- Email: [email protected]
On the computational aspects of Zernike moments
Image and Vision Computing 25 (2007) 967–980 www.elsevier.com/locate/imavis
On the computational aspects of Zernike moments Chong-Yaw Wee, Raveendran Paramesran
*
Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Lembah Pantai, 50603 Kuala Lumpur, Malaysia Received 18 June 2005; received in revised form 30 June 2006; accepted 12 July 2006
Abstract The set of Zernike moments belongs to the class of continuous orthogonal moments which is defined over a unit disk in polar coordinate system. The approximation error of Zernike moments limits its applications in real discrete-space images. The approximation error of Zernike moments is divided into geometrical and numerical errors. In this paper, the geometrical and numerical errors of Zernike moments are explored and methods are proposed to minimize them. The geometrical error is minimized by mapping all the pixels of discrete image inside the unit disk. The numerical error is eliminated using the proposed exact Zernike moments where the Zernike polynomials are integrated mathematically over the corresponding intervals of the image pixels. The proposed methods also overcome the numerical instability problem for high order Zernike moments. Experimental results prove the superiority and reliability of the proposed methods in providing better image representation and reconstruction capabilities. The proposed methods are also not lacking in preserving the scale and translation invariant properties of Zernike moments. 2006 Elsevier B.V. All rights reserved. Keywords: Zernike moments; Approximation error; Geometrical error; Numerical error; Square-to-circular mapping; Exact Zernike moments
1. Introduction The set of orthogonal Zernike moments was first introduced for image analysis by Teague [1]. It is a set of complex orthogonal functions with a simple rotational invariant property which forms a complete orthogonal basis over the class of square integrable functions named as Zernike polynomials which is defined over the unit circle [2]. Although it is computationally complex if compared to other moment functions such as geometric and Legendre moments, Zernike moments had been proven to be superior in terms of their feature representation capability [3,4], image reconstruction capability [5,6] and low noise sensitivity [7,8]. Besides that, the orthogonal property also enables the separation of the individual contributions of each order moment to the reconstruction process. This will make the *
Corresponding author. Tel.: +60 3 7967 5253; fax: +60 3 7967 5316. E-mail addresses: [email protected] (C.-Y. Wee), [email protected] (R. Paramesran). 0262-8856/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2006.07.010
image reconstruction or inverse transform process of Zernike moments much more easier to be performed if compared to other moment functions. Zernike moment is rotational invariant by nature and the translation and scale invariants of Zernike moment are achieved easily through the normalization of its orthogonal polynomials [9]. These invariant properties enable the Zernike moments to form a robust set of image feature representation. Due to these superiorities, the set of Zernike moments is widely applied in image analysis includes invariant pattern or object recognition [10,11,12,13,14], image reconstruction [5,15], edge detection [16], image segmentation [17], optimal corneal surfaces modeling [18,19], watermarking [20], face recognition [21,22], content-based retrieval [23] and palmprint verification [24]. The definition of Zernike moments has a form of mapping the discrete-space image function which is normally in square or rectangular shape onto Zernike polynomials over a unit disk. The set of Zernike polynomials needs to be approximated by sampling at fixed interval when it is applied to a discrete-space image. Hence, these
968
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
approximate values of Zernike moments introduce the approximation error which in somehow degraded its information representation ability. Not like Legendre moments, the approximation error of Zernike moments can be divided into two categories, namely the geometrical error and numerical error [25]. The geometrical error is caused by the projection of a square discrete image onto the unit disk of Zernike polynomials. In the traditional approach, the pixel where its center falls outside the border of unit disk is not included in Zernike moments computation [25]. This approach will not introduce any error only when the input image is also circular in shape. For square image, information at the four corners of the image will be excluded in computation and hence information lost occurs. However, this geometrical error generally decreases as the moment order used is increased [15]. The numerical error is caused by the needs of