all other points
as ¯
p= argminp∈S(n−1) 1
mPm
j=1 d2
g(pj,p) (Pennec et al.,2019).
However, an E-SlO is a swept region with the relative curvature condition (RCC) that
assures that the cross-sections do not intersect within the object region (Damon,2008;Ma
et al.,2018b). To the best of our knowledge, PDM analysis (as discussed by an example in
Supplementary Materials) and almost all common shape analysis methods such as elastic
shape analysis (Jermyn et al.,2017), functional shape analysis (Srivastava and Klassen,
2016), Euclidean distance matrix analysis (Lele and Richtsmeier,2001), persistence homol-
ogy (Gamble and Heo,2010), and even common skeletal-based methods (Fletcher et al.,
2004;Pizer et al.,2013;Taheri and Schulz,2022) do not take into account the important
property of RCC for calculating the mean shape.
Figure 2: Illustration of the curvature tolerance in 2D. By increasing the object’s width from left to right,
the cross-sections intersect within the object by violating RCC, as depicted in the right figure.
As depicted in Figure 2, the RCC defines curvature tolerance for a swept region (e.g.,
in 2D as κ < 1
r) such that the curvature (κ) of the center curve at each point cannot be
larger than the inverse of the object’s width (r) in the direction of the curve’s normal.
Analogously, in E-SlO, the slicing planes intersect within the object if the center curve
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violates the RCC (see Figures 3 and 5). The ultimate ob jective of this work is to define
the shape and shape space for E-SlOs by considering the RCC such that the shape is an
element of the shape space, and each element of the space is associated with a valid unique
E-SlO that satisfies the RCC. In this sense, by moving from one element to another element
of the space along a path, the object corresponding to the first element transforms into the
object corresponding to the second element.
Figure 3: Top row: Extrinsic transformation b etween the 3D E-tubes ΩE1and ΩE2. Bottom row:
Intrinsic transformation between E-tubes ΩE1and ΩE2. The middle column depicts the mean ob ject.
Without considering the RCC, the path can be defined extrinsically. However, we are
interested in determining the path intrinsically such that transformation associated with
the path does not violate the RCC and the path complies with the structure of the shape
space. For a better intuition, imagine two (valid) 3D E-tubes ΩE1and ΩE2(represented
by a sequence of elliptical disks), as depicted in Figure 3. The top row shows the extrinsic
transformation between ΩE1and ΩE2without considering the RCC based on an extrinsic
path between their shapes. The path is invalid because the transformation violates the RCC
with an obvious self-intersection. In other words, a part of the extrinsic path is located
outside the underlying shape space. The bottom row depicts an intrinsic transformation
based on an intrinsic path following the RCC that avoids self-intersection. We discuss the
underlying shape space, corresponding paths, and transformations in Section 4. Based on
the definition of the path, we can define the (extrinsic or intrinsic shape) distance and the
mean shape as a shape with the minimum sum of squared distances to the shapes of ΩE1
and ΩE2. We call the object corresponding to the mean shape the mean object. The middle
column of Figure 3 depicts the E-tube associated with the mean object of ΩE1and ΩE2.
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Note that the extrinsic mean object might be a valid object, but it might be an inap-
propriate representative of the population as its shape does not comply with the boundary
condition of the underlying shape space (as discussed by a simple example in Supplemen-
tary Materials). Also, we do not consider arbitrary object modifications to remove self-
intersections from the mean object, for example, by elongating or narrowing the objects or
by smoothing the boundary (Ma et al.,2018a). Such modifications can be considered as
arbitrary projections of the invalid shape into the underlying shape space.
To define the shape space and intrinsic path, we need a robust shape representation.
Since, for an E-SlO, there is a moving plane along the spine that sweeps the object’s skele-
tal structure, Taheri et al. (2023) introduced locally parameterized discrete swept skeletal
representations (LPDSSRep) based on a sequence of orthonormal material frames (Yang
et al.,2022;Giomi and Mahadevan,2010) (representing a moving frame) on the E-SlO’s
spine with one element tangent to the spine and one element normal to the skeletal sheet
(see Figure 4). The LPDSSRep is a powerful shape representation as it is invariant to the
act of rigid transformation and alignment-independent (i.e., translation and rotation do not
affect the shape representation). Besides, the E-SlO’s boundary can be reconstructed pre-
cisely based on the LPDSSRep’s implied boundary (i.e., the envelope of the skeletal spokes’
tips). However, considering the RCC, defining the LPDSSRep shape space and intrinsic
transformation is not straightforward because of the complex structure of the cross-sections
that are 2D generalized tubes. Since each cross-section of an E-SlO can be inscribed and
approximated by an ellipse, it seems reasonable to use a simplified version of the LPDSSRep
called elliptical tube representation (ETRep), where the 2D generalized tube cross-sections
are replaced with elliptical disks (see Figures 1 and 4). Thus, the ETRep is an E-SlO
representation as a discrete E-tube compatible with the object’s swept skeletal structure,
which enables us to explain the shape space and intrinsic transformation explicitly.
The structure of this work is as follows. Section 2 discusses basic terms and definitions
regarding the swept skeletal structure. Section 3 explains the discrete material frame that
we use to define the ETRep space based on the RCC. Section 4 shows how to calculate the
mean shape by demonstrating ETRep model fitting, RCC for ETReps, the ETRep space,
skeletal coordinate system, and extrinsic and intrinsic transformations. Section 5 discusses
statistical shape analysis of ETRep by applying global and partial hypothesis testings on a
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real data set to compare hippocampal differences between patients with Parkinson’s disease
versus a control group. Finally, Section 6 summarizes and concludes the work.
2 Basic terms and definitions
In this section, we review some basic terms and definitions regarding E-SlO and the swept
skeletal structure essential for defining the ETReps.
The (Cartesian) product space over sets O1, ..., Odis {(o1, ..., od)|∀i∈ {1, ..., d};oi∈Oi}
denoted by O1×... ×Odor if the sets are identical by (O1)d. The d-dimensional Eu-
clidean space is the product space (R)ddenoted by Rd, where Ris the set of real num-
bers. A point or a vector is an element of Rd>1denoted by small and bold letters. The
set Bd
r(p) = {x∈Rd|∥x−p∥ ≤ r}defines a d-dimensional closed ball in Rdwith center
p∈Rdand radius r∈R+, where R+= (0,∞) . Similarly, a d-dimensional open ball is
defined by Bd
<r(p) = {x∈Rd|∥x−p∥< r}.
Let the unit closed ball Bd
1(0) be denoted by Bd. We consider the set Ω ⊂Rdas
ad-dimensional object if it is homeomorphic to Bd(i.e., there is a continuous invertible
mapping between Ω and Bd(Gamelin and Greene,1999)). A point p∈Ω is an interior
point of Ω if ∃r∈R+such that Bd
r(p)⊂Ω. Let Ωin be the set of all interior points of Ω.
Then, the boundary of Ω is ∂Ω = Ω/Ωin (Siddiqi and Pizer,2008).
Assume an interior point p∈Ωin and a unit direction u∈Sd−1. If we start at pand
move straight forward in the direction of u, we ultimately reach the boundary at a point
q∈∂Ω. Such an interior path is a line segment called spoke, which can be seen as the
vector −→
pq. The set of skeletal spokes of Ω is a set of (non-crossing) spokes emanating from
the object’s skeleton denoted by MΩ. Thus, the skeletal structure of Ω is a field of skeletal
spokes Son MΩdenoted by (MΩ, S). The envelope of the skeletal spokes’ tips forms the
object’s (implied) boundary as ∂Ω = {p+S(p)|p∈MΩ}, where S(p) is the set of all
skeletal spokes with tail on p(Damon,2003;Pizer et al.,2013).
Now, assume Γ(λ) as a parameterization of a curve Γ in Rdsuch that λ∈[0,1], where
Γ(0) and Γ(1) denote the curves’ endpoints. Let Π(λ) denote a (d−1)-dimensional plane
crossing Γ(λ) normal to Γ at Γ(λ). In this work, a d-dimensional object Ω is a swept
region with the center curve Γ if Ω is a disjoint union of cross-sections Ω(λ)=Ω∩Π(λ)
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such that ∀λ; Ω(λ) is a (d−1)-dimensional object and Γ(λ) is a central point (e.g., the
barycentroid (Rustamov et al.,2009)) of Ω(λ). The skeletal structure (MΩ, S) is a swept
skeletal structure if for each p∈Ω(λ)∩MΩwe have S(p)⊂Ω(λ) (Damon,2008).
