Proceedings of 2010 IEEE 17th International Conference on Image Processing September 2629, 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 preselection 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 twodimensional (2D) 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 [36]. 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 loworder moments are not
p, x = 0, 1,…, N–1. (2)
satisfying, they continue searching for highorder moments.
Here (a) is the Pochhammer symbol
k
The preselecting moments depend on the types of
()aa(a 1)(a 2) (ak 1) , k 1 and (a) = 1, (3)
0
k
symmetric images (Nfold rotation symmetry, NFRS). In
and the squarednorm (p, N) is given by
fact, complex moment c with noninteger (pq)/N equals
pq
() Np !
zero for Nfold rotation symmetric images and it can not be
(, pN) . (4)
selected as the preselecting moment [5]. This preselecting
(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 highorder moments are considered. In [3],
where
the highest 30th order moment was used.
9781424479931/10/$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 precomputed and stored in
k 0
a lookup 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 2D (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 2D 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 noisefree 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 righthand 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
Noisefree 100% 100%
the complex moment invariants (CMIs) as reported in [36],
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 7.4E3 for CMIs and 3.3E5 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
saltandpepper 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. 25302542, 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 preselecting (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. 179187, 1962.
Thus, we have
[2] P.T. Yap, X.D. Jiang, A.C. Kot, “TwoDimensional 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. 12591270,
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 CloseForm
mn
gt
I (1)
Solution,” IEEE Trans. Pattern Anal. Mach. Intel., vol. 21, no. 5, pq
st
mn00s0t0i0 j0
pp. 466476, 1999.
ns t m t s N N N N g
[4] T. Suk and J. Flusser, “Affine normalization of symmetric (cos ) (sin ) cc d d T
g gp,,mqnst,imnst,j ij
(A3)
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NN11
3708, pp. 100107, 2005.
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
vol. 15, no. 12, pp. 37843790, 2006.
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. 913920, 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. 13571364, 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. 11521146, 2010.
g f
I I
[9] V. Adamchik, “On Stirling number and Euler sums,” J. Comput.
pqpq
Appl. Math., vol. 79, no. 1, pp. 119130, 1997.
The proof of Theorem 1 is now completed.
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