Abstract: Zernike moments are complex moments with the orthogonal Zernike polynomials as kernel function, compared with other moments; Zernike moments have greater advantages in image rotation and low noise sensitivity. Because of the Zernike moments have image rotation invariance, and can construct arbitrary high order moments, it can be used for target recognition. In this paper, the Zernike moment algorithm is improved, which makes it having scale invariance in the processing of digital image. At last, an application of the improved Zernike moments in image recognition is given. Copyright © 2014 IFSA Publishing, S. L

Min Huang

Qiuping Gong

Yaqiong Ma

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Sensors & Transducers, Vol. 166, Issue 3, March 2014, pp. 219-223 
  219
     
S S Se e en n ns s so o or r rs s s & & & T T Tr r ra a an n ns s sd d du u uc c ce e er r rs s s
© 2014 by IFSA Publishing, S. L. 
http://www.sensorsportal.com 
 
 
 
 
 
Image Recognition Using Modified Zernike Moments 
 
Min HUANG, Yaqiong MA and Qiuping GONG 
School of Computer and Communication Engineering, Zhengzhou University of Light Industry, 
Zhengzhou 450001, China 
Tel.: 13783493389 
E-mail: myq1988726@163.com 
 
 
Received: 20 December 2013   /Accepted: 28 February 2014   /Published: 31 March 2014 
 
 
Abstract: Zernike moments are complex moments with the orthogonal Zernike polynomials as kernel function, 
compared with other moments; Zernike moments have greater advantages in image rotation and low noise 
sensitivity. Because of the Zernike moments have image rotation invariance, and can construct arbitrary high 
order moments, it can be used for target recognition. In this paper, the Zernike moment algorithm is improved, 
which makes it having scale invariance in the processing of digital image. At last, an application of the 
improved Zernike moments in image recognition is given. Copyright © 2014 IFSA Publishing, S. L. 
 
Keywords: Zernike moments, Rotation invariance, Image recognition, Zernike polynomial, Normalization. 
 
 
 
1. Introduction 
 
Image recognition is a main branch in the field of 
pattern recognition; it is according to the image 
feature extracted from the image to classify the 
target  [1]; the image feature generally including 
color, texture, shape, etc, at present the recognition 
technology based on the shape feature is the most 
widely used. In the two-dimensional image space, 
shape is generally considered to be the region 
surrounded by a closed contour curve, so the 
description of the shape involves contour boundary 
and the region surrounded by the boundary. The 
recognition methods based on shape mostly revolve 
round contour feature and regional feature of the 
shape. Descriptions of the shape contour feature 
mainly include: spline fitting curve, straight line 
segment description, Gaussian parameter curves and 
Fourier descriptor, etc. But these methods have 
limitations when being used, the descriptions of some 
shape features are not independent of the position, 
size and orientation of the shape, and they also lose a 
large amount of information, so the identification 
effects are not ideal. Descriptions of the shape 
regional feature mainly include the area of the region, 
shape aspect ratio and shape invariance, etc. In 1961, 
M. K. Hu [2] proposed seven geometric invariant 
moments to measure the shape feature; it is a highly 
concentrated image feature, and not sensitive to the 
translation, rotation and scaling of the object, which 
plays a significant role in promoting the image 
recognition, after that researches on the moments had 
a rapid development, and successively appeared 
complex moments, rotation moments as well as other 
invariant moment analysis methods, the discovery of 
orthogonal moments is a big step of the moment 
technology, the orthogonal moments have proved to 
be better in terms of their strong ability to resist 
noise, good sampling performance, absolute 
independence, and no information redundancy; they 
are suitable for description and identification of the 
image. In 1979, M. R. Teague [3] proposed the 
Zernike moments based on the theory of orthogonal 
polynomials, they can be constructed easily into an 
arbitrary high order, and have good rotation 
invariance as well as simple calculation. C. H. Teh, 
Article number P_1930Sensors & Transducers, Vol. 166, Issue 3, March 2014, pp. 219-223 
  220
R.  T.  Chin [4] and A. Khotanzad [5] compared 
Zernike moments with many other moments, the 
results show that Zernike moments have the best 
performance in noise sensitivity, information 
redundancy, image description ability, etc. 
 
