WESDWeighted Spectral Distance for Measuring Shape Dissimilarity.
ABSTRACT This paper presents a new distance for measuring shape dissimilarity between objects. Recent publications introduced the use of eigenvalues of the Laplace operator as compact shape descriptors. Here, we revisit the eigenvalues to define a proper distance, called Weighted Spectral Distance (WESD), for quantifying shape dissimilarity. The definition of WESD is derived through analyzing the heat trace. This analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects. We analyze the resulting distance definition, present and prove its important theoretical properties. Some of these properties include: 1) WESD is defined over the entire sequence of eigenvalues yet it is guaranteed to converge, 2) it is a pseudometric, 3) it is accurately approximated with a finite number of eigenvalues, and 4) it can be mapped to the $([0,1))$ interval. Last, experiments conducted on synthetic and real objects are presented. These experiments highlight the practical benefits of WESD for applications in vision and medical image analysis.
 Radhika Mani Madireddy, Pardha Saradhi Varma Gottumukkala, Potukuchi Dakshina Murthy, Satyanarayana Chittipothula[Show abstract] [Hide abstract]
ABSTRACT: The complexity in shape context method and its simplification is addressed. A novel, but simple approach to design shape context method including Fourier Transform for the object recognition is described. Relevance of shape context, an important descriptor for the recognition process is detailed. Inclusion of information regarding all the contour points (with respect to a reference point) in computing the distribution is discussed. Role of similarity checking the procedure details regarding the computation of matching errors through the alignment transform are discussed. Present case of shape context (for each point with respect to the centroid) descriptor is testified for its invariance to translation, rotation and scaling operations. Euclidean distance is used during the similarity matching. Modified shape context based descriptor is experimented over three standard databases. The results evidence the relative efficiency of the modified shape context based descriptor than that reported for other descriptor of concurrent interests.SpringerPlus 11/2014; 3:674.  SourceAvailable from: Rongjie LaiYonggang Shi, Rongjie Lai, Danny J. J. Wang, Daniel Pelletier, David Mohr, Nancy Sicotte, Arthur W. Toga[Show abstract] [Hide abstract]
ABSTRACT: In this paper we present a novel approach for the intrinsic mapping of anatomical surfaces and its application in brain mapping research. Using the LaplaceBeltrami eigensystem, we represent each surface with an isometry invariant embedding in a high dimensional space. The key idea in our system is that we realize surface deformation in the embedding space via the iterative optimization of a conformal metric without explicitly perturbing the surface or its embedding. By minimizing a distance measure in the embedding space with metric optimization, our method generates a conformal map directly between surfaces with highly uniform metric distortion and the ability of aligning salient geometric features. Besides pairwise surface maps, we also extend the metric optimization approach for groupwise atlas construction and multiatlas cortical label fusion. In experimental results, we demonstrate the robustness and generality of our method by applying it to map both cortical and hippocampal surfaces in population studies. For cortical labeling, our method achieves excellent performance in a crossvalidation experiment with 40 manually labeled surfaces, and successfully models localized brain development in a pediatric study of 80 subjects. For hippocampal mapping, our method produces much more significant results than two popular tools on a multiple sclerosis study of 109 subjects.IEEE transactions on medical imaging. 03/2014;  [Show abstract] [Hide abstract]
ABSTRACT: Segmentationbased scores play an important role in the evaluation of computational tools in medical image analysis. These scores evaluate the quality of various tasks, such as image registration and segmentation, by measuring the similarity between two binary label maps. Commonly these measurements blend two aspects of the similarity: pose misalignments and shape discrepancies. Not being able to distinguish between these two aspects, these scores often yield similar results to a widely varying range of different segmentation pairs. Consequently, the comparisons and analysis achieved by interpreting these scores become questionable. In this paper we address this problem by exploring a new segmentationbased score, called normalized Weighted Spectral Distance (nWSD), that measures only shape discrepancies using the spectrum of the Laplace operator. Through experiments on synthetic and real data we demonstrate that nWSD provides additional information for evaluating differences between segmentations, which is not captured by other commonly used scores. Our results demonstrate that when jointly used with other scores, such as Dices similarity coefficient, the additional information provided by nWSD allows richer, more discriminative evaluations. We show for the task of registration that through this addition we can distinguish different types of registration errors. This allows us to identify the source of errors and discriminate registration results which so far had to be treated as being of similar quality in previous evaluation studies.IEEE transactions on medical imaging. 09/2012;
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WESD  Weighted Spectral Distance for Measuring
Shape Dissimilarity
Ender Konukoglu, Ben Glocker, Antonio Criminisi and Kilian M. Pohl
Abstract—This article presents a new distance for measuring
shape dissimilarity between objects. Recent publications intro
duced the use of eigenvalues of the Laplace operator as compact
shape descriptors. Here, we revisit the eigenvalues to define
a proper distance, called Weighted Spectral Distance (WESD),
for quantifying shape dissimilarity. The definition of WESD is
derived through analysing the heattrace. This analysis provides
the proposed distance an intuitive meaning and mathematically
links it to the intrinsic geometry of objects. We analyse the
resulting distance definition, present and prove its important
theoretical properties. Some of these properties include: i) WESD
is defined over the entire sequence of eigenvalues yet it is
guaranteed to converge, ii) it is a pseudometric, iii) it is accurately
approximated with a finite number of eigenvalues, and iv) it can
be mapped to the [0,1) interval. Lastly, experiments conducted
on synthetic and real objects are presented. These experiments
highlight the practical benefits of WESD for applications in vision
and medical image analysis.
Index Terms—Shape Distance, Spectral Distance, Laplace Op
erator, Laplace Spectrum, Segmentations, Label Maps, Medical
Images
I. INTRODUCTION
Quantifying shape differences between objects is an im
portant task for various areas in computer science, medical
imaging and engineering. In manufacturing, for example, one
may wish to characterize the difference in shape of two
fabricated tools. In radiology, a doctor frequently diagnoses a
disease based on anatomical and pathological shape changes
over time. In computer vision, discriminative shape models are
used for automated object recognition, [1], [2].
In order to define measurements of shape dissimilarity, sci
entists rely on descriptors of objects that capture information
on their geometry [1]. These descriptors can be in the form of
parametrized models (e.g. point clouds, surface patches, space
curves, medial axis transforms) or in the form of geometric
properties (e.g. volume, surface area to volume ratio, curvature
maps). Once a descriptor is formulated the distance between
two shapes can be defined as the difference between the
associated descriptors. The exact definition of the distance
however, is a critical issue. In order to define an intuitive and
theoretically sound distance, one should ensure that it takes
Corresponding author: E. Konukoglu (email:ender.konukoglu@gmail.com) is
with MGH and Harvard Medical School, MA 02129, USA.
B. Glocker and A. Criminisi are with Microsoft Research Cambridge, CB3
0FB, UK.
K.M. Pohl is with University of Pennsylvania, PA 19104, USA.
Authors would like to thank Dong Hye Ye from University of Pennsylvania
for his valuable comments as well as the support in part by Grant Number
UL1RR024134 and by the Institute for Translational Medicine and Therapeu
tics (ITMAT) Transdisciplinary Program.
into account the nature of the descriptor. For instance, the
descriptor might be an infinite sequence of positive values, in
which case we should be careful not to define a distance that
diverges for every nonidentical pair of shapes.
Shape descriptors based on the eigensystems of Laplace
and LaplaceBeltrami operators, called spectral signatures,
have recently gained popularity in computational shape anal
ysis [3], [4], [5], [6], [7], [8]. These descriptors leverage the
fact that the eigenvalues and the eigenfunctions of Laplace
operators contain information on the intrinsic geometry of
objects [9], [10], [11]. A visual analogy useful for an intuitive
understanding is to think of an object (e.g. in 2D) as the
membrane of a drum. In this case the eigenvalues correspond
to the fundamental frequencies of vibration of the membrane
during percussion, and the eigenfunctions correspond to its
fundamental patterns of vibration. Both the eigenvalues and
the eigenfunctions depend on the shape of the drum head and
thus can be used as shape descriptors for the object.
Despite recent progress by [3], [4], [5], [6], [7], [8], design
ing meaningful shape distances based on spectral signatures
remains challenging. Difficulties arise from the nature of the
eigensystems. The eigenfunctions of a shape mostly provide
localized information on the geometry of small neighborhoods.
