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2D Shape Model Selection via Efﬁciency Measures: An Empirical Study

Kathryn Leonard

Dept. of Applied and Computational Mathematics

California Institute of Technology

Abstract

In related work, we propose efﬁciency of representation

as a criterion for evaluating 2D shape models. In this work,

we apply the criterion to three shape databases of shape

contours: Mokhtarian’s 1100 ﬁsh, Kimia’s 1003 shapes,

and a new database we extracted from the BSDS300 train-

ing set. In particular, we determine which shapes are more

efﬁciently modeled by the boundary curve and which are

better modeled by the medial axis. Surprisingly, we ﬁnd that

nearly every shape is more efﬁciently represented by the me-

dial axis, and that this superiority is reasonably robust. We

offer an explanation for these results based on the geomet-

ric relationships between the medial axis and the boundary

curve.

1. Introduction

The study of 2D shapes and their similarities plays a

central role in the ﬁeld of computer vision, relating to

tasks such as object detection, classiﬁcation and recogni-

tion [4,10,14,17,18,25]. It is not clear what the best shape

model is, or if a best model exists, as different tasks rely on

different qualities of a shape [16]. Reasons for choosing a

shape model may be compatibility with a larger image anal-

ysis structure (e.g., [19,27,7,26]), qualities of an image

(e.g. the dalmation dog versus a line drawing), or simply

religious belief. In [1], we propose that efﬁciency of repre-

sentation offers a useful, quantitative selection criterion for

shape models, and we derive such a criterion based on the

intrinsic geometry of each shape. In this paper, we provide

an empirical study of that efﬁciency criterion.

At the heart of our work is the idea that regardless of

the model used for a particular task, understanding efﬁcient

representation gives insight into the objects being modeled.

For linear spaces, this observation has led to the invention of

wavelet-type constructions, the study of sparse representa-

tion, and the subsequent revolution of signal processing. We

are working toward similar successes in the nonlinear space

of 2D shapes. We begin by considering two of the most

popular shape models, the boundary curve and the medial

axis.

While the boundary curve is an obvious choice for a

shape model, the medial axis is controversial because the

axis structure is not stable under small perturbations of the

boundary curve. The reverse, however, is not true, as the

medial axis is a very stable representation of the boundary

curve: small perturbations of a branch of the medial axis re-

sult in small perturbations of the boundary curve [2]. From

a theoretical perspective, the axis is appealing because of

the way it encodes geometric properties of the boundary

curve into its own geometry. It is equally appealing from a

practical perspective because of its ability to decompose an

object into parts. Consequently, it has been applied to many

shape-related problems, such as 2D- and 3D-recognition

[29,12], animation [22], medical imaging [6,28], shape

reconstruction [3,9], and shape matching [23,21].

We analyze the efﬁciency of boundary and medial axis

representations of naturally occurring shapes drawn from

three databases: the Kimia 1003 [20], Mokhtarian’s 1100

ﬁsh [15], and a new database of 219 shapes we have ex-

tracted from the hand-segmented BSDS300 training set

[13]. We selected the benchmark BSDS300 dataset because

our results give insight into the role of shape in detection

and recognition tasks; we wanted our data to be the same

data used in such research.

We begin with a brief overview of the results presented

in [1], then apply our efﬁciency criterion to the shape

databases to see which of the two models is more efﬁcient

for each shape. We deﬁne a relative efﬁciency measure µto

be the ratio of bit length using the boundary to bit length us-

ing the medial axis, then examine its behavior as the bound-

ary is smoothed or sub-sampled. Because our computations

involve delicate estimates of second derivatives associated

to the medial axis, we also analyze stability of µafter per-

turbations of the medial points.

2. Model Comparison Theory

For us, a shape is a planar region with boundary curve

having ﬁnite length and curvature—a silhouette. We place

a metric on the space of curves that reﬂects the strong signif-

icance of orientation in human perception [11], a Hausdorff

1

metric on both location and tangent orientation:

ρ(γ1, γ2)

= sup

i=1,2

sup

sj

j6=i

inf

si

1

λ|γi(si)−γj(sj)|+|θi(si)−θj(sj)|

,

where λis a dimension-normalizing constant.