 
2. Zernike Moments 
 
2.1. Zernike Polynomials 
 
In 1934, Zernike [6] proposed a set of orthogonal 
polynomials defined on the unit circle, namely 
orthogonal Zernike polynomials, their definition form 
is as follows: 
 
  (,) (,) ()
jm
nm nm nm Vx yV R e
θ ρθ ρ == ,  (1) 
 
  2 (| | ) / 2
0
(1 ) [ ( ) ! ]
()
|| ||
!!!
22
sn s nm
nm
s
ns
R
nm nm
s ss
ρ
ρ
− −
=
−−
=
+−   −−  
 

, 
(2) 
 
where  () nm R ρ  is the orthogonal radial polynomial, 
(,) nm Vx y  is the orthogonal Zernike polynomial, it 
is a set of complex-valued orthogonal functions with 
completeness defined over the unit disk 
22 1 xy +≤ ,n  and m  are the orders of the 
orthogonal Zernike polynomials, where n  is a 
positive integer or zero, m  is a positive or negative 
integer, they are subject to the conditions 
|| n m even −=  and  || nm ≥  [7]. 
 
 
2.2. Definition of Zernike Moments 
 
In 1980, based on the orthogonal Zernike 
polynomials, Teague proposed the definition of 
Zernike moments of an image function  (,) f xy in 
two dimensions for the first time [8]. 
 
 
22
*
1
1
(,) (,) nm nm xy
n
Z fxy V xy d x d y
π
+≤
+
=  ,  (3) 
 
Zernike moments in polar coordinates can be 
defined as: 
 
21
*
00
21
00
1
(,) (,)
1
(,) () e x p ( )
nm nm
nm
n
Zf V d d
n
f Rj m d d
π
π
ρθ ρθρ ρθ
π
ρ θ ρ θ ρρθ
π
+
=
+
=−


,  (4) 
 
where 
* (,) nm V ρ θ  is the complex conjugate of the 
Zernike polynomials  (,) nm V ρ θ . Making the real 
and imaginary parts of  nm Z  respectively denoted as 
nm C and nm S , then: 
 
12
00
22
() c o s ( )(,) nm nm
n
CR m f d d
π
ρ θ ρ θ ρρθ
π
+
= 
,  (5) 
 
 
12
00
22
() s i n ( )(,) nm nm
n
SR m f d d
π
ρ θ ρ θ ρρθ
π
+
= 
,  (6) 
 
For digital images, the integrals are replaced by 
summations, and then the Zernike moments are 
rewritten as: 
 
  *2 2 1
(,) (,) , 1 nm nm
xy
n
Zf x y V x y x y
π
+
=+ ≤  ,  (7) 
 
Because the images processed in practical 
application are usually digital images, the real and 
imaginary parts of their Zernike moments need to be 
discretized. Mukundan and Ramakrishnan proposed 
the following transformation [9]: for any image 
(,) f xy  of size NN × , making its coordinate 
origin located at the center of the image, then 
2, 2 Nx y N −≤≤ . For pixel(,) x y , introducing 
two parameters η  and σ  correspond to the pixel 
uniquely, it is defined as: 
 
  max(| |,| |) x y η = ,  (8) 
 
If  || x η = , then  2( ) | | x yy x y ση η =− + . 
If  || y η = , then  2yx y ση =− . 
From the formula we can see, the value of η  is 
from 1 to 2 N , the value of σ  is from 1 to 8η , by 
the parameters η  and σ we can define the 
corresponding polar coordinates as follows: 
 
  2 N ρ η = ,   4 θπ ση = ,  (9) 
 
Through the above transformation, the real and 
imaginary parts of the Zernike moments  nm Z  can be 
written in the discrete form as follows: 
 
2 8
2
11
22
(2 / ) cos ( , )
4
N
nm nm
Nm
CR N f
N
η
ησ
πσ
η ησ
η ==
+
=  ,  (10) 
 