calculating the double integral over a fixed sampling intervals. The traditional approach for solving the double integral is based on the zeroth order approximation. Another approach proposed by Liao and Pawlak in [25] to minimize this error is based on the extended version of Simpson’s rules (multipledimension cubature rules). As opposed to the geometrical error, this numerical error is increasing as the moment order used is increased [15]. In this paper, the computational aspect of Zernike moments in terms of error analysis is carried out in details. The inherent limitation in precision of computing the Zernike moments due to geometric nature of a circular domain is concerned. In order to minimize this geometrical error, a square to circular mapping technique is used where all pixels of a discrete square image are mapped inside the unit disk of Zernike polynomials and hence included in moment computation. This approach is very similar to the method proposed by Chong and et al. [9,26] excepts the way in computing the transformed image coordinate. This transformed image coordinate is important for the computation of exact Zernike moments through simple mathematical integration. The numerical error of Zernike moments is solved by mathematically integrating the Zernike polynomials over the corresponding intervals of the image pixels. This method will provide the theoretical exact values of the double integral of Zernike polynomials only under the aforementioned square to circular mapping method. If these exact Zernike moments are computed with traditional mapping approach, the moment values computed are not exact anymore due to information lost. The set of Zernike moments can be expressed as the linear combinations of geometric moments [7]. Hence, it can be implied that by first computing the exact values of the set of monomials and then combining them linearly, we can formulate the set of exact Zernike moments. The effectiveness and advantages of the proposed methods are assessed through several numerical experiments by
means of the moment computation, image reconstruction, numerical instability and invariant property. The organization of the rest of this paper is given as follow. The theoretical overview of Zernike moments is provided in Section 2. Section 3 provides the error analysis of Zernike moments when applied to a two-dimensional discrete image. The square to circular mapping method used for minimizing the geometrical error is discussed in detail in Section 4. Section 5 gives the derivation of the exact Zernike moments. Section 6 provides the experimental validation of the proposed mapping technique in minimizing the geometrical error. Section 7 provides the experimental validation of the proposed exact Zernike moments in eliminating the numerical error. Section 8 concludes the study. 2. Zernike moments The two-dimensional Zernike moments of order p with repetition q of an image intensity function f(x, y), are defined as Z Z pþ1 Z pq ¼ V pq ðx; yÞf ðx; yÞ dx dy ð1Þ p x2 þy 2 61
The pth order Zernike polynomials are defined as ð2Þ V pq ðx; yÞ ¼ Rpq ðrÞejqh ; r 2 ½1; 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y 2 is the length of the vector from the origin to the pixel (x, y), and h = tan1(y/x) is the angle between the vector r and the principle x-axis. The Zernike real-valued radial polynomials are given by Rpq ðrÞ ¼
ðpqÞ=2 X
ð1Þ
s¼0
ðp sÞ!
s
sÞ!ðpjqj sÞ! s!ðpþjqj 2 2
rp2s
ð3Þ
where p jqj is even, 0 6 jqj 6 p and p P 0. By letting s fi (p k)/2, Zernike polynomials in (2) can be represented as V pq ðx; yÞ ¼
p X
Bpqk rk ejqh
ð4Þ
k¼q
with the polynomial coefficient, Bpqk is defined as pk
Bpqk ¼
ð1Þ 2 ðpþk Þ! 2 Þ!ðkþq Þ!ðkq Þ! ðpk 2 2 2
ð5Þ
Hence, the definition of Zernike moments in (1) can be rewritten in polar form as Z Z p pþ1 1 p X Z pq ¼ Bpqk rk ejqh f ðr; hÞr dr dh ð6Þ p 1 p k¼q with dx dy = rdr dh and p 6 h 6 p. Graphs for the first four orders of radial polynomials, Rpq(r) are shown in Fig. 1.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
with Zpq is computed over the same unit disk and kp = (p + 1)dA/p is the normalizing constant. dA is the elemental area of the normalized square image in discrete form when the square image is mapped over the unit disk of Zernike polynomials. This elemental area provides different values with different square to circular mapping techniques. Therefore, proper mapping technique plays an crucial role in minimizing the geometrical error. If only Zernike moments of order 6Pmax are given, then the function f(x, y) can be approximated by a continuous function in a truncated series as
Orthogonal Radial Polynomials, Rpq(r) R00
1 0.8
R40
Radial Polynomials Values
0.6
R11
0.4
R22 R33
0.2
R44
0 R20 R31
R42
969
f^ ðx; yÞ ¼
p P max X X
kp Z pq V pq ðx; yÞ
ð12Þ
p¼0 q¼0
0
0.1
0.2
0.3
0.4
0.5 r
0.6
0.7
0.8
0.9
1
Fig. 1. Graphs for the first four orders of radial polynomials, Rpq(r).
