A 2D shape structure for decomposition and part similarity

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A 2D Shape Structure
for Decomposition and Part Similarity
Kathryn Leonard
CSU Channel Islands
kleonard.ci@gmail.com
Geraldine Morin
Univ. of Toulouse
morin@n7.fr
Stefanie Hahmann
Univ. of Grenoble Alpes
stefanie.hahmann@inria.fr
Axel Carlier
Univ. of Toulouse
carlier@n7.fr
Abstract—This paper presents a multilevel analysis of 2D
shapes and uses it to find similarities between the different parts
of a shape. Such an analysis is important for many applications
such as shape comparison, editing, and compression. Our robust
and stable method decomposes a shape into parts, determines a
parts hierarchy, and measures similarity between parts based on
a salience measure on the medial axis, the Weighted Extended
Distance Function, providing a multi-resolution partition of the
shape that is stable across scale and articulation. Compari-
son with an extensive user study on the MPEG-7 database
demonstrates that our geometric results are consistent with user
perception.
I. INTRODUCTION
Performing integrated tasks with a shape such as shape
generation, animation, editing, or partial matching requires
structure-aware shape processing. A full shape structure con-
sists of a decomposition into parts, understanding of parts
hierarchy, and the ability to measure relative part similarity.
As noted in [18], hierarchy is often the most difficult to attain.
This paper proposes a geometric approach to shape analysis
based on the Blum medial axis that realizes a full shape
structure in a way that is robust to noise and stable under
changes of scale, rotation, and part articulation. Moreover, it
matches well with human perception of shape structure.
An initial decomposition into parts provides the foundation
of any shape structure. Characteristics of part decompositions
have been considered extensively (see, for example, [13],
[27]), but most follow generally the ideas first outlined in [11]:
(1) two distinct parts will meet transversally, (2) parts should
be divided along minima of curvature, (3) decompositions
should be reliable, versatile, and computable. Our method
meets these criteria for most shapes. What our method offers
that most parts decompositions do not is the ability to retain
geometric relationships between parts instead of returning a
chain of regions whose connections have been lost.
Our method relies on functions defined on the interior
Blum medial axis that capture shape importance. In [12],
importance measures of parts based on existing functions on
the medial axis are derived and a new function, the Weighted
Extended Distance Function (WEDF), is proposed to improve
the importance measure.
The WEDF value at a point on the medial axis measures
the area of the shape subtended by that medial point. Our
preliminary results in [12] using the WEDF on extremely
simple tube-like shapes to distinguish between “parts” and
“main shape” form the basis of our more comprehensive work
here to generate a full shape structure tested on the MPEG-7
database. Because the medial axis effectively encodes shape
geometry, our method retains not only geometric relationships
between parts but also their relative importance within a
multiscale parts hierarchy.
The primary contribution of this paper is an unsupervised,
robust skeleton-based shape structure that decomposes shapes
into parts, provides a parts hierarchy, and measures part
similarity all while maintaining geometric relationships be-
tween parts. This shape structure is stable under rigid motion,
noise, and articulation of parts. Our method does not require
denoising, as noisy points emerge naturally in the hierarchy.
Section III gives an overview of the relevant shape importance
measures. Section IV describes our parts decomposition and
hierarchy process. We compare performance of our decom-
position and hierarchy to results from a user study of shape
structure in Section V, and give a brief introduction to our
parts similarity measure in Section VI.
II. RE LATE D WOR K
Parts decomposition. Much recent work on parts decompo-
sition builds on the idea of decomposing a shape into convex
regions using a greedy algorithm to minimize length of cuts
between parts [13], [21]. These decompositions tend to be
unstable in the presence of small boundary curve deformations,
and cannot handle parts such as animal tails that have been
articulated into non-convex shapes. Some methods to address
those shortcomings have been explored to some success [21],
but most fight against geometry (for example, artificially
straightening articulated non-convex parts) and do not allow
for the retention of geometric relationships between parts.