2 8
2
11
22
(2 / ) sin ( , )
4
N
nm nm
Nm
SR N f
N
η
ησ
πσ
η ησ
η ==
+
=  ,  (11) 
 
 
2.3. Invariance of Zernike Moments 
 
Zernike moments are region-based shape 
descriptors, and their bases are the orthogonal radial 
polynomials, compared with other shape descriptors, 
Zernike moments have better rotation 
invariance [10]. 
If an image is rotated through angle α , 
according to the definition of Zernike moments, the Sensors & Transducers, Vol. 166, Issue 3, March 2014, pp. 219-223 
  221
relationship between the Zernike moments of the 
rotated image 
*
nm Z and the original one  nm Z  is: 
 
  * () nm nm Z Ze j m α =−, (12) 
 
From the formula we can see, the Zernike 
moments before and after the image rotation only 
have phase changes, their magnitudes remain 
unchanged, so the magnitude || nm Z  of the Zernike 
moments can be taken as a rotation invariant feature 
of the target [11]. 
 
 
3. Improved Zernike Moments 
 
In the discrete case, because the rotation and scale 
transformation will cause the resampling and 
requantization of digital image, thereby causing 
certain error, which will makes its invariance not 
keep strictly invariant. Therefore considering first to 
carry on shape normalization to the target area of the 
image, and then normalize the Zernike moments,   
in order to get a better rotation and scale 
invariance [12]. 
The steps are as follows: 
1) Preprocess the image: using median filter to 
remove the noise and retain the image’s edge detail, 
turning the true color image into grayscale image of 
256 levels, selecting the appropriate threshold for 
binarization processing and then doing negated 
operation, so that the target image and the 
background are separated; thereafter the target image 
is converted to the unit circle in polar coordinates, 
that is taking the barycenter of the image as the 
center of the polar coordinates, and then taking the 
distance between the center and the outermost pixel 
in the target area as the radius, thus the pixels in the 
target area are resampled to the unit circle. 
2) Calculate the zeroth order geometric moment 
of the target, namely: 
 
00 (,) mf x y = ,  (13) 
 
3) Calculate each order of the Zernike moments in 
the unit circle: 
 
 
* 1
(,) (,) nm nm
xy
n
Z fxy V xy
π
+
=  ,  (14) 
 
4) Use  00 m  to normalize the Zernike moments: 
 
 
'
00
nm
nm
Z
Z
m
= ,  (15) 
 
5) Evaluate the magnitudes 
' || nm Z  of 
'
nm Z  as the 
invariant feature of the target’s improved moments.  
Next testing the improved Zernike moments, 
taking the following Fig. 1 for example. 
 
 
 
 
Fig. 1. The example image. 
 
 
Preprocessing the image first; then rotating the 
image by 30
°, 60
° , 90
° and 135
°; at last calculating 
the improved Zernike moments of the rotated images, 
Table 1 shows the results: 
Scaling Fig. 1 by 0.2 times, 0.8 times, 2 times and 
4 times, and then calculating the improved Zernike  
Moments of these scaled images, the results are 
represented in Table 2. 
For the sake of comparison, calculating the 
original Zernike moments of the scaled images, the 
results are shown in Table 3. 
 
 
Table 1. Improved Zernike moments of the rotated images. 
 
Rotated degree 
 
Feature 
0
°
  30
°
  60
°
  90
°
  135
°
 
'
20 || Z   0.4903 0.4904 0.4904 0.4903  0.4899 
'
22 || Z   0.0592 0.0592 0.0592 0.0592  0.0593 
'
31 || Z   0.0895 0.0894 0.0894 0.0895  0.0893 
'
33 || Z   0.0331 0.0330 0.0330 0.0331  0.0332 
'
40 || Z   0.1713 0.1717 0.1717 0.1713  0.1709 
'
42 || Z   0.1507 0.1507 0.1508 0.1507  0.1508 
'
44 || Z   0.0805 0.0805 0.0804 0.0805  0.0806 
 Sensors & Transducers, Vol. 166, Issue 3, March 2014, pp. 219-223 
  222
Table 2. Improved Zernike moments of the scaled images. 
 