In this setting, we construct an ε-lossy encoding of gen-

eral plane curves that reduces to the optimal encoding for

compact classes of curves [1], with bit rate as follows. The

ε-lossy encoding of a plane curve γof bounded curvature

κγand length L, with κγcontinuous a.e., requires a bit

length Bγsatisfying:

BγR[0,L]|κγ|ds +δ

ε,(1)

where δis an error term that can be made arbitrarily small.

Equation 1results from the fact that the leading term of

the bit rate for the encoding of a curve comes entirely from

encoding the tangent angle function. Reﬁning the encoding

to correct for location error requires only lower order terms.

Note that Equation 1gives a bit rate for encoding both

the boundary curve and the medial curve. A simpliﬁcation

of the encoding gives a parallel result for functions, with

leading term in the bit rate dependent only on the function’s

derivative. We will employ this in Section 2.2.

2.1. Properties of the medial axis

The medial axis, deﬁned in 1970 by Blum [5] in the con-

text of mathematical morphology, can be thought of as the

skeleton of a region in the plane. It captures the local sym-

metries of a shape, thereby encoding the boundary geome-

try in its own geometry. There is a one-to-one correspon-

dence between medial axis pairs and boundary curves, and

there are explicit formulas to go back and forth between

medial geometry and boundary geometry. We include the

essential relationships here, but refer to [8,2,24] for more

complete explanations.

The medial axis pair,(m(t), r(t)), of a closed planar

region consists of m(t), the curve deﬁned by the closure of

the locus of centers of maximal circles contained within the

region, and r(t), the associated radii. The medial curve m

will consist of several branches with a degree of smooth-

ness determined by the smoothness of the boundary curve

[8], meeting at branch points. We will denote a branch con-

tained in mby m. Throughout, the symbol ’ will be re-

served for derivatives with respect to an arclength parameter

of m.

For other notation, refer to Figure 1. Because the max-

imal circles are bitangent to the boundary curve, the radius

joining a medial point to a corresponding boundary point

will be parallel to the normal to the boundary. The angle

between the tangent Tmto the medial curve and the out-

ward normal to the boundary is denoted by φ. Because the

medial axis sweeps out both sides of the boundary at once,

each medial point corresponds to two boundary points: one

to the left of the medial curve, γ+, and one to the right,

γ−. The subscripts ±will refer to these two portions of the

boundary, noting that γ−will have the same orientation as

mwhile γ+will have the opposite.

(s )

+

γ

−

−

T

m

φ

m(v)

γ

(s )

m

r

r

γ

T

T

+

−

++

−

γ

Figure 1. Notation for the medial axis.

The key geometric relationship underlying our work is

the expression of the tangent angle function to the boundary,

θ±, in terms of medial data:

θ±=θm±φ+π/2.(2)

In other words, to reconstruct the tangent angle function for

the boundary, one needs only the tangent to the medial curve

and the angle φ. Note that because r′=−sin φ, knowledge

of the angle φis equivalent to knowledge of r′.

As mentioned in the introduction, the medial axis is a

stable representation of the boundary curve. In [2], we de-

rive geometric and analytic bounds on how far the boundary

curve can wander given medial data at two nearby points on

a medial branch. We include Figure 2here as a demonstra-

tion of the geometric bounds.

2.2. Adaptive Coding and Model Comparison

We now present the criterion for comparing efﬁciency of

the boundary curve with that of the medial axis [1]. The me-

dial axis is more efﬁcient than the boundary curve over do-

mains Ithat are decomposable into subdomains Ijwhere:

supIj|κm|

supIj|φ′|>2 + √3 or (3)

supIj|φ′|

supIj|κm|>2 + √3.(4)

We are able to obtain these results because of the inti-

macy between the boundary and medial geometry. From

+

2

m12

m

r

r

1

2

γ

γ1

γ2

1

C

R

C

+

+

+

−−

−

R

−

γ

Figure 2. Possible regions R±for γbetween γ±

1and γ±

2, given

medial data at point m1,m2on a medial branch.