The set of Zernike polynomials, Vpq(r, h) forms a complete orthogonal set on the interval [1, 1] as Z 1 Z p V pq ðr; hÞ½V p0 q0 ðr; hÞ ; r dr dh 1 p ð7Þ p d 0 d 0 p ¼ p 0 ; q ¼ q0 ; pþ1 pp qq ¼ 0 otherwise and its radial polynomials also satisfy the orthogonality relation as ( Z 1 1 dpp0 p ¼ p0 ; ð8Þ Rpq ðrÞ½Rp0 q ðrÞ ; r dr ¼ 2ðpþ1Þ 0 otherwise 1 where dij is the Kronecker delta. The set of Zernike moments possesses the inherit rotational invariant property. The invariant features are obtained by taking the magnitude values of Zernike moments since these values are remain identical to those image functions before and after rotation. The set of Zernike moments of an image which is rotated by a angle is given as Z rpq
¼ Z pq e
jqa
ð9Þ
where Z rpq are the Zernike moments of rotated image and Zpq are the Zernike moment of original image. The rotation invariant Zernike moments are extracted by considering only their magnitude values as jZ rpq j ¼ jZ pq ejqa j ¼ jZ pq j
ð10Þ
Only magnitude of Zernike moments with q P 0 are concerned since Z p;q ¼ Z pq and jZp,qj = jZp,qj [27]. The piece-wise continuous and bounded image intensity function f(x, y) can be expressed as an infinite series of Zernike polynomials expansion over the unit disk as f ðx; yÞ ¼
p 1 X X p¼0 q¼0
kp Z pq V pq ðx; yÞ
ð11Þ
3. Approximation error of Zernike moments The aforementioned favorable properties of the Zernike moments are valid as long as one uses a truly analog image function. Since most of the computer images are discrete in format and square or rectangle in shape, there is an inherent error when computing Zpq due to the circular nature and the approximation of the double integral term in Zernike polynomials. For a general two-dimensional image, this approximation error, Epq between the continuous original Zpq and discrete approximated Z~ pq can be decomposed into geometrical and numerical errors. The geometrical error is caused by the square to circular mapping procedure while the numerical error is caused by the discrete approximation of double integral [25]: ðnÞ Epq ¼ Z pq Z~ pq ¼ EðgÞ pq þ E pq
ð13Þ
ðnÞ where EðgÞ pq denotes the geometrical error and E pq denotes the numerical error. The details of the approximation error is explored in following subsections.
3.1. Geometrical error Since the set of Zernike polynomials is defined in terms of polar coordinate (r, h) with jrj 6 1 as given in (2), then the computation of Zernike moments requires a linear mapping process to map the image coordinates (i, j) to a suitable unit circular domain (r, h) 2 R2. The general form of mapping technique is given as xi ¼ c1 i þ c2 ;
y j ¼ c1 j þ c2
ð14Þ
where the coefficients c1 and c2 are given as c1 ¼
2h ; N 1
c2 ¼ h
ð15Þ
with 0 6 h 6 1 and h 2 R. The value of h is determined based on the applied square to circular mapping technique. The traditional mapping approach is showed in Fig. 2. This approach maps the (N · N) square image plane onto a unit disk by first taking the center of the image as the origin.
970
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
pþ1 Z~ pq ¼ p
N 1 X N 1 X i¼0
f ðxi ; y j ÞV pq ðxi ; y j ÞDxi Dy j
ð17Þ
j¼0
This approximation of double integral term causes error in the computed Zernike moments and the error can be determined as ~ EðnÞ pq ¼ Z pq Z pq ¼ Fig. 2. The traditional square to circular mapping technique.