Medial-axis-based parts decomposition and hierarchies.
Skeletons have been viewed as unreliable because the branch
structure can change significantly in the presence of noise on
the boundary of the shape [1]. Much of the previous work
on skeletal part hierarchies has focused on identifying noise
in order to prune noisy points [15], [16], [19], [25]. In an
early work, Ogniewicz [19] proposes a multiresolution repre-
sentation of a 2D shape consisting of iteratively computing a
real valued function on the medial axis based on the boundary
length, generating a robust filters for noise. The resulting de-
composition into hierarchy levels is achieved through selecting
arbitrary thresholds: the user decides what is noise. Pizer et al.
[20] compares the Ogniewicz approach with other multiscale

representations of a shape based on shock graphs [23] or cores.
More generally, pruning algorithms typically erode points at
the ends of important medial branches, thereby smoothing out
sharp and meaningful shape features.
Recent work looks more explicitly at parts decompositions
and hierarchies, but relies on the boundary curve geometry
for key part detection cues. In [16], the authors use the
medial axis to extract an abstraction of shapes, a simplified
model to clarify the structure of a shape. Using user-specified
thresholds, they move from smallest to largest branches in the
medial axis, extracting the part associated to the current branch
and replacing it with a smooth connection along the boundary.
Doing so preserves part structure and level of hierarchy. Parts
below the threshold are discarded, and what remains is the
abstraction. The work in [30] approaches shape decomposition
by introducing a measure of protrusion strength that looks
at the ratio between the radius of the medial circle and the
length along the boundary of the resulting part to determine if
a potential part cut is desired. The parts decomposition occurs
only after significant denoising of the skeletal structure.
Shape similarity There has been substantial work to iden-
tify similarities between 2D shapes for applications such as
shape retrieval. Structural approaches have used skeletons [2],
[28], shock graph [26], or shape context [3] to that end. These
approaches match both the topology and the geometry of the
two shapes to be compared. These approaches are global,
however, and therefore time consuming, and they only match
between two shapes. It is still an open problem how to apply
these methods for selecting several sets of similar shape parts
from within one shape. In 3D, recent work has begun to
address the identification of local part similarities within a
single shape (see [17] for a survey), but most of these methods
rely on a local analysis of the surface. Curve skeletons have
been proposed for shape matching, but again for a global
matching and not for intra-shape similarity detection. Without
a reliable method for intra-shape similarity detection in 3D,
manual user-interaction is still required [10].
III. FUNCTIONS ON THE BLUM MEDIAL AXIS
Following notation in [9], we define the (interior) Blum
medial axis. See Figure 1.
Definition 1: The (interior) Blum medial axis pair (m, r)
of a simple, closed plane curve γis the locus of centers mof
maximally inscribed circles of γtogether with their radii r.
For many reasons, the medial axis is a desirable shape
representation. It encodes the geometry of the shape boundary
in its own geometry [9] and is more efficient in shape
compression [14]. Each branch m∈mcorresponds to a
coherent part within the shape, and the branch connections
provide information about adjacency of these parts. Well-
studied discrete formulations, the Delaunay triangulation and
Voronoi diagram, give an approximation to the medial axis
for a discretized boundary curve that converges to the true
medial axis as the density of the boundary sampling increases
[8]. When two medial axes are close, the resulting boundary
curves must also be close [14].
Fig. 1. A 2D shape (grey) and its internal medial axis (green and blue curves).
An arbitrary point x(in red) on the internal medial axis is associated to its
EDF (x)and W E DF (x)values. Left: EDF (x)measures the distance on
the medial axis to the closest extremity of the longest path (in blue) through
x. Right: W EDF (x)follows the same principle, but considers the area
corresponding to a medial axis segment rather than its length. Here the blue
medial axis segment corresponds to the path on the medial axis through the
point xwith the larger area, while the dark blue part’s area is WE DF (x).
