SHAPE COMPARISON BASED ON SKELETON ISOMORPHISM

L. Domakhina and A. Okhlopkov

Moscow State University, Moscow, Russia

Keywords:

Shape comparison, Continuous skeleton, Skeleton isomorphism, Hausdorff metrics, Shape metrics.

Abstract:

A new approach to shape comparison problem is presented in this work. The approach is based on skeleton

isomorphism. We propose a shape metrics construction instrument which is based on ﬁnding close shapes

having isomorphic continuous skeletons. We propose several metrics based on this instrument that can be

used for shape comparison. The main advantage over existing approaches is mathematically correctly deﬁned

shape metrics via Hausdorff distance. The efﬁciency of the proposed approach is conﬁrmed on the shapes

recognition problem.

1 INTRODUCTION

In this paper we report on an approach to comparing

two-dimensional shapes by constructing close shapes

with isomorphic skeletons. The problem of shape

comparison is useful in many document processing

applications (like organizing and querying an image

database, recognition and computer-vision problems,

medical structure comparison etc.)

The goal of the present paper is to develop an ef-

fective instrument for metrics construction. Metrics

obtained via this instrument should accord with vi-

sual intuition and at the same time be correct, i.e. be

a distance. We present such an instrument as an al-

gorithm for constructing new shapes close to a given

pair of shapes but having isomorphic skeletons. Close

shapes with isomorphic skeletons give us an oppor-

tunity to construct correct metrics based on this al-

gorithm. We also present experimental results as a

shape recognition application conﬁrming correctness

and effectiveness of the proposed solution.

2 SHAPE METRICS BASED ON

SKELETONS

2.1 Previous Work in using Skeletons to

Compare Shapes

Existing shape comparison methods are based on the

border of the shape or its interior. The latter often use

skeletons. These methods are compared in (Sebastian

and Kimia., 2001). It is shown that skeletons are bet-

ter to be used when solving general object recognition

problems.

Ideas that skeletons may be used as an instrument

to compare shapes were mentioned many times, for

example in (Tanase, 2005) and (Klein et al., 2001).

The main drawback of all known methods is the lack

of mathematically correct distances. Visual intuition

is a good criterion but it’s too subjective and often not

sufﬁcient to solve recognition problems.

Most of approaches use skeletons and boundaries

matching to compare shapes. (Liu and Geiger., 1999)

use an algorithm to match shape axis trees, which are

computed by ﬁnding a correspondence between the

shape outline and its mirror image. Their algorithm

does not preserve ordering of edges at nodes which

can result in matches that do not preserve coherence

of the shapes. (Klein et al., 2001) solved this prob-

lem by proposing an idea of using edit-distance when

skeletons are matched to compare shapes. Their idea

is based on observing discrete changes in the shock

graphs as a shape is being morphed to another. Two

drawbacks of the edit-distance are:

- It is an heuristic similarity measure.

- The edit-distance may suffer from noisy bound-

ary and noisy skeleton’s edges.

237

Domakhina L. and Okhlopkov A. (2009).

SHAPE COMPARISON BASED ON SKELETON ISOMORPHISM.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 237-242

DOI: 10.5220/0001807502370242

Copyright

c

SciTePress

2.2 Problems using Skeletons to

Compare Shapes

It is well known that continuous skeleton reﬂects the

structure of a shape. However, a skeleton may of-

ten have noise branches that have nothing in common

with general shape’s structure. Noise branches cutting

methods are proposed in many sources. For example

in (Tanase, 2005) the deletion of all terminal skele-

ton edges is proposed. Most of cutting methods are

heuristic. The only global cutting criterion is given

in (I.Reyer and Mestetski, 2003) as obtaining a base

skeleton with a ﬁxed accuracy.

Another problem of shape presentation via skele-

ton lays in the area of serious shape structure changes

affected by small boundary variations. Here is the

question: are the two shapes so different enough if

they look similar except one has noisy boundary while

another has a smooth one (ﬁg. 1). Another example

of two similar shapes is two human ﬁgures having dif-

ferent hands and legs positions (ﬁg. 2).

Figure 1: Different or similar shapes?