Aslab is a 3D object whose skeleton is a smooth 2D topological disk called the skeletal
sheet. An SlO is a slab with a swept skeletal structure such that each cross-section is a
2D generalized tube (Pizer et al.,2022;Taheri et al.,2023). Thus, any E-tube is also an
SlO with a developable3skeletal sheet where the center curve of each elliptical cross-section
is its major axis (based on the chordal locus of Brady and Asada (1984)). A semi-flat
SlO is an SlO whose entire skeletal sheet has relatively low curvature everywhere (i.e., the
total curvature of the sheet is close to zero, like an entirely flat surface). For example, a
mandible (without considering the coronoid process) is an SlO but not semi-flat, while a
hemimandible is a semi-flat SlO (Taheri et al.,2023).
An SlO Ω is an E-SlO if it can be inscribed inside an (optimal) E-tube ΩEsuch that
∀λ∈[0,1]; Ω(λ)⊂ΩE(λ) and the SlO’s spine be the E-tube’s spine Γ. Additionally, at
each spinal point Γ(λ), the material frame of both objects coincides (i.e., ∀λ∈[0,1] the
skeletal sheets of both objects are tangent at Γ(λ)), and the area of ΩE(λ) approximates
the area of ΩE(λ), i.e., the Jaccard index J(Ω(λ),ΩE(λ)) = |Ω(λ)∩ΩE(λ)|A
|Ω(λ)∪ΩE(λ)|A≈1, where |.|A
measures the area.
Based on the definition, the swept skeletal structure of an SlO provides insight into
whether the object qualifies as an E-SlO. However, calculating the swept skeletal structure
of SOs (like the mandible) that are not semi-flat is an open question. Therefore, in this
work, we provide examples and study E-SlOs regarding semi-flat SOs. Nevertheless, the
theoretical discussions are valid for E-SlOs in general.
3 Discrete moving frame
For statistical shape analysis, we usually consider the swept skeletal structure in the discrete
format to establish a meaningful correspondence (Van Kaick et al.,2011) among a popula-
tion of E-SlOs. The correspondence between different objects can be established by using
the curve registration (Srivastava and Klassen,2016) of the spine and a discrete moving
3A developable surface is a ruled surface that can be swept out by moving a line (Abbena et al.,2017).
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frame along the spine. Essentially, this method represents the spine as a finite set of points
that correspond with those of other objects. At each spinal point, there is an orthonor-
mal frame that defines the orientation of the cross-section at that point. Consequently,
we have a set of corresponding cross-sections. The size, skeletal structure, and orientation
of the corresponding cross-sections can be used for comparisons and statistical inferences.
Note that without considering the local frames, cross-sectional analysis analogous to the
approach of Apostolova et al. (2012) is not desirable as it is alignment-dependent (Taheri
and Schulz,2022). In this work, we define and use discrete Frenet frames4and discrete
material frames to deal with the RCC in E-SlO analysis.
Assume λias n+1 equally spaced points on [0,1] as λi=i
n, where i= 0, ..., n. Thus, the
spine of an E-SlO can be registered by a sequence of n+ 1 points pi= Γ(λi). We consider
the discrete Frenet frame associated with the point pias Ti= (ti,ni,n⊥
i)∈SO(3), where
ti=pi+1−pi
∥pi+1−pi∥,n⊥
i=ti−1×ti
∥ti−1×ti∥,ni=n⊥
i×ti, and i= 1, ..., n −1 (Lu,2013). Thus, niis
the normal of the spine at pi, which is coplanar with the triangle △pi−1pipi+1. For the
spine’s endpoints (i.e., p0and pn) we assume T0=T1, and Tn=Tn−1. Since the Frenet
frame is sensitive to flipping (specifically when the spine has zero curvature), as noted by
Carroll et al. (2013), we use the material frame for establishing correspondence.
Let γibe the center curve of the ith cross-section Ω(λi). Based on the definition of E-
SlO, we have a smooth sequence of cross-sections. Thus, the union ∪n
i=0γidefines a discrete
surface representing the orientable smooth skeletal sheet MΩwith two sides, namely the
positive and negative sides. We consider a discrete material frame on Γ(λi) consistent with
the orientation of MΩas Fi= (ti,ai,bi)∈SO(3), where tiis the first element of the ith
discrete Frenet frame, aiis the unit vector tangent to γiat pi, and ai×bi=ti. Therefore,
in an E-tube, aiand biare along the semi-major and semi-minor axes of the ith elliptical
cross-section, respectively (see Figure 4). There are two possible options to choose ai
and bi(as an ellipse has two semi-major and two semi-minor axes). We choose Fisuch
that ∀i;bilies on the positive side of the MΩ. In other words, (based on the right-hand
rule) the orientation induced by the material frame is consistent with the orientation of
MΩ(Guillemin and Pollack,2010). We consider Fi−1as the parent frame of Fi, where
4The Frenet frame, alternatively referred to as the TNB frame, is formed by the combination of the
tangent vector, normal vector, and binormal vector of a curve (do Carmo,2016).
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i= 1, ..., n, and we assume F0as the parent of itself.
To calculate the frame orientation based on its parent frame, we use the spherical
rotation matrix R(x,y) = I3+ (sin ψ)(y⊗z−z⊗y) + (cos ψ−1)(y⊗y+z⊗z), where
x,y∈S2,z=x−y(x·y)
∥x−y(x·y)∥,ψ=dg(x,y), I3= (e1,e2,e3) is the identity matrix based on
the unit vectors e1= (1,0,0), e2= (0,1,0), and e3= (0,0,1), and ⊗denotes the outer
product. Therefore, the rotation R(x,y) transfers xto yalong the shortest geodesic path
on S2such that R(x,y)x=y(Amaral et al.,2007). Assume frame F†= (t,a,b) be the
parent of ˜
F. We align F†to I3by R2R1F†=I3, where R1=R(t,e1), and R2=R(R1a,e2).
Thus, ˜
F∗=R2R1˜
Frepresents ˜
Fin its parent coordinate system. Let frame F∗be the
orientation of Fbased on its parent. To find Fbased on the global coordinate system I3,
we align F†to I3by R2R1F†=I3. Thus, F= [R2R1]−1F∗(Taheri and Schulz,2022).
The vector −−−−→
pi−1picalled the ith spinal connection with length xi=
−−−−→
pi−1pi
, connects
frame Fito its parent Fi−1. Thus, the spine can be locally parameterized by a sequence
of tuples as ((F∗
i, xi)i)n
i=0, where F∗
iis the ith material frame’s orientation based on its
parent frame. Note that by having a locally parameterized spine, we can reconstruct the
spine in 3D. Thus, the parameterization also provides Frenet frames. Based on the locally
parameterized spine, we define ETRep.
4 ETRep
In Sections 2 and 3, we discussed E-tubes. In this section, we define ETRep as an alignment-
independent shape representation for an E-tube. We explain ETRep model fitting based
on an optimal E-tube. Then, we discuss the ETRep shape space by explaining the extrinsic
and intrinsic paths. To define the intrinsic path, we discuss the RCC for ETReps. Further,
we explain the intrinsic sample mean shape based on the intrinsic skeletal coordinate system
of the shape space.
4.1 ETRep model fitting
An ETRep is a shape representation. Therefore, we need to discuss the ETRep model
fitting procedure to explain how an ETRep represents the actual object.
Let ΩE(λi) be the ith cross-section of the E-tube ΩEwith principal radii aiand bi.
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Since in the discrete format ΩE=∪n
i=0ΩE(λi), we consider the ETRep of ΩEas the
sequence s= (ωi)n
i=0, where ωi= (F∗
i, xi, ai, bi) is the representation of ΩE(λi) . Based on
the definition, an E-SlO Ω can be inscribed inside an optimal E-tube, as we defined in
Section 2. Let ΩEbe the optimal E-tube of Ω. Thus, we consider the shape representation
for the Ω as s= (ωi)n
i=0. In this sense, there is an optimal E-tube for Ω that its shape
approximates the shape of Ω.
To find the optimal E-tube of an E-SlO Ω, we calculate the swept skeletal structure of
Ω, as discussed by Taheri et al. (2023). Then, we inscribe Ω inside an E-tube ˜
ΩEsuch that
the spine and material frames of Ω and ˜
ΩEare the same and ∀λi; Ω(λi)⊂˜
ΩE(λi). Let
aiand bibe the principal radii of ˜
ΩE(λi). The area of each cross-section ˜
ΩE(λi) is πaibi.