Fig. 2. L: The EDF for a stingray shape with EDF values indicated with
a heat map. Note that there is one long curve (axis) through the shape with
continuous EDF variation while all other branches show a jump in EDF from
the value on the connecting point. R: The WEDF for the same shape. Note
how the maximum WEDF value (red dot) is in the center of the shape whereas
the maximum EDF value is toward the tail.
Unfortunately, any noise on the boundary will produce a
new branch of the medial axis. Changes in boundary sampling
may change the branching structure, which makes any naive
parts decomposition based on branches unstable.
Originally proposed in [15] as a significance measure, the
Extended Distance Function (EDF) introduces a measure of
shape depth on the medial axis.
Given a medial point xcontained in a continuous path fin
the medial axis, rf(x)is the shortest distance to a boundary
point through the axis via f. The EDF for xis the largest
such rf(x)among all paths f⊂mcontaining x. See Figure
1 where the fgiving the largest rf(x)is indicated in blue and
the EDF value rf(x)is the shortest distance to the boundary
along that f.
Definition 2: A trunk T⊂mis a path in mwith the
property that, for each branch point b∈T,limx→b=ED F (b)
for x∈T.
In other words, a trunk is a path in the medial axis with
continuous EDF values. The EDF measures how deep into a
shape a point is, which gives us important information about
the shape. Unfortunately, adding long thin protuberances can
change the EDF values significantly. See Figure 2, L, where
the tail of the stingray has shifted the maximum EDF values

Fig. 3. A shape with ET (L) and ST (R) values shown as a heat map. Note
how the ST maxima are in the tube-like legs, whereas the ET maximum occurs
in the much more blob-like torso region.
away from the core of the shape. To obtain a more stable shape
depth measure, we turn to the Weighted Extended Distance
Function (WEDF) [12].
Viewing ED F (x)as the length of the most significant
shape part subtended by x, we view the Weighted Extended
Distance Function (WEDF) as the area of the most significant
shape part subtended by x. See Figure 1.
For discrete shape boundaries, areas can be computed by
simply summing the areas of the Delaunay triangles along the
desired path f. Because the WEDF is area-based, its values are
robust to the addition of long but insignificant protuberances.
See Figure 2 where the maximum WEDF values are in the
core of the stingray despite the long tail.
EDF and WEDF give information about part importance.
Two quantities based on the EDF provide measures of “blob-
biness” and “tube-iness” of a shape part. These will help us
define part similarity. See Figure 3.
Erosion thickness, first introduced in [24], measures the
difference at a point xbetween the distance to the boundary
along the medial axis and the Blum radius r(x):
ET (x) = E DF (x)−r(x).(1)
.
Shape tubularity, introduced in [15], measures the ratio of
the ET to the EDF:
ST (x) = E DF (x)−r(x)
ED F (x)= 1 −r(x)
ED F (x).(2)
See Figure 3. Note that ET is monotonic from the inside to
the outside of the shape, but ST is not.
The ET and ST are extremely useful when used together:
Theorem 1: Let T1(v)and T2(v)be two medial trunks
parametrized over the same domain, with associated ra-
dius functions r1and r2, and ET, ST and EDF values
ET1, ST1, EDF1and ET2, ST2, EDF2. If ET1≡E T2
and ST1≡ST2, then r1≡r2,EDF1≡EDF2and, if
parametrizations are constant speed, the curves must also have
the same length.
To see this, note that ET1≡ET2implies that EDF1=
EDF2+ (r1−r2). Inserting that identity into the equality
for the ST, we find either r1=r2or r2=EDF2. Since the
radius function equals the EDF only at endpoints of branches
(for y∈∂T ), we conclude that r1=r2. We immediately
obtain EDF1=EDF2and, remembering that E DF (v) =
l−v+r(l)for van arclength parameter on [0, l], we find that
l1=l2when vis an arclength parameter. The result holds true
in approximation as well: when the ET and ST values of two
trunks are close, values of the two EDF and radius functions
must also be close.