Aggregating such local information into an overall shape
dissimilarity measure is nontrivial. On the other hand, the
eigenvalues provide information about the overall shape, so
they are ideal for defining global distances. However, they
form a diverging sequence making it difficult to define a
theoretically sound metric. Here, we tackle this latter problem
and propose a new shape distance based on the eigenvalues,
which is technically sound, intuitive and practically useful.
In the remainder of this section, we first review in further
detail the literature on spectral signatures and shape distances
related to eigenfunctions and eigenvalues. Then, we provide a
brief overview of our new shape distance.
A. Eigenfunctions
The eigenfunctions of an object constitute an infinite set of
functions. Each function depends on the shape of the object
and is different than the rest of the set. Figure 1 illustrates
this for two example objects where a few eigenfunctions are
shown. The values these functions attain at each point capture
the local geometry around the point, i.e. of its neighborhood.
Inspired from this geometric information, methods define local
shape signatures [4], [5], [6], [12] for each point on an object
by evaluating a subset of eigenfunctions at that specific loca
tion. Global shape distances are then defined using such local
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Fig. 1: Starfish and tarantula. The objects represented as binary
maps are shown on the left, followed by the 1st, 2nd, 5th,
20th, and 100th eigenfunction. The values increase from blue
(negative) to red (positive) with green being zero.
signatures. Such distance definitions rely on correspondences.
These correspondences should hold both in terms of points and
the subset of eigenfunctions used in the signatures, a condition
hard to satisfy in practice [12]. Explicitly searching for such
correspondences leads to expensive algorithms [12], [13], [14],
[15], [16]. On the other hand, computing distances between
distributions of local signatures obtained by aggregating all
the points, as in [5], [6], [17], might implicitly construct false
correspondences. In summary, defining a global distance based
on local signatures is not an easy task.
Instead of extracting local information from an eigenfunc
tion, one can also think of capturing its global pattern by
looking at regions where its values are all positive or all
negative. Such regions are called nodal domains. Different
eigenfunctions induce different patterns and, in turn, have
different number of nodal domains, called nodal counts [11].
For a given object, the ordered sequence of nodal counts
contain information on its overall geometry [18], [19]. Inspired
by these observations, authors in [20] used this sequence as
a global shape signature. They further defined the associated
shape distance between two objects as the Euclidean norm
of the vector difference between their nodal count sequences.
However, it is not intuitively clear what the nodal counts
represent. Furthermore, the entire sequence is diverging so
that, in practice, one first chooses a finite subset and then
computes the distance for that subset. These difficulties make
it hard to define an intuitive and sound shape distance based
on nodal counts.
B. Eigenvalues
Signatures based on eigenvalues, on the other hand, have
a clearer geometric interpretation. The set of eigenvalues
contains information on the overall geometry of the object.
Specifically, the ordered sequence is analytically related to
the intrinsic geometry by the heattrace, [21], [22], [23], [24],
[25]. Hence, more intuitive distances can be constructed using
the eigenvalues. However, similar to the sequence of nodal
counts, the eigenvalue sequence is also divergent. This makes
the distance definition theoretically challenging. Inspired by
the sequence’s link to the geometry, Reuter et al. in [3],
used the smallest N eigenvalues as a shape signature, called
shapeDNA. As the associated shape distance, the authors
proposed the Euclidean norm of the vector difference between
the shapeDNAs of objects. Although this is a very good first
attempt the divergent nature of eigenvalue sequence results
in important theoretical limitations for this distance, as also
pointed out in [14]. The main problems are i) defining a
distance on the entire sequence does not yield a proper metric,
ii) the differences between the higher components of two
sequences dominate the final distance value, even though these
components do not necessarily provide more information on
the geometry, and iii) the distance value is sensitive to the
choice of the signature size N. These theoretical problems
also cause practical drawbacks as we demonstrate later.
This article proposes a new shape distance, called Weighted
Spectral Distance (WESD), using the sequence of eigenvalues
of the Laplace operator. We derive WESD from the functional
relationship between the eigenvalues and the geometric in
variants as given by the heattrace. This derivation provides
WESD a clear geometric intuition as a shape distance. It also
links WESD to the distance defined by Reuter et al. in [3]
as well as to the local signature defined in [5]. The resulting
formulation of WESD differs from other previously proposed
scores based on eigenvalues, whether in shape analysis or
other fields [26], both in its formulation and in the fact that
it is defined over the entire sequence. This latter point, as
we will show later, alleviates the critical importance of the
choice of the signature. We furthermore analyse and prove
theoretical properties of WESD showing that it does not share
some of the fundamental problems the distance proposed
in [3] has. Specifically, we prove that WESD: i) converges
despite the fact that it is defined over the entire eigenvalue
sequence, ii) can be mapped to the [0,1) interval, iii) is
accurately approximated with a finite number of eigenvalues
and the truncation error has an analytical upper bound and
iv) is a pseudometric. These theoretical properties also yield
important practical advantages such as being less sensitive
to the signature size (truncation parameter) N, providing a
principled way of choosing this parameter, providing more
stable lowdimensional shape embedding and simplicity in
combining with other distances as WESD can be normalised.
Applying to synthetic and real objects, we further demonstrate
the benefits of WESD in comparison to the other eigenvalue
based distance defined in [3].
The remainder of this article is structured as follows.
Section II presents a brief overview of the Laplace operator,
the eigenvalue sequence and its role in shape analysis. In
Section III we define WESD and derive its theoretical prop
erties. Section IV presents an extensive set of experimental
analysis on 2D objects extracted from synthetic binary maps,
shapebased retrieval results for 3D objects using the SHREC
dataset [27], low dimensional embeddings of real 3D data such
as subcortical structures in brain scans and 4D analysis of
binary maps extracted from cardiac images.
II. SPECTRUM OF LAPLACE OPERATOR
This section provides a brief background on the Laplace
operator, its eigenvalue sequence, called spectrum, and its role
in shape analysis. We first relate an object’s intrinsic geometry
to the spectrum of the corresponding Laplace operator. We
then provide some details on the previously proposed shape
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DNA [3] and discuss the associated issues. For further details
we refer the reader to [11], [24], [25] and [3].
We denote an object as a closed bounded domain Ω ⊂ Rd
with piecewise smooth boundaries. In the case of binary maps,
Ω would correspond to the foreground representing the object.
For a given Ω, the Laplace operator on this object is defined
with respect to a twice differentiable realvalued function f as
∆Ωf ?
d
?
i=1
∂2
∂x2
i
f, ∀x ∈ Ω
where x = [x1,...,xd] is the spatial coordinate. The eigenval
ues and the eigenfunctions of ∆Ωare defined as the solutions
of the Helmholtz equation with Dirichlet type boundary con
ditions1, [11],
∆f + λf = 0 ∀x ∈ Ω, f(x) = 0, ∀x ∈ ∂Ω,
where ∂Ω denotes the boundary of the object and λ ∈ R
is a scalar. There are infinitely many pairs {(λn,fn)}∞
satisfying this equation and they form the set of eigenvalues
and eigenfunctions respectively. The ordered set of eigenvalues
is a positive diverging sequence such that 0 < λ1 ≤ λ2 ≤
··· ≤ λn≤ .... This infinite sequence is called the Dirichlet
spectrum of ∆Ω, which we refer simply as the “spectrum”. In
addition, each component of the spectrum is called a “mode”,
e.g. λnis the called nthmode of the spectrum
The spectrum contains information on the intrinsic geometry
of objects. Weyl in [9] showed the first spectrumgeometry link
by proving that the asymptotic behavior of the eigenvalues is
given as
?
where VΩis the volume of Ω and Bdis the volume of the unit
ball in Rd. Later works, as [21], [22], [23], [24], extended this
result by studying the properties of the Green’s function of the
Laplace operator, and showed that a more accurate spectrum
geometry link is given by the heattrace, which in Rdis given
as
∞
?
The coefficients of the polynomial expansion, as/2, are the
components carrying the geometric information. These coef
ficients are given as sums of volume and boundary integrals
of some local invariants of the shape, [22], [23], [25]. For
instance, as given in [22], the first three coefficients are:
1
(4π)d/2VΩ
n=1
λn∼ 4π2
n
BdVΩ
?2/d
, n → ∞,
Z(t)
?
n=1
e−λnt=
∞
?
s=0
as/2t−d/2+s/2, t > 0. (1)
a0
=
a1/2
=
−
1
4(4π)d/2−1/2SΩ,
1
6(4π)d/2
a1
=
−
?