Since shrinkage of ˜
ΩE(λi) does not violate the RCC in ˜
ΩE, we optimize the size of ˜
ΩE(λi)
by maximizing the Jaccard index J(˜
ΩE(λi),ΩE(λi)) (or alternatively by maximizing the
multiplication of the Jaccard index and the eccentricity of ˜
ΩE(λi)). This can be done by
(iteratively) reducing the value of aiand bito minimize aibias much as possible such that (in
each iteration) bi< aiand ˜
ΩE(λi) contains Ω(λi) (see Figure 1 (right)). By optimizing all
the cross-sections ∀λi, the ˜
ΩEbecomes an optimal E-tube ΩEfor Ω. Note that if Ω cannot
be inscribed inside an E-tube or after the optimization ∃isuch that J(˜
ΩE(λi),ΩE(λi)) is
relatively small (e.g., less than 0.8), then we do not consider Ω as an E-SlO. Figure 4
illustrates the ETRep of a hippocampus and a caudate nucleus.
Figure 4: ETRep of a hippocampus (left) and a caudate nucleus (right) in two angles. The spine is
depicted by a dark curve equipped with material frames. The cross-sections are non-intersecting elliptical
disks. The skeletal sheet is the union of the cross-section’s center curves (i.e., thier major axes).
4.2 Extrinsic approach
Based on the defined shape representation, we are in the position to discuss shape space and
shape distance. In this section, we ignore the RCC and explain the shape space equipped
with an extrinsic distance. Afterward, the intrinsic approach will be discussed.
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Without considering the RCC, ETRep s= (ωi)n
i=0 = ((F∗
i, xi, ai, bi)i)n
i=0 is living
on the product space (SO(3) ×(R+)3)n+1. Assume two ETReps s1= (ω1i)n
i=0 and
s2= (ω2i)n
i=0 associated with two E-tubes ΩE1and ΩE2, where ωji = (F∗
ji, xj i, aj i, bj i)i,
and j= 1,2. Let f∗
ji ∈S3be the unit quaternion representation of the
frame F∗
ji (Huynh,2009). Thus, s= (ωi)n
i=0 = (f∗
ji, xj i, aj i, bj i)iis living on
Sn+1 = (S3×(R+)3)n+1, which is a differentiable manifold because it is the product space
of finite sets of differentiable manifolds (Lee,2013;do Carmo,2016). The extrinsic
distance between s1and s2can be defined as ds(s1, s2) = Pn
i=0 d2
ω(ω1i, ω2i)1
2, where
dω=Pn
i=0 d2
g(f∗
1i,f∗
2i) +∥x1i−x2i∥2+∥a1i−a2i∥2+∥b1i−b2i∥21
2. Thus,
ds(s1, s2) =
n
X
i=0
d2
g(f∗
1i,f∗
2i) +∥x1i−x2i∥2+∥a1i−a2i∥2+∥b1i−b2i∥2
1
2
.(1)
We know the geodesic path between f∗
1iand f∗
2ion the unit sphere is
ζF(λ;f∗
1i,f∗
2i) = 1
sin(ψ)[sin(ψ(1 −λ))f∗
1i+ sin(λψ)f∗
2i],(2)
where ψ=dg(f∗
1i,f∗
2i), and λ∈[0,1] (Srivastava and Klassen,2016). We define the
extrinsic path from s1to s2as ζs(λ;s1, s2)=(ωλi )n
i=0 such that
ωλi = (ζF(λ;f∗
1i,f∗
2i), ζ(λ;x1i, x2i), ζ (λ;a1i, a2i), ζ(λ;b1i, b2i))i,(3)
where ∀x,y∈Rd;ζ(λ;x,y) = (1 −λ)x+λyis a straight path in the Euclidean space
Rd. Thus, the extrinsic transformation is based on an extrinsic path that converts each
cross-section of ΩE1to its corresponding cross-section of ΩE2.
In the discrete format, we assume k+ 1 steps for the transformation. The λcan be
considered as the arithmetic sequence λj=j
k∈[0,1] (i.e., 0,1
k, ..., k−1
k,1), where j= 0, ..., k.
Let sλj=ζs(λj;s1, s2). Therefore, the distance between s1and s2can be expressed as
ds(s1, s2) = limk→∞ Pk
j=1 ds(sλj−1, sλj) (which is identical to Equation (1)).
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Given a set of mobservations s1,..., sm∈Sn+1 . The extrinsic mean shape is
¯sext = argmin
s∈Sn+1
m
X
j=1
d2
s(s, sj).(4)
Following the approach of Taheri and Schulz (2022), the extrinsic sample mean can be
calculated as ¯sext = ( ¯ωi)n
i=0 such that ¯ωi= ( ¯
f∗
i,¯xi,¯ai,¯
bi)i, where ¯
f∗
iis the mean frame of
{¯
f∗
ji}m
j=1 that is obtainable based on principal nested sphere analysis (PNS) of Jung et al.
(2012) (or alternatively based the approach of Moakher (2002)), and ¯xi, ¯ai, and ¯
biare the
arithmetic (or geometric) means of {xji}m
j=1,{aji }m
j=1, and {bji }m
j=1, respectively. Thus, the
mean ETRep represents an E-tuple based on the mean of the corresponding cross-sections.
As we mentioned in Section 1, if the extrinsic mean object is invalid, it is almost always
possible to make it valid based on object modifications (e.g., by narrowing or elongation).
Such arbitrary modifications result in an arbitrary mean shape and unreliable statistics. In
other words, any shape in the shape space can be considered the mean shape. On the other
hand, even if the mean shape is valid, it does not necessarily comply with the structure of
the space, as the extrinsic path ignores the RCC. In fact, based on the extrinsic path, the
actual ETRep space is not convex. That is to say, ∃s1, s2∈ An+1 , and ∃λ∈(0,1) such that
ζs(λ;s1, s2)/∈ An+1 (O’Searcoid,2006), where An+1 denotes the actual space of ETReps
(that we discuss later) by considering RCC.
Figure 5: Self-intersecting issue in the mean shape. Left column: Three E-tub es representing three
hippocampi. The skeletal sheets are flattened for better visualization. Middle column: The extrinsic
mean exhibiting a self-intersection problem. Right column: The intrinsic mean without self-intersection.
Figure 5 (middle column) illustrates the extrinsic mean of three ETRep of three hip-
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pocampi with an obvious self-intersection. Although such self-intersections are rare and
often considered as artifacts, they reflect that An+1 is more complicated than Sn+1. By
providing an explicit definition for An+1, we can follow the idea of Rustamov et al. (2009)
to map An+1 to a convex space, calculate the mean and paths, and transfer them back
to An+1.Figure 5 (right column) and Figure 3 (bottom row) illustrates such an intrinsic
mean and path. To define An+1 and to explain the intrinsic approach, we discuss the RCC.
4.3 The RCC for discrete E-tubes
In this section, we discuss the RCC for discrete E-tubes based on the fact that each cross-
section of an E-tube is an elliptical disk. For simplicity in writing, we denote Π(λi), Ω(λi),
and Γ(λi) by Π(i), Ω(i), and Γ(i), respectively.
For a discrete E-tube ΩE, we have a sequence of slicing planes Π(i),i= 0, ..., n. We
consider a local self-intersection at spinal point Γ(i)if Π(i)intersects with Π(i−1) or Π(i+1)
within ΩE. Since any two slicing planes of a swept region are adjacent or not adjacent, it is
sufficient to discuss the RCC for each slicing plane relative to its previous one. Note that
by considering non-local self-intersections, the E-tube space becomes highly complex as we
need to define the shape space and an intrinsic path on the shape space of tangled knot-
like objects such that object transformation (that can be seen as transforming a tangled
knot to another tangled knot) associated with the path does not produce non-local self-
intersections. In Supplementary Materials, we provide an iterative algorithm to calculate
the Fr´echet mean for a class of E-tubes called simply straightenable by considering the
problem of non-local self-intersection. In this work, we focus on the problem of local self-
intersections that can be explored via the RCC, and we leave a detailed study of non-local
self-intersections to our future research.
To define shape space based on the RCC, we need to consider the spine curvature. Since
there are various ways to define the curvature of a discrete curve (Vouga,2014), we take a
closer look at the structure of the slicing planes to provide a clear definition of the RCC in
the discrete format.
