image description with charlier and meixner moments · image description with nonseparable...

IMAGE DESCRIPTION WITH

NONSEPARABLE TWO-DIMENSIONAL

CHARLIER AND MEIXNER MOMENTS

HONGQING ZHU*, MIN LIUy and YU LIz

Department of Electronics and Communications Engineering

East China University of Science and TechnologyShanghai 200237, P. R. China

*[email protected]@[email protected]

HUAZHONG SHUx and HUI ZHANG{

Laboratory of Image Science and TechnologySchool of Computer Science and Engineering

Southeast University

Nanjing 210096, P. R. [email protected]

{[email protected]

This paper presents two new sets of nonseparable discrete orthogonal Charlier and Meixnermoments describing the images with noise and that are noise-free. The basis functions used by

the proposed nonseparable moments are bivariate Charlier or Meixner polynomials introduced

by Tratnik et al. This study discusses the computational aspects of discrete orthogonal Charlier

and Meixner polynomials, including the recurrence relations with respect to variable x andorder n. The purpose is to avoid large variation in the dynamic range of polynomial values for

higher order moments. The implementation of nonseparable Charlier andMeixner moments does

not involve any numerical approximation, since the basis function of the proposed moments isorthogonal in the image coordinate space. The performances of Charlier andMeixnermoments in

describing images were investigated in terms of the image reconstruction error, and the results of

the experiments on the noise sensitivity are given.

Keywords : Bivariate discrete orthogonal polynomials; nonseparable discrete orthogonal

moments; Charlier; Meixner; three-term recurrence relations; second order linear partialdi®erence equations.

1. Introduction

Moments have been extensively used in image processing, pattern recognition and

computer vision.1,8,13,19,24,25,28 Teague proposed continuous orthogonal polynomials

‡Author for correspondence

International Journal of Pattern Recognitionand Arti¯cial Intelligence

Vol. 25, No. 1 (2011) 37�55

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0218001411008506

37

http://dx.doi.org/10.1142/S0218001411008506

as the basis functions to calculate continuous moments.16 These orthogonal poly-

nomials include Zernike polynomials, Legendre polynomials, and pseudo-Zernike

polynomials. Practical implementations of continuous orthogonal moments in digital

image analysis involve two main dilemmas9: (i) discrete approximation of the con-

tinuous integrals, and (ii) transformation of the image coordinate system into the

domain of polynomials. Consequentially, the development of moments led to their

discrete versions. Recently, Mukundan et al. and Yap et al. de¯ned discrete

orthogonal moments based on the discrete Tchebichef and Krawtchouk orthogonal

polynomials respectively.10,22 For the discrete orthogonal moments to be of use in

image processing, it must be extensible to two dimensions. Two forms of two-

dimensional moments have been suggested: a separable form and a nonseparable

form. A standard method computing two-dimensional separable form of moments is

via the row-column decomposition, which takes advantage of the separability of basis

functions of the moments.10,22,23 That is, the two-dimensional moment is broken up

into two one-dimensional moments computed in row-column or column-row wise

format. The nonseparable form moments do not work the same way; however, it can

be written using bivariate polynomials as the basis function of moments. This

requires less computation than for the separable moments. To the best of our

knowledge until now, no nonseparable form of two-dimensional moments has been

de¯ned in pattern recognition community. This is mainly due to the fact that

bivariate orthogonal polynomials are too di±cult to use. The purpose of this paper is

to introduce bivariate discrete orthogonal polynomials as basis functions of the

moments, and to expect that they have potential applications in image processing

and pattern recognition.

Actually, bivariate orthogonal polynomials have been investigated by many

authors.2�5,14,15,20,21 Bivariate orthogonal polynomials have been studied as the

generalization of one variable's orthogonal polynomials.3 However, comparing to the

theory of one variable, the structure of bivariate discrete orthogonal polynomial is

much more complicated. In 1982, Kowalski introduced a matrix notation for poly-

nomials applied in multiple variables, and showed the matrix notation with the

properties similar to three-term recurrence relations.4,5 Xu also presented a second-

order di®erence equations and bivariate discrete orthogonal polynomials in Refs. 20

and 21. More recently, Rodal et al. made a systematic study of the orthogonal

polynomials solutions of a second order partial di®erence equation with hypergeo-

metric type on two variables in Refs. 14 and 15. Although bivariate orthogonal

polynomials play an important role in applications such as numerical analysis, optics,

and quantum mechanics,11,12,18 much less attention has been paid to the application

of bivariate orthogonal polynomials in image analysis.

In this study, we have introduced the bivariate Charlier and Meixner polynomials

derived from the methods proposed by Tratnik.17 Then, the nonseparable form

bivariate moments, Charlier and Meixner, are constructed using their orthogonal

polynomials, respectively. Due to the new sets of moments being orthogonal in the

discrete domain of the image coordinate space, the proposed moments completely

38 H. Zhu et al.

eliminate the two problems referred to above. The image can be reconstructed from a

su±ciently large number of computed moments. In order to reduce the numerical

errors caused in the computation of Charlier and Meixner moments, this study

presents the three-term recurrence relations about variable x and polynomials

order n. The simulated results demonstrate that the proposed two-dimensional

nonseparable form moments Charlier and Meixner have better image description

capabilities.