∂Ω
κd∂Ω,
where SΩ is the surface area (circumference in 2D) and
κ is the mean (geodesic) curvature on the boundary of Ω.
1Other boundary conditions yield different eigensystems. Here we are only
interested in the Dirichlet type. Please refer to [11] for the other types.
The functional relationship between the eigenvalue sequence
and the coefficients as/2can be seen in Equation (1). This
connection relates the spectrum to the intrinsic geometry,
which is the reason why Laplace spectrum is important for
the computational study of shapes.
In addition to the spectrumgeometry link, the eigenvalues
of the Laplace operator have two other properties which make
them useful for shape analysis, [11]. These are: 1) the Laplace
operator is invariant to isometric transformations and 2) the
spectrum depends continuously on the deformations applied to
the boundary of the object. The advantage of the first property
is obvious since isometric transformations do not alter the
shape. In addition to this, the second property states that there
is a continuous link between the differences in eigenvalues
and the difference in shape, which makes eigenvalues ideal
for measuring shape differences.
Unfortunately, it has also been shown that there exists
isospectral noncongruent objects, i.e. objects with different
shape but the same spectrum [28]. Therefore, theoretically
the Laplace spectrum does not uniquely identify shapes.
However, as stated in [3], practically this does not cause
a problem mostly because the constructed isospectral non
congruent objects in 2D and 3D are rather extreme examples
with nonsmooth boundaries.
The spectral signature, shapeDNA, proposed in [3] is in
spired from the properties given above. For a given shape Ω, its
shapeDNA is the first N modes of the spectrum of the Laplace
operator defined on Ω: [λ1,λ2,...,λN]. In addition to the
properties the shapeDNA inherits from the eigenvalues, the
authors also proposed several normalisations to obtain almost
scale invariance2. The normalisations used in the experiments
in [3], [7], [27], [29] are given as λn → λnV2/d
λn→ λn/λ1.
In [3], the authors also defined a shape distance based on
shapeDNA. Either using the original or its scale invariant
version, this distance is given as
?N
n=1
where Ωξdenotes the object with the spectrum {ξn}∞
ing ρN
distinct shapes [27], construct shape manifolds based on the
pairwise distances and perform statistical comparisons [7],
[29].
However, as also pointed out in [14], due to the diverging
nature of the spectrum, ρN
drawbacks limiting its usability: i) differences at higher modes
of the spectrum have higher impacts on the final distance value
even though they are not necessarily more informative about
the intrinsic geometry, ii) the distance is extremely sensitive
to the signature size N, while the choice of this parameter
is arbitrary, and iii) the distance cannot be defined over the
entire spectrum because it does not yield a proper metric in
Ω
and
ρN
SD(Ωλ,Ωξ) ?
?
(λn− ξn)2
?1/2
,
(2)
n=1. Us
SD(Ωλ,Ωξ), the authors were able to distinguish between
SDsuffers from three essential
2We use the term “almost” because scale invariance is an application
dependent concept and the definition of scale difference between arbitrary
objects is a mathematically vague notion. A further discussion of scale
invariance is outside the scope of this article and we refer the reader to [3].
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4
that case. Therefore, defining a sound and intuitive distance
based on the spectrum is still an open question for which we
propose a solution in the next section.
III. WEIGHTED SPECTRAL DISTANCE  WESD
This section presents the proposed spectral distance, WESD,
the analysis of the heattrace leading to its definition and its
theoretical properties. The structure of presentation aims to
separate the definition of the distance, which is essential for
its practical implementation, from the details related to its
derivation and theoretical properties. In this light, we first
present the definitions and mention the associated proper
ties with appropriate references to the following subsections,
which contain further details.
We define the Weighted Spectral Distance  WESD  for two
closed bounded domains with piecewise smooth boundaries,
Ωλ,Ωξ⊂ Rdas
?∞
n=1
with p ∈ R and p > d/2. Unlike the distance given in
Equation (2), WESD is defined over the entire eigenvalue
sequence and the factor p is not fixed to 2. In addition, the
difference at each mode contributes to the overall distance
proportional to λn− ξn/λnξn instead of λn− ξn. The
additional λnξn factor (seeming like a simple addition to
Equation 2) actually arises from analysing the relation between
the nthmode of the spectrum and the heattrace, which will be
presented in Section IIIA. This analysis also provides WESD
with a geometric intuition. Furthermore, for p > d/2 the
infinite sum in the definition is guaranteed to converge to a
finite value for any pair of shapes. Hence, WESD exists. In
addition to its existence, WESD also satisfies the triangular
inequality making it a pseudometric. These points are proven
in Section IIIB. Moreover, the pseudometric WESD has a
multiscale aspect with respect to p. In Section IIIC we show
that adjusting p controls the sensitivity of WESD with respect
shape differences at finer scales, i.e. with respect to geometric
differences at local level such as thin protrusions or small
bumps. Thus, for higher values of p the distance becomes less
sensitive to finer scale differences.
In addition to WESD, we define the normalised score for
shape dissimilarity nWESD as
ρ(Ωλ,Ωξ)
W(Ωλ,Ωξ)∈ [0,1),
which maps ρ(Ωλ,Ωξ) to the [0,1) interval using the shape
dependent normalisation factor
?
The factors C and K are the shape based coefficients defined
in Corollary 1, and ζ(·) is the Riemann zeta function [30].
Being confined to [0,1), nWESD allows us to i) compare
dissimilarities of different pairs of shapes and ii) easily use
the shape dissimilarity in combination with scores quantifying
ρ(Ωλ,Ωξ) ?
?
?λn− ξn
λnξn
?p?1/p
,
(3)
ρ(Ωλ,Ωξ) ?
(4)
W(Ωλ,Ωξ) ?
C + K ·
?
ζ
?2p
d
?
− 1 −
?1
2
?2p
d??1
p
.
other type of differences between objects such as volume
overlap in case of matching or Jacard’s index in case of
accuracy assessment.
One important issue in defining a distance or a score using
the entire eigenvalue sequence is computational limits. In prac
tice we can only compute a finite number of eigenvalues and
therefore, can only approximate such distances. Considering
this, here we define the finite approximations of WESD and
nWESD using the smallest N eigenvalues as
?N
n=1
ρN(Ωλ,Ωξ)
W(Ωλ,Ωξ)∈ [0,1),
where N is a truncation parameter. Previous works, such
as [3], [5], [17], [20], [26], also define distances based on
finite number of modes. However, their view on the distance
definition was first to construct finite shape signatures and
then to define a distance on the signatures. Therefore, the
signature size was a critical component of the definition itself.
Furthermore, the effects of the choice of the signature size
on the distance values have not been carefully analysed in
these works. The view presented here defines the distance
directly using the entire sequence without constructing a finite
signature. This alleviates the importance of the signature size
on the distance. The finite computation given in Equations 5
and 6 are viewed as approximations to the distance and N as
the truncation parameter. In this conceptually different setting,
unlike previous works, we provide in Section IIID a careful
analysis of the choice of N on the spectral distance. Specifi
cally, we prove that limN→∞ρ(Ωλ,Ωξ) − ρN(Ωλ,Ωξ) = 0
and limN→∞ρ(Ωλ,Ωξ)−ρN(Ωλ,Ωξ) = 0. Furthermore, we
provide a theoretical upper bound for these errors that shows
how fast they decrease in the worst case leading to a principled
strategy for choosing N.
Section IIIE ends the section by focusing on the invari
ance of WESD and nWESD to global scale (relative size)
differences between objects. Specifically, we discuss how an
“approximate” scale invariance can be attained for WESD and
nWESD by following the same strategy proposed in [3].
ρN(Ωλ,Ωξ)
?
?
?λn− ξn
λnξn
?p?1/p
(5)
ρN(Ωλ,Ωξ)
?
(6)
A. Analysis of the HeatTrace and Derivation of WESD
We derive WESD by analysing the mathematical link be
tween the spectrum of an object and its geometry. This link is
given by the heattrace defined in Equation (1). Let us consider
the heattrace as a function of both t and the spectrum,
Z(t,λ1,λ2,...). The main question we answer is how much
the Z(t,·) function changes when we change the nthmode
of the spectrum from λn to ξn. Considering the polynomial
expansion equivalent to Z(t,·) given in Equation (1), one can
see that the change in the value Z(t,·) is directly related
to the changes in the coefficients as/2 and so to changes
in the integrals over the local invariants. By analysing the
influence of the change in the nthmode on Z(t,·), we actually
analyse the influence of this change on the integrals over
local geometric invariants. Following this line of thought, we
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5
quantify the influence of the change from λnto ξnon Z(t,·)
in terms of λnand ξn. This can be done by defining
?∞
−Z(t,...,λn−1,ξn,λn+1,...)dt,
which is simply the L1norm of the difference between the
functions that is linked to the difference between λnand ξn.