Theorem 1 has strong implications: when two trunks have
equal values for both ET and ST functions, the associated
shape parts must be equal up to curvature of the trunk—in
other words, up to articulation.
IV. SHAPE DECOMPOSITION AND HIERARCHY
In the next sections, we present procedures based on the
medial axis for unsupervised parts decomposition, hierarchy,
and similarity. We developed and refined our methods on
23 shapes randomly selection from the Kimia 1001 database
[22]. Our test set is the MPEG-7 database [4] interpreted
to consist of simply connected regions, together with some
artificial shapes designed without semantic information. We
also present results from a user study that we use to assess
performance of our parts decomposition and hierarchy.
A. Computation of the hierarchy
We compute hierarchies for each shape using intrinsic
properties of the medial axis and functions defined thereupon.
Within the medial axis, a new branch begins at every shape
protrusion. Discontinuities in WEDF values occur only at
branch junctions, and the character of the discontinuities gives
insight into the salience of the associated shape part.
For a discrete shape boundary, we compute its Delaunay tri-
angulation, taking centers of the circumcircles of the triangles
to approximate m. We discard any medial points outside the
boundary. The distances from the interior circle centers to the
triangle vertices approximate r, producing a discrete medial
axis (ˆm,ˆr). Each x∈ˆm is associated to a triangle. Taking
the sum of the areas of triangles along the WEDF-defining
axis fgives an approximation for the WEDF value at x.
Next, we determine the core of the shape using WEDF
values on its medial points. If a shape’s medial axis has
enough branches, we determine shape core points using the
subset of the medial axis points consisting of branch points
where three or more branches meet, and neighbors of branch
points. Otherwise, we use all internal medial points. Denote
this initial clustering set by ICS. We find the core of the shape
by performing a seeded k-means clustering with two clusters
on the WEDF values for the ICS. Seeds are chosen as the
minimum and maximum WEDF values of the medial points
being clustered. Once clusters have been determined on the
ICS, any remaining medial points are assigned to the cluster
with centroid closest to the associated WEDF value. Points
closest to the centroid with largest WEDF value are selected
as the core and assigned to the coarsest level.
Once we have the shape core, we determine the additional
levels in the shape hierarchy through a two-step process. The

Fig. 4. (L) A bird shape after k-means clustering on WEDF values at
branch points. The black region is the core from the 2-cluster clustering.
The remaining levels are assigned during a second clustering where number
of clusters is automatically determined. (C) The bird shape after assigning the
hierarchy levels. (R) The previous hierarchy viewed as a three-level hierarchy,
where “core” and “details” are inherited from the previous hierarchy (details
being the finest level) and all other levels are assigned to “part”.
first step mimics the process for determining the core, but
with an automatically determined number of clusters. We
begin by operating on non-core ICS points, performing gap
analysis on the associated WEDF values to determine the
intrinsic number of clusters, K∈ {2,3,...,12}. Because
gap analysis can be sensitive to initial conditions, we repeat
the process several times and select Kto be the mode of
the result. We then perform a seeded k-means clustering on
the WEDF values with Kclusters. Seeds are chosen as the
minimum, maximum, and 100
Kth percentiles of the WEDF
values of the ICS points being clustered. Once clusters have
been determined, any remaining non-core medial points are
assigned to the cluster with centroid closest to the associated
WEDF value. See Figure 4, left.
In the second step, we use information about the branching
structure of the medial axis to refine the clusters and determine
their hierarchy levels. At the end of the first step, each medial
point belongs either to the core or to one of the Kclusters.
Points in the core cluster are assigned as level 1. Now suppose
we have matched points to levels up to level k. To determine
which cluster should define level k+1, we consider all branch
points contained in levels 1 to kand find the branch point
neighbor with the largest WEDF value vk+1 where vk+1 <
vk(we may take v1=∞). We assign all points x∈m
with W ED F (x)≤vk+1 to cluster k+ 1. Repeating until all
branch points have been considered gives the complete shape
hierarchy. See Figure 4, center.