The organization of this paper is given as follows: in Sec. 2, we recall the de¯nition

of the classical discrete orthogonal polynomials of one variable. Section 3 shows two

recurrence relations about univariate Charlier and Meixner polynomials. It is fol-

lowed by bivariate discrete orthogonal polynomials in Sec. 4. In Sec. 5, we de¯ne the

nonseparable discrete orthogonal moments with bivariable orthogonal polynomials

as basis functions. Some experimental results are presented and discussed in Sec. 6.

Section 7 concludes the study.

2. The Classical Discrete Orthogonal Polynomials of One Variable

The classical discrete orthogonal polynomials of one variable can be de¯ned as the

polynomial solutions of the following di®erence equation11,12

�ðxÞ�rpnðxÞ þ �ðxÞ�pnðxÞ þ �npnðxÞ ¼ 0 ð1Þwhere �pnðxÞ ¼ pnðxþ 1Þ � pnðxÞ, rpnðxÞ ¼ pnðxÞ � pnðx� 1Þ denote the forward

and backward ¯nite di®erence operators, respectively. �ðxÞ and �ðxÞ are functions ofsecond and ¯rst degrees respectively, �n is an appropriate constant. The solution of

this partial di®erence equation can be expressed by Rodrigues formula as follows:

pnðxÞ ¼Bn

wðxÞ rn½wnðxÞ� ð2Þ

where wðxÞ is weight function, wnðxÞ in the case of n ¼ 0. Thus the polynomials

solutions of Eq. (1) are determined by Eq. (2) depending on the normalizing factors

Bn. For the backward di®erence operator O we have the property11,12

rnfðxÞ ¼Xnk¼0

n

k

� �ð�1Þkfðx� kÞ ð3Þ

Combining Rodrigues formula and Eq. (3) one can obtain an explicit expression for

the polynomials pnðxÞ.The classical orthogonal polynomials of one discrete variable satisfy the following

three-term recurrence relation

xpnðxÞ ¼ �npnþ1ðxÞ þ �npnðxÞ þ �npn�1ðxÞ ð4ÞThe polynomials pnðxÞ satisfy an orthogonality relation of the form

Xsx¼0

pnðxÞpmðxÞwðxÞ ¼ d2n � �mn; 0 � m; n � s ð5Þ

Image Description with Nonseparable Two-Dimensional Charlier and Meixner Moments 39

where d2n denotes the square of the norm of the corresponding orthogonal poly-

nomials, and �mn denotes the Dirac function. The scaled orthogonal polynomials can

be obtained by utilizing the square norm and weighted function,

~pnðxÞ ¼ pnðxÞffiffiffiffiffiffiffiffiffiffiffiwðxÞd2n

sn ¼ 0; 1; s ð6Þ

Therefore, the orthogonal property of normalized orthogonal polynomials of Eq. (5)

can be rewritten as

Xsx¼0

~pmðxÞ~pnðxÞ ¼ �mn 0 � m; n � s ð7Þ

For univariate Charlier and Meixner polynomials, Nikiforov et al.11 introduced

some basis information listed in Table 1. Their weight functions are de¯ned in the

following discrete domain

G ¼ fx j 0 � x � sg ð8Þ

3. Computational Aspects of Univariate Charlierand Meixner Polynomials

In the following two subsections, we have attempted to present two recurrence

relations to calculate univariate Charlier and Meixner polynomials.

Table 1. Data for univariate Charlier ca1n ðxÞ, and univariate

Meixner $ð�;�Þn ðxÞ polynomials, (a1 > 0 for Charlier, and

0 < � < 1 for Meixner).

pnðxÞ ca1n ðxÞ $ð�;�Þn ðxÞ

s 1 1�ðxÞ x x

�ðxÞ a1 � x �� xð1� �Þ�n n nð1� �ÞBn 1

an1

1

�n

wnðxÞ e�a1axþn1

x!

�xþn�ðnþ � þ xÞ�ð�Þx!

d2n

n!

an1

n!ð�Þn�nð1� �Þ�

�n �a1�

�� 1

�n nþ a1 nþ �ðnþ �Þ1� �

�n �n nðn� 1þ �Þ�� 1

40 H. Zhu et al.

3.1. Recurrence relations with respect to n

(i) Univariate Charlier polynomials

The Charlier polynomials of one variable ca1n ðxÞ satisfy the following ¯rst partial

di®erence equation of the form

x�rca1n ðxÞ þ ða1 � xÞ�ca1n ðxÞ þ nca1n ðxÞ ¼ 0 ð9Þwhere a1 is restricted to a1 > 0.

The nth Charlier polynomial is also de¯ned by using hypergeometric function as

follows:

ca1n ðxÞ¼2F0ð�n;�x; ;�1=a1Þ ð10ÞThe normalized discrete Charlier polynomials satisfy the following orthogonal

condition:

X1x¼0

~c a1mðxÞ~c a1

n ðxÞ ¼ �mn m � 0; n � 0 ð11Þ

The discrete Charlier polynomials satisfy the following three-term recurrence

relations:

A~c a1n ðxÞ ¼ B �D~c a1

n�1ðxÞ þ C � E~c a1n�2ðxÞ ð12Þ

where the coe±cients A� E in Eq. (12) are listed in Table 2.