The pixel coordinates are then normalized over the range of the unit disk as shown in Fig. 2(b). In this traditional approach, the h value is equal to the radius of unit disk; i.e., h = 1. Hence, the values of c1 and c2 are obtained as 2/(N 1) and 1, respectively. Therefore, the elemental area of this projection, dA, which is the ratio of the area of unit disk to the total number of pixels in image plane is given as p/N2. Unfortunately, this traditional approach introduces the geometrical error during Zernike moments computation. The pixels which their center fall outside the border of the unit disk are ignored during the moment computation. All the information carried by these pixels will be lost and the condition becomes worse if these pixels contain important information of the image. The lost of these information will degrade the image representation capability and also the quality of reconstructed image. 3.2. Numerical error ðnÞ The numerical error, Epq is caused by the needs of computing the double integral term in the definition of Zernike moments as shown in (1) and (6). Let the image coordinates of a discrete-space (N · N) square image, f(x, y) is defined as a discrete set of points (xi, yj). Then, the set of Zernike moments in (1) can be rewritten as Z xi þDx2 i N 1 X N 1 pþ1 X f ðxi ; y j Þ Z pq ¼ Dx p i¼0 j¼0 xi 2 i
Z
yj þ
Dy j 2
Dy yj 2j
V pq ðx; yÞ dx dy
ð16Þ
where (Dxi = xi+1 xi) and (Dyj = yj+1 yj) are the intervals between two successive image pixels along x and y Cartesian axis, respectively. In the traditional approach, the double integral term is often evaluated using the zeroth order approximation method where the values of Zernike polynomials are assumed to be constant over the intervals ½xi Dx2 i ; xi þ Dx2 i Dy Dy and ½y j 2 j ; y j þ 2 j , and the value of each interval is obtained by sampling the Zernike polynomials at the central point of these intervals. Hence, the set of approximated Zernike moments computed using the zeroth order approximation order is given as
N 1 X N 1 pþ1 X f ðxi ; y j Þ p i¼0 j¼0 "Z Dxi Z # Dy j xi þ 2 yj þ 2 V pq ðx; yÞ dx dy V pq ðxi ; y j ÞDxi Dy j Dy xi
Dxi 2
yj
j
2
ð18Þ It should be noted that this error increases as the number of sampling points decreases and increases further if the order of moments is increased [15]. The increment of numerical error is caused by the sampling approximation of Zernike radial polynomials. The Zernike radial polynomials are oscillating like sine and cosine functions within the interval [1, 1] as shown in Fig. 1. As the moment order increases, the shape of polynomial will oscillate at higher spatial-frequency. Since the moment kernels of the approximated Zernike moments are approximated by sampling the Zernike polynomials, the moment kernels are susceptible to information loss if it is undersampled. The number of sampling points is determined by the size of the image, (N · N). Hence, if the moment order of p < N is considered, then the sampling of the Zernike polynomials becomes insufficient and the resulting moment kernels are undersampled. Consequently, the set of Zernike moments computed based on these undersampled Zernike polynomials cannot get rid from the approximation error. In order to minimize the error caused by this double integral approximation, Liao and et al. had proposed several cubature formulas based on the extended Simpson’s rules which use a two-dimensional approximation approach for the double integral term [6,15,25]. This approach successfully improves the image reconstruction performance. However, the computed moment values is still not exact and the reconstruction error is increasing after certain moment order. For certain images, high order Zernike moments are required to represent the finest information of the images. However, numerical instability is arisen when the moment order used is over certain limit due to the involvement of factorial terms in polynomial definition. The numerical instability causes the computed moment values deviate from the actual values and hence causes tremendous degradation in the reconstructed images. In order to overcome this problem, Liao et al. had proposed a modified set of Zernike moments to increase the maximum moment order before the occurrence of numerical instability. This modified set of Zernike moments is computed by restricting the radius as r = x2 + y2 < 1. This
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
971
approach successfully increases the maximum moment order from around 25th to around 50th order [15]. However, same numerical instability problem occurs when more than 50th order of Zernike moments are computed. 4. Geometrical error minimization In order to minimize the geometrical error and provide the essential conditions for the exact Zernike moments computation, a square to circular mapping technique which is very similar to the technique proposed by Chong et al. [9,26] is used in this paper. In this mapping technique, the image projection operation used is same with the technique in [26] but the operation of computing the transformed coordinate is slightly different. Both technique map the square image into the unit disk of Zernike polynomials as shown in Fig. 3. By using this technique, the problem of information lost in traditional approach is overcome and hence the geometrical error is minimized. As shown in Fig. 3, the entire square image is bounded inside the unit disk. This approach ensures that there is no pixel loss during moment computation and thus all information of the square image can be preserved during moment computation. The transformed coordinate (xi, yj) of this mapping technique is computed as below. Firstly,pthe ffiffiffi h value of this mapping technique is obtained as 1= 2. The coefficients c1ffiffiffiand c2 are determined p pffiffiffi based on (15) and are equal to 2=ðN 1Þ and 1= 2, respectively. The new elemental area is derived from the ratio between the area of normalized square image to the total pixels of the image and is given as dA = 2/N2. For the technique in [26], the transformed coordinate is computed using the traditional formula in (14); i.e., the computed transformed coordinate is located at the edges of the transformed image pixel as shown in Fig. 4. However in our technique, the computed transformed coordinate is located at the center of the transformed image pixel as shown in Fig. 5. This is to ensure that the mapping technique used can provide the essential conditions for the exact Zernike moment computation through the simple mathematical integration. Hence, the area of the computed
a
Fig. 4. The transformed coordinate and intervals computed in [26].
