Proceedings of 2010 IEEE 17th International Conference on Image Processing September 26-29, 2010, Hong Kong

SYMMETRIC IMAGE RECOGNITION BY TCHEBICHEF MOMENT INVARIANTS

1 1 1 2 1

Hui Zhang , Xiubing Dai , Pei Sun , Hongqing Zhu , Huazhong Shu

1

Laboratory of Image Science and Technology, School of Computer Science and Engineering, Southeast

University, 210096 Nanjing, China

2

Department of Electronics and Communications Engineering, East China University of Science and

Technology, 200237 Shanghai, China

ABSTRACT (3) Calculation of complex moment which requires a

mapping transformation leads to additional CPU time.

In this paper, we proposed a set of translation and Recently, Mukundan [7] introduced the discrete

rotation invariants extracted from Tchebichef moments. A orthogonal Tchebichef moments. Since the Tchebichef

set of Tchebichef moment invariants is derived from the polynomials satisfy perfectly the orthogonal property in the

relationship between Tchebichef moments of the original discrete domain of image coordinate space and the

image and those of the transformed image. These invariants computation of Tchebichef moments does not require any

are then used for symmetric image recognition. Contrarily discrete approximation, these properties make them superior

to the methods based on the complex moments in symmetric to the conventional continuous orthogonal moments in

image analysis, our method does not need the pre-selection terms of image representation capability. Moreover, due to

of moment values. Experimental results show that the the fact that every Tchebichef polynomial contains both

proposed method achieves better performance compared to even order and odd order terms, Tchebichef moments obtain

the existing methods. none zero values for symmetric images. Consequently, it

could be expected that the utilization of Tchebichef

moments provides more powerful invariants in recognition

Index Terms—Moment invariants, symmetric images,

Tchebichef moments. of symmetric objects.

1. INTRODUCTION 2. TCHEBICHEF MOMENTS

Moment invariants were firstly introduced by Hu [1] and The two-dimensional (2-D) Tchebichef moment of order

successfully used in a number of applications such as (p+q) of an image intensity function f(x, y) with size N N

pattern recognition, image normalization, pattern matching,

is defined as [7]

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image registration and contour shape estimation [2].

Tt ()xt (y)f(xy , ) , (1)

pq p q

Recently, many researchers are interested in using moment

xy00

invariants in symmetric image analysis. Especially, the

where t (x) is the pth order orthogonal Tchebichef

p

invariants based on the complex moments have been

polynomial defined by

extensively investigated [3-6]. However, their methods have

p

(1 N )

()px( )(1p)

p

following disadvantages. kk k

tx () ,

p

2

(1) Their methods need selecting nonzero complex moments (!kN ) (1 )

(, pN) k 0

k

for symmetric image. When low-order moments are not

p, x = 0, 1,…, N–1. (2)

satisfying, they continue searching for high-order moments.

Here (a) is the Pochhammer symbol

k

The pre-selecting moments depend on the types of

()aa(a 1)(a 2) (ak 1) , k 1 and (a) = 1, (3)

0

k

symmetric images (N-fold rotation symmetry, N-FRS). In

and the squared-norm (p, N) is given by

fact, complex moment c with non-integer (p-q)/N equals

pq

() Np !

zero for N-fold rotation symmetric images and it can not be

(, pN) . (4)

selected as the pre-selecting moment [5]. This pre-selecting

(2pN 1)( p 1)!

strategy limits the pattern recognition application.

Equation (2) can be rewritten as

(2) Compared to orthogonal moments, complex moments

p

Nk

contain redundant information and are sensitive to noise tx () c x , (5)

pp ,k

k 0

especially when high-order moments are considered. In [3],

where

the highest 30th order moment was used.

978-1-4244-7993-1/10/$26.00 ©2010 IEEE 2273 ICIP 2010

p pr

Equation (15) shows that the translation invariants of

(1) (pr)!(N r 1)!

N

cS (,rk) . (6)

pk,1

Tchebichef moments can be expressed as a linear

2

(, pN)(p r)!(r!)(N p 1)!

rk

combination of those of the original image. Compared to

It can be deduced from (5) that

Zhu’s method [10], the advantage of our method is that the

p

coefficients on the right hand side of (15) are explicitly

pN

x dt ()x , (7)

pk , k

given. These coefficients can be pre-computed and stored in

k 0

a look-up table to accelerate the computational speed.