Assume two consecutive cross-sections Ω(i−1) and Ω(i)associated with the spinal points
pi−1= Γ(i−1) and pi= Γ(i)of an E-tube with their corresponding material frames Fi−1
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and Fi, where Fi= (ti,ai,bi) as described in Section 3. Assume φi=dg(ti−1,ti)∈
[0,π
2] as the curvature angle of Ω(i)based on Rn⊥
i(φi)ti−1=ti, where n⊥
i=ti−1×ti
∥ti−1×ti∥
and Ru(φ) = I3+ sin φ[u]×+ (1 −cos φ)(u⊗u−I3) is the rotation around axis uby φ
degrees (Lu,2013). Further, let θi∈[−π, π] be the twisting angle between aiand nirelative
to tiaccording to the right-hand rule as Rti(θi)ai=ni, as depicted in Figure 6.
Assume φi∈(0,π
2], i.e., Π(i−1) ∥ Π(i)(as φi= 0). Thus, Π(i−1) and Π(i)intersect where
the intersection is a line orthogonal to ti−1and tiand parallel to ti×ti−1(Georgiades,
1992). Let point qbe a point on the intersection line with minimum Euclidean distance
to pi(i.e.,
−→
piq
is the radius of a circle on Π(i)centered at pitangent to Π(i−1)). Since
the slicing planes cannot intersect within the object, qmust be on the boundary or outside
the E-tube ΩE. Let q′be an arbitrary point on the intersection line and let θi=∠qpiq′.
Obviously, θi∈[−π
2,π
2] and the width of Ω(i)in the direction of −−→
piq′cannot exceed
−−→
piq′
.
In other words, if q′′ is the boundary point of ωialong −−→
piq′, then
−−→
piq′′
cos θi≤
−→
piq
.
Let κi=1
∥−→
piq∥. Then, condition
−−→
piq′′
cos θi≤1
κiis compatible with RCC in a continuous
format (i.e., rcos θ < 1
κ) (Damon,2008). Therefore, the RCC at pican be strictly defined
as the maximum possible size of the Ω(i)such that Ω(i)does not intersect with Π(i−1). Also,
we overlook the size of the Ω(i−1) because if Ω(i)intersects with Π(i−1) without intersecting
Ω(i−1), then the area bounded by the envelope of Ω(i−1) and Ω(i)may not be a convex
region, which is problematic for defining a smooth implied boundary as the issue produces
thorn shape bumps on the implied boundary.
Figure 6: Visualization of two consecutive slicing planes Π(i−1) and Π(i). Left: Planes crossing pi−1and
piwith normals ti−1and ti. Middle: Elliptical cross-sections of different sizes. The largest cross-section
violates the RCC. Right: Illustration of a slice of the left and middle figures crossing pi−1,pi, and q.
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The ith cross-section Ω(i)is an elliptical disk in Π(i)centered at pi. We apply orthogonal
projection to map Ω(i)on the line ←→
piqi. Obviously, the projection is a line segment centered
at piwith a maximum length of 2
−→
piq
. Let ribe half the length of the line segment
divided by pi. Therefore, we have ri≤
−→
piq
. Let φ′
i=∠pipi−1q, and φi=π
2−φ′
ibe
the complement of φ′
i. Thus,
−→
piq
=xi
cos φ′
i, and we have ri≤xi
cos φ′
i, where xi=
−−−−→
pi−1pi
.
The RCC of Ω(i)can be expressed as ri≤xi
sin φi(because sin φi= cos φ′
i). Therefore,
given φiand xi, to check whether ωicomplies with the RCC, it is sufficient to calculate ri.
Hence, analogous to the approach of Vouga (2014), the curvature of the discrete curve at
pirepresenting the rate of frame rotation can be considered as κi=sin φi
xi.
The boundary of Ω(i)is an ellipse that can be parameterized as ∂Ω(i)= (aicos t, bisin t),
where t∈[0,2π]. By rotating ∂Ω(i)with θidegree clockwise relative to Fiwe have
cos θi−sin θi
sin θicos θi
aicos t
bisin t
=
aicos tcos θi−bisin tsin θi
aicos tsin θi+bisin tcos θi
.
Thus, aicos tcos θi−bisin tsin θiis the parameterized projection of ∂ωionto the line ←→
piqi.
Assume we have the function h(t;a,b, θ ) = acos tcos θ−bsin tsin θ. For given ai,bi, and
θi, the maximum value of h(t;ai,bi, θi) defines ri. Thus, based on ∂ h(t;ai,bi,θi)
∂t = 0 we have
−aisin tcos θi−bicos tsin θi= 0 ⇒sin t
cos t=−bi
ai
sin θi
cos θi⇒t= tan−1(−bi
ai
tan θi).
Hence,
ri=|aicos (tan−1(−bi
ai
tan θi)) cos θi−bisin (tan−1(−bi
ai
tan θi)) sin θi|.(5)
Let s= (ωi)n
i=0 = ((f∗
i, xi, ai, bi)i)n
i=0, where ωirepresents Ω(i). Based on f∗
i, we can
calculate F∗
iof ωiand consequently the curvature and twisting angles φiand θi. Thus, by
considering the RCC, An+1 can be seen as a subspace of Sn+1 such that if s∈ An+1, then
∀i;ri≤xi
sin φi, where ricomes from Equation (5). For a circular tube (C-tube) ∀i, Ω(i)is a
circle, and we have ri=ai=bi.Figure 6 illustrates two slicing planes Π(i−1) and Π(i)with
an elliptical cross-section Ω(i)of different sizes.
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4.4 Semi-intrinsic approach
In this section, we discuss a simple and straightforward approach called the semi-intrinsic
approach for defining the ETRep shape space and mean shape. This approach is a modified
version of the extrinsic approach discussed in Section 4.2, incorporates the RCC.
Let s= (ωi)n
i=0 = ((f∗
i, xi, ai, bi)i)n
i=0 ∈ An+1 and let τi=bi
ai∈(0,1]. Equation (5)
becomes
ri=|aicos(tan−1(−τitan(θi))) cos(θi)−τiaisin(tan−1(−τitan(θi))) sin(θi)|.(6)
Since ri≤xi
sin φi, the ratio ρi=tan−1(ri)
tan−1(xi
sin φi)is in (0,1], where tan−1(x) is a control function
that maps R+to the finite interval [0,π
2]. Thus, τireflects the eccentricity of Ω(i), and ρi
reflects the ratio of its size relative to its maximum possible size.
Obviously, ri= tan(ρitan−1(xi
sin φi)). By replacing riin Equation (6) we have
ai=tan(ρitan−1(xi
sin φi))
|cos(tan−1(−τitan θi)) cos θi−τisin(tan−1(−τitan θi)) sin θi|,(7)
and bi=τiai.
In fact, we map the non-convex space An+1 to a convex space, namely A†n+1, by the
invertible mapping F†(f∗
i, xi, ai, bi)i= (f∗
i, xi, τi, ρi)i, where A†=S3×R+×(0,1]2. Thus,
ωican be represented as ω†
i= (f∗
i, xi, τi, ρi)iin A†n+1. Assume s†
1= ((f∗
1i, x1i, τ1i, ρ1i)i)n
i=0
and s†
2= ((f∗
2i, x2i, τ2i, ρ2i)i)n
i=0. The convexity of A†n+1 can be explained based on a straight
path between s†
1and s†
2. It is sufficient to replace τji and ρji with aj i and bji , in Equation (3),
where j= 1,2. Thus, the straight path between s†
1and s†
2is
ζs†(λ;s†
1, s†
2) = ((ζF(λ;f∗
1i,f∗
2i), ζ(λ;x1i, x2i), ζ (λ;τ1i, τ2i), ζ(λ;ρ1i, ρ2i))i)n
i=0.
Thus, F−1(ζs†(λ;s†
1, s†
2)) is a valid path in An+1 corresponding to ζs†(λ;s†
1, s†
2) in A†n+1.
The rest of the discussion regarding distance and sample mean is the same as the extrinsic
distance and sample mean from Equation (1) and Equation (4). Thus, we have
ds†(s†
1, s†
2) =
n
X
i=0
d2
g(f∗
1i,f∗
2i) +∥x1i−x2i∥2+∥τ1i−τ2i∥2+∥ρ1i−ρ2i∥2
1
2
,
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and
¯s†= argmin
s†∈S†
n+1
m
X
j=1
d2
s†(s†, s†
j).