The zero-order and the ¯rst-order normalizedCharlier can be calculated as follows:

~c a10 ðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiwðxÞd20

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie��x

x!

r

~c a11 ðxÞ ¼ �� x

�


s¼ �� x

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie��xþ1

x!

r ð13Þ

(ii) Univariate Meixner polynomials

Meixner polynomials of one variable $ð�;�Þn ðxÞ satisfy the following ¯rst partial

di®erence equation of the form

x�r$ ð�;uÞn ðxÞ þ ð�� xð1� �ÞÞ�$ ð�;uÞ

n ðxÞ þ nð1� �Þ$ ð�;uÞn ðxÞ ¼ 0 ð14Þ

where � and � are restricted to 0 < � < 1, and � > 0.

Table 2. Data for three-term recurrence relations of univariate Charlier, and univariate Meixner

polynomials (a1 > 0 for Charlier, and � > 0, 0 < � < 1 for Meixner).

~pnðxÞ A B C D E

~c a1n ðxÞ �a1 x� nþ 1� a1 n� 1

ffiffiffiffiffiffia1n

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21

nðn� 1Þ

s

~$ð�;�Þn ðxÞ �

�� 1x� x�� nþ 1� �nþ ��

1� �

ðn� 1Þðn� 2þ �Þ1� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

nð� þ n� 1Þr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2

nðn� 1Þð� þ n� 2Þð� þ n� 1Þ

s


The nth Meixner polynomial is also de¯ned by using hypergeometric function as

follows:

$ ð�;uÞn ðxÞ ¼ ð�Þn2F1ð�n;�x;�; 1� 1=�Þ ð15Þ

The discrete normalized Meixner polynomials satisfy the following orthogonal

polynomials condition:

X1x¼0

~$ ð�;uÞm ðxÞ ~$ ð�;uÞ

n ðxÞ ¼ �mn m � 0; n � 0 ð16Þ

The discrete Meixner polynomials satisfy the following three-term recurrence

relations:

A ~$ ð�;�Þn ðxÞ ¼ B �D ~$

ð�;�Þn�1 ðxÞ þ C � E ~$

ð�;�Þn�2 ðxÞ ð17Þ

where the coe±cients A� E in Eq. (17) are listed in Table 2. The zero-order and the

¯rst-order normalized Meixner can be calculated as follows:

~$ð�;�Þ0 ðxÞ ¼


s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�xð� þ x� 1Þ!

x!ð� � 1Þ! ð1� �Þ�s

~$ð�;�Þ1 ðxÞ ¼ � þ x� x

�

� � ffiffiffiffiffiffiffiffiffiffiffiwðxÞd21

s¼ � þ x� x

�

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�xð� þ x� 1Þ!

x!ð� � 1Þ!�ð1� �Þ�

�

s

ð18Þ

According to Table 1, the above recurrence relations Eqs. (12) and (17) can

be used to calculate the weighted univariate Charlier and Meixner polynomials,

respectively.

3.2. Recurrence relations with respect to x

Mukundan discussed the reasons in Ref. 7 that the recursive nature of polynomial

evaluation can lead to numerical problems when the required moment order is large.

An e®ective solution to this problem is changing the recurrence relation to avoid

cumulative multiplication of large values. In this subsection, we present x recurrence

relations to eliminate the numerical instability while computing polynomials or high-

order moments. The general frame of their recurrence relations with respect to x is as

follows:

Considering the properties of the operators O and 4, we have

4OpnðxÞ ¼ pnðxþ 1Þ � 2pnðxÞ þ pnðx� 1Þ ð19Þ

42 H. Zhu et al.

Thus, the recursive relation of discrete orthogonal polynomials with respect to x can

be obtained according to Eqs. (1) and (19) as follows:

pnðxÞ ¼2�ðx� 1Þ þ �ðx� 1Þ � �n

�ðx� 1Þ þ �ðx� 1Þ pnðx� 1Þ � �ðx� 1Þ�ðx� 1Þ þ �ðx� 1Þ pnðx� 2Þ ð20Þ

The recurrence relation of normalized discrete orthogonal polynomials is also written

using the following general form:

~pnðxÞ ¼ffiffiffiffiffiffiffiffiffiffiwðxÞp

�ðx� 1Þ þ �ðx� 1Þ

� 2�ðx� 1Þ þ �ðx� 1Þ � �nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðx� 1Þp ~pnðx� 1Þ � �ðx� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wðx� 2Þp ~pnðx� 2Þ" #

ð21Þ

Thus, one can obtain the recurrence relations of discrete Charlier, and Meixner

with respect to x according to Eq. (21) and Table 1 as follows:

~c a1n ðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiwðxÞp


� 2�ðx� 1Þ þ �ðx� 1Þ � �nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðx� 1Þp ~c a1

n ðx� 1Þ � �ðx� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðx� 2Þp ~c a1

n ðx� 2Þ" #

ð22Þ

~$ ð�;�Þn ðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiwðxÞp


� 2�ðx� 1Þ þ �ðx� 1Þ � �nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðx� 1Þp ~$ ð�;�Þ

n ðx� 1Þ � �ðx� 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwðx� 2Þp ~$ ð�;�Þ

n ðx� 2Þ" #

ð23ÞUsing Eqs. (2), (3), (6) and Table 1, we obtain the initial values of recurrence

relations with respect to x listed in Table 3. The above equations can be used to

e®ectively calculate the weighted univariate Charlier and Meixner polynomials'

values. Figure 1 shows the plots of the ¯rst few orders of these polynomials with

di®erent parameter values.

Table 3. The initial values of three-term recurrence relations with

respect to x for univariate Charlier and Meixner polynomials. (a1 > 0for Charlier, and 0 < � < 1; � > 0 for Meixner).

~pnðxÞ x ¼ 0 x ¼ 1

~c a1n ðxÞ ffiffiffiffiffiffiffiffiffiffi

wð0Þd2n

sa1 � n

a1

ffiffiffiffiffiffiffiffiffiffiwð1Þwð0Þ

s~c a1n ð0Þ

~$ð�;�Þn ðxÞ ð�Þn

ffiffiffiffiffiffiffiffiffiffiwð0Þd2n

s�ðnþ �Þ � n

��

ffiffiffiffiffiffiffiffiffiffiwð1Þwð0Þ

s~$ ð�;�Þ

n ð0Þ


4. Bivariate Discrete Orthogonal Polynomials

4.1. Bivariate Charlier polynomials

The bivariate Charlier polynomials were introduced by Tiatnik17 and Rodal

et al.14,15 For a bivariable, let n1;n2 2 N0, one has

ca1;a2n1;n2ðx; yÞ ¼ ~$ ð�x�y;�a1=a2Þ

n1ðxÞ~c ða1þa2Þ

n2ðxþ y� n1Þ ð24Þ

where ~$ð�;�Þn ðxÞ is the normalized univariateMeixner polynomials de¯ned by Eq. (23),

and ~c anðxÞ is the normalized univariate Charlier polynomials de¯ned by Eq. (22).

Charlier polynomials of two variables satisfy the following orthogonality relation,

X1x¼0

X1y¼0

ca1;a2n1;n2ðx;yÞca1;a2m1;m2

ðx;yÞ ax1a

y2

x!y!e�a1�a2

¼ n1!n2!a2a1

� �n1

ða1 þ a2Þn1�n2�n1;m1�n2;m2

ð25Þ

0 50 100 150 200 250−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

Poly

nom

ials

Val

ues

Charlier Polynomials (a1 = 128, Order:10 )

(a)

0 50 100 150 200 250−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

Poly

nom

ials

Val

ues

Charlier Polynomials (a1 = 240, Order:10 )

(b)

0 50 100 150 200 250−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

Poly

nom

ials

Val

ues

Meixner Polynomials (β = 60, µ = 0.5, Order:10 )

(c)

0 50 100 150 200 250−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

x

Poly

nom

ials

Val

ues

Meixner Polynomials (β = 150, µ = 0.5, Order:10 )

(d)

Fig. 1. Plots of univariate polynomials when N ¼ 256, Order ¼ 10, (a) Charlier (a1 ¼ 128), (b) Charlier

(a1 ¼ 240), (c) Meixner (� ¼ 60; � ¼ 0:5), and (d) Meixner (� ¼ 150; � ¼ 0:5).

44 H. Zhu et al.

The bivariate polynomials uðx; yÞ ¼ ca1;a2n;m ðx; yÞ satisfy the following di®erence

equation

x�1r1uðx; yÞ þ y�2r2uðx; yÞ þ ða1 � xÞ�1uðx; yÞþ ða2 � yÞ�2uðx; yÞ þ ðnþmÞuðx; yÞ ¼ 0 ð26Þ

Figure 2 shows the plots of bivariate polynomials along y direction, the para-

meters a1 and a2 vary from 20 to 50.

4.2. Bivariate Meixner polynomials

The multivariate Meixner polynomials were introduced by Tiatnik17 and Rodal

et al.14,15 For a bivariable, let n1;n2 2 N0, we have

$�;�1;�2n1;n2

ðx; yÞ ¼ ~$ ð�x�y;��1=�2Þn1

ðxÞ ~$ ð�þn1;�1þ�2Þn2

ðxþ y� n1Þ ð27Þ

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Bivariate Charlier Polynomials (a1= 20, a

2= 50, Order= 6 )

y

u(x,

y)

(a)

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08


2= 25, Order= 6 )

y

u(x,

y)(b)

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06


2= 20, Order= 6 )

y

u(x,

y)

(c)

Fig. 2. Plots of bivativate Charlier polynomials uðx; yÞ ¼ ca1 ;a2n;m ðx; yÞ along y direction, when N ¼ 60,