N

where , Dd ( )0 k p N–1, is the inverse matrix of

Np,k

N N

the lower triangular matrix Cc . The elements d

3.2. Rotation invariants

Np,k p,k

of the inverse matrix D are given by [8]

N

We first establish a relationship between the Tchebichef

2

p

(, kN)(2k 1)(m!) (N k 1)!

N

moments of the transformed image and those of the original

dS (,km) . (8)

pk,2

(1 mk)!(m k)!(N m1)!

mk image. The transformation of an image f(x, y) into another

image g(X, Y) can be represented

Here S (i, j) and S (i, j) are respectively the first kind and

1 2

second kind of Stirling numbers [9]. X aa x

11 12

. (16)

Yy aa

21 22

3. TCHEBICHEF MOMENT INVARIANTS

The 2-D (p+q)th order Tchebichef moment of the

transformed image g(X, Y) is defined by

3.1. Translation invariants

NN11

() g

TA t() axayt(axay)f(x,) y . (17)

pq,1 p112 q2122

The translation invariance of 2-D Tchebichef moments xy00

can be obtained by evaluating their central moments T ,

with A aa a a .

pq

11 22 12 21

which are defined by

In the following, we discuss the way to express the

NN11

Tchebichef moments of the transformed image defined by

T t() xx t(y yf)(x,y) , (9)

pq p 00 q

(16) in terms of Tchebichef moments of the original image.

xy00

By replacing the variable x by a x+a y in (5), we have

11 12

where (x , y ) denotes the image centroid coordinates given

0 0

p m

m

by Ns m ss m s

ta() x a y c aa xy . (18)

pp 11 12 ,m 11 12

NN N N

s

cT c T cT c T

ms00

00 10 10 00 00 01 10 00

xy , . (10)

00 N N

Similarly,

cT cT

11 00 11 00

p n

n

By replacing the variable x by x x in (5), we obtain Nt n t t n t

0

ta() x a y c a a xy . (19)

qq 21 22 ,n 21 22

p

t

nt00

Nm

tx()x c() x x

pp 0, m 0

Substituting (18) and (19) into (17), we obtain

m 0

. (11)

pq

mn stmnst

p

m mn

m () g NN

Ns m s

TA cc

cx (x ) pq p,, m q n

pm,0

st

. (20)

s mn00s00t i0 j0

ms00

sms t nt st mnst

From (7), we have aa a a x y f (x,y)

11 12 21 22

s

sN Using (7), we have

x dt()x . (12)

si , i

st

i 0

st N

x dt()x , (21)

st ,i i

Substituting (12) into (11) yields

i 0

p ms

mn s t

m

NN m s

mns t N

tx()x c d( x) t(x) . (13)

yd ty () . (22)

pp 0,ms,i0 i

mns t,j j

s

ms00i0

j 0

Similarly,

Substitution of (21) and (22) into (20) yields

q nt

pq

mn stmnst

n

mn

NN n t

() g sm s

ty() y c d( y) t(y) . (14)

TA ()a (a)

qq 0,nt,j0 j

pq 11 12

t

st

nt00j0 . (23)

mn00s0t0i0 j0

tn tNNN N ()f

Substituting (13) and (14) into (9) leads to

(aa ) ( ) c cd d T

21 22 pm , qn , s t,i m n s t,j ij

pq mn s t

mn

NM N M

Equation (23) shows that the Tchebichef moments of the

T cc d d

pq pm,, qn s,i t,j

st transformed image can be expressed as a linear combination

mn00s00t i0j0

. (15)

of those of the original image. Let a = a = cos , a = sin ,

ms n t 11 22 12

(xy ) ( )T

00 ij

a = sin , we derive the following theorem:

21

2274

(a) (b) (c) (d) (e) (f) (g) (h)

Fig 1. Eight symmetrical images for pattern recognition

Theorem 1. Let number invariants with lower order which are more robust

pq mn stmnst

to noise than the higher order ones. For this reason, the

mn

f t

I (1)

pq same six invariants of CMIs and TMIs as in the previous

st

, (24)

mn000s t0i0 j0

experiment are used and the Euclidean distance is used here

ns t m t s N N N N f

(cos ) (sin ) cc d d T

f fp,,mqnst,imnst,j ij as the classification measure. Table 2 shows the

classification results using the different methods. One can

where

ff NN N N 2

observe from this table that both CMIs and TMIs achieve

1 uT vT 22 cc c(c )

11 00 22 00 22 10

arctan( ),uv , .

high recognition rates in the noise-free case. It is noted that

f ff NN 2 N 2

2 TT ()cc ()c

20 02 11 00 11

the CMIs recognition accuracy decreases with increasing

f

then I is invariant to image rotation.

noise level. However, TMIs perform perfectly in terms of

pq

recognition for noisy images, whatever the noise type.

