SHAPE RECOGNITION USING THE LEAST SQUARES

APPROXIMATION

Nacéra Laiche and Slimane Larabi

Department of Computer, USTHB University, Algiers, Algeria

Keywords: Shape, Curvature Points, Approximation, Least-Squares method, Matching.

Abstract: This paper represents a novel algorithm to represent and recognize two dimensional curve based on its

convex hull and the Least-Squared modeling. It combines the advantages of the property of the convex hulls

that are particularly suitable for affine matching as they are affine invariant and the geometric properties of

a contour that make it more or less identifiable. The description scheme and the similarity measure

developed take into consideration technique for shape similarity. According to this method, the contours are

extracted and decomposed into portions of curves. Each portion curve is approximated by some explicit

curve using the Least Squares approximation. The obtained cubic curves are normalized in order to make

the method invariant to scale change. Finally the resulting curves are used to compare and to compute

similarity between shapes in images database using the Hausdorff distance. The proposed algorithm has

been tested and its performance is found favourable as compared to other matching techniques.

1 INTRODUCTION

Object representation and recognition is a very

difficult problem with many applications including

computer vision. Computer vision researchers aim to

capture image information in feature vectors which

describe shape, texture and color properties

databases of the image. These vectors are indexed or

compared to one another during query processing to

find similar images from the database. Considerable

amount of information exists in two dimensional

boundaries of objects since humans can readily

recognize an object using the shape of its boundary.

As a result, shape similarity retrieval plays an

important role in content based image database

systems.

Many techniques have been developed in the

literature to represent the shape of a free form 3D

object based on 2D silhouettes and most of them can

be classified into two categories: surface-based

methods and contour- based methods. Surface-based

methods extract features from the whole shape

region and are usually easy to compute and resistant

to noise and shape distortions. Different moments,

such as Zernike moments (Hwang et al., 2006)

(Chong et al., 2003) and Legendre moments (Yang

et al., 2006) have been demonstrated to achieve

excellent performance. These methods are not

suitable for object recognition in the presence of

occlusion. Unlike the contour-based methods

explore boundary shape information and are more

complicate requiring sophisticated implementations.

They are low, but more suitable than surface-based

methods for recognizing partially visible objects. In

this category we find: invariant features extracted

from boundaries of the object silhouette (Matusiak et

al, 1998), 2D boundary curves of silhouette using

Curvature Scale Space (CSS) (Mokhtarian et al.,

1992) and (Dudek et al., 1997). The polygonal

approximation has been used as a representation for

recognizing objects (Carmona-poyato et al., 2010).

Shape context (Belongie et al., 2002) is a method for

describing shapes and finding the correspondence

between point sets. Another shape descriptor is the

Medial Axis Transform, which was presented by

Blum (Blum, 1967) and later Sebastian and al

(Sebastian et al., 2004) used this descriptor for shape

recognition. Other techniques consist of approximate

the shape contour by Fourier descriptors (Zahn et al.,

1972) and B-spline (Paglieroni, 1985).

The notion of a part-based representation has

played an important role in object recognition. In

(Argawal et al, 2004), informative patches in the

images are derived from the training examples and

are used as fragments. Daliri and Torre (Daliri et al.,

2010) proposed a representation for shape-based

572

Laiche N. and Larabi S. (2012).

SHAPE RECOGNITION USING THE LEAST SQUARES APPROXIMATION.

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pages 572-575

DOI: 10.5220/0003778405720575

 SciTePress

recognition based on the extraction of the

perceptually relevant fragments. Therefore, our aim

is to develop a recognition system which requires

two components: part-based representation and

matching method. The part-based silhouette

representation we use is built only on curves. Our

shape matching algorithm is done by introducing the

convex hull of the shape.

2 GEOMETRIC DESCRIPTION

Shape representation is one of the most challenging

aspects of computer vision because shapes are often

more complex than color and texture. The problem

remains difficult in similarity retrieval in image

databases. In This section, we present our approach

for representing shape by using Least Squares

approximation. The shape contour is first segmented

into several curve segments. Each curve segment is

then approximated by a cubic explicit curve using

the Least- Squares method.

2.1 Extracting the Local Boundary

Features (Parts)

In this section we describe how to extract the

different curve segments. The decomposition

process can be started by taking into account some

features of boundaries which exert a crucial role in

attracting the attention of an observer. Examples of

such features, closely related to those considered in

an early version of this paper are the high curvature

points which give important clues for shape

representation and analysis.

The basic step in our proposed algorithm is to

extract the curvature points using the Chetverikov

algorithm (Chetverikov, 2003) and locate the main

points that may preserve the object shape as concave

and convex points. Using the selected concave

points, the shape boundary is then decomposed into

a set of portions of curves as illustrated in Fig. 1.

Figure 1: Shape with different parts.

2.2 Curves Modelling

Our approach consists first in the use of the

minimum rectangle MR that encloses the outline

shape (Graham, 1972). OXY is the referential

attached to MR chosen such as the origin O is the

left top edge of MR and the OX (resp. OY) axis

corresponds to the width (resp. length) of the outline

shape (see Fig. 2).

Figure 2: The minimum rectangle MR including the shape.

Given a shape contour

. We assume that the

original contour

is close, Ω is traversed in a

counter-clockwise sense (the object is to the left).