Order ¼ 4, (a) a1 ¼ 20, a2 ¼ 50, (b) a1 ¼ 25, a2 ¼ 25 and (c) a1 ¼ 50, a2 ¼ 20.


where �1 and �2 are nonzero real parameters satisfying

j�1j þ j�2j < 1 ð28Þand ~$

ð�;�Þn ðxÞ is the univariate Meixner polynomials de¯ned by Eq. (23), orthogonal

with respect to

wðxÞ ¼ �xð�Þx�ðxþ 1Þ ; � > 0 and 0 < � < 1 ð29Þ

Meixner orthogonal polynomials uðx; yÞ ¼ $�;�1;�2n1;n2

ðx; yÞ of two variables satisfy

the following orthogonal relation

X1x¼0

X1y¼0

$�;�1;�2n1;n2

ðx; yÞ$�;�1;�2m1;m2

ðx; yÞ �x1�

y2ð�Þxþy

x!y!ð1� �1 � �2Þ�

¼ ��;�1;u2n1;n2

�n1;m1�n2;m2

ð30Þ

where,

��;�1;�2n1;n2

¼ n1!n2!ð�Þn1þn2ð�1 þ �2Þn1�n2

�2

�1ð1� �1 � �2Þ� �

n1

ð31Þ

The bivariate Meixner polynomials satisfy the following di®erence equation

ð1� �2Þx�1r1uðx; yÞ þ ð1� �1Þy�2r2uðx; yÞþ �1y�1�2uðx; yÞ þ �2x�2r1uðx; yÞþ ðð�1 þ �2 � 1Þxþ ��1Þ�1uðx; yÞþ ðð�1 þ �2 � 1Þyþ ��2Þ�2uðx; yÞ� ð�1 þ �2 � 1Þðn1 þ n2Þuðx; yÞ ¼ 0 ð32Þ

Figure 3 shows the plots for the ¯rst few orders of the bivariate Meixner polynomials

with the parameters �, �1 and �2 varying from 0.15 to 0.35. The bivariate Charlier

and Meixner polynomials with di®erent parameters are listed in Table 4.

5. Two-Dimensional Discrete Orthogonal Moments

Given a digital image fðx; yÞ with size N �N , i.e. x 2 ½0;N � 1� and y 2 ½0;N � 1�,the (mþ n)th order moments with bivariate orthogonal polynomials as basis func-

tion of an image is de¯ned as follows

Mmn ¼XN�1

x¼0

XN�1

y¼0

fðx; yÞuðx; yÞ ð33Þ

Here, bivariate polynomials uðx; yÞ ¼ $�;�1;�2n1;n2

ðx; yÞ and uðx; yÞ ¼ ca1;a2n;m ðx; yÞ are

used to construct the nonseparable Charlier and nonseparable Meixner moments.

46 H. Zhu et al.

Using the orthogonal property of normalized orthogonal polynomials, Eq. (33)

also leads to the following inverse moment transform.

fðx; yÞ ¼Xsm¼0

Xs

n¼0

Mmnuðx; yÞ ð34Þ

where s > 0 for bivariate Charlier and Meixner polynomials. It indicates that an

image can be completely reconstructed by calculating its discrete orthogonal

moments. If moments are limited to an order K, we can approximate f by f̂ .

f̂ ðx; yÞ ¼XKm¼0

XK�m

n¼0

Mmnuðx; yÞ ð35Þ

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08Bivariate Meixner Polynomials (β = 60, µ

1= 0.15, µ

2= 0.35, Order:6 )

y

u(x,

y)

(a)

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06


1= 0.25, µ

2= 0.25, Order:6 )

y

u(x,

y)(b)

0 10 20 30 40 50 60−0.06

−0.04

−0.02

0

0.02

0.04

0.06


1= 0.35, µ

2= 0.15, Order:6 )

y

u(x,

y)

(c)

Fig. 3. Plots of bivariate Meixner polynomials uðx; yÞ ¼ $�;�1 ;�2n1 ;n2

ðx; yÞ along y direction, when N ¼ 60,

Order ¼ 6, � ¼ 60, (a) �1 ¼ 0:15, �2 ¼ 0:35, (b) �1 ¼ 0:25, �2 ¼ 0:25, and (c) �1 ¼ 0:35, �2 ¼ 0:15.


6. Experimental Results

In this section, three experiments have been carried out to verify the image rep-

resentation capability of the proposed moments when they are used for gray-level

image or binary image. The test gray-level images of the size 128� 128 pixels used

for this study are shown in Fig. 4. The di®erence between an image and a version

reconstructed from a ¯nite set of its moments is a good measure of the image rep-

resentation capability of the set of moments considered. Hence, the ¯rst experiment

reconstructed the images by the moments. The di®erence between the original image

and the reconstructed image is measured using the mean squared error (mse). The

Table 4. Bivariate orthogonal polynomials with di®erent parameters.