Figure 2: Different or similar shapes?

The similarity of shapes in both cases could be

seen with a naked eye. This similarity could be

described using skeletons. However classically de-

ﬁned continuous skeletons (Mestetski, 1998) of simi-

lar shapes could be strongly different (ﬁg. 3). We can

see ”common” parts of skeletons of similar shapes.

But how to describe these common parts correctly?

Figure 3: Different skeletons of similar shapes.

We tried to describe strictly shapes visual simi-

larity using common skeleton parts. We considered

skeletons as a graph and used graph isomorphism to

deﬁne a similarity measure between two shapes. The

main advantage of our approach towards all known

methods is that still using skeletal graphs we don’t

forget about the boundary and deﬁne mathematically

correct shape distance.

2.3 Skeleton Isomorphism

In this section we give several basic deﬁnitions used

in our approach.

Medial axis (skeleton) of the shape Ω (Mestetski,

1998) is a set of all maximal circles inscribed in the

shape Ω.

Medial axis can be represented as a planar graph

(Choi et al., 1997), i.e.a skeletal graph. Vertices of the

skeletal graph are the centers of maximal inscribed

circles that touch the shape’s boundary ∂Ω in three or

more points. Edges of the skeletal graph touch the

shape’s boundary ∂Ω in two or more points. A skele-

ton vertex that has only one incident edge is called a

terminal vertex, more than one edge — a knot. An

edge that is incident to a terminal vertex is called a

terminal edge.

Two graphs are isomorphic G

∼

=

H if there is a

vertex mapping between them that keeps edge adja-

cency. Graph isomorphism searching is an NP-full

problem (E. M. Reingold and Deo, 1977).

Two skeletons are isomorphic if their skeletal

graphs are isomorphic and the traversal order of ter-

minal vertices is the same in both graphs.

Skeleton isomorphism can’t be used directly to

compare shapes because similar shapes have strongly

different, i.e. not isomorphic skeletons (ﬁg. 3). Thus

we decided to ﬁnd better solution.

2.4 New Approach to Compare Shapes

Using Skeletons

We propose an approach based on a simple idea

(ﬁg. 4). For any two given shapes we ﬁnd two new

shapes that are close to the given ones but have iso-

morphic skeletons.

Figure 4: Shapes with isomorphic skeletons.

Let’s denote MA(Shape) — the medial axes of the

shape (or its continuous skeleton), M — an algorithm

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

238

that transforms the two given shapes into two new

shapes with isomorphic skeletons. Thus an algorithm

M should provide the following solution for any two

shapes Sh

1

and Sh

2

:

M(Sh

1

, Sh

2

) → (Sh

′

1

, Sh

′

2

) (1)

MA(Sh

′

1

)

∼

=

MA(Sh

′

2

) (2)

Moreover, new shapes Sh

′

1

and Sh

′

2

(2) should

minimally deviate from the given shapes Sh

1

and Sh

2

.

(

D

H

(Sh

1

, Sh

′

1

) → min

D

H

(Sh

2

, Sh

′

2

) → min

(3)

Where D

H

is Hausdorff distance.

This constraint (3) is very important for several

reasons. It narrows the set of solutions. We don’t need

any shapes with isomorphic skeletons. Without this

constraint any two simple shapes (not depending on

input) with isomorphic skeletons (for example, rect-

angles) could provide a solution which at least is not

correct and at most is not applicable. It deﬁnes what

we mean by ”close shapes”. It provides an opportu-

nity to estimate shape similarity as a correct distance.

Finding new shapes that satisfy both constraints

(2) and (3) among all possible plain shapes is a very

complex problem. However looking at this problem

from the skeleton side we may ﬁnd an effective and

correct solution.

Let’s consider a shape representation as a set of

all maximal inscribed circles, so-called boundary-

skeletal representation (Mestetski, 1998). Using this

kind of shape representation we may directly deal

with the skeleton, change its structure and corre-

sponding radiuses. Thus we may alter the given shape

with operations affecting boundary-skeletal represen-

tation.