Considering as a part each connected component of the
shape at each hierarchy level, we obtain a decomposition of
the shape into parts that are already assigned a hierarchy. This
fully automated process, based on intrinsic properties of the
medial axis and functions defined on it, gives two of the three
components of a shape structure analysis.
B. Results
Our results for automated shape hierarchy for some MPEG-
7 shapes are presented in Figure 5. When the boundary curve
is sampled densely enough to capture all salient shape features,
the hierarchy is quite stable across sampling rates. Addition of
noise that is of lower amplitude than the finest shape details
will not affect existing hierarchies but may introduce a finer
hierarchy level to capture the noise or may allow for hierarchy
Fig. 5. Top row: Intrinsic hierarchy levels for each shape part for a few of
the MPEG-7 shapes, two animals and two abstract shapes. Triangles with the
same color have been assigned to the same hierarchy level. These levels do
not take into account the levels of the other parts within the shape. Center
row: Full hierarchy levels for the same shapes. Triangles with the same color
have been assigned to the same hierarchy level. Core triangles are in black.
These levels take into account the intrinsic levels of other parts within the
shape, so that sub-parts at the same level can be compared across all parts.
Bottom row: Full hierarchies for the same shapes with added white noise on
the boundary. Note that despite a medial axis with many additional branches,
the proposed hierarchy remains stable.
Fig. 6. Hierarchies for similar shapes (dancers) in different poses to show
that the proposed hierarchy is stable under articulation. Coarser levels of the
hierarchy are consistent even if finer levels are added in the presence of finer
details. Also, note that the hierarchy is retained even with occlusion: The pink
level of the left arm of the first dancer is occluded, but the blue level begins
as it should.
breaks to occur at slightly different locations. See Figure 5,
bottom row.
V. US ER ST UDY
Because a parts hierarchy for a shape is only useful insofar
as humans agree with it, we require a baseline understanding
of human perception in order to evaluate our work. We
launched a user study as part of GISHWHES 2015, a one-
week, international online scavenger hunt with thousands of
participants. For each shape, users were asked to label each
triangle of a shape’s Delaunay triangulation as belonging to
main shape, part, or detail by coloring it black, magenta, or
green using a web-based interface. Users could choose to add
additional levels to the parts hierarchy, but very few did. By
the end of the week, 2,861 users had annotated 41,953 shapes
and every shape had been annotated at least 24 times [6].

Fig. 7. Annotations of the four MPEG7 shapes by the users (top) and by our
algorithm with the number of clusters set to 3 (bottom).
A. Methods
In the user study, participants categorized each Delaunay
triangle in a shape as belonging to one of three hierarchy
levels. Triangles in those few shapes where users added an
additional parts level were relabeled as the second level. Our
shape hierarchy gives as many as twelve levels for hundreds of
medial points per shape. To obtain a three-level hierarchy from
our automated algorithm, we perform the process described in
Section IV, then designate the finest hierarchy level as details
and all levels between the main shape and details as parts. See
Figure 4, right.
We define the similarity Dbetween the two annotations,
a1(S)and a2(S), of a shape Swith Delaunay triangulation
{t1, . . . , tn}as:
D(a1(S),a2(S)) =
n
X
i=1
δ(a1(ti),a2(ti)) Ai
AS
(3)
where δ(a, b)=1if a=b,0if a6=b.
We use a simple majority vote to obtain a representative user
annotation for each shape. Figure 7 compares our algorithm
to the user study on two animals and two abstract shapes.
B. Results
We use the similarity measure defined in equation 3 to
compare the shape hierarchy of our algorithm against the
shape hierarchy of the users. We find an average similarity of
0.764 with a median value of 0.800 and a standard deviation
of 0.147. Our hierarchy and the user-annotated hierarchy
therefore agree on more than 75% of shape area.