Finally, the sample mean shape is F−1(¯s†).
The main concern about the semi-intrinsic approach is defining the control function.
There is a variety of functions that can be considered as control functions. For instance,
the numerator of Equation (7) changes to the term (tan(ρitan−1(xi
sin φi)η))1
ηby considering
the control function as tan−1(xη), where η∈(0,∞). Therefore, as depicted in Figure 7,
the semi-intrinsic transformation and the mean object are very sensitive to the selection
of the control function. On the other hand, in the semi-intrinsic approach, the rotation of
the cross-section is not commensurate with its size during the transformation. Besides, the
structure of the shape space is not explicit.
Figure 7: Top row: The semi-intrinsic transformation between E-tubes ΩE1and ΩE2based on the control
function tan−1(xη), wherein the top row ηis 0.8, and in the bottom row ηis 2.
To explain the ETRep space in the semi-intrinsic approach, we considered the RCC as
the maximum size of the cross-section Ω(i)based on its curvature angle φi. That is to say,
what the maximum size of Ω(i)could be based on a given φi. To commensurate the rotation
of Ω(i)(i.e., rotation of the material frame) with the size of Ω(i), we propose a shape space
and intrinsic path by considering the RCC as the maximum value of φibased on the given
Ω(i). To provide an explicit intuition about the structure of the space, first, we discuss the
shape space for C-tubes.
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4.5 C-tube shape space
This section discusses the shape space of C-tubes (with circular cross-sections) based on the
fact that a C-tube can be seen as an E-tube with infinitesimal cross-sectional eccentricity.
Afterward, Section 4.6 generalizes the discussion of this section to propose an intrinsic
approach for calculating the mean shape of a sample of ETReps.
As discussed in Section 4.3, in an ETRep s= ((F∗
i, xi, ai, bi)i)n
i=0,∀i∈ {1, ..., n}vectors
ti,ti−1,niare coplanar. Thus we have, sinφi=∥vi∥, where viis the projection of ti
on Π(i−1), as depicted in Figure 6 (Left and Right). The RCC of Ω(i)can be expressed as
∥vi∥ ≤ min{1,xi
ri}, where ricomes from Equation (5). If xi≥rithen∥vi∥can take any value
in [0,1]; also, if θi= 0 then ∥vi∥ ≤ min{1,xi
ai}, and if θi=±π
2then ∥vi∥ ≤ min{1,xi
bi}, as
discussed in Section 4.3. In this sense, based on the size of riwe impose bending restriction
on cisuch that ∥vi∥cannot exceed xi
ri.
By considering very small but not zero eccentricity for all the cross-sections of a discrete
E-tube, we have ∀i;ai≃bi≃ri(but not equal). In fact, a C-tube can be seen as an E-
tube with a skeletal sheet equipped with material frames. Thus, the ETRep becomes the
representation of a C-tube that we call CTRep, denoted by sc= ((F∗
i, xi, ri)i)n
i=0, where ri
is the radius of the ith cross-section. Thus, the RCC is ∥vi∥ ≤ min{1,xi
ri}.
Figure 8: Left: Illustration of Ac2. Middle: The swept skeletal structure of a slice of Ac2based on xi= 1.
Right: A slice of Acbased on xi= 1 located inside a convex cylinder with radius 1.0.
In 2D we have F∗
i= (ti,ai)∈SO(2). Thus, vihas one scalar element vi∈[−1,1] =
B. The ith cross-section Ω(i)can be represented by the vector ωi= (vi, xi, ri)∈ Ac2,
where Ac2⊂R3is a 3D space with hyperbolic boundary such that if (v, x, r )∈ Ac2, then
v∈[−1,1], r, x ∈R+, and |v| ≤ min{1,x
r}. In 3D, Ω(i)can be represented as ωi=
(vi1, vi2, xi, ri)∈ Ac, where Ac⊂R4is a 4D space with hyperbolic boundary such that if
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(v1, v2, x, r)∈ Ac, then (v1, v2)∈B2,r, x ∈R+, and
(v1, v2)
≤min{1,x
r}. Thus, the
CTRep with n+ 1 cross-sections lives on the product space (Ac)n+1 , which we denote by
An+1
c.Figure 8 (left) illustrates Ac2.Figure 8 (middle) depicts a slice of Ac2by considering
xi= 1 (with its swept skeletal structure). The visualization of Acis challenging as it is a
4D space. Analogous to the middle figure, by considering xi= 1, Figure 8 (right) depicts
a slice of Ac, which is a non-convex 3D hyperbolic area located inside a convex region as a
cylinder with radius 1.0.
Each slice of Acis symmetric relative to the riaxis. Therefore, each slice is a C-tube
with a swept skeletal structure. In this sense, the product space An+1
cis a combination of
swept regions with swept skeletal structures. Since any point inside a swept region with a
swept skeletal structure has an intrinsic skeletal coordinate, analogous to the approach of
Pizer et al. (2022), we can consider an intrinsic skeletal coordinate system for An+1
c. To
elaborate further, we discuss the skeletal coordinate system.
4.5.1 Skeletal coordinate system
The skeletal coordinate system is an intrinsic coordinate system that can be established
based on the skeletal structure of the space (Pizer et al.,2022). This section explains the
skeletal coordinate system for spaces with swept skeletal structures.
Let ΩEbe a 3D E-tube with the parameterized spine Γ(λ) based on the curve length
parameterization. Thus, λcan be seen as the proportion of the curve length at Γ(λ)
from the starting point Γ(0) relative to the total length of the curve. Assume pλ= Γ(λ),
and let Fλ= (tλ,aλ,bλ) be the material frame at pλ. For any interior point of a swept
region with a swept skeletal structure, there is a unique skeletal spoke crossing the point
and connecting the Γ to the boundary (Damon,2008). Let pbe an interior point of ΩE.
Without loss of generality, let −−→
pλqbe the direction of the skeletal spoke crossing psuch
that q∈∂ΩE. Thus, p=pλ+ς1−−→
pλq, where ς1=∥−−→
pλp∥
∥−−→
pλq∥∈[0,1]. Therefore, pcan
be represented by the tuple (λ, ς1−−→
pλq). Since vector ς1−−→
pλqis coplanar with aλand bλ,
by assuming ς2as the clockwise angle between aλand ς1−−→
pλqrelative to tλbased on the
right-hand rule, the direction of vrelative to Fλcan be represented by ς2∈[−π, π] such
that ς1−−→
pλq=Rtλ(ς2)aλ. Thus, given the spine Γ, any interior point pcan be represented
as a unique vector (λ, ς1, ς2) in [0,1] ×[0,1] ×[−π, π]. Besides, for the given (λ, ς1, ς2),
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there is a unique skeletal spoke with the tail at Γ(λ) = pλand the direction Rtλ(ς2)aλ,
which is identical to the direction of ς1−−→
pλq. Thus, the tip of the skeletal spoke is at q.
Therefore, (λ, ς1, ς2) corresponds to the point p=pλ+ς1−−→
pλq∈Ω. Hence, there is a
bijective mapping Fs:ΩE→[0,1] ×[0,1] ×[−π, π]. Note that in 2D, we do not have ς2
as the local frame associated with λbelongs to S O(2). Thus, pcan simply be represented
as the vector (λ, ±ς1) in the product space [0,1] ×[−1,1]. The same discussion is valid for
the generalized tubes by considering the local frames as the Frenet frames.
Figure 9: Left: A distribution of p oints inside a 2D C-tube with a swept skeletal structure. The bold dot
is the extrinsic mean, which is outside the region. Middle: The mapped distribution to the product space
and the Euclidean mean µinside the product space. Right: Intrinsic mean as F−1
s(µ) depicted as a bold
dot which is inside the C-tube.
Since the product space is convex, given a distribution of points p1, ..., pmin ΩE,
the intrinsic mean can be considered as F−1
s(µ), where µis the Euclidean mean of
Fs(p1), ..., Fs(pm). Figure 9 (left) depicts a distribution of points inside a 2D C-tube
with the Euclidean mean (as the extrinsic mean) that is located outside the region. Fig-
ure 9 (middle) illustrates the mapped distribution to the product space and the Euclidean
mean µinside the product space. Figure 9 (right) shows the intrinsic mean as F−1
s(µ)
inside the C-tube.