Fig. 5. The transformed coordinate and intervals computed in this paper.
intervals, Dxi and Dyj are different in location for both techniques. In order to make sure that the center of each pixel is used in moment computation, the value of ‘0.5’ is included in computing every (xi, yj). Hence, the general formula used to compute the transformed coordinate (xi, yj) in (14) is modified as xi ¼ c1 ði þ 0:5Þ þ c2 ;
where the coefficients c1 and c2 are given as pffiffiffi 1 2 ; c2 ¼ pffiffiffi c1 ¼ N 2
N-1
ð19Þ
ð20Þ
Nevertheless, this technique is not a linear transformation because a pixel in the unit disk represents more than one pixel of a square image. It causes the area of square image has been scaled down by 50% after mapping inside the unit disk. Furthermore, this approach has lower reconstruction power and requires higher order of Zernike moments for image reconstruction if compared to the aforementioned traditional approach. This is because the reconstructed coarse information of the image through the low order
b 0
y j ¼ c1 ðj þ 0:5Þ þ c2
y
i 1
h
x
N-1 j
Fig. 3. The square to circular mapping approach where whole square image is mapped inside a unit disk.
972
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Zernike moments is degraded by the scaling process. Nevertheless, this approach performs well in reconstructing the fine information using the high order Zernike moments. It also increases the maximum moment order before the occurrence of numerical instability besides maintaining all the information carried by the square image.
Notice that Hp(xi) and Hq(yj) are independent of the image and can always be precomputed, stored and recalled later to avoid repetitive computation. Based on (25) and (26), (22) can be rewritten as N 1 X N 1 X f ðxi ; y j ÞH p ðxi ÞH q ðy j Þ ð27Þ M pq ¼ i¼0
5. Numerical error minimization In the previous section, we have discussed that the numerical error is due to the zeroth order approximation of the double integral term as shown in (18). This problem can be solved by mathematically integrating the Zernike polynomials over the corresponding intervals of the image pixels. By using this method, the computed Zernike moments are exactly equal to the theoretical values. The exact computation of Zernike moments is derived based on the exact geometric moments and the relationship between geometric moments and Zernike moments. 5.1. Exact computation of geometric moments The geometric moments of order (p + q) for (N · N) square image f(x, y) are defined within the interval of [1, 1] · [1, 1] as Z 1 Z 1 M pq ¼ xp y q f ðx; yÞ dx dy ð21Þ 1
1
If the image is defined only for a discrete set of points (xi, yj), then (21) can be rewritten as Z xi þDx2 i Z y j þDy2 j N 1 X N 1 X M pq ¼ f ðxi ; y j Þ xp y q dx dy Dy i¼0
¼
xi
j¼0
N 1 X N 1 X i¼0
Dxi 2
yj
j
2
f ðxi ; y j Þhp ðxi Þhq ðy j Þ
ð22Þ
j¼0
with hp(xi) and hq(yj) are given as Z xi þDx2 i xp dx hp ðxi Þ ¼ xi
Dxi 2
ð23Þ
hq ðy j Þ ¼
Z
yj þ
Dy j 2
Dy j yj 2
y q dy
ð24Þ
The exact computation of hp(xi) and hq(yj) based on simple mathematical integration rule are given as pþ1 xi þDx2 i Z xi þDx2 i x p x dx ¼ ¼ H p ðxi Þ ð25Þ Dx p þ 1 xi Dxi xi 2 i 2
and Z
yjþ
yj
Dy j 2
Dy j 2
y qþ1 y dy ¼ qþ1 q
Note that there is no approximation involved in (27), so the moments computed using this method is named as the exact geometric moments. 5.2. Exact computation of Zernike moments Since different sets of polynomials up to the same order define the same subspace, any complete set of moments up to a given order can be obtained from any other set of moments up to same order, at least in theory [7,8]. Hence, most of the moment functions include Zernike moments can be computed exactly or approximately from the simple geometric moments. The set of Zernike moments can be computed using the geometric moments up to the same order as [7] p q s X X s q pþ1 X Z pq ¼ Bpqk wn M k2mn;2mþn p k¼q m n m¼0 n¼0 ð28Þ where i q > 0; w¼ þi q 6 0
y j þDy2 j yj
Dy j 2
¼ H q ðy j Þ
ð26Þ
ð29Þ
with pffiffiffiffiffiffiffi 1 s ¼ ðk qÞ; and i ¼ 1 2 Based on (28) and (27), the set of Zernike moments can be computed exactly as p q s X X s q pþ1 X Z pq ¼ Bpqk wn p k¼q m n m¼0 n¼0 ð30Þ N 1 X N 1 X f ðxi ; y j ÞH k2mn ðxi ÞH 2mþn ðy j Þ i¼0
and
j¼0
j¼0