The proof of Theorem 1 is given in Appendix A.

Table 1. The TMIs values for rhombus image

4. APPLICATION TO SYMMETRIC IMAGES

The first experiment was carried out to test the

performance of the Tchebichef moment invariants (TMIs).

I(2,0) 3.0951 3.0953 3.0952 3.0953 3.0953

Circular image shown in Fig. 1(a) was selected and then

I(0,2) 3.0557 3.0558 3.0558 3.0557 3.0558

translated and rotated as the test images. The reason to

I(3,0) -7.8727 -7.8735 -7.8733 -7.8740 -7.8734

choose this image is that it has -FRS which represents the

I(2,1) -5.3194 -5.3197 -5.3196 -5.3198 -5.3197

most special symmetry. Six TMIs I , I , I , I , I , I

20 02 30 21 12 03

I(1,2) -7.6411 -7.6419 -7.6421 -7.6417 -7.6419

were computed for the original image and the translation

I(0,3) -5.2512 -5.2514 -5.2514 -5.2513 -5.2514

and rotation images. TMIs were calculated by (15) and (24)

where the Tchebichef moments on the right-hand side of (24)

Table 2. The recognition rates of CMI and TMI for

are replaced by T computed in (15). TMIs values are

pq

symmetrical images

summarized in Table 1. These results validate the invariant

CMIs TMIs

and discriminative capabilities of the TMIs. Different from

Noise-free 100% 100%

the complex moment invariants (CMIs) as reported in [3-6],

S&P with noise density = 0.02 93.06% 100%

the TMIs show none zero values. We use the values of /

S&P with noise density = 0.03 87.96% 100%

to evaluate the performance of the invariants, where

S&P with noise density = 0.04 80.09% 100%

denotes the mean of the invariants and the the standard

S&P with noise density = 0.08 70.37% 88.89%

deviation. CMIs [5] were also computed for this image, note

WG with STD = 5 94.91% 100%

that six complex moment invariants C , C , C , C , C ,

11 22 33 44 55

WG with STD = 10 86.11% 100%

C were used for circular image. The average value of /

66

WG with STD = 15 79.17% 100%

is 7.4E-3 for CMIs and 3.3E-5 for TMIs. It is noted that

WG with STD = 20 68.98% 87.5%

TMIs of the other symmetric images have the similar

Computation time 573.9s 106.1s

performance and the values are omitted here.

The second experiment was carried out to illustrate the

We also compare the computational speed of the CMIs

discrimination power of the CMIs [5] and TMIs. Eight

and the TMIs in the above experiment. The computation

symmetric images whose size is 128 128 pixels (Fig. 1) are

time required for all recognition tasks to CMIs and TMIs

used for the recognition task. The testing set is generated by

are respectively 573.9 seconds and 106.1 seconds as shown

rotating the original images from 0° to 360° with interval 5°

in Table 2. Note that the program was implemented in

forming a set of 576 images. This is followed by adding

MATLAB 7 on a PC Core2 3.0 GHZ, 2G RAM. It can be

salt-and-pepper noise (S&P) with different noise densities

seen that the TMIs computation is much faster than the

and white Gaussian noise (WG) with different standard

CMIs. This is due to the fact that the computation of the

deviations (STD). The feature selection prefers the same

2275

[10] H.Q. Zhu, H.Z. Shu, T. Xia, L.M. Luo, and J.L. Coatrieux,

complex moments requires a mapping transformation which

“Translation and scale invariants of Tchebichef moments,” Pattern

is time consuming.

Recognit., vol. 40, no. 9, pp. 2530-2542, 2007.