A C-tube can be seen as an E-tube with infinitesimal eccentricity in all the cross-
sections, i.e., ∀λ;aλ≃bλ. Let Ac(xi) be a slice of Acat xi, as depicted in Figure 8 (middle
and right). Ac(xi) is a C-tube with a straight spine, namely the axis ri. Thus, the material
frames similar to the Frenet frames can be considered as I3= (e1,e2,e3) along the spine
(i.e., the material frames coincide with the Frenet frames). By considering a large number
R >> 1 such that ∀i;ri< R,Ac(xi) has a finite spine with an intrinsic skeletal coordinate
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system. However, Ac(xi) is not finite. Nevertheless, it can be seen as an infinite swept
region with a straight infinite spine such that each cross-section is a circular disk.
Therefore, each point p= (v1, v2, r)∈ Ac(xi)⊂R3is located on a skeletal spoke −−→
prq
with tail at pr= (0,0, r) and tip at q= (min{1,x
r}u1,min{1,x
r}u2, r) with direction u=
v
∥v∥, where v= (v1, v2,0). Thus, p=pr+ς1−−→
prq, where ς1=∥v∥
∥−−→
prq∥=√v2
1+v2
2
min{1,x
r}∈[0,1]. Hence,
pcan be represented based on a cylindrical coordinate as Fs(p) = (ς1cos ς2, ς1sin ς2, r),
where ς2∈[−π, π] is the clockwise angle between e2and v. Since uand vhave the same
direction, we have Fs(p)=(ς1u1, ς1u2, r ) or more explicitly as Fs(p)=(ς1u, r) in the
convex cylinder B2×R+, as illustrated in Figure 8 (right). In other words, Fscan be seen
as the bijective mapping Fs:Ac(xi)→B2×R+.
Based on the fact that Acis a combination of its slices and by considering that the
mapping Fs(p) acts on the first two elements of p, any arbitrary point ω= (v1, v2, x, r)
(representing a cross-section) inside Accan be represented as an element of the convex 4D
product space
>
Ac=B2×(R+)2as Fs(ω)=(ς1u, x, r ) (based on the skeletal coordinate
system of the slice that ωbelongs to), i.e., Fs:Ac→
>
Acis a bijective mapping. The same
discussion is valid if we consider the slices based on the r-axis. In this sense, the plane
expanded by the xand raxes can be seen as the skeletal sheet of Ac(see Figure 8 (left)).
4.6 Intrinsic approach
Let Ae⊂R6be the space of elliptical cross-sections ω= (v1, v2, θ, x, a, b)∈ Ae, where
v= (v1, v2)∈B2,a, b, x ∈R+,θ∈[−π , π], a≥b, and the condition ∥v∥ ≤ min{1,x
r}
associated with RCC is satisfied, where rcomes from Equation (5) (by considering (ai, bi, θi)
as (a, b, θ)). Thus, ωcan be considered as a representation of an elliptical cross-section.
Obviously, the concavity of Aecomes from the condition ∥v∥ ≤ min{1,x
r}as by ignoring
this condition ωcan be represented by ω= (v, θ, x, a, b) belongs to the convex product
space
>
Ae=B2×[−π, π]×(R+)3. Therefore, analogous to Ac, we consider an intrinsic
skeletal coordinate for ω(based on the mapping Fs) as Fs(ω)=(ς1u, θ, x, a, b)∈
>
Ae,
where ς1=∥v∥
min{1,x
r}and u=v
∥v∥(intuitively, there is a skeletal spoke with tail and tip at
(0,0, θ, x, a, b) and (min{1,x
r}u1,min{1,x
r}u2, θ, x, a, b)). Thus, Fs(ω):Ae→
>
Ae, and we
consider intrinsic coordinate systems for Aeand for the product space (Ae)n+1. We denote
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(Ae)n+1 by An+1
eand its corresponding convex space (Fs(Ae))n+1 by
>
Aen+1.
By representing ETRep as Fs(s) = ((ς1iui, θi, xi, ai, bi)i)n
i=0 in
>
Aen+1, we can reconstruct
the E-tube associated with sin 3D. Since ∀i; (vi1, vi2) = min{1,xi
ri}ς1iui, we calculate the
ith tangent vector based on its parent frame (which is the (i−1)th material frame) as ti=
(p1−v2
i1−v2
i2, vi1, vi2). Thus, by considering the geodesic path on the unit sphere from
Equation (2), we obtain the ith Frenet frame based on its parent frame as T∗
i= (ti,ni,n⊥
i),
where n⊥
i=1
sin ψ(−e1+ sin(ψ−π
2)ti), ψ=dg(−e1,ti), and ni=n⊥
i×ti. Consequently,
the ith material frame based on its parent is F∗
i= (ti,ai,bi), where ai=Rti(−θi)ni, and
bi=ti×ai. Since xiprovides the distance between the frames, by assuming T0=F0=I3,
we can obtain the spine by calculating the material frames in the global coordinate system
as discussed in Section 3. Finally, aiand bidefine the size of the ith cross-section.
Assume ETReps s1= (ω1i)n
i=0 and s2= (ω2i)n
i=0 in An+1
e. We define the intrinsic path
between s1and s2as ζI(λ, s1, s2)=(F−1
s(ζ(λ, Fs(ω1i),Fs(ω2i))))n
i=0. Let s1, ..., smas a
set of ETReps. By assuming the intrinsic distance as the Euclidean distance in the convex
space
>
An+1
e, the intrinsic mean is the inverse map of the Euclidean mean of the mapped
shapes as ¯s=F−1
s(1
mPm
j=1 Fs(sj)). Figure 10 illustrates the intrinsic mean object of a set
of arbitrary 3D E-Tubes. For better visualization, the E-Tubes are aligned based on their
initial material frames.
Figure 10: The intrinsic mean object of ten 3D E-tubes, which need not be aligned.
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5 Application
One significant application of sample mean shape is in studying shape differences based
on hypothesis testing. In this section, we discuss ETRep hypothesis testing based on the
intrinsic approach of Section 4.6 using a hippocampal data set provided by the Stavanger
University Hospital from the ParkWest study (Alves et al.,2009). In the Supplementary
Materials, the same data set is used to examine hypothesis testing through the semi-intrinsic
approach of Section 4.4.
The ETRep intrinsic representation is invariant to the act of rigid transformation.
For the statistical shape analysis, we need to remove the scale. Assume an ETRep
s= ((ς1u1i, ς1u2i, θi, xi, ai, bi)i)n
i=0 ∈
>
An+1
e. Following Taheri and Schulz (2022), we have
ℓ= exp ( 1
3n+2 (Pn
i=1 ln xi+Pn
i=0 ln ai+Pn
i=0 ln bi)) as the LP-size of s. Thus, the scaled
and vectorized form of sis a vector s= (ς1u1i|n
i=0, ς1u2i|n
i=0, θi|n
i=0,xi
ℓ|n
i=0,ai
ℓ|n
i=0,bi
ℓ|n
i=0) living
on the convex feature space (B)2(n+1) ×([−π, π])n+1 ×(R+)3(n+1) ⊂R6(n+1).
Let A={sAj}m1
j=1 and B={sBj }m2
j=1 be two groups of scaled ETReps of sizes m1and
m2. Since each ETRep is represented as a vector. The global test can be considered as
H0:µA=µBversus H1k:µA=µBwhere µAand µBare the observed Euclidean sample
means in the convex feature space. For the hypothesis testing, we consider the permutation
test with minimal assumption. Given the pooled group of two data sets (here A∪B), the
permutation method randomly partitions the pooled group into two groups of sizes m1and
m2without replacement many times and measures the test statistic between the paired
groups. The empirical p-value is η=1+PL
h=1 χI(|th|≥|to|)
L+1 (Rizzo,2007), where tois the
observed test statistics (e.g., Hotelling’s T2metric T2= (¯
x−¯
y)Tˆ
Σ−1(¯
x−¯
y), where ˆ
Σ
is the common covariance matrix (Mardia et al.,1982)), this the hth permutation test
statistic, L is the number of permutation (usually greater than 104), and χIis the indicator
function (i.e., χI(φ) = 1 if φis true, otherwise χI(φ) = 0) (Taheri and Schulz,2022). Since
in ETRep analysis, the feature space is a high dimensional space, for the global test, we
considered the direction projection permutation (DiProPerm) method of Wei et al. (2016).
