SHAPE COMPARISON OF FLEXIBLE OBJECTS

Similarity of Palm Silhouettes

Leonid Mestetskiy

Department of Mathematical Methods of Forecasting

Moscow State university, Moscow, Russia

Keywords: Shape analysis, continuous skeleton, skeletal base, matching, circular tree, biometrical identification.

Abstract: We consider the problem of shape comparison for elastic objects presented by binary bitmaps. Our approach

to similarity measuring of such objects is based on the conception of a flexible object. A flexible object is

defined as a planar graph with a family of circles centered on graph edges. A set of admissible deformations

is connected with each flexible object. These deformations are described as a group of planar graph vertices

transforms. We define the flexible objects similarity through matching and alignment within the group of

admissible deformations. The regular method for approximation of the binary bitmap shape by the flexible

object is presented. The flexible object is designed as a subgraph of continuous skeleton of the binary

bitmap. The proposed approach is applied to a problem of palm shape recognition for personal biometrical

identification.

1 INTRODUCTION

The problem of binary bitmap shapes comparison

arises in many applications. In particular, the

problem of the palm shape recognition for

biometrical identification is reduced to it. The shape

of human hand (palm geometry) is an important

feature used for personal identification in access

control systems.

The color or the grayscale image of a palm

received by means of any device (for example, of

the web camera or the scanner) may be transformed

to the monochrome bitmap in which an object (a

palm) is presented by black pixels, and a background

– by white. But a human palm is a difficult object to

classify. The person can’t repeat the same position

of a palm even if he wants to. Two photos of the

same palm and two photos of two different palms

can have differences of the same range (Figure 1).

Known approaches to shape comparison of the

objects based on alignment of their outline contours

(Sederberg and Greenwood, 1992) are unsuitable for

solving these problems. These methods don’t

preserve important invariants of the palm shape –

finger’s width and curvature.

Another approach to compare general structure

features of object shapes is based on their skeletons

(Sebastian and Kimia, 2001). A skeleton is more

detailed description of topological structure of a

shape. However, it isn’t enough for comparison of

palms. The topology of skeletons of palms anyway

is almost identical. There are only 5 topologically

different skeletons of palm. But skeletons don’t

allow comparing such important features as palm’s

outlines.

The proposed approach solves this problem. We

propose a measure for human palms comparison

(and other similar elastic objects) using both this

Figure 1: The silhouettes of palms (the first row – the

same person, the second row – different persons).

390

Mestetskiy L. (2007).

SHAPE COMPARISON OF FLEXIBLE OBJECTS - Similarity of Palm Silhouettes.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 390-393

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features. Alignment of skeletons allows comparing

general structure of objects and deforming of their

outlines. Then measure of difference between

objects is obtained by matching of outlines.

The basis of this approach is the model of the so-

called flexible object. Such object has a shape which

can change within certain limits. The limits of these

changes are described by special group of

transforms. The transforms are organized in such a

way that some elements of the shape are fixed and

constant, and others can vary. With reference to a

palm constant elements are metacarpus and

phalanges of fingers, but changeable elements are

joints. The problem of flexible object comparing is

reduced to selection of such admissible transforms

of these objects, in which their shapes will be the

closest to one another. The difference of their shapes

in this (the closest) position is accepted as a measure

of the distinction of objects.

2 A FLEXIBLE OBJECT

Let's consider a set of points

on the Euclidean

plane

, which is presented by a planar graph with

tree type structure. This graph has a finite set of

vertices, and its edges are continuous lines.

We bind with each point

Tt ∈ in graph

some circle

c with the center in this point. Let’s

call this family of circles

{}

TtcC

∈

= , a

circular tree. A graph

we call an axial graph of

the circular tree. And a union

∈

= of all

circles of family

C (as point sets) we call a

silhouette of the circular tree. The silhouette of the

circular tree represents a closed connected set of the

Euclidean plane

RS ⊂ . The outline of this set is

an envelope of the whole family of circles

C . There

is an example of the circular tree on Figure 2. The

set

is infinite, bat we use its finite subset for

visualization on Figure 2 and other Figures.