Replacing Z(t,·) with its definition leads to
?∞
Without loss of generality let us assume ξn≥ λn. Then
e−λnt≥ e−ξntfor t > 0.
We can then evaluate the integral in Equation (7) as
?∞
∆n
on Z(t,·). Now, aggregating these influences across all modes
leads to the definition of WESD
?∞
n=1
∆n
Z
?
0
Z(t,...,λn−1,λn,λn+1,...)
∆n
Z
=
0
??e−λnt− e−ξnt??dt
(7)
∆n
Z=
0
e−λnt− e−ξntdt =λn− ξn
λnξn
.
Zcaptures the influence of the difference at the nthmode
ρ(Ωλ,Ωξ) =
?
(∆n
Z)p
?1/p
=
?∞
n=1
?
?λn− ξn
λnξn
?p?1/p
.
Surprisingly, the formulation of WESD, which results from
the analysis presented above, also has very beneficial proper
ties that makes it theoretically sound and useful in practical
applications. These properties will be analysed in the follow
ing.
Before delving into this analysis though let us make two
remarks. The first relates ρSD(·,·) (Equation (2)) to the
analysis of the heattrace presented above.
Remark 1. Let us define
∆n,m
Z
?
????
?∞
0
dm
dtmZ (t,...,λn−1,λn,λn+1,...)
−dm
dtmZ (t,...,λn−1,ξn,λn+1,...)dt
????.
One can see that ∆n,0
∆n,m
Z
=
n
derived as follows
Z
= ∆n
??. By setting m = 2 ρSD(·,·) can be
?∞
n=0
Z. Evaluating this integral yields
??λm−1
− ξm−1
n
ρSD(Ωλ,Ωξ) =
?
?
∆n,2
Z
?2?1/2
.
This relationship not only relates WESD to ρSD(·,·) but also
provides the link between ρSD(·,·) and the heattrace. One
notices that the functional difference definition used in this
remark differs from the previous one used to derive WESD,
see Equation 7. This is because ρSDcannot be derived from
the L1distance definition used previously but can be derived
from the less ideal definition used in this remark. The existence
of an alternative derivation of ρSDthat would start from an
appropriate functional difference is an open question.
The second remark notes the link between WESD and
Global Point Signatures (GPS), a local shape descriptor,
presented in [5].
Remark 2. GPS, as presented in [5], is defined for each
point in an object Ωλ as the infinite series GPSΩλ(x) ?
{Φλ,n(x)} ?
the ntheigenfunction. GPS has a connection to WESD arising
from the following elementwise integrals
?
where the equality arises from the fact that eigenfunctions form
an orthonormal basis in Ωλ[11], i.e.?
integral, WESD can also be regarded as a distance between
GPS’ of two objects as
?∞
n=1
Ωλ
Ωξ
This link also provides an alternative view on the normali
sation factor λ−1/2
n
used in GPS. In [5] author justifies this
normalisation factor by noting that for an object the Green’s
function can be written as an inner product in the GPS
domain, see Section 4 in [5]. This is later used to argue the
geometric meaning of GPS as authors point out the use of
Green’s function in different shape processing tasks. Our link
between GPS and WESD provides an alternative view on the
normalisation factor as it connects this local signature to the
heattrace Z(t).
?
λ−1/2
n
fn(x)
?∞
n=1, where x ∈ Ωλand fn(x) is
Ωλ
Φ2
λ,n(x)dx =
?
Ωλ
?
λ−1/2
n
fn(x)
?2
dx = λ−1
n,
Ωλfn(x)fm(x)dx =
δ(n − m) with δ(·) being the Dirac’s delta. Considering this
?
??
Φ2
λ,n(x)dx −
?
Φ2
ξ,n(x)dx
?p?1
p
= ρ(Ωλ,Ωξ).
B. Existence of the Pseudometric WESD
WESD is defined as the limit of an infinite series as given in
Equation (3). For such a distance to be a proper one, actually
a pseudometric in this case, the limit of the infinite series
should exist for any two spectra. In the case of WESD, this
is not evident because it is defined over the entire spectra
and each spectrum is a divergent sequence. The first corollary
presented below proves that when p > d/2 WESD indeed
satisfies this condition, i.e. the infinite series converges. The
corollary further provides an upper bound for this limit, which
is used to construct nWESD. We would like to note that for the
ease of presentation, the proofs for all the following corollaries
and lemmas are given in Appendix B in the supplemental
material.
Corollary 1. Let Ωλ⊂ Rdand Ωξ⊂ Rdbe any two closed
domains with piecewise smooth boundaries and {λ}∞
{ξ}∞
distance
?∞
n=1
converges for p >d
?
n=1and
n=1be their Laplace spectrum. Then the weighted spectral
ρ(Ωλ,Ωξ) =
?
?λn− ξn
λnξn
?p?1/p
2. Furthermore,
?
ρ(Ωλ,Ωξ) <C + K ·
ζ
?2p
d
?
− 1 −
?1
2
?2p
d??1
p
, (8)
Page 6
6
where ζ(·) is the Riemann zeta function and the coefficients
C and K are given as
i
?d + 2
ˆV
?
max(V (Ωλ),V (Ωξ)), µ ? max(λ1,ξ1),
where V (·) denotes the volume (or area in 2D) of an object.
The Inequality (8) states that WESD has a shapedependent
upper bound. We thus can map the WESD to the [0,1) interval
through normalising it with this upper bound. The nWESD
score, given in Equation 4 is constructed based on this strategy.
Since its existence is established next we prove that WESD
is a pseudometric, i.e. satisfies the other criteria to be a
pseudometric, such as the triangle inequality.
C
?
?
i=1,2
d + 2
d · 4π2·
d · 4π2·
?
?
?2
BdˆV
?2
d
−1
µ·
?
d
d + 4
?p
?i−1
p
K
?
BdˆV
d−1
µ·
d
d + 2.64
Corollary 2. ρ(Ωλ,Ωξ) is a pseudometric for d ≥ 2.
We note that WESD is not a metric because the spectrum
is invariant to isometries, which is a desirable property for
shape analysis. However, in addition to this, the spectrum is
also invariant to isospectral noncongruent shapes. This is not
desirable but does not cause problems in practice as discussed
in Section II and also confirmed in our experiments.
C. On the multiscale aspect of WESD
The previous section highlighted the role of p on the
convergence properties of WESD and therefore on its exis
tence. We now demonstrate that p also provides WESD a
multiscale characteristic. The sensitivity of WESD to the
shape differences at finer scales depends on the value of p.
Specifically, we show that the higher the value p the less
sensitive WESD is to finer scale details and its sensitivity
increases as p gets lower.
The multiscale aspect of WESD arises from the rela
tionship between the Laplace operators and heat diffusion
processes [31]. We first present an intuitive summary of this
relationship, which is about the multiscale aspect of Z(t) and
t in particular. For a more mathematical treatment we refer the
reader to [6]. As stated in [6] and [14], t can be interpreted as
the time variable in a heat diffusion process within an object.
A useful visual analogy to consider here is the Laplacian
smoothing of a surface where t would correspond to the
amount of smoothing. Similar to the surface smoothing, as
t increases, the local geometric details of an object, such
as sharp ridges or steep valleys, lose further their influence
on the Z(t) value. As a result Z(t) becomes somewhat
insensitive to these local geometric details, in other words
shape details at finer scales. From an alternative view, the
value of Z(t) loses its information content with regards to
local geometric details. This effect intuitively summarizes the
multiscale characteristic of the heattrace with respect to t.
Having explained the multiscale aspect of Z(t), we now
analyse how this aspect is reflected upon the eigenvalues. To
do so let us define the influence ratio D(n,t) ?e−λnt
Z(t). This
ratio captures the influence of the nthmode on the heattrace.
In other words, the higher the ratio, the higher the influence
of λnon the value of Z(t) at that specific t. The following
lemma compares the influence ratios of different modes and
how this comparison depends on t.