     Scaled  times 
Feature 0.2 0.8  1  2  4 
'
20 || Z   0.4649 0.4888 0.4903 0.4934  0.4950 
'
22 || Z   0.0624 0.0594 0.0592 0.0588  0.0586 
'
31 || Z   0.0916 0.0897 0.0895 0.0892  0.0891 
'
33 || Z   0.0350 0.0332 0.0331 0.0328 0.0326 
'
40 || Z   0.1450 0.1696 0.1713 0.1748 0.1766 
'
42 || Z   0.1506 0.1507 0.1507 0.1507 0.  1506 
'
44 || Z   0.0887 0.0811 0.0805 0.0794  0.  0789 
 
 
Table 3. Zernike moments of the scaled images. 
 
      Times 
Feature  0.2  0.8  1 2 4 
20 || Z   92.3278 1553.1  2434.4  9799.3  39320 
22 || Z   12.4000 188.7619  294.0115  1168.2  4657.2 
31 || Z   18.1891 284.9703  444.3653  1771.8  7075.7 
33 || Z   6.9424 105.6076  164.2716  650.6338 2589.3 
40 || Z   28.8039 539.0052  850.7314  3472.4  1403.0 
42 || Z   29.9081 478.8410  748.2657  2992.1  11966 
44 || Z   17.6094 257.5578  399.6595  1577.5  6267.7 
 
 
Through the experimental data, we can find that 
the improved Zernike moments have good rotation 
invariant features, the moment values of the rotated 
images of the same kind are very close, for the scaled 
images, in addition to the change of the moment 
values are larger when they are scaled by smaller 
multiples, in other cases, the deviation is small, 
compared with the original Zernike moments, the 
modified moments have certain scale invariance, thus 
proving their feasibility. 
 
 
4. Application 
 
Next, we will introduce the application of 
improved Zernike moments in image recognition, 
assuming that there are the following three kinds of 
butterfly images in the image library: 
 
 
        
 
(a)                            (b)                      (c) 
 
Fig. 2. Three standard images in the library. 
Doing some rotation and scale transforms to the 
images, calculating their each order Zernike moment 
nm Z  according to the definition of Zernike moments, 
and conducting the normalized processing to  nm Z  
using the improved moment method proposed in this 
paper, then evaluating their magnitudes, we will get a 
new feature vector 
' || nm Z , after that averaging the 
feature vector of the image of the same kind and 
using the obtained feature value represents it. 
Afterwards we use the block distance method to 
recognize the image to be identified. Block distance 
is defined as: supposing  1 P  and  2 P  are two pixels 
of a given image, their position coordinates in the 
image respectively are (1 ,1 ) x y ,  (2 ,2 ) x y , 
assuming that D  is a distance measure function, 
then the block distance between  1 P   and  2 P   is 
(1 ,2 )|1 2 | |1 2 | DP P x x y y =− + −  [13]. According 
to the formula, calculating the block distance 
between the feature values of the target to be 
identified and each kind of the known targets, the 
target to be identified belongs to the one having the 
minimum calculation result. 
According to the above method to identify 15 of 
these three kinds of images, from the result we can 
see: the recognition rate of some order of the 
improved Zernike moments can reach one hundred 
percent, to the other order, even if it can’t correctly Sensors & Transducers, Vol. 166, Issue 3, March 2014, pp. 219-223 
  223
identify all of the targets, its recognition rate is 
relatively high. 
 
 
5. Conclusion 
 
The improved Zernike moments introduced in the 
paper not only have rotation invariant feature, but 
also have certain scale invariance compared with the 
original Zernike moments, but they still exist 
shortcomings, namely when the scaling times of the 
image are smaller, the deviation of the resulting 
improved Zernike moment values are relatively large, 
which results in certain recognition errors, in order to 
have widespread use we need to improve them by 
further researches.  
 
 
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Image Recognition Using Modified Zernike Moments

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