We obtain a similarity score superior to 0.8 on 38 classes out
of 69, and between 0.6 and 0.8 on 26 classes. The five lowest
scoring classes include four devices classes and butterflies
and have a similarity score between 0.4 and 0.6. In those
categories, the geometry makes it difficult to determine what
is main shape and what are parts and details, since large parts
(wings) are attached to a small central part (body). In fact,
users themselves do not agree.
VI. PART SIMILARITY
Given a shape hierarchy, we can compare similarity of
parts within and across hierarchy levels. For the shapes in
our database, the medial structure corresponding to a part at
a given level will be an embedded tree. Efficient matching of
embedded trees is costly [5], [7], [29]. To improve efficiency,
we use the trunk introduced in Definition 2, which gives us
the most significant path through the tree, allows us to apply
Theorem 1, and provides a road map for attached parts at finer
hierarchy levels.
Given a shape part at hierarchy level k∈ {1, . . . , K}where
Kis the finest level in the parent shape, we generate the
trunk traversing the part from level kto the finest level in
the part. Resampling the trunk to obtain equidistant sample
points, we interpolate corresponding ET and ST values. Based
on Theorem 1, closeness of these values indicates a strong
similarity of shape.
We complete our shape structure analysis by establishing
part similarity. Each shape contains multiple trunks. Taking
ET and ST values into a single feature vector for each trunk,
we perform k-means clustering on the resulting collection
of feature vectors. The mode of several gap analyses again
determines the number of clusters. This automated process
determines trunk similarity within the shape across trunks of
all levels, which we may then join together to determine a total
similarity measure for a given hierarchy level. Some clusters
of parts are shown in Figure 8.
Because a trunk at a coarser level often contains trunks
at finer levels, we may use the trunk similarity clustering to
determine a measure of similarity persistence between two
trunks. Each point in a trunk has a natural level, the finest level
at which it appears. With our part clustering process, points
that appear in multiple trunks may be assigned to multiple
clusters based on their membership in multiple trunks. We
now assign a unique cluster label to each point by selecting
the cluster label given to the trunk at that point’s natural level.
For example, the points in the fork tines in Figure 8 would
be assigned their natural level values as indicated displayed
in fork in the bottom row, even though the tips of the tines
have been assigned to four distinct level values corresponding
to the four trunks shown above.
Computing the proportion of shape part area assigned to a
given cluster label for each trunk gives a similarity persistence
value between two trunks:
P S(T1, T2) = d(v1, v2),(4)
for vj= (vj1, . . . , vjm )where vjk gives the proportion of
shape part area for trunk Tjwith cluster label k.
VII. DISCUSSION
We have presented a robust and stable method for shape
structure analysis using the Blum medial axis that circumvents
the usual issues with skeletal shape representations. Our parts
decomposition agrees remarkably well with human perception
and is interwoven with a hierarchy that captures the range of
scales of parts within a shape. Our parts similarity measure re-
liably and accurately identifies similar parts within and across
levels of the shape hierarchy, and is stable under articulation

Fig. 8. Bottom: Four shapes of the MPEG-7 datashape with all levels dis-
played simultaneously. Above: Parts in the same similarity clusters highlighted
with the same color.
and rigid motion. Our methods should apply equally well to
between-shape part matching.
Interestingly, the interaction between the core of the shape
and the next level (pink) shows that for some shapes the
pink level indicates a connector between shape core blobs and
narrow parts, while for other shapes it is the first level within
a clearly distinct part. Further study is required in order to
determine automatically when each case holds.
These methods have important implications for graphics
applications once they are extended to 3D. Current work is
underway to determine a 3D version of the EDF and WEDF,
and to find 3D analogies for ET and ST.
Acknowledgments The authors gratefully acknowledge the
support of Marie-Paule Cani and ERC Advanced Grant 291184
EXPRESSIVE, NSF award IIS-0954256, and CIMI, program
ANR-11-LABX-0040-CIMI.
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