Lemma 1. Let Ωλ⊂ Rdrepresent an object with piecewise
smooth boundary and D(l,t) ?
influence ratio of mode l at t. Then for any two spectral indices
m > n > 0
D(n,t) > D(m,t), ∀t > 0
and particularly for two t values such that t1> t2
D(m,t1)
e−λlt
Z(t)be the corresponding
D(n,t1)<D(m,t2)
D(n,t2).
The first inequality of the lemma indicates that the lower
modes in the spectrum have more influence on the value of
Z(t) than the higher modes. The second inequality shows that
the influence of the higher modes become more prominent
as t decreases. Considering that for lower t values Z(t) is
more informative with regards to shape details at finer scales,
Lemma 1 suggests that the higher modes are more important
for finer scales than for coarser scales. We illustrate this
observation on a synthetic example shown in Figure 2 with
the pair (a) + (b) being an example showing coarser shape
differences and the pair (a) + (c) showing finer differences.
The plots given in Figure 2 (d) and (e) show the corresponding
spectral differences observed at modes between 1 and 150.
Between (a) and (b) the shape differences are at the coarse
level. According to Lemma 1 these differences should show up
at the very first modes. On the other hand, between (a) and (c)
the differences are at a finer scale and furthermore the objects
are very similar at the coarse level. Lemma 1 states that these
differences therefore, should show up at higher modes and
the differences at the lower modes should be low. Satisfying
these expectations, the differences at the first few modes shown
in plot (d) have relatively large values compared to the ones
in plot (e). Furthermore, the amplitude of the differences at
higher modes are generally larger in plot (e) than in plot(d),
especially after 100.
In order now to connect these findings to WESD and p let
us present the following corollary, which studies the influence
of p on the components inside the infinite sum defining the
distance.
Corollary 3. Let Ωλand Ωξbe two objects with piecewise
smooth boundaries. Then for any two scalars with with p >
d/2, q > d/2, p ≥ q and for all n with λn− ξn > 0 there
exists a M > n so that ∀m ≥ M
?
λn−ξn
λnξn
λm−ξm
λmξm
?
?p
?p ≤
?
λm−ξm
λmξm
?
?q
λn−ξn
λnξn
?q
Thus, the relative contributions of the higher spectral modes
on ρ(Ωλ,Ωξ) with respect to the contributions of the lower
modes depend on the value of p. Specifically, the higher spec
tral modes become more influential as p decreases. Combining
this finding with the result of Lemma 1, we follow that as p
Page 7
7
(a)(b)(c)
(d) between (a) and (b) (e) between (a) and (c)
Fig. 2: Multiscale characteristics of different spectral
modes:(a), (b) and (c) show three synthetic shapes. In (d)
we plot the absolute differences between the corresponding
modes of (a) and (b) with respect to the spectral index. In (e)
we plot the same difference for the shapes in (a) and (c). The
shape difference between (a) and (b), which is at a coarser
level, is already captured at the lower spectral modes. The
difference between (a) and (c) results in lower differences in
lower spectral modes because these objects are more similar
at a coarser level. At the higher spectral modes though the
difference between (a) and (c) becomes more prominent since
these objects differ more substantially at the finer scales. The
plots in (d) and (e) demonstrate that the higher modes for a
given object are more important for finer scale shape details.
increases WESD gives less importance to differences at higher
spectral modes and therefore becomes less sensitive to the
shape differences at finer scales. This provides WESD with a
multiscale aspect with respect to p and also provides us the
intuition for choosing a proper value for p.
D. Finite Approximations of WESD and nWESD
One of the important practical questions regarding spectral
distances is the number of modes to be included in the
calculation of the distance. The computation of eigenvalues
and eigenfunctions can be expensive and inaccurate especially
for the higher modes. Therefore, spectral distances require
the user to set a finite number of modes to be used. This
parameter is often referred to as the signature size. Having
defined the distance over the entire sequence, we refer to it as
the truncation parameter. This actually provides a different
perspective on the number of modes used to compute the
distance. In previous works, such as [3], [5], [20], the value of
this parameter, viewed as the signature size, is often set arbi
trarily and its effect on the distances have not been carefully
analysed. Here, viewing it as a truncation parameter, we study
its influence. Specifically, we formulate the difference between
using the entire spectra to only using a finite number of modes
as an approximation/truncation error. So we analyse how
this error changes with respect to the truncation parameter.
We specifically show in the next corollary that the errors
in approximating WESD and nWESD by the first N modes
converges to zero as N increases. Furthermore, we provide an
upper bound for both errors as a function of N.
Corollary 4. Let ρN(Ωλ,Ωξ) be the truncated approximation
of ρ(Ωλ,Ωξ) based on the first N modes and ρN(Ωλ,Ωξ) of
ρ(Ωλ,Ωξ). Then ∀p > d/2
lim
N→∞ρ − ρN = 0
and
lim
N→∞ρ − ρN = 0.
Furthermore, for a given N ≥ 3 the truncation errors ρ−ρN
and ρ − ρN can be bounded by
?
?
The above corollary has important practical implications.
First of all, the sensitivities of ρNand ρNwith respect to
N decreases as N increases. For any application relying on
the shape distances, such as constructing low dimensional
embeddings, this reduced sensitivity is particularly important
as it provides stability with respect to N both for the distance
and for the application using the distance. We note that
the opposite is true for ρN
disadvantages of this distance.
In addition, Corollary 4 can guide the choice for the number
of modes N and the norm type p. Specifically, the error upper
bounds given in Equations 9 and 10 provide the worst case
errors for a given pair of shapes without the need to compute
the eigenvalues. So for instance, once a number of modes are
computed then based on the distance value obtained so far and
the worst case error computed using the upper bounds, one can
decide whether to compute more modes or not. Furthermore.
these upper bounds are shapespecific as they depend on C and
K. One can go one step further and define a shapeindependent
residual ratio for N ≥ 3 and p > d/2 as
ζ?2p
that satisfies R(N,p) > ρ − ρN, for which the proof is given
in Proposition 1 in Appendix B. Based on this, R(N,p) can
be used to select the parameters N and p as it quantifies the
quality of the approximation for a given pair of (N,p) in terms
of the error upper bounds.
In Figure 3, we plot R(N,p) versus N for different settings
of p and d = 2,3. Besides the obvious point that the error
upper bound decreases for increasing N we also notice that
??ρ − ρN??
< C + K ·
?
ζ
?2p
?N
n=3
d
?
?1
??N
ζ?2p
− 1 −
?1
2
?2p
d??1
p
(9)
−
C + K ·
?
n
?2p
d??1
p
ρ − ρN
<
1 −
C + K ·
n=3
?1
n
?2p
d
?
?2p
C + K ·
?
d
?− 1 −?1
2
d
?
1
p
(10)
SD, which is one of the main
R(N,p) ? 1 −
?N
d
n=3
?− 1 −?1
?1
n
?2p
d
2
?2p
d
1
p
.
(11)
Page 8
8
Fig. 3: Choosing N: The figures plot the residual ratio R(N,p)
versus N for different p values in 2D (left) and in 3D (right).
As expected the error upper bound drops with increasing N.
The rate of decrease also becomes faster with increasing p.
This inverse relation suggests the tradeoff between N and
the sensitivity of WESD to finer scale shape differences since
WESD becomes less sensitive as p increases, see Section IIIC.
i) the behavior in 2D and 3D are similar and ii) the rate of
decrease of the error upper bound is much faster for higher
p. Considering the multiscale aspect of WESD captured in p,
this behavior is interesting. It demonstrates that the choice of p
and N are correlated and suggests a tradeoff between the rate
of decrease of the truncation error and the sensitivity of WESD
to shape differences at finer scales. In theory, the choice of
these parameters depends on the application and the expected
shape differences. If one expects coarse scale differences then
choosing a large p and small N might be sufficient. However,
if one is interested in finer scale differences then a small p
value will be required, which in turn will require a large N
value to have a decent approximation. The important aspect
of R(N,p) is that it is universal, i.e. it does not depend on the
objects. So it can be used in any type of application to choose
the parameter pair N,p and to have a rough estimate of the
computational costs for computing the distance WESD. We
note once again, the specific values should be chosen based
on the application and the shapes at hand.
E. Invariance to global scale differences
We end this section studying the impact of global scale
differences on WESD and how invariance to such differences
can be attained. We would like to note that the notion of global
scale in this section refers to the relative size of an object,
which is not to be confused with the notion of multiscale used
in Section IIIC. The spectrum of an object depends on the ob
ject’s size, i.e. a global scale change alters all the eigenvalues
by a constant multiplicative factor [11]. As a result, the global
scale difference between two objects contributes to the spectral
shape distance WESD. In some applications this contribution
might not be desirable, for instance in an object recognition
task, where objects in the same category have varying sizes.