Among all vertices of a circular tree we extract

some subset of the points

P ⊂

called bend

points. We connect a range of angles between each

couple of edges incident to bend point. We forbid

the changing a relative position of edges for the rest

vertices of the graph.

A change of angles in bend points in an

admissible range is called a deformation of a

circular tree. Such deformation implies not only the

change of an axial graph, but also moving of a

family of circles and a respective alteration of a

circular silhouette of a tree.

Let

be a group of deformations of a circular

tree

C . The couple of circular tree and its group of

deformations we call a flexible object

),( VCG = .

If some deformations of two circular trees make

their silhouettes coincide then these trees are called

equivalent. A shape of the flexible object is

described by a set of silhouettes of all its equivalent

circular trees.

3 THE COMPARISON OF

FLEXIBLE OBJECTS

We introduce the metrics on the set of flexible

objects in the following way. We define distance

),(

between two flexible objects

),(

111

VCG

and ),(

222

VCG

as the minimal

distance between their circular trees on the whole set

of admissible deformations, i.e.

)](),([inf),(

221121

CvCvGG

∈

Here

)(

and

)(

are circular trees

and

C transformed by means of deformations

and

v , and is a measure of distance of circular

trees.

)\\(),(

122121

SSSSAreaCC ∪

Here

and

are silhouettes of circular trees

C and

C . And ),(

is equal to the area of

Figure 2: An axial graph, a family of circles and a

silhouette of the circular tree.

SHAPE COMPARISON OF FLEXIBLE OBJECTS - Similarity of Palm Silhouettes

391

a symmetric difference of

and

minimized for

all possible variants of matching.

Thus, the problem of an estimation of similarity

of flexible objects consists in their matching on each

other and a choice of such deformation of these

objects and such matching at which the value of the

distinctions of their shapes will be minimal. With

reference to human palms it means, that it is

necessary to apply palms to each other and move

fingers in such a way that silhouettes of palms have

coincided at the greatest degree.

4 THE CONSTRUCTION OF A

FLEXIBLE OBJECT

Let it be a binary bitmap (Figure 1). A construction

of a flexible objects which approximating this

bitmap includes the following steps:

1. An approximation of the binary bitmap outline by

the minimal perimeter polygon. The polygon is a

closed path of the minimal length separating black

and white pixels on the binary bitmap (Figure 3a).

2. A construction of the continuous skeleton of the

polygon (Mestetskiy, 1998, 2006). The skeleton of

the polygon is a locus of the centers of its inscribed

maximal circles (Figure 3a). The skeleton of the

polygon with its circles forms a circular tree, and the

polygon itself is a silhouette of this tree.

3. A pruning of a skeleton to get the so-called

skeletal base of the polygon (Figure 3b).

Let

be a polygon, and S - a silhouette of

circular tree of a connected subgraph of its skeleton.

We will call the minimal subgraph of the skeleton, at

which a silhouette of a circular tree differs from a

polygon in the Hausdorf metrics no more than on the

given value

, a skeletal base of the polygon

i.e.

≤

),( SMH

A skeletal base has a much more simple

structure, than the skeleton of a polygon (Figure 3b)

and is more stable to noise distortions connected

with the source binary bitmap.

5 THE COMPARISON OF

SILHOUETTES OF PALMS

Let’s choose a third degree vertex of a skeletal base

graph which is incidence with the branch of the

thumb (the vertex

on Figure 3b). The branches of

the thumb and a wrist are crossing in this vertex. The

image of a wrist is extraneous information for palm

shape description. Therefore we delete the branch of

a wrist in the skeletal base. The obtained graph is an

axial graph of a circular tree of a flexible object

(Figure 3c). The vertex

A is a root of the circular

tree. Its circle is called “root” circle. The next third

degree vertex

we call “center” of a palm (Figure

3c) and its circle is called “middle” circle.