Bivariate Charlier polynomials

a1= 15, a

2= 15 a

1= 15, a

2= 30 a

1= 15, a

2= 60

a1= 30, a

2= 15 a

1= 30, a

2= 30 a

1= 30, a

2= 60

a1= 60, a

2= 15 a

1= 60, a

2= 30 a

1= 60, a

2= 60

µ1= 0.15, µ

2= 0.15 µ

1= 0.15, µ

2= 0.25 µ

1= 0.15, µ

2= 0.35

µ1= 0.25, µ

2= 0.15 µ

1= 0.25, µ

2= 0.25 µ

1= 0.25, µ

2= 0.35

µ1= 0.35, µ

2= 0.15 µ

1= 0.35, µ

2= 0.25 µ

1= 0.35, µ

2= 0.35

Bivariate Meixner polynomials ( =50) β

(a) (b) (c)

Fig. 4. Test images (128� 128): (a) Lena, (b) Flower, (c) Girl.

48 H. Zhu et al.

mean squared error is de¯ned as follows:

mse ¼ 1

N 2

XN�1

x¼0

XN�1

y¼0

½fðx; yÞ � f̂ ðx; yÞ�2 ð36Þ

where fðx; yÞ and f̂ ðx; yÞ denote the original image and the reconstructed image,

respectively.

The reconstructed images by the nonseparable Charlier and Meixner moments are

shown in Fig. 5. In order to set a global feature extraction model and obtain better

reconstruction results, we set � ¼ 70, �1 ¼ �2 ¼ 0:25 for nonseparable Meixner

moments, and a1 ¼ a2 ¼ 60 for nonseparable Charlier moments. In this case, non-

separable Charlier and Meixner polynomials shall emphasize the central region of the

image. Due to the in°uence of parameters on orthogonal polynomials, the global and

local information of an image can be emphasized by choosing the right values of the

parameters. In our previous study, we had discussed how to obtain a global feature

extraction model by choosing appropriate parameters.26,27 Owing to the limit of the

space, this study omits the discussion of choosing proper parameters for the proposed

nonseparable moments. From Fig. 5, it can be seen that the images reconstructed by

Order=30 Order=60 Order=90 Order=128

Fig. 5. Reconstruction of the gray-level image Flower and Girl. The order's numbers from left to right are

30, 60, 90, and 128, respectively. The ¯rst two rows are reconstructed images using nonseparable Meixner

moments (� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25). The last two rows are reconstructed images using nonseparableCharlier moments (a1 ¼ a2 ¼ 60).


the nonseparable Charlier are clearly of better quality than those by nonseparable

Meixner. Figure 6 shows the plotting of mse values for nonseparable Charlier and

Meixner moments in a maximum order up to 128. The reconstruction errors decrease

monotonically with the increase of the order's values as predicted. It also demon-

strates that the images reconstructed by the nonseparable Charlier moments are

much more like the original images with the increasing of moment's order. From

these results, one can conclude that moments constructed by bivariate Charlier

polynomials have better global extraction capability than the moments constructed

by bivariate Meixner polynomials.

The aims of the second experiment are to compare the performances of the pro-

posed nonseparable Charlier and Meixner moments with that of the Legendre

moments16 and Tchebichef moments.10 Image reconstruction processes are repeated

for the gray-level image Lena shown in Fig. 7. The images' reconstruction errors mse

by using di®erent methods are shown in Fig. 8. Both Figs. 7 and 8 show that

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Moment Order

Rec

onst

ruct

ion

Err

or

110 115 120 125 130

1.5

2

2.5

3

3.5

4

x 103

Bivariate Meixner MomentsBivariate Charlier Moments

(a)

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

Moment Order

Rec

onst

ruct

ion

Err

or

110 115 120 125 130

2

3

4

5

6

7x 10

3

Bivariate Meixner MomentsBivariate Charlier Moments

(b)

Fig. 6. Comparative study of reconstruction errors by using nonseparable Charlier moments

(a1 ¼ a2 ¼ 60), and nonseparable Meixner moments (� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25). (a) Image Flower,

(b) image Girl.

(a) (b) (c) (d)

Fig. 7. Reconstruction of the gray-level image Lena, order is 128, (a) Legendre moments, (b) Tchebichefmoments, (c) nonseparable Charlier moments (a1 ¼ a2 ¼ 60), (d) nonseparable Meixner moments

(� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25).

50 H. Zhu et al.

Legendre moments give the worst results comparing to both Tchebichef moments

and the proposed moments. One also observes that the reconstruction results based

on nonseparable Charlier and Meixner moments are not as good as Tchebichef

moments.

Sensitivity to noise is a critical issue for image moments. Only when the values of

the moments are not sensitive to noise can they be ideal features of an image. In the

¯nal experiment, we have investigated and compared the behaviors of the proposed

moments in noisy images. The discussion will concentrate on binary images Hen and

Snow°ower which are used in the MPEG-7 Core Experiment CE-Shape-1 part B.6

(This database consists of 70 shape categories with 20 objects per category.) How-

ever, the extension to gray-level images is straightforward. Table 5 presents the

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

Moment Order

Rec

onst

ruct

ion

Err

or

110 115 120 125 130

4

6

8

10

12

14

16x 103

LegendreTchebichefCharlierMeixner

Fig. 8. MSE comparison of images reconstructed using Legendre, Tchebichef, nonseparable Charlier

(a1 ¼ a2 ¼ 60), and nonseparable Meixner moments (� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25), respectively.

