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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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