Equation 2, encoding θγwith εerror is the same as encod-

ing θm±φ+π/2to within ε, and so one may encode θm

with error η∈[0, ε]and encode φwith error ε−η, allo-

cating precision where it is most necessary. This gives the

bit rate (ignoring the slop term) for an encoding of γvia the

medial axis as:

R|κm|dv

η+R|φ|dv

ε−η.(5)

We can also derive an expression in termsof medial data

for the bit rate for encoding θγdirectly. For sarclength

on γ,varclength on a medial branch, and corresponding

domains Dand [0, l]:

ZD|κγ|ds =Z[0,l]|κm|+|φ′|+||κm| − |φ′|| dv. (6)

Choosing the optimal ηadaptively, one may compare the

expressions in Equations 5and 6to determine when the me-

dial axis is more efﬁcient. Doing so gives the criterion pre-

sented at the beginning of the section.

Equation 2shows that the medial axis decouples the ge-

ometry of the boundary into a portion arising from the me-

dial curve and a portion arising from radial rays. Equations

3&4indicate that if the boundary curvature comes from

either the curvature of the medial curve or the variation of

the angle φ, the medial axis is more efﬁcient. When the

curvature of the boundary relies heavily on both sources,

however, the boundary curve is more efﬁcient; Equations 3

&4give the threshold for the transition.

3. Experimental Methods

We apply these ideas to shape data, in order to learn

which naturally occurring shapes are better modeled by the

boundary and which are better modeled by the medial axis.

For all experiments, the approximation error is ε= 0.01.

Recall that our relative efﬁciency measure µis the ratio of

the boundary bit rate to the medial axis bit rate.

3.1. Shape Data

To ensure diversity of data, we analyzed shapes from

three databases. The ﬁrst, Kimia’s 1003-shape database

[20], consists of ﬂoating-point shape contours in several

shape categories, including humans, ﬁsh, animals, cups,

bones, airplanes, tools, and others. See Figure 3. The

second is the 1100 ﬁsh database of Mokhtarian, et al [15],

which consists of integer-valued contours of many strange

and wonderful ﬁsh extracted from marine biology texts.

See Figure 4. The ﬁnal database consists of 219 integer-

valued shape contours which we extracted from the Berke-

ley BSDS300 hand-segmented training set [13]. See Figure

5.

Figure 3. Random samples from Kimia’s 1003-shape database.

We operated on the raw data from the Kimia dataset,

but the integer-valued contours have large pixelation ef-

fects. We therefore smoothed the boundaries of the other

two datasets using 5 time steps of width .01 in a discrete

implementation of the geometric heat equation, with cur-

vature weighted by 0.3. This gives minimal smoothing, as

exempliﬁed by Figure 6, where even the sharp corners re-

main clear.

3.2. Medial Data Extraction and Curvature Ap-

proximation

To extract medial axis data from the contours, we use the

Delaunay triangulation of the boundary points to approx-

imate the centers of the medial circles. Because the De-

launay triangulation corresponds to a circle through three

boundary points, whereas most medial circles touch only

two boundary points, the resulting approximation to a me-

dial branch will artiﬁcially zig-zag as the third boundary

Figure 4. Random samples from the Mahktarian 1100-ﬁsh

database.

Figure 5. Sample shapes from the hand-segmented BSDS300

training set.

point is alternately chosen from γ+and γ−. We apply the

same smoothing used for the pixelated boundaries to these

rough medial branches, again for 5 time steps. From the

smoothed medial curve, we approximate rusing the ray Nγ

joining the medial point to the singleton of the three bound-

ary points.

We use the standard 3-point approximations for deriva-

tives. The tangent to a medial point miis approximated by

Ti=mi+1 −mi−1. The angle between Tiand Nγfor the

corresponding boundary point then offers an approximation

Figure 6. A shape before and after smoothing.

φiof φ, from which we ﬁnd an approximation φ′

i. The cur-

vature at miis approximated by the 3-point formula for the

derivative of θmwith respect to arclength.

3.3. Stability of Results

To test the robustness of our results, we manipulated

the data in a few ways. First, we investigated the effects

of smoothing the boundary curve, looking at bit rates after

smoothing for 5, 10, 15, and 20 time steps for the pixelated

data, and for 0, 5, 10, and 15 time steps for the ﬂoating-

point data. Next, we experimented with sub-sampling the

boundary points, analyzing bit rates for 25%, 50%, 75%,

and 100% of the data points. Note that because different

choices of εmerely scale the resulting bit rates and leave µ

ﬁxed, there is no need to vary ε.