Table 5. Reconstructed images of noisy images with Legendre, Tchebichef, Charlier (a1 ¼ a2 ¼ 60)

and Meixner moments (� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25). 5% salt-and-pepper noises for image Hen, �2 ¼0:1 Gaussian white noises for image Snow°ower.

Original Images Noisy Images Legendre Tchebichef Charlier Meixner


reconstructed results for noisy images, Hen and Snow°ower, sized 128� 128. The

¯rst column represents the original binary images. The second column shows two

noisy images. Here, binary image Hen is corrupted by 5% salt-and-pepper noises, and

�2 ¼ 0:1 Gaussian white noises are added in image Snow°ower. Figure 9 shows the

mse as a function of the total number of moments used in the reconstruction. From

Fig. 9, one can see that the reconstruction error decreases monotonically with the

increase of the moments' orders. The images reconstructed from Tchebichef and

nonseparable Charlier moments have much better quality than those from the

nonseparable Meixner and Legendre moments. The nonseparable Meixner moments

also yield lower reconstruction error than Legendre moments. These experimental

results are approximately consistent with those obtained by the second experiment.

7. Conclusions

Although moment functions had been extensively utilized for a number of image

processing tasks, no bivariate discrete orthogonal polynomials had been used as basis

function sets to directly de¯ne the two-dimensional moments. In this paper, we

introduced two nonseparate discrete orthogonal moments: Charlier and Meixner

moments. The image representation capability of the proposed moments had been

veri¯ed. We did the comparison of two most recent orthogonal moments: continuous

orthogonal Legendre moments and discrete orthogonal Tchebichef moments. Ex-

perimental results showed that the proposed moments were comparable to Legendre

moments and moderately poorer than Tchebichef moments.

The main contribution of this study is as follows: (1) This work has presented two

formulae to obtain x and n based recurrence relations calculating the univariate

Charlier and Meixner polynomials. (2) We de¯ned two-dimensional nonseparable

discrete orthogonal moments based on bivariate discrete orthogonal polynomials in

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Moment Order

Rec

onst

ruct

ion

Err

or

110 115 120 125 130

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07LegendreTchebichefCharlierMeixner

(a)

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Moment Order

Rec

onst

ruct

ion

Err

or

110 115 120 125 130

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1LegendreTchebichefCharlierMeixner

(b)

Fig. 9. Reconstruction errors for noisy images using Legendre, Tchebichef, Charlier (a1 ¼ a2 ¼ 60), and

Meixner moments (� ¼ 70; �1 ¼ 0:25; �2 ¼ 0:25), respectively. (a) Image Hen with 5% salt-and-pepper

noises; (b) image Snow°ower with �2 ¼ 0:1 Gaussian white noises.

52 H. Zhu et al.

the image processing and pattern recognition ¯eld. (3) The proposed method here

could be easily extended to obtain other bivariate orthogonal polynomials and their

corresponding moments.

Due to the nonsepariability of bivariate orthogonal polynomials, Charlier and

Meixner moments' values could not be computed in two steps by successive one-

dimensional operations on rows and columns of an image. Thus, pixel by pixel

calculation was needed to obtain the moments' values, and this process was very

expensive. Further on, our work will focus on the fast algorithm to calculate

nonseparable Charlier or Meixner moments.

Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments

and suggestions. This work has been supported by National Natural Science Foun-

dation of China under Grant No. 60975004.

References

1. X. B. Dai, H. Z. Shu, L. M. Luo, G. N. Han and J. L. Coatrieux, Reconstruction oftomographic images from limited range projections using discrete Radon transform andTchebichef moments, Patt. Recogn. 43 (2010) 1152�1164.

2. C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia ofMathematics and Its Applications (Cambridge University Press, Cambridge, 2001).

3. T. Koornwinder, Two-variable Analogues of the Classical Orthogonal Polynomials, inTheory and Application of Special Functions (Academic Press, 1975).

4. M. A. Kowalski, The recursion formulas for orthogonal polynomials in n variables, SIAMJ. Math. Anal. 13 (1982) 309�315.

5. M. A. Kowalski, Orthogonality and recursion formulas for polynomials in n variables,SIAM J. Math. Anal. 13 (1982) 316�376.

6. MPEG7 CE Shape-1 Part B: http://www.imageprocessingplace.com/root ¯les V3/image databases.htm

7. R. Mukundan, Some computational aspects of discrete orthogonal moments, IEEETrans. Image Process. 13 (2004) 1055�1059.

8. R. Mukundan, Fast computation of geometric moments and invariants using Schlick'sapproximation, Int. J. Patt. Recogn. Artif. Intell. 22 (2008) 1363�1377.

9. R. Mukundan, S. H. Ong and P. A. Lee, Discrete vs. continuous orthogonal moments forimage analysis, Int. Conf. Imaging Science, Systems and Technology-CISST'01 (2001),pp. 23�29.