Further, to detect local dissimilarities, we compare ETReps element-wise based on
partial permutation tests H0k:µA(k) = µB(k) versus H1k:µA(k)=µB(k) based on the
test statistic as t-statistic (i.e., t=¯x−¯y
ˆσpq1
m1+1
m2
), where µA(k) and µB(k) are the observed
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sample mean of the kth feature, and k= 1, ..., 6(n+ 1). Since the partial tests are multiple
comparisons, to control false positives, we adjust p-values using the false discovery rate
(FDR) of Benjamini and Hochberg (1995). Figures 11 and 12 illustrate the result of the
Figure 11: Global test based on DiProPerm. Left: DiProPerm projection score of the observed
distributions. Class labels -1 and 1 are associated with CG and PD. Right: The permutation statistics.
Figure 12: Hypothesis testing based on the significant level α= 0.1. The top row depicts significant
features based on raw p-values. The bottom row shows significant features after p-value adjustment with
F DR = 0.1. Significant features are highlighted in bold where columns (a-d) depict significant sizes
associated with principal radii aiand bi, significant spinal connections’ lengths associated with xi,
significant twisting degrees associated with θi, and significant rotation of the material frame according to
the tangent vectors associated with v1iand v2i, respectively.
global and partial hypothesis tests on ParkWest data comparing hippocampi of patients
with Parkinson’s disease (PD) versus a healthy control(CG), where in PD and CG we
have 182 and 108 samples, respectively. As depicted in Figure 11, there is a statistically
significant difference between PD and CG based on the obtained p-value, which is 0.048,
and the significant level of α= 0.05. Further, Figure 12 visualizes significant features that
reflect the size, spinal elongation, degree of twist, and cross-sectional rotation with (bottom
row) and without (top row) p-value adjustment.
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6 Conclusion
In this work, we proposed a novel framework aimed at analyzing swept regions, particularly
addressing the issue of self-intersection. We focused on a specific class of swept regions
called elliptical slabular objects (E-SlOs). We introduced elliptical tube representation
(ETRep) as a robust method for representing E-SlOs. The ETRep is invariant to rigid
transformation and can be seen as a sequence of elliptical cross-sections along the object’s
center curve. To deal with the self-intersection problem in the ETRep analysis, we discussed
the relative curvature condition (RCC). By considering the RCC, we defined ETRep space
with an intrinsic skeletal coordinate system. We explained ETRep intrinsic distance and
mean shape based on the introduced space. We introduced an intrinsic distance measure
for the underlying shape space to avoid self-intersection and ensure valid statistics. We
briefly outlined the issue of non-local intersections and proposed a solution for a specific
class of ETReps that are simply straightenable. The advancement of the proposed solution,
detailed in the Supplementary Materials, could serve as a focal point for future research.
Finally, we demonstrated the application of ETRep analysis by comparing the hippocampi
of patients with Parkinson’s disease to those of a healthy control group.
Acknowledgments
This work is funded by the Department of Mathematics and Physics of the University of
Stavanger (UiS). We thank Prof. James Damon (late of UNC) for the inspiration of this
work. We also thank Prof. Guido Alves (UiS) for providing the ParkWest data.
SUPPLEMENTARY MATERIALS
Supplementary: Supplementary Materials referenced in this work are available as a pdf.
R-code: In Supplementary.zip, R codes and files are placed. (zip)
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Supplementary Materials of “The Mean
Shape under the Relative Curvature
Condition”
Mohsen Taheri∗, Stephen M. Pizer†and J¨orn Schulz‡
March 31, 2024
GPA mean
In this section, we provide a simple example to show the problem of PDM analysis without
considering the RCC in swept regions. As depicted in Figure 1, the boundaries of two 2D
C-tubes (as two swept regions) are represented by two PDMs based on an equal number
of corresponding boundary points (as the tip of the skeletal spokes). The mean shape
obtained by generalized Procrustes analysis (GPA) is not a swept region, as it violates the
RCC with an obvious self-intersection.
Figure 1: Illustration of the GPA mean shape of two C-tubes Ω1and Ω2.
∗Mohsen Taheri, Department of Mathematics and Physics, University of Stavanger (UiS), Email:
mohsen.taherishalmani@uis.no
†Prof. Stephen M. Pizer, Department of Computer Science, University of North Carolina at Chapel
Hill (UNC), Email: pizer@cs.unc.edu
‡Assoc. Prof. J¨orn Schulz, Department of Mathematics and Physics, University of Stavanger (UiS),
Email: jorn.schulz@uis.no
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Extrinsic mean versus intrinsic mean
In this section, we compare extrinsic and intrinsic means based on a simple example to
provide an intuition about the compatibility of the means with the structure of the space.
Imagine two distributions of points inside a non-convex space, as depicted in Figure 2
(left). The extrinsic mean based on the extrinsic Euclidean distance of the blue distribution
is not acceptable as it is outside the region, and the extrinsic mean of the red distribution is
very close to the boundary, while most of the red points are not. In contrast, by considering
an intrinsic distance based on the skeletal coordinate system, as discussed in the main
manuscript, the intrinsic means of both distributions are inside and relatively far away
from the boundary. In other words, despite the extrinsic distance and mean, the intrinsic
distance and mean comply with the structure of the space. Figure 2 (right) shows a closed
path as a circle in [0,1] ×[−1,1] and its corresponding closed path inside the non-convex
region based on the bijective mapping Fs.
Figure 2: Left: Illustration of the intrinsic and extrinsic means of two distributions in a non-convex
region in red and blue. The intrinsic mean of each distribution is bold with the same color. The extrinsic
means are bold with the cross. Right: Visualization of a closed path as a circle in [0,1] ×[−1,1] and its
corresponding closed path inside the non-convex region.
Non-local self-intersection
In the main manuscript, we discussed the problem of local self-intersections in calculating
the mean ETRep based on the RCC. However, in addition to the local self-intersections,
we need to consider the problem of non-local self-intersection as the mean object with a
non-local self-intersection is also an invalid object. In this section, we discuss the issue and
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propose a possible solution for a class of ETReps called simply straightenable. We leave
the generalization of the solution to future research.
The definition of a suitable space by considering the condition of non-local self-
intersections for ETRep is a complex problem because the corresponding object can be
twisted and intertwined like a tangled knot such that it is exceedingly difficult or even
impractical to unravel it. Consequently, even by considering the RCC, the mean shape of
a population of valid ETReps could be a shape with no local self-intersections but with a
non-local self-intersection issue. Figure 3 provides an intuition about the problem as the
mean object associated with the mean shape of two ETReps s1and s2(that are relatively
symmetric) has no local self-intersection, but it is self-intersected non-locally.
Figure 3: Illustration of non-local self-intersection in the mean object obtained based on the intrinsic
mean shape of two ETReps s1and s2.
A valid transformation between two tangles E-tubes is like transforming a tangled
knot into another tangled knot without producing non-local self-intersections, which is
not straightforward. In fact, the actual ETRep space is a subspace of An+1
e. However, for
simply straightenable ETReps, we can define a valid transformation that avoids non-local
self-intersections, in addition to a suitable distance and, consequently, the mean shape.
Let s1→s2denote the path ζI(λ, s1, s2). We call s1→s2a linear path (or trans-
formation) between s1and s2. A linear path is valid if there are no occurrences of local
or non-local self-intersections. Note that the non-local self-intersections are intersecting
ellipses that are detectable based on the discussion of Eberly (2000). We recognize sas
simply straightenable if s→s⋆is valid where s= ((vi,θi, xi, ai, bi)i)n
i=1 and s⋆is its straight-
ened version s⋆= (((1,0),0, xi, ai, bi)i)n
i=1, i.e., s⋆is not twisted or bent. Thus, the space of
simply straightable ETReps is a subspace of the ETReps’ space that we denote by A⋆(n+1)
e.
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We consider any non-linear path s1⇝s2between s1, s2∈ A⋆(n+1)
eas valid if it is
a combination of a finite number of valid linear paths. Let sλ0,...,sλmbe sequence of
ETReps, where s1=λ0and sm=λmsuch that ∀j∈1, ..., m;sλj−1→sλjis valid. Thus,
there is a non-linear valid path s1⇝s2as sλ0→sλ1→... →sλm, with the length
d⇝
s(s1, s2) = Pm
j=1 ds(sλj−1, sλj). Assume ˆ
d⇝
s(s1, s2) is the path length of the shortest non-
linear valid path between s1and s2. The sample mean of s1, ..., sm∈ A⋆(n+1)
ecan be defined
as ¯s= argmins∈A⋆(n+1)
ePm
j=1(ˆ
d⇝
s(s, sj))2. Calculating the shortest path analytically is not
trivial. However, since s⋆
1→s⋆
2is valid (because the curvature angles in both ETReps are
zero), we know there is at least one evident valid non-linear path as s1→s⋆
1→s⋆
2→s2.