Therefore, it is a useful property of a shape distance to allow
invariance to global scale differences.
Reuter et al.[3] proposed different approximations for
normalising the effects of scale differences on the spectrum. In
particular, the authors use two different normalisations in their
experiments in [7], [27], [29]. Both normalisations directly act
on the eigenvalues. The first one normalises the eigenvalues
with respect to the volume (area in 2D or surface area for
Riemannian manifolds) and is given as λn → λnV2/d
second one normalises the eigenvalue with respect to the first
eigenvalue in the sequence as λn → λn/λ1. Both of these
strategies can be used when computing distances with WESD.
Furthermore, since these strategies do not alter the mathe
matical characteristics of the entire spectrum the theoretical
properties of WESD and nWESD hold either way. For our
experiments we adopt the first strategy, volume normalisation,
using the volume as defined in Euclidean geometry. When
using the volume normalised eigenvalues, the only change
that applies to the technical details presented so far isˆV in
Equation 8 becomesˆV = 1. The rest applies directly without
any modification.
We would also like to note that estimating the global scale
difference between two arbitrary objects is not always a well
posed problem. It is especially hard when the objects are of
different category, e.g. an octopus and a submarine. Further
more, the scale normalisation is application dependent and it
might not be desirable for all applications. In Section IVC2
we present such an example where we analyse the temporal
change of the left ventricle shape during a heart cycle. In this
case, the volume change is essential for analysing the heart of
the same patient so that scale invariance is not appropriate.
Ωλ. The
IV. EXPERIMENTS
This section presents a variety of experiments on synthetic
and real data highlighting the strengths and weaknesses of
WESD and nWESD. We start by briefly explaining the de
tails of the numerical implementation of WESD used in the
experiments presented here. Then in Section IVB, the pro
posed distances are applied to synthetically generated objects
demonstrating that
(i) Ordering objects with respect to their shapes using
nWESD results in a visually coherent series (Sec
tion IVB1),
(ii) WESD is useful for constructing low dimensional em
beddings, in particular it yields stable embeddings with
respect to the signature size N, (Section IVB2) and
(iii) WESD is a suitable distance for shape retrieval, which is
shown through experiments on the SHREC dataset [27]
(Section IVB3).
Lastly, in Section IVC WESD is applied to real objects
extracted from 3D medical images. We focus on two examples
from a wide variety of applications WESD and nWESD can
be beneficial to: population studies of brain structures and
analysis of 4D cardiac images.
A. Implementation Details
There are two different aspects in the implementation of
WESD: the numerical computation of the Laplace spectra and
the parameter settings. First, any numerical method tailored
towards computing the eigenvalues of the Laplace operator
can be used to compute WESD. Examples of such method
are listed in [3], [32]. Our specific implementation represents
objects simply as binary images with the foreground defining
Ω. Using the Cartesian grid of the image, it discretizes
Page 9
9
∆Ω through finite difference scheme (see also Chapter 2 of
[32]). This step yields a sparse matrix of which we compute
the eigenvalues via Arnoldi’s method presented in [33] and
implemented in MATLABR ?. We choose this specific imple
mentation as 1) it is simple 2) it does not introduce any
additional parameters and 3) when working with images it
avoids any extra preprocessing steps, such as surface extraction
or mesh construction.
With regards to the second implementation aspect, we set
the parameters N and p empirically. Based on Section IIID,
we set p = 1.5 or p = 2.0 in 2D and p = 2.0 in 3D. These
values result in a relatively fast diminishing upper bound of
the truncation error with respect to N (see Figure 3) while
being sensitive to shape differences at finer scales. In both 2D
and 3D, we chose N = 200 for the number of modes as the
truncation error seemed to vanish at that point. Furthermore, in
addition to the theoretical considerations on the effects of N
and p on WESD given in Section IIID, in Sections IVB2 and
IVB3 we experimentally study the effects of these parameters
on applications using WESD, specifically on constructing low
dimensional embeddings and shape retrieval.
B. Synthetic Data
We conduct three experiments: first two are on 2D objects
and the last one is on 3D objects. For all of the experiments,
we use the scale invariant versions of the spectra obtained
by normalising the eigenvalues with the object’s volume as
described in Section IIIE. As a result the distances WESD and
nWESD become “almost” invariant to global scale differences.
1) Ordering of Shapes: For the first experiment we created
two synthetic datasets. Each dataset consists of a reference
object and random deformations of this reference. These
deformed versions are generated by transforming the reference
via random deformations of varying magnitude and amount
of nonlinearity. As a result the datasets contain objects that
are very similar to the reference ones and objects that are
substantially different. Figures 4(a) and (b) show some ex
amples from these datasets where the binary images to the
very left show the reference objects. In the first dataset, the
reference object is a disc and in total there are 500 random
deformations of this reference disc. The first 400 are generated
via nonlinear deformations while the last 100 are isometric
transformations. In the second dataset, the reference is a
slightly more complicated object (see Figure 4(b)) and in total
there are 400 random transformations of this reference. The
first 300 are generated by nonlinear deformations and the
last 100 produced via isometric transformations. All objects
are discretized as binary maps with a size of 200×200 pixels.
The numerical computations are performed on these image
grids as discussed earlier.
We computed the nWESD scores (ρNwith p = 1.5 and
N = 200) between the reference and the deformed objects
in each dataset. Based on these scores, we then ordered the
deformed objects according to their similarity in shape to the
reference. Figures 4 (c) and (d) show examples of the resulting
orderings. We notice that the orderings are visually meaningful
, i.e. the further the deformed objects visually deviate from
(a)
(b)
(c)
(d)
Fig. 4: Shapebased ordering of objects: We generate two
artificial datasets each consisting of a reference object and its
random deformations. Samples from the datasets are shown
in (a) and (b). The binary images to the very left show the
reference objects for each dataset. We then ordered all the
deformed objects with respect to the nWESD scores between
the object and the reference. The graphs in (c) and (d) plot
these orderings. Based on visual inspection the ordering is
quite reasonable.
the references, the higher their nWESD score is. Furthermore,
all the objects generated via isometric transformations yielded
scores close to zero as a result of the invariance of the proposed
scores to this type of transformation.
2) Low Dimensional Embeddings: In the second experi
ment we focus on creating low dimensional embeddings. We
compare the embeddings constructed by WESD with the ones
constructed using ρN
in [3]. We do so based on the TOSCA dataset (toolbox for
surface comparison and analysis), [34], [35]. This dataset
contains binary segmentations of 5 human, 5 centaurs and
5 horses as shown in Figure 5(a). We compute the pairwise
affinity matrices between objects via ρN
with p = 2.0). We then apply the ISOMAP algorithm [36] to
these matrices, which maps the 15 objects to a 2D plane based
on the pairwise shape distances. We repeat this experiment for
affinity matrices computed using different number of spectral
modes, i.e. N = 50,100,200, to demonstrate the effect of the
signature size (truncation parameter) on both distances.
The plots in Figures 5(b),(d) and (f) present the resulting
2D embeddings of the dataset using ρN
substantially different for different N. This variation arises
due to high sensitivity of ρN
N. Moreover, the embeddings obtained using higher N are
less satisfactory in terms of separating the three different
object classes. This is actually as expected since the spectral
SD(Equation (2)), the distance proposed
SDand WESD (ρN
SD. The embeddings are
SDtowards the signature size
Page 10
10
(a)
(b) ρN
SD, N=50
(c) ρN, N=50
(d) ρN
SD, N=100 (e) ρN, N=100
(f) ρN
SD, N=200 (g) ρN, N=200
Fig. 5: Low dimensional embeddings: (a) The 15 objects used
in this experiment. The graphs plot the 2D embeddings of the
objects based on the affinity matrices constructed by ρN
WESD (ρN). Each row presents the results based on different
N: 50, 100 and 200 from top to bottom respectively. The struc
tures of the 2D embedding based on ρN
for different N. WESD however, produces embeddings that
are similar. This demonstrates the stability of the embedding
with respect to N when WESD is used.