Consists of a finite number of an ordered list of

parts that define the shape of the object silhouette

(see Fig. 1). A curve modelling should be applied in

order to facilitate matching and recognizing object

shapes. In this paper the Least Squares model is

employed to approximate each cut (part)

C by an

explicit cubic curve. The least squares curve of order

3 is defined by

01 2 3

()

Cx ax a ax ax ax

==+++

∑

(1)

Where the polynomial factors

)

3,2,1,0=i

a are

computed such that the sum of the quadratic errors

between the discrete data and their corresponding

least squares curve is minimized.

2.2.1 Normalization

In this subsection, we introduce the convex hull of

the shape to generate an invariant representation.

Convex hulls have some properties that make them

suitable for recognition and representation tasks

(Preparata et al., 1985).

The boundary

of any object consists of a

finite number of an ordered sequence of points that

define the shape of an object:

{

}

ppp ,.......,,

consisting of n two-

dimensional points. Let

C denotes the convex hull

for the set

. Let

(

)

yx , ,

mi ,.....,2,1=

be the

ordered vertices forming the convex hull. Using the

Green’s theorem (Gope et al., 2007) the centroid of

C denoted by

),(

yxg

ccC

can be expressed as:

SHAPE RECOGNITION USING THE LEAST SQUARES APPROXIMATION

573

()( )

111

ii ii ii

cxxxyxy

cyyxyxy

−

+++

−

+++

⎧

=+ −

⎪

⎨

⎪

=+ −

⎪

⎩

∑

(2)

Where

is the area of

C given by

()

ii i i

Axyxy

−

=−

∑

(3)

In order to make the representation invariant to scale

change we carry out a transformation on the

approximation points. This can be accomplished by

converting each approximation point

)','(

yx of

each curve to another point by the transformation:

⎧

→

⎪

⎨

⎪

→

⎪

⎩

(4)

Where

{

}

),(max

igMax

pCdd =

represents the

maximal distance from the centroid of the convex

hull of the shape to the boundary curve.

This transformation allows us to bring back the

different cubic curves approximating the original

boundary curve at the different sizes on the same

neighborhood

3 SHAPE MATCHING

In this section, we describe the basic concepts of our

matching algorithm which compares images of the

database with a query image. Consider that the

features here are related to the convex hull and the

normalized curves.

3.1 Boundary Signature Matching

The boundary signature

extracted from an object’s

boundary that characterizes the shape of an object is

defined as the ratio of the minimal Euclidean

distance between the centroid of the convex hull and

the boundary shape to the maximal Euclidean

distance

Max

d . Matching between query shape to

models is accomplished by comparing their

boundary signatures.

3.2 Computing Shape Similarity

A necessary condition to match two shapes (query

and model shapes) is the similarity between of their

all normalized curves.

3.2.1 Matching using the Normalized

Curves

In this section, we explain how to match two curves.

Hausdorff distance is used for matching two

different curves.

Given two normalized curves C

and

'C of a query shape

and a reference shape

respectively, the Hausdorff distance is defined

as:

( , ') max( ( , '), ( ', ))HCC hCC hC C

(5)

Where

(, ') maxmin '

hCC c c

∈

−

(6)

and

. is a norm defined on the curve, such as the

norm.

A valid match between two normalized curves is

found if the maximal difference between them (the

similarity measure defined above) is under a

threshold defined experimentally

E ; otherwise

they are different.

4 EXPERIMENTAL RESULTS

Our method is tested on ETH-80 database of 80

objects built by Leibe and B.Schiele (Leibe et al.,

2003). Each object is represented by some views

spaced evenly over the upper viewing hemisphere.

The method possesses the important property of

rotation and scale change. Invariance to rotation is

achieved by computing using the referential defined

by the minimum rectangle that encloses the shape.

Using the maximal distance from the centroid of the

convex hull, the representation is invariant to scale

change. Some of the matching results are shown in

Figure 3. The queries shapes are in the first row (at

the left of each row). The similar shapes that have

been matched by the proposed algorithm are shown

in the rest rows. In the examples shown in Figure 3

there is a difference in the view angle between the

query shape and the similar shapes. This examples

show the robustness of our approach to orientation

changes of the shapes.

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

574

Figure 3: More matching of results.

We have summarized recognition rates for some

different approaches which are tested on the ETH-80

database and cited in (Daliri et al., 2009).

Table 1: Some recognition rates for different algorithms

tested on ETH-80 database.

Algorithm Recognition rate (%)

SC greedy 86.40

Decision tree 93.02

Fragment-based

approach

86.40

Kernel-edit-distance 91.33

Robust symbolic

representation

89.03

Proposed algorithm 92.50

5 CONCLUSIONS

In this paper, we have presented a new approach to

represent the shape of the projection of a 3D object

which enables similarity search. A key characteristic

of our approach is the use of the geometric

description of different parts constituting the outer

closed boundary of the shape using a set of cubic

curves. These curves enable us comparisons between

different shapes. A shape matching technique, using

the Hausdorff distance between two curves has been

proposed. In our experiments, we have demonstrated

invariance to similarity transformations: rotation and

scaling. The results are encouraging. The proposed

approach achieves a recognition rate equal to 92.5%.

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