Analogous to the approach of Srivastava and Klassen (2016, ch.6) in geodesic computation,
it is possible to generate the evident path and then optimize it to become as short as
possible. For this purpose, we propose Algorithm 1.
Algorithm 1 Valid path
Require: s1, s2∈ A⋆(n+1)
e, and number of steps n, m ∈N
Ensure: Ss1⇝s2as the optimized non-linear path between s1and s2with length ˆ
d⇝
s(s1, s2).
Generate n+ 1 shapes sλjsuch that sλj=ζI(λj, s1, s⋆
1), j= 0, ..., n
Ss1⇝s2← {},ˆ
d⇝
s(s1, s2)← ∞, and j←0
while j < n do
Generate m+ 1 shapes sλ′
ksuch that sλ′
k=ζI(λ′
k, sj, s2), where k= 0, ..., m.
Calculate d=Pj
i=0 ds(sλi, sλi+1) + Pm−1
i=0 ds(sλ′
i, sλ′
i+1)
if ∀k;sλ′
kis valid (with no self-intersection) and d < ˆ
d⇝
s(s1, s2)then
Ss1⇝s2← {sλ1, ..., sλj, sλ′
1, ..., sλ′
m}
ˆ
d⇝
s(s1, s2)←d
end if
j←j+ 1
end while
(←indicates the assigning operator.)
Note that Algorithm 1 checks the validity of Ss1⇝s2based on its elements, which depends
on nand m. Thus, theoretically, they must be large enough so that Ss1⇝s2represents a
smooth transformation between s1and s2. However, selecting extremely large nand mis
unreasonable because of the computational expenses. Figure 4 depicts six distinct paths
between objects A and B that involve different amounts of straightening. Paths #1 and
#2 are not valid, and #3 is valid with the minimum path length as mentioned in Table 1.
Sample #5 on path #3 (depicted inside an ellipse) can be considered as the sample mean
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with the minimum sum of squared distances to s1and s2.
Figure 4: Illustration of six paths from s1to s2. The first and the second paths are not valid. All shapes
in the bottom row (are similar and) represent s2.
Steps Path 1 Path 2 Path 3 Path 4 Path 5 Path 6
14.89 4.89 4.89 4.89 4.89 4.89
23.39 5.28 5.28 5.28 5.28 5.28
32.57 3.49 6.39 6.39 6.39 6.39
42.57 2.70 4.00 8.15 8.15 8.15
53.39 2.82 3.16 4.89 10.52 10.52
64.89 3.63 3.27 4.00 6.33 11.10
74.89 3.93 3.97 5.46 10.52
84.89 4.30 4.97 8.15
94.89 4.75 6.39
10 4.89 5.28
11 4.89
Table 1: The sum of squared distances between shapes associated to Figure 4. Path #3 is the shortest
valid non-linear path as depicted in Figure 4. Sample #5 on path #3 can be considered as the sample
mean as it has the minimum sum of squared distances to s1and s2.
To calculate the sample mean shape, we can consider the gradient descent Algorithm 2.
We leave the detailed discussion regarding the algorithm’s convergence to our future re-
search. Figure 5 illustrates the convergence of the algorithm in R2.
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Algorithm 2 Sample mean
Require: Observations s1, ..., sn∈ A⋆(n+1)
e, and the threshold δ
Ensure: Mean shape ¯s
1: ¯s←s1;Dnew ← ∞;Dold ←0
2: while |Dnew −Dold| ≥ δdo
3: Dold ←Pn
i=0(ˆ
d⇝
s(¯s, si))2
4: S← {S¯s⇝si}n
i=0 (NB: Sis the union of all paths from ¯sto the observations.)
5: s∗←argmins∈SPn
i=0(ˆ
d⇝
s(s, si))2
6: Dnew ←Pn
i=0(ˆ
d⇝
s(s∗, si))2
7: if Dnew < Dold then
8: ¯s←s∗
9: end if
10: end while
(←indicates the assign operator.)
Figure 5: Convergence of Algorithm 2 in the Euclidean space R2. The red dot is the arithmetic mean.
The small dots are the generated samples (regarding set S) based on the paths between observations. The
black line is the non-linear path as a combination of several linear paths toward the mean.
Hypothesis testing regarding the semi-intrinsic approach
Section 5 of the main manuscript discussed the ETRep hypothesis testing based on the
intrinsic approach. This section discusses global and partial hypothesis testing regarding
the semi-intrinsic approach.
Let {s†
j}m
j=1 be a set of scaled (but not vectorized) ETReps with n+ 1 cross-sections.
Based on the semi-intrinsic approach, we have s†
j= (ω†
ji)n
i=0 where ω†
ji = (f∗
ji, xj i, τj i, ρj i)i∈
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S3×(R+)3, where xji , τji , ρji are obtained after the act of scaling. Since we have non-
euclidean data regarding the frames on the unit sphere S3, we use PNS Euclideanization
as discussed by Jung et al. (2012); Taheri and Schulz (2022) to Euclideanize the spherical
data. The PNS defines a mapping from the unit sphere to a cube as FSd:Sd→([−π , π])d
based on a given spherical distribution on Sdsuch that the sample mean of the spherical
data is mapped to the center of the cube (i.e., the origin of the Euclidean space 0). Thus, we
can map {f∗
ji}m
i=1 by PNS into ([−π, π])3. Let FS3(f∗
ji) = fj i = (f1ji , f2ji , f3ji ). Therefore,
the jth ETRep can be represented as s†
j= ((f1ji, f2j i, f3j i, xj i, τj i, ρji )i)n
i=0 (such that ∀j;
1
mPm
i=1(f1ji , f2ji , f3ji ) = (0,0,0)). Thus, s†
jcan be represented as the vector
s†
j= vec(s†
j) = (f1ji|n
i=0, f2ji |n
i=0, f3ji |n
i=0, xji |n
i=0, τji |n
i=0, ρji |n
i=0),
living in the convex product space ([−π, π])3(n+1) ×(R+)n+1 ×([−1,1])2(n+1) ⊂R6(n+1) .
Let A={s†
Aj}m1
j=1 and B={s†
Bj }m2
j=1 be two groups of scaled ETReps of sizes m1and
m2. Assume the pooled group A∪Bas {s†
j}m
j=1, where m=m1+m2. By applying PNS on
{s†
j}m
j=1 and vectoring its elements, we obtain two groups of vectorized ETReps as {s†
Aj}m1
j=1
and {s†
Bj }m2
j=1. The rest of the discussion regarding the global test and the partial tests is
the same way as we discussed in the main article.
Figure 6 shows the result of the global test on PD and CG based on the DiProPerm test.
Despite the intrinsic approach, PD and CG are not significantly different as the p-value
is greater than 0.1. Further, Figure 7 depicts the result of the partial tests before and
after p-value adjustment. Analogous to the intrinsic approach, it seems the main difference
comes from the length of the spine, as depicted in column (c).
Figure 6: Global DiProPerm test based on the semi-intrinsic approach. Left: DiProPerm projection score
of the observed distributions. Class labels -1 and 1 are associated with CG and PD, respectively. Right:
Plot of the permutation statistics.
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Figure 7: Shape analysis. Significant level α= 0.1. The top row depicts significant features based on raw
p-values. The bottom row shows significant features after p-value adjustment with FDR = 0.1.
Significant features are depicted in red. Columns (a-d) illustrate significant eccentricities associated with
ρi, significant relative sizes associated with τi, significant spinal connections’ lengths associated with xi,
and significant frames’ orientations associated with Fi, respectively.
References
Eberly D (2000) Intersection of ellipses. Geom Tools 200:1998–2008
Jung S, Dryden IL, Marron J (2012) Analysis of principal nested spheres. Biometrika
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Srivastava A, Klassen E (2016) Functional and Shape Data Analysis. Springer Se-
ries in Statistics, Springer New York, URL https://books.google.no/books?id=
0cMwDQAAQBAJ
Taheri M, Schulz J (2022) Statistical analysis of locally parameterized shapes. Journal of
Computational and Graphical Statistics 0(ja):1–28, URL https://doi.org/10.1080/
10618600.2022.2116445
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