SDand
SDare quite different
modes with higher indices dominate the value of ρN
though they are not informative with regards to the overall
geometry and thus, negatively impact the outcome. The plots
in Figures 5(c), (e) and (g) present the embeddings obtained
using WESD. The embeddings obtained at different N are
very similar. This shows that the construction of the low
dimensional embedding is stable with respect to N when
WESD is used. This is a direct consequence of the convergent
behavior of WESD discussed in Sections IIIB and IIID. As
illustrated by this experiment, this property has very important
SDeven
Fig. 6: The theoretical analysis of the effects of truncating the
computation of WESD at N modes on the lowdimensional
embeddings: Points (markers) indicate the lowdimensional
embeddings obtained by using ρN(·,·). The rectangle around a
point denotes the theoretical maximum extent that point might
move to if infinite number of modes were used to compute the
distance, i.e. if ρ(·,·) were used. These bounds are computed
using using Equation 9 without the need to compute more
eigenvalues than N.
practical implications.
The experimental analysis presented above requires com
puting a high number of eigenvalues in advance. Besides this
option, through Corollary 4, WESD provides us the opportu
nity to perform the same analysis without the need to compute
eigenvalues in advance. Once N modes are computed, one can
use Equation 9 to analyse the stability of the embedding with
respect to N and ultimately use this analysis to decide whether
N is enough. In the following we perform such an analysis
for this experiment.
At three different N = (100,150,200), we computed the
error upper bounds for each element of the affinity matrix
using Equation 9. Let us refer to these bounds as EN(Ω1,Ω2),
so we can write ρ(·,·) ∈ [ρN(·,·),ρN(·,·) + EN(·,·)). For
the analysis we assume ρ(·,·) can lie anywhere in this range,
i.e. uniformly distributed. By randomly sampling these ranges
for each element, we generate 5000 different affinity matrices
and build lowdimensional embeddings for each matrix. In
Figure 6 we plot the embeddings obtained with ρN, with
markers, along with the minimum and the maximum low
dimensional coordinates each object attained during random
sampling  indicated with a rectangle around each point. These
rectangles show the maximum error on the lowdimensional
embedding that we might be introducing by truncating the
computation of WESD at N. Observing these graphs: i)
one can provide a guarantee that adding more modes after
N = 200 will not change the embedding much, ii) at N = 150
one could have stopped computing more modes because the
embedding cannot change substantially and iii) at N = 100 the
embedding theoretically can change so more modes might be
necessary. An important point to note is that our experimental
findings suggest that the theoretical upper bounds given in
Corollary 4 are very conservative. In this experiment, for
instance, the embedding does not change much between using
N = 100 and N = 200, i.e. the maximum absolute coordinate
change vector for all the objects is (0.24×10−4,0.22×10−4).
3) Shapebased Retrieval of 3D Objects: In this last experi
ment with synthetic data, we focus on the application of shape
based object retrieval, i.e. given a test object identifying other
Page 11
11
NNFTSTE DCG
WESD (ρN)
p=3.15, N=100
WESD (ρN)
p=2.0, N=100
ρN
SD
N=12, norm1
ρN
SD
N=12, normA
ρN
SD
N=15, norm1
MDSCMBOF
SDGDMmeshSIFT
0.99330.9020 0.9305 0.69000.9706
0.9933 0.89230.9238 0.68240.9691
0.99670.88960.9521 0.6959 0.9748
0.99170.91530.9569 0.70470.9783
0.99330.86830.9431 0.6895 0.9705
0.9950
1.0000
0.9127
0.9720
(a)
0.9691
0.9901
0.7166
0.7358
0.9822
0.9955
(b) p = 3.15 and N = 100
(c) p = 3.15
(d) N = 100
Fig. 7: Shapebased Object Retrieval Results on SHREC Dataset. a) Retrieval scores obtained by WESD for two different sets
of N and p values along with the scores obtained by the distance proposed in [3] (values taken from [27].) b) PrecisionRecall
curves obtained for shape retrieval via WESD for the entire dataset. c) Effect of the signature size N on the retrieval scores
obtained by WESD for a fixed p = 3.15. d) Effect of the norm type p on the same scores for a fixed N = 100.
“similar” objects within a dataset using shape information.
Similarity in this context can be defined in various ways but
the definition used here is semantic similarity, meaning that
objects that are of the same semantic category (e.g. human
bodies, aeroplanes, etc) are similar and objects of different
categories are not. Shapes of similar objects have similar traits
and properties. Shape distances used for retrieval purposes
should be able to capture these traits yielding the lowest values
between similar object pairs. Here, WESD’s value for shape
based retrieval is evaluated using the publicly available dataset
SHREC presented in [27]3.
SHREC dataset consists of 600 3D nonrigid objects from
30 different categories, i.e. 20 objects per category. Ob
jects from the same category differ with substantial non
linear deformations, which makes retrieval in this dataset
challenging. To evaluate the retrieval accuracy of WESD,
first each object was converted from its original watertight
surface mesh discretization to a 3D binary image using the
Iso2mesh software package4. Then pairwise shape distances
across the entire dataset were computed using WESD and
the 600 × 600 affinity matrix was constructed, where each
entry is a pairwise distance. This affinity matrix was then
evaluated using the software provided with the dataset3. The
evaluation consists of a variety of retrieval accuracy scores
such as Nearest Neighbor (NN), FirstTier (FT), SecondTier
(ST), EMeasure (E), Discounted Cumulative Gain (DCG) and
PrecisionRecall curve. The first two rows of the table in
Figure 7(a) list these scores obtained using WESD for two
different settings of the p and N values. Additionally, the
next three rows of the same table show the results obtained
3Available at http://www.itl.nist.gov/iad/vug/sharp/contest/2011/NonRigid/
4http://iso2mesh.sourceforge.net/cgibin/index.cgi
using ρN
different types of scale normalisation (norm1: normalising
with respect to the first eigenvalue, normA: area normalisation,
see Section IIIE for further details). These accuracy scores
show that WESD and ρN
from the SHREC dataset. The plot given in Figure 7(b) shows
the precisionrecall curve of WESD (p = 3.15 and N = 100)
for the entire dataset. The curve is very similar to the best
curve obtained using ρN
confirms that both distances perform similarly.
For a complete comparison, the table given in Figure 7(a)
also reports the results of the two methods that achieved
the highest retrieval scores in the SHREC challenge: MDS
CMBOF ( [37]) and SDGDMmeshSIFT ( [38], [39]). Both
of these methods construct multistep pipelines that provide
powerful tools for retrieval. However, one has to note that
these methods also have added complexities in contrast to
WESD and ρSD, such as: i) the need for a training dataset,
ii) the correspondence requirements and the resulting need
for remeshing, iii) computation of local keypoints for feature
computation, iv) high computational cost of geodesic distance
matrix, and iv) local feature or point matching to compute a
global distance using local features. Nevertheless, their high
accuracies demonstrate the advantage of combining different
fundamental components to achieve powerful retrieval tools.
Lastly, the graphs shown in Figure7(c) and (d) provide an
analysis of the retrieval results with respect to the parameters
N and p. Graphs in Figure 7(c) plot the change of different
retrieval scores with respect to the number of modes used
N, i.e. signature size, keeping p fixed at 3.15. Graphs in
Figure 7(d) plot the changes with respect to the norm type p
SD(Equation 2, [3]), as listed in [27]5, using two
SDperform very similar in retrieval
SDshown in [27]. Once again, this
5We note that for these latter results a slightly different notation is used
here than in [27] to conform to the overall notation of this article.
Page 12
12
keeping N fixed at 100. These graphs show that as N increases
the scores seem to increase slowly and then converge. On the
other hand, p has a stronger effect on the results than N,
particularly on FT, ST and E scores. However, the changes in
the scores with respect to changes in N or p are rather small
especially compared to the relatively larger fluctuation of the
FT score of ρN
provided in the table in Figure 7(a).
The experiment presented above showed that the retrieval
power of WESD is similar to that of the distance ρN
by Reuter et al. [3]. The soundness and theoretical properties
of WESD do not come at the expense of lower retrieval power.
On the contrary, WESD is able to leverage the descriptive
power of the spectra while its properties guarantee that it does
not suffer from similar drawbacks as other distances, such as
sensitivity to signature size.
SDwith respect to the two sample N values
SDproposed
C. Real Data
The experiments on real data are conducted on segmen
tations of 3D structures obtained from magnetic resonance
images (MRI). First, we apply WESD to subcortical brain
structures. The experiment demonstrates WESD’s capabilities
to differentiate categories of objects even in the presence of
high intraclass variability. In the second experiment, we focus
on temporal analysis of cardiac images. We apply nWESD to
delineations of the blood pool of the left ventricle obtained
from 3D + time cardiac MRI. The experiment shows that
the shape dissimilarity measurements between time points
correlates with the dynamic processes of the beating heart.