2.5 The Main Algorithm

We propose an algorithm M that changes the struc-

ture of two given shapes to obtain new shapes with

isomorphic skeletons (2). Two shapes are changed at

each step of the algorithm so that the distance between

the given and changed shapes is minimal (3).

The input of the main algorithm M is two

shapes Sh

1

and Sh

2

and their continuous skeletons

MA(Sh

1

) ≡ ma

1

and MA(Sh

2

) ≡ ma

2

. The output is

two new shapes Sh

′

1

and Sh

′

2

having isomorphic skele-

tons (2) and close input shapes (3).

2.5.1 Skeleton Operations

One or several operations of two types are executed at

each step of the algorithm (detailed description may

be found in (Domakhina and A.Okhlopkov, 2008)):

1. Terminal skeleton edges cutting (”cutting”). Cut-

ting means terminal edge’s deletion. When the

skeleton’s edge is cut all corresponding circles are

deleted as well. Cutting operation affects the in-

put ﬁgure ”angles round-up” (as shown in ﬁg 5a).

2. Close skeleton knots merging (”merging”). This

operation merges two adjacent knots of the shape,

i.e. deletes an internal skeleton’s edge and all cor-

responding circles as well. We assign the radius

of the new knot’s circle as arithmetic mean of two

merged circles radiuses. Local shape’s changing

under merging operation is shown in ﬁgure 5b.

Figure 5: Figure’s changes during cutting and merging.

We describe an algorithm M in terms of isomor-

phic skeletons construction for two given skeletons

ma

1

and ma

2

. Remember that shape is changed dur-

ing each operation execution as described above.

2.5.2 Main Algorithm

1. Primary cutting (removing noise) of both skele-

tons ma

1

and ma

2

for a ﬁxed valueε, i.e. obtaining

the base skeleton with a ﬁxed accuracy ε (I.Reyer

and Mestetski, 2003). The level of noise may be

estimated depending on a shape. It may be as-

sumed to 1 if the shape is given as a raster object.

2. Equalizing the number of terminal vertices (graph

isomorphism necessary condition (E. M. Reingold

and Deo, 1977)) - secondary cutting. Let the ﬁrst

skeleton (ma

1

) have more terminal vertices than

the second one (ma

2

). Skeleton’s ma

1

terminal

vertices are cut until both skeletons ma

1

and ma

2

have the same number of them.

3. Primary merging (removing small accidental

structure defects) for a ﬁxed value ε. Internal

edges are deleted sequently while Hausdorff dis-

tance between new and input shape is less than

ﬁxed value ε.

4. Equalizing the number of skeleton knots (graph

isomorphism necessary condition (E. M. Reingold

and Deo, 1977)) - secondary merging. This step

is executed like the second step but internal edges

are cut.

5. Sequent single operations (cutting and merging)

execution until skeleton graphs become isomor-

SHAPE COMPARISON BASED ON SKELETON ISOMORPHISM

239

phic. To satisfy algorithm’s M constraint to min-

imize shape’s deviation (3) we need to choose the

operation sequently so that deletion of the corre-

sponding edge affects the less on a shape.

2.5.3 Computational Complexity

It is easy to prove that each operation affects only lo-

cal shape changes. The computational complexity for

all algorithm’s steps is shown in table 1.

Table 1: Computational Complexity Estimation.

Step Est. The unit complexity estimated

1 and 2 O(

n

2

2

) the number of terminal vertices

3 and 4 O(

n

2

2

) the number of internal edges

the number of skeletal graph

5 O(

n

2

2

) vertices remained after

main algorithm steps 1-4

Thus maximal computational complexity could be

at most quadratic by the total number of skeleton

edges. However real complexity becomes close to lin-

ear when we use the fact that medial axis is not ab-

stract graph but a tree with the realization on a plane.

2.6 Shape Metrics

We propose two metrics based on the main algorithm.

The ﬁrst one (Naive Edit-Cost) looks like edit dis-

tance (Klein et al., 2001). However we use the global

stop criterion and another algorithm. The second

(Adapted Hausdorff Metrics) is our main result. It

is a classic distance and at the same time agrees with

visual intuition. Efﬁciency of both metrics has been

conﬁrmed on experiments as well.