The analysis of a real skeleton of a human palm

shows, that it is enough to consider six bend points:

two points of the thumb and one point of each of the

rest four fingers (Figure 3d). We choose two bend

points (0 and 1) of maximal curvature at thumb

branch and four bend points (2-5) as crossing of

finger branches and the “root” circle.

We can estimate the distance

),(

between two palms by more simple measure:

≤

∈

)](),([inf),(

221121

CvCvGG

),(

)](,[inf

21221

GGCvC

≤

∈

The estimation

),(

represents a measure

of distance of flexible objects

and

, received

by a static position of

G and a deformation of

G .

Alignment of two palms is carried out by the

following steps.

1. Coincidence of centers of “middle” circles

(vertices

in Figure 3d).

2. Coincidence of directions from the centers of

“middle” circles to the centers of “root” circles

(vectors

in Figure 3d).

(a)

(b)

(c)

(d)

Figure 3: Approximation of a palm by flexible object: (a)

the minimal perimeter polygon and its skeleton, (b) the

skeletal base, (c) the circular tree, (d) banding points.

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

392

Figure 6: Left – distance distribution to the nearest palm,

right – classification errors (FRR – left curve, FAR – right

curve).

3. Deformation of the axial graph of the second palm

for coincidence with the axial graph of the first palm

(Figure 4). For that we “rotate fingers” of the second

palm (branches of axial graph) around bending

points. The Hausdorf metrics can be used as a

measure of coincidence of these branches.

4. Construction of circular tree silhouettes as

envelopes of a family of circles.

5. Comparing of silhouettes (Figure 5). The effective

algorithm for computation of the areas of a

symmetric difference is designed with the help of

methods of computational geometry.

The computing experiment was carried out for

testing of proposed method. The data base of 1662

bitmaps of 320 palms (4-6 images per person) has

been used in this experiment. All images 640×480

were obtained for the same conditions (camera,

distance, brightness). The approximating flexible

objects have been constructed for each of these

bitmaps. The measure of distance between

silhouettes

S and

S was computed as

1000

)(

)\\(

),(

1221

⋅

∪

SArea

SSSSArea

The left diagram on Figure 6 shows the

distribution of distances to the nearest sample of the

same person (left curve), and of different people

(right curve). Such distance enables to construct a

classification rule by the nearest neighbor. The

diagram of classification errors for different values

of the threshold is shown on Figure 6 (right).

The running time for binary bitmap

approximation of one bitmap by the flexible object

is 15 msec, and for two palms comparison is 0.5

msec using Intel processor 1.3 GHertz.

6 CONCLUSIONS

The combination of two constructions – an outline

and a skeleton – opens up opportunities for the

comparison of objects which don’t have strictly

fixed shapes using a matching method. The

proposed method is well adjusted with common

sense, is easily visualized and allows efficient

implementation.

ACKNOWLEDGEMENTS

The author thanks the Russian Foundation for Basic

Researches, which has supported this work (grant

05-01-00542).

REFERENCES

Mestetskiy, L., 1998. Continuous skeleton of binary raster

bitmap. In Graphicon’98, International Conference on

computer graphics, Moscow, in Russian.

Mestetskiy, L., 2006. Skeletonization of a multiply

connected polygonal domain based on its boundary

adjacent tree. In Siberian journal of numerical

mathematics, vol.9, N 3, 2006, 299-314, in Russian.

Sebastian, T., Kimia, B., 2001. Curves vs skeletons in

object recognition. In Proceedings of International

Conference on Image Processing, Thessaloniki,

Greece.

Sederberg, T., Greenwood, E., 1992. A physically based

approach to 2-D shape blending. In Computer

Graphics 26(2), 25-34.

(а) (б)

Figure 4: Deformation of the circular tree:

(a) rotation of branches, (b) moving of circles.

Figure 5: Comparison of silhouettes: (a) images of the

same palm, (b) palms of different persons.

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