1) Clustering SubCortical Structures:
frequently relies on morphometric studies analysing anatom
ical shapes from medical images [40]. In this experiment
we construct a low dimensional embedding of subcortical
structures extracted from Magnetic Resonance Image (MRI)
scans of different individuals based on WESD as well as
shapeDNA based distance, ρN
For this experiment, we use the publicly available LPBA40
dataset [41]6. The dataset contains manual segmentations of
various subcortical structures from MRI brain scans of 40
healthy subjects. Figures 8(a), (b) and (c) show some examples
from these structures. The are two main difficulties associated
with such datasets. First, the structures have very large intra
class (intersubject) variability, i.e. the shape of an anatomical
structure is often very different across subjects. Second, the
segmentations were obtained by manually delineating the 3D
objects on successive 2D slices. This creates inconsistencies
between segmentations in two successive slices. Such inconsis
tencies in the end manifest themselves as local artefacts on the
object. The protrusion that can be seen on the top of the second
hippocampus in Figure 8(a) is an example of such an artefact.
These artefacts can influence shape distances negatively.
We select six structures for each patient: left/right cau
date nucleus, left/right putamen and left/right hippocampus,
resulting in 240 structures in total. We then create pairwise
affinity matrices of the 240 structures first using ρN
N = 200, as proposed in [3], and then WESD (ρNwith p = 2
Medical research
SD, as proposed in [3].
SDwith
6website:http://www.loni.ucla.edu/Atlases/LPBA40
and N = 200). Finally, we use the ISOMAP algorithm [36]
to construct 2D embeddings of the structures. Figures 8(d)
and (e) show the resulting embeddings. We observe that
the embedding obtained via WESD well clusters the data
with respect to the anatomical structures. The separation of
the clusters for the SD case, however, is more ambiguous,
especially between putamen and hippocampus.
The embeddings presented above were obtained by directly
using the manual segmentations without any preprocessing.
A natural question is how do these embeddings change if
the effects of various artefacts are reduced say via surface
smoothing. To answer this question, we smooth the surface of
the anatomical 3D models and recomputed the embeddings,
which are shown in Figures 8(f) and (g). The embedding
obtained with ρN
from similar ambiguity as in Figure 8(d). The new embedding
based on WESD on the other hand, compared to Figure 8(e),
even more clearly separates different anatomical structures.
However, we also note that this type of preprocessing can
also produce undesirable artefacts such as altering the topology
of the anatomical object. This is the case for one caudate in
Figures 8(f) and (g), which ends up as an outlier that is clearly
separated from the other data points. Considering this, the fact
that WESD is able to produce visually pleasing embeddings
without the need of preprocessing is an advantage.
2) Analysing HeartFunction
dimensional imaging of patient anatomy is gaining interest in
the medical community. The temporal analysis of anatomical
structures is used to extract the characteristics of related dy
namic processes, which often indicate certain pathologies [42].
Furthermore, in the recent work [43] authors show that shape
information, in addition to volumetric measurements, improve
the accuracy of pathology related classification tasks in such
dynamical analyses. In this section, we apply nWESD to the
shapes of the hearts extracted from four dimensional cardiac
images of five different patients. The scan of each patient cap
tures a full cycle of one heartbeat as a series of 20 3D images.
Each image shows the left ventricle (LV) at a specific point in
the cycle, from which we manually segment the corresponding
blood pool. Our reference is the blood pool extracted from the
first frame (diastole). We compute the nWESD scores between
this reference and all other shapes extracted from the series
of images. Here, we do not normalise the eigenvalues with
respect to the global scale since size change is an important
aspect of the heartbeat dynamics. The graph given in Figure 9
shows the results of these measurements over time across the
five patients. The figure also shows some exemplary images
and shapes. We observe that the symmetry of the heartbeat
along the systolic (as the blood pumps out of the LV pool)
and the diastolic phases (as the blood fills in the pool) is
well captured with the nWESD score. Furthermore, the end
systolic phase (the time point with the largest distance w.r.t.
the reference) is at different time points for different patients,
which is to be expected since the different patient scans are not
synchronized in time. In summary, WESD well captures the
dynamics of the beating heart, which is to be expected given
the continuous link between the differences in eigenvalues and
the difference in shape (see Section II).
SD, although to a lesser extent, still suffers
in 4D MRI:Four
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13
(a) Four Hippocampi (b) Four Caudate Nuclei(c) Four Putamen
(d) ρN
SD no preprocessing
(e) WESD  no preprocessing (f) ρN
SD surface smoothing
(g) WESD  surface smoothing
Fig. 8: 2D embedding of subcortical structures: 240 structures (80 caudate nucleus, 80 putamen and 80 hippocampus) are
extracted from MR scans of 40 different individuals. (a),(b) and (c) show some example structures from this dataset. Note the
high intraclass variability and the artefacts due to finite resolution and manual segmentations. (d) and (e) plot 2D embeddings
of these 240 structures obtained based on the affinity matrices computed via ρN
are computed without any preprocessing applied to the structures. The embedding obtained with WESD distinctly clusters the
objects with respect to the anatomical structures. The embedding in (d) however, shows some ambiguities in the separation.
Graphs in (f) and (g) plot the similar embeddings obtained after smoothing the surfaces of the structures to remove artefacts.
The embedding obtained by ρN
on the other hand, now even between better separates the groups.
SDand WESD respectively. These embeddings
SD, although better than (d), still suffer from similar problems. The embedding based on WESD
Fig. 9: Analysing 3D + time (4D) cardiac images: First column
shows corresponding 2D slices of a 4D MRI dataset at time
points t = {0,6,12}. The second column, 3D shapes extracted
at each of the time points. For five patients, we compute the
nWESD shape dissimilarity score of the LV blood pool at each
time point with respect to its shape at t = 0. The graph plots
these scores. We note that the proposed shape distance is able
to capture the dynamic process of the LV shape changes and
furthermore, the symmetry between the two phases of an heart
beat: diastole and systole.
V. CONCLUSION
This article proposed WESD, a new spectral shape distance
defined over the eigenvalues of the Laplace operator. WESD
is a theoretically sound shape metric that is derived from
the heattrace. The theoretical analysis given in this article
presented and proved the properties of WESD related to its
existence, computability and multiscale aspect. The presented
experiments showed that the theoretical properties of WESD
have many practical advantages over previous works. These
experiments further highlighted that WESD is beneficial for
various applications.
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In Proceedings of Medical Image Computing and Computer Assisted
Intervention, 2012.
E nder Konukoglu was born in Istanbul, Turkey,
in 1981. He received a Ph.D. degree in computer
science specializing in medical image analysis from
Universit´ e de Nice and INRIA Sophia Antipolis,
France. He is now a Research Fellow at Martinos
Center for Biomedical Imaging, MGH / Harvard
Medical School. His research interests are medical
image analysis, biophysical models, machine learn
ing and applications of partial differential equations.
B en Glocker was born in Goettingen, Germany,
in 1980. He received a Ph.D. degree in medi
cal computer science from Technische Universitaet
Muenchen, Germany. He is currently a Postdoctoral
Researcher at Microsoft Research Cambridge, UK
where he is a member of the Machine Learning
and Perception group. His main research focus is
on the application of machine learning techniques
for medical image analysis.
A ntonio Criminisi was born in 1972 in Italy. He
received a Degree in Electronics Engineering at
the University of Palermo and obtained a PhD in
Computer Vision at the University of Oxford. He is
currently a Senior Researcher at Microsoft Research
Cambridge, UK. His research interests are in the
area of medical image analysis, object category
recognition, image and video analysis and editing,
onetoone teleconferencing, 3D reconstruction from
single and multiple images with application to vir
tual reality, forensic science and history of art.
K ilian M. Pohl received a M.S. degree from the
Department of Mathematics, University of Karl
sruhe, and his PhD from the Computer Science
and Artificial Intelligence Laboratory, Massachusetts
Institute of Technology. Dr. Pohl is now an Assis
tant Professor at the Department of Radiology and
holds a secondary appointment at the Bioengineering
Graduate Group, University of Pennsylvania. His
research focuses on creating algorithms for automat
ically quantifying and generalizing the information
latent in medical images.
Supplementary resources (1)

wesd konukoglu appendix file