2.6.1 ”Naive Edit-Cost” Shape Metrics

We propose the ”Naive Edit-Cost” (D

cost

) similarity

measure as a sum of all operations of the main algo-

rithm that should be executed to obtain isomorphic

skeleton. Noises of the border as well as small struc-

ture ﬂuctuations are not taken into account. Therefore

all operations from steps 1 and 3 of the main algo-

rithm should be eliminated from the sum.

D

cost

=

∑

operations of steps 2, 4, 5 (alg. M) (4)

Figure 6 shows an example of ”Naive edit-cost” as

Shape Metrics on 12 ﬁgures of 3 classes: cats, birds

and dogs. We must mention that we added one more

stop criterion to the main algorithm. The main algo-

rithms exits when maximal of Hausdorff distances be-

tween input and changed shapes exceeds ﬁxed value

η, i.e. max(D

H

(Sh

1

, Sh

′

1

), D

H

(Sh

2

, Sh

′

2

)) ≥ η Thus

the main algorithm may exit when isomorphism is

not found. For a pair of shapes with no found iso-

morphism we assign the distance equal to inﬁnity ∞

which is denoted as

′

##

′

in fugure 6. It’s easy to see

that the distance between objects from one class is

less than the distance between objects from different

classes. In most cases the latter is equal to inﬁnity

(

′

##

′

in a ﬁg. 6).

Figure 6: An example of ”Naive edit-cost” shape metrics.

Despite the visually good results the similarity

measure deﬁned in such a way has several essential

drawbacks:

1. Strong dependency on the algorithm’s parameters

(as we make the sum of the algorithm’s steps);

2. Discontinuity (D

cost

equals to 0,1,2,...,∞);

3. Not a distance.

Therefore we propose the better similarity mea-

sure that avoids these drawbacks. We call Adapted

Hausdorff Metrics.

2.6.2 ”Adapted Hausdorff Shape Metrics”

Hausdorff metrics (D

H

) between two shapes S

1

and

S

2

is deﬁned as follows:

D

H

(S

1

, S

2

) = max{ρ

x∈S

1

(x, S

2

), ρ

y∈S

2

(S

1

, y)} (5)

Statement: Hausdorff Metrics (5) is a classic dis-

tance .

We deﬁne ”Adapted Hausdorff Metrics” (D

AH

)

between two shapes S

1

and S

2

as follows:

D

AH

(S

1

, S

2

) = max{D

H

(S

1

, S

2

), D

H

(S

′

1

, S

′

2

)} (6)

Where S

′

1

and S

′

2

are shapes as results of the main

algorithm M(Sh

1

, Sh

2

) → (Sh

′

1

, Sh

′

2

).

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

240

Statement: Adapted Hausdorff Metrics (6) is a

classic distance.

Figure 7 shows an example of ”Adapted Haus-

dorff Metrics”. We must mention that we added one

more stop criterion to the main algorithm. The main

algorithm exits when the maximal of Hausdorff dis-

tances between input and changed shapes exceeds

ﬁxed value η, i.e. D

AH

(S

1

, S

2

) ≥ η Thus the main

algorithm may exit when isomorphism is not found.

For a pair of shapes with no found isomorphism we

assign the distance D

AH

equal to inﬁnity ∞ which is

denoted as

′

##

′

in ﬁgure 7.

Statement: Adapted Hausdorff Metrics (6) with

additional stop criterion still remains a classic dis-

tance.

Figure 7: An example of Adapted Hausdorff Shape Metrics.

3 RECOGNITION APPLICATION

In this section we report on applying our algorithm

and proposed naive edit-cost metrics (4) and adapted

Hausdorff metrics (6) to shape recognition problems.

We took a database of total 142 binary shapes

(ﬁg. 8). The database consists of shapes of three

classes: mice (69), hands (22) and birds (51).

Figure 8: Test data set.

We construct a feature vector for each shape by

comparing it with a set of templates. We assume that

each class has at least one template shape. These tem-

plates can be chosen accidentally or by expert.

The following 8 templates are taken: T

1

, ..., T

8

(ﬁg. 9)

Figure 9: Test templates.