10. R. Mukundan, S. H. Ong and P. A. Lee, Image analysis by Tchebichef moments, IEEETrans. Imag. Process. 10 (2001) 1357�1364.

11. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics (Boston,Bessel, 1988).

12. A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical Orthogonal Polynomials of aDiscrete Variable (Springer, New York, 1991).

13. G. A. Papakostas, E. G. Karakasis and D. E. Koulouriotis, Novel moment invariants forimproved classi¯cation performance in computer vision applications, Patt. Recogn. 43(2010) 58�68.


14. J. Rodal, I. Area and E. Gogoy, Orthogonal polynomials of two discrete variables on thesimplex, Integral Transf. and Spec. Func. 16 (2005) 263�280.

15. J. Rodal, I. Area and E. Gogoy, Linear partial di®erence equations of hypergeometrictype: Orthogonal polynomials solutions in two discrete variables, J. Comput. Appl. Math.200 (2007) 722�748.

16. M. R. Teague, Image analysis via the general theory of moments, J. Opt. Soc. Amer. 70(1980) 920�930.

17. M. V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discretefamilies, J. Math. Phys. 32 (1991) 2337�2342.

18. A. Vercin, Ordered products, W1-algebra, and two-variable, de¯nite-parity, orthogonalpolynomials, J. Math. Phys. 39 (1998) 2418�2427.

19. B. Wang and Y. Q. Chen, An invariant shape representation: Interior angle chain, Int. J.Patt. Recogn. Artif. Intell. 21 (2007) 543�559.

20. Y. Xu, On discrete orthogonal polynomials of several variables, Adv. Appl. Math. 33(2004) 615�632.

21. Y. Xu, Second order di®erence equations and discrete orthogonal polynomials of twovariables, Int. Math. Res. Notices. 8 (2005) 449�475.

22. P.-T. Yap, R. Paramesran and S.-H. Ong, Image analysis by Krawtchouk moments,IEEE Trans. Image Process. 12 (2003) 1367�1377.

23. P.-T. Yap, R. Paramesran and S.-H. Ong, Image analysis using Hahn moments, IEEETrans. Patt. Anal. Mach. Intell. 29 (2007) 2057�2062.

24. P.-T. Yap and P. Raveendran, An e±cient method for the computation of Legendremoments, IEEE Trans. Patt. Anal. Mach. Intell. 27 (2005) 1996�2002.

25. F. Zhang, S.-Q. Liu, D.-B. Wang and W. Guan, Aircraft recognition in infrared imageusing wavelet moment invariants, Imag. Vis. Comput. 27 (2009) 313�318.

26. H. Zhu, H. Shu, J. Zhou, L. Luo and J. L. Coatrieux, Image analysis by discreteorthogonal dual-Hahn moments, Patt. Recogn. Lett. 28 (2007) 1688�1704.

27. H. Zhu, H. Shu, J. Liang, L. Luo and J. L. Coatrieux, Image analysis by discreteorthogonal Racah moments, Sign. Process. 87 (2007) 687�708.

28. J. Žunić, K. Hirota and P. L. Rosin, A Hu moment invariant as a shape circularitymeasure, Patt. Recogn. 43 (2010) 47�57.

54 H. Zhu et al.

Hongqing Zhu obtainedher Ph.D. in 2000 fromShanghai Jiao Tong Uni-versity. From 2003 to2005, she was a post-doctoral fellow at theDepartment of Biologyand Medical Engineering,Southeast University.Currently she is anassociate professor at East

China University of Science & Technology.Her current research interests mainly focus

on image reconstruction, image segmentation,image compression, and pattern recognition.

Min Liu received theM.Sc. degree in data tele-communications and net-works from University ofSalford, U.K. in 2001.After carrying outresearch, in 2005 sheobtained the Ph.D. degreein wireless laser com-munications. Currentlyshe is a lecturer at the

East China University of Science and Technol-ogy (ECUST), China.

Her research interests include optical com-munications and pattern recognition.

Yu Li received his Ph.D.degree from ZhejiangUniversity in 2007. In2007, he joined EastChina University of Sci-ence and Technology.Now, he is a teacher in theDepartment of Electronicand CommunicationEngineering.

His research interestsinclude image signal processing, array signalprocessing, etc. He has published 11 papersincluding nine journal papers. He is a member ofIEICE.

Huazhong Shu (M'00-SM'06) received the B.S.degree in applied math-ematics from WuhanUniversity, China, in1987, and a Ph.D. innumerical analysis fromthe University of Rennes(France) in 1992. He isnow with the Departmentof Computer Science and

Engineering of Southeast University, China.His recent work has focused on image anal-

ysis, pattern recognition and fast algorithms ofdigital signal processing. Dr. Shu is a seniormember of IEEE.

Hui Zhang received theB.S. degree in radioengineering in 2003 andM.S. degree in biomedicalengineering in 2006 fromSoutheast University, re-spectively. He is currentlypursuing the Ph.D. at theDepartment of ComputerScience of Southeast Uni-versity.

His research interests are mainly focused onpattern recognition, image watermarking, imageregistration and image processing.


image description with charlier and meixner moments · image description with nonseparable...

Documents