5. CONCLUSION

7. APPENDIX A

The main contribution of this paper is to present a set of

Let f and g be two images displaying the same pattern but

translation and rotation invariants based on orthogonal

with distinct orientation , i.e., g(X, Y) = f(xcos +ysin ,

Tchebichef moments. These invariants are useful for

xsin +ycos ). Let us first prove = :

f g

recognition tasks, especially for recognizing the symmetric

With the help of (23), we have

objects. Thanks to the property of Tchebichef moments, we

22

g ff22f f

have successfully applied the TMIs to symmetric image

TT sin cos sinT cosT sin cos T

11 02 11 11 20

uu

recognition. The advantage of our method compared to the

vv v

methods using complex moments is that TMIs obtain more ff22f

TT sin cosT (A1)

00 00 00

features than CMIs for a fixed order. Moreover, the

uu u

gg 22 f f

proposed invariants are directly calculated from the

TT (cos sin )T 2u cos sinT

20 02 20 11

Tchebichef moments, which do not require the pre-selecting (A2)

22ff

(cos sin )Tv 2 cos sin T

02 00

process. Thus, it is more suitable for pattern recognition

applications. With the help of (A1) and (A2), we have

gg

uT vT

11 00

tan 2

g

Acknowledgements gg

TT

20 02

ff f f

sin2Tucos2T sin2T vcos2 T

We would like to thank Professors Flusser and Suk and 02 11 20 00

ff f f

Dr. Hosny for offering the standard symmetrical images for cos 2Tu sin 2T cos 2T vsin 2 T

20 11 02 00

pattern recognition experiments. This work was supported

ff f f

tan 2TuT tan 2T vT

02 11 20 00

by the National Natural Science Foundation of China under

fff f

Tu tan 2T T v tan 2T

20 11 02 00

Grant 60873048 and 60975004, by National Basic Research

ff

Program of China under grant No. 2010CB732503.

uT vT

11 00

tan 2

ff

TT

20 02

tan(2 2 )

6. REFERENCES

f

ff

uT vT

11 00

1t an2

ff

TT

[1] M.K. Hu, “Visual pattern recognition by moment invariants,”

20 02

IRE Trans. Inf. Theory, vol. 8, no. 2, pp. 179-187, 1962.

Thus, we have

[2] P.T. Yap, X.D. Jiang, A.C. Kot, “Two-Dimensional Polar

=

f g

Harmonic Transforms for Invariant Image Representation,” IEEE

Then we give the proof of the Theorem 1. The rotation

Trans. Pattern Anal. Mach. Intel., vol. 32, no. 7, pp. 1259-1270,

Tchebichef moment invariants of the image intensity

2010.

function g(X, Y) is defined as

[3] D. Shen, H.S. Ip, K.T. Cheung, E.K. Teoh, “Symmetry

pq

mn stmnst

Detection by Generalized Complex (GC) Moments: A Close-Form

mn

gt

I (1)

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st

mn00s0t0i0 j0

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ns t m t s N N N N g

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g gp,,mqnst,imnst,j ij

(A3)

objects,” Advanced Concepts for Intelligent Vision Systems, vol.

NN11

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tX (cos sinY )

pg g

[5] J. Flusser, and T. Suk, “Rotation Moment Invariants for

xy00

Recognition of Symmetric Objects,” IEEE Trans. Image Process.,

tX ( sin cosY )g(X ,Y )

qg g

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With the help of triangle formula and = , we have

[6] S. Pei and C. Lin, “Normalization of rotationally symmetric f g

shapes for pattern recognition,” Pattern Recognit., vol. 25, no. 9,

tX (cos sinY )

pg g

pp. 913-920, 1992.

tx cos ( cosy sin ) sin (xsiny cos )

[7] R. Mukundan, S.H. Ong, and P.A. Lee, “Image analysis by

pg g

Tchebichef moments,” IEEE Trans. Image Process., vol. 10, no. 9,

tx (cos siny) (A4)

pf f

pp. 1357-1364, 2001.

[8] X.B. Dai, H.Z. Shu, L.M. Luo, G.N. Han, and J.L. Coatrieux, Similarly, we have

“Reconstruction of tomographic images from limited range

tX (sin cosY)t(sinx cosy) (A5)

qg g q f f

projections using discrete Radon transform and Tchebichef

Substituting (A4) and (A5) into (A3) yields

moments,” Pattern Recognit., vol. 43, no. 3, pp. 1152-1146, 2010.

g f

I I

[9] V. Adamchik, “On Stirling number and Euler sums,” J. Comput.

pqpq

Appl. Math., vol. 79, no. 1, pp. 119-130, 1997.

The proof of Theorem 1 is now completed.

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