Similar to the exact geometric moments computation, there is no approximation involved in computing the double integral in exact Zernike moments computation. By using Eq. (30), the numerical error occurs in Zernike moment computation is eliminated because it provides the exact Zernike moment values. 6. Experimental study on geometrical error In this section, the proposed square to circular mapping technique is compared with the traditional mapping technique. The performance of both techniques are evaluated through the image reconstruction of several grayscale images. The testing images used are five 30 · 30 grayscale Chinese character images as shown in Fig. 6.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
973
Fig. 6. Grayscale Chinese character images used in computer simulation. From left to right are C1, C2, C3, C4 and C5.
The reconstruction error between the original image and the reconstructed image is used as the measurement of the reconstruction capability for the traditional and proposed mapping techniques. The reconstruction error, e is determined as ( ) 2 N 1 X N 1 X ½f ði; jÞ f^ ði; jÞ e¼ ð31Þ ½f ði; jÞ2 i¼0 j¼0 where f(i, j) and f^ ði; jÞ are the original and reconstructed images, respectively. Average reconstruction error for five Chinese character images with different maximum order of Zernike moments are shown in Fig. 7. From Fig. 7, it is very clear that the proposed square to circular mapping technique outperforms the traditional mapping technique. The average reconstruction error of traditional technique up to 45th order is decreasing from 0.85 to 0.40. The minimum reconstruction error that can be achieved using the traditional technique is about 0.40 at 45th order. The large reconstruction error when using the traditional technique is caused by the geometrical error where the pixels which their center located outside the unit disk are excluded in Zernike moment computation. All the information carried by these pixels is lost after reconstruction. These excluded pixels are about 21.5% of the overall pixels of a (N · N) image. The reduction of reconstruction error using the proposed technique is significant. The average reconstruction
error is reduced to as low as 0.05 at 45th order. The significant reduction is due to the inclusion of all image pixels in the Zernike moment computation. The effect of geometrical error is reduced to its minimum level by using the proposed mapping technique. This experimental result also proves that the geometrical error dominates the reconstruction error due to the inherit property of Zernike polynomials which are defined over a unit disk. The increment of moment order used causes the decrement in reconstruction error (geometrical error) which is congruence with the findings in [15]. The reconstructed images for five testing images with different maximum order using the traditional and proposed techniques are shown in Fig. 8. For traditional technique, the reconstructed images facing the problem of missing information at their four corners. The intensity values after reconstruction also smaller as shown by the lighter illumination of the reconstructed characters. These problems are not faced by the proposed technique. Therefore, the reconstruction error for proposed technique is much smaller and better reconstruction capability is shown. 7. Experimental study on numerical error In this section, computer simulations are performed to validate the proposed theoretical framework in providing the exact Zernike moments and hence minimizing the numerical error. The reconstruction power of the proposed method also been explored. 7.1. Moment computation
Average Reconstruction Error vs. Moment Order 1 Proposed Traditional
The set of Zernike moments is computed from simple artificial images using several Zernike moment computation methods with two mapping approaches, i.e., the traditional and proposed approaches. The computation methods used included the proposed exact Zernike moments (EZM), zeroth order approximation (ZOA) and several extended Simpson’s rules (ESR). The ESR used are two five-dimensional cubature formulas and two nine dimensional cubature formulas. The five-dimensional cubature formulas for first type, C I5 and second type, C II5 were proposed by Liao and et al. and are given as [25]
0.9
Average Reconstruction Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
45
50
Moment Order
Fig. 7. Average reconstruction error for five Chinese character images with different maximum order of Zernike moments using the traditional and proposed mapping methods.