Finally, we analyzed effects of the sensitivity of the sec-

ond derivative approximation to noise and round-off er-

ror. We focused on the Kimia database because of its size

and variety of shape categories, and because the values are

ﬂoating-point. We perturbed the smoothed medial points

by adding a random vector to each point. The coordinates

of the random vector were generated from a uniform dis-

tribution, scaled so that the length of the vector would not

exceed 1/c ·lfor some c, where lis the minimum distance

between two medial points in a given branch. We experi-

mented with the value of c, recomputing the bit rates after

each perturbation. For each choice of c, adding a random

vector can change the curvature approximations by at most

α=π−2 tan−1c

l. Note that the maximum curvature change

occurs when c= 1, giving α≈1.57/l.

4. Results

4.1. Primary Experiment

We ﬁnd overwhelmingly that the medial axis is more ef-

ﬁcient. Out of the 2,322 shapes analyzed, only three were

more efﬁciently represented by the boundary curve. Fig-

ure 7shows a scatter plot of the bit rate for the medial axis

against the bit rate for the boundary curve for each shape.

The median value for µis 1.476; see the leftmost box plot

displayed in Figures 10 and 12. Figure 8shows the three

shapes and corresponding bit rates for which the boundary

was more efﬁcient (µ < 1); we believe these suffer from

insufﬁcient boundary smoothing. Figure 9shows the shape

and corresponding bit rates from each database for which µ

was the largest.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

0.5

1

1.5

2

2.5

3

3.5

x 104

Figure 7. Scatter plots of bit rates using the medial axis (x-axis)

and boundary curve (y-axis). The line shown is y=x.

Boundary bits: 4800

Axis bits: 5944

Boundary bits: 11640

Axis bits: 27774

Boundary bits: 39109

Axis bits: 82692

Figure 8. Shapes for which the boundary curve is more efﬁcient

than the medial axis, and their medial axes. The third shape, a

park bench from BSDS, has an anomaly in the medial axis due to

the hand segmentation.

4.2. Robustness Experiments

The efﬁciency of the medial axis is reasonably robust to

the manipulations performed, though we ﬁnd evidence of

some susceptibility to down-sampling.

Figure 10 displays boxplots for µas smoothing increases

from left to right. The median value for µdecreases slightly

as the boundary is smoothed, but its variance decreases sig-

niﬁcantly. The notches in the box plot indicate the 95%

conﬁdence interval for the value of the median. In all cases,

it is well away from the value 1, indicating that the efﬁ-

ciency of the medial axis is statistically signiﬁcant. Figure

Boundary bits: 7733

Axis bits: 4120

Boundary bits: 16489

Axis bits: 9990

Boundary bits: 21369

Axis bits: 11979

Figure 9. Shapes from each of the three databases for which the

medial axis was most efﬁcient, their medial axes, and respective

bit rates. From top to bottom: Mokhtarian, BSDS, Kimia.

1234

0.8

1

1.2

1.4

1.6

1.8

2

Values

Column Number

Figure 10. Box plots of values for µas the boundary curve is

smoothed by (left to right) 5, 10, 15, and 20time steps.

11 displays the corresponding scatter plots; note that after

only 10 time steps of smoothing, all shapes are better repre-

sented by the medial axis, suggesting further investigation

into the effects of pixelation are necessary.

Sub-sampling the boundary points, the median value for

µdecreased with each increase in subsampling. See Figure

12. Interestingly, µremained stable for the Kimia database.

Figure 13 shows results for sub-sampling with the Kimia

data removed. Even without the Kimia data, however, it

is not until 75% of the original boundary points have been

removed that the conﬁdence interval for the median of µ

contains the value 1. Again, the difference of results for the

Kimia dataset suggests pixelation is a likely cause.