The feature vector for an object S is:

{D

cost

(S, T

1

), ..., D

cost

(S, T

8

), D

AH

(S, T

1

), ..., D

AH

(S, T

8

)}

Where D

cost

is Naive Edit-Cost (4) and D

AH

is

Adapted Hausdorff Metrics (6).

The goal was to solve a classic recognition prob-

lem: having a number of precedents (training sample)

divide testing sample objects into 3 classes.

We solve the problem using following steps:

1. Feature vector construction for each test data

shape, i.e. obtaining the feature space.

2. Accidental dividing all objects into 2 groups for

cross validation: training and testing sample.

3. Using standard methods to learn on a training

sample and estimate method’s accuracy on a test-

ing sample.

4. Repeat steps 2 and 3 to obtain impartial accuracy

estimation.

Figure 10 shows the feature space projection on a

plane with the highest dispersion.

Figure 10: Projection on a plain with the highest dispersion.

We chose several recognition methods for our ex-

periments (full methods descriptions can be found in

(Zhuravlev et al., 2005)):

Q-nearest Neighbors. The method was used as a

simple method that gives proper results when objects

are in compact groups (table 2).

Logical Regularities. The basis of Logical regular-

ities method is searching for logical regularities in

data (table 3).

Support Vector Machines. Support vector ma-

chines method is based on construction of optimal

separating hyperplane between each pair of classes.

The method is ﬂexible and often gives the best result

comparing to other methods (table 4).

SHAPE COMPARISON BASED ON SKELETON ISOMORPHISM

241

Combined Committee Method. Committee meth-

ods use voting schemes. It’s the best solution in

case of standard methods provide errors on differ-

ent objects. We used maximum of afﬁliation esti-

mates for each class of the algorithms: Q-nearest

neighbors, Logical regularities and Support vector

machines. The results are perfect (table 5).

Table 2: The results of ”Q-nearest neighbors”.

Class Correct Errors Correct in classes (%)

(%) (%)

1 2 3

1 93.2 6.8 98.6 0.0 9.8

2 100.0 0.0 0.0 100.0 0.0

3 97.9 2.1 1.4 0.0 90.2

Total 95.8 4.2

Table 3: The results of ”Logical regularities”.

Class Correct Errors Correct in classes (%)

(%) (%)

1 2 3

1 95.8 4.2 98.6 0.0 5.9

2 91.7 8.3 0.0 100.0 3.9

3 97.8 2.2 1.4 0.0 88.2

Total 95.1 4.2

Table 4: The results of ”Support vector machines”.

Class Correct Errors Correct in classes (%)

(%) (%)

1 2 3

1 94.4 5.6 98.6 9.1 3.9

2 100.0 0.0 0.0 90.9 0.0

3 98.0 2.0 1.4 0.0 96.1

Total 96.5 3.5

As a result the only incorrectly classiﬁed object

is a mouse that has been referred to a ”birds” class.

Thus we proved that our approach has been imple-

mented successfully. Reported experiments showed

very good results in recognition application.

Our future work includes enlarging the test data

base and ﬁnding a real application for our approach.

4 CONCLUSIONS

A new approach to shape comparison problem is pre-

sented in the paper. An approach is based on skeleton

isomorphism, in particular, on ﬁnding close shapes

with isomorphic skeletons. We proposed the shape

comparison algorithm and two skeleton metrics based

Table 5: The results of combined committee method.

Class Correct Errors Correct in classes (%)

(%) (%)

1 2 3

1 100.0 0.0 98.6 0.0 0.0

2 100.0 0.0 0.0 100 0.0

3 98.1 1.9 1.4 0.0 100.0

Total 99.3 0.7

on shapes with isomorphic continuous skeletons con-

struction. The proposed shape comparison algorithm

differs from existing ones by correctly deﬁned dis-

tance corresponding with visual intuition. The exper-

iments showed very good results in recognition appli-

cations. Thus we conﬁrmed that our theoretical result

does not contradict practical experiments.

ACKNOWLEDGEMENTS

Work is supported by the RFBR project

08− 01 − 00670. We thank our supervisor pro-

fessor L. Mestetski for his help.

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