For the perturbation study, the value of the constant c

determining the maximum length of the perturbation vector

is crucial. We found that for c≥21, the efﬁciency of the

medial axis is stable. At c= 20, a few shapes begin to

be better modeled by the boundary, and for c≤19, the

boundary is almost always more efﬁcient. We performed 25

trials in our perturbation study, taking c= 20 as the center

0 5 10

x 104

0

2

4

6

8

10

12 x 104

0 5 10

x 104

2

4

6

8

10

12 x 104

0 5 10

x 104

0

2

4

6

8

10

12

x 104

0 5 10

x 104

0

2

4

6

8

10

12 x 104

Figure 11. Scatter plots of bit rates using the medial axis (x-axis) and boundary curve (y-axis) after smoothing for (left to right) 5, 10, 15,

20 time steps.

1234

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Values

Column Number

Figure 12. Box plots of values for µas the boundary curve is sub-

sampled: (left to right) 100%, 75%, 50%, 25% of points sampled.

of the transitional range. We give an idea of the effect on

curvature as cvaries: for c= 19,α≈0.105/l; for c= 20,

α≈0.099/l; and for c= 21,α≈0.095/l.

For each trial, the boundary curve became more efﬁcient

for between 4 and 12 out of the 1003 Kimia shapes. Figure

14 shows the box plot for µafter perturbations. The median

value has decreased slightly, but is still signiﬁcantly above

µ= 1.

On a shape-by-shape basis, the outcome of these trials

was remarkably consistent. A few shapes appeared only

once or twice (see Figure 16), but the 19 shapes depicted in

Figure 15 consistently appeared in several of the trials, with

one appearing in every single trial (noisy bowtie, upper left

corner). Note that the collection of shape categories among

the perturbed shapes for which the boundary is more efﬁ-

1234

0.4

0.6

0.8

1

1.2

1.4

1.6

Values

Column Number

Figure 13. Box plots of values for µas the boundary curve is sub-

sampled, with Kimia data removed: (left to right) 100%, 75%,

50%, 25% of points sampled.

cient is quite small—noisy bowtie, tool, bone, human, and

goblet categories account for all but two of these shapes—

and that these shapes share some qualitative properties. The

median value for µwithout perturbation for the 19 shapes

in Figure 15 is 1.65, whereas the median for the larger pop-

ulation is 1.476. This indicates that, instead of being shapes

for which the efﬁciency of the boundary and the medial axis

are close, these shapes are more sensitive to perturbation

than others. This merits further exploration to determine

the source of the increased sensitivity of their axis curves.

1

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Values

Column Number

Figure 14. Box plot for values for µafter perturbation of the medial

points.

Figure 15. Shapes for which a small perturbation made the bound-

ary more efﬁcient in three or more of the 25 perturbation trials,

displayed in decreasing order of number of trials. The shape in the

upper left corner appeared in every trial.

5. Discussion

We began these experiments with the expectation that the

boundary would be more efﬁcient for certain categories of

shape and the medial axis would be more efﬁcient for oth-

ers. Certainly, such a result is implied by the criterion stated

in Equations 3&4. Surprisingly, we found that the medial

axis is more efﬁcient in almost all settings for almost all of

the 2,322 shapes analyzed. While it is not true that the me-

dial axis is more efﬁcient for every branch associated to a

particular shape, for the majority of the branches, it is sig-

Figure 16. Shapes for which a small perturbation made the bound-

ary more efﬁcient in one or two of the 25 perturbation trials.

niﬁcantly more efﬁcient.

Our results show that for most of the branches in a shape,

|κm|is very small, especially compared to |φ′|. In other

words, the medial axis is tremendously successful in decou-

pling the geometry of the boundary, so that the medial curve

supplies a near-linear scaffold upon which φbuilds bound-

ary curvature. The robustness of these results suggests that

despite the delicacy of the second-order estimates and de-

spite the sensitivity of the medial axis to small changes in

the boundary, the greater efﬁciency of the medial axis is

meaningful.

Knowledge of the efﬁciency of the medial axis is ex-

tremely signiﬁcant for the vision community. Certainly, our

work provides justiﬁcation for the medial axis as a shape

model. In addition, such knowledge can be used to enhance

shape detection and recognition: skeletal preferences can

be built into priors on feature detectors; knowledge of local

symmetries can reduce the search for large wavelet coefﬁ-

cients; branch structures can guide construction of constel-

lation models and account for missing features. We hope

our work will inﬂuence these tasks, as well as inspiring fur-

ther investigations into model efﬁciency.

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