1 C I5 ðV Þ ¼ f8V ð0; 0Þ þ V ð0; 1Þ þ V ð1; 0Þ þ V ð0; 1Þ 3 þ V ð1; 0Þg ð32Þ 4 C II5 ðV Þ ¼ fV ð0; 0Þ þ V ð0; 0:5Þ þ V ð0:5; 0Þ þ ð0; 0:5Þ 3 þ V ð0:5; 0Þg ð33Þ
974
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Fig. 8. The reconstructed images using traditional and proposed mapping approaches with different maximum order. (Trad. = Traditional approach, Prop. = Proposed approach.)
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
The nine-dimensional cubature formulas of first type, C I9 and second type, C II9 are given as [28] C I9 ðV Þ ¼
1 f16V ð0; 0Þ þ 4½V ð0; 1Þ þ V ð1; 0Þ þ V ð0; 1Þ 36 þV ð1; 0Þ þ ½V ð1; 1Þ þ V ð1; 1Þ þ V ð1; 1Þ
þV ð1; 1Þg ð34Þ n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 C II9 ðV Þ ¼ 64V ð0; 0Þ þ 40½V ð0; 0:6Þ þ V ð 0:6; 0Þ 324 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi þV ð0; 0:6Þ þ V ð 0:6; 0Þ þ 25½V ð 0:6; 0:6Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi þV ð 0:6; 0:6Þ þ V ð 0:6; 0:6Þ þ V ð 0:6; pffiffiffiffiffiffiffi o 0:6Þ ð35Þ (1) Artificial image I: The artificial image used is a (8 · 8) image with all its pixel intensity values are constant, i.e., f(i, j) = K, K = constant, "(i, j). Small artificial image is used in order for the approximation error to be observed readily by directly compared with the theoretical values. In traditional mapping technique, it is evident that all Zernike moments should provide the value of zero, except Z00. This can be shown mathematically by performing the Zernike moments computation through the following equation p q s X X X s q Z pq ¼kp Bpqk wn m n k¼q m¼0 n¼0 " # pffiffiffiffiffiffiffi Z 1 Z 1x2 ðk2mnÞ ð2mþnÞ x dy dx pffiffiffiffiffiffiffi y K ¼ 0
1x2
1
if p ¼ q ¼ 0;
ð36Þ
otherwise:
975
Providing that if (k 2m n) and/or (2m + n) is odd, then (36) will give zero values. Therefore, only the even order of Zernike moments are taken into the consideration in this case. For case K = 1, only the zeroth order Zernike moment gives the nonzero value, i.e., Z00 = 1 while others give zero values. The results for exact Zernike moments (EZM, Zpq (30)), the zeroth order approximation (ZOA, Z~ pq (17)) and several extended Simpson’s rules (ESR, C I5 (32), C II5 (33), C I9 (34), C II9 (35)) are shown in Table 1. The values given by the EZM are quite far away from the theoretical values calculated using (36). The C II5 of ESR provides closest values and followed by ZOA, C I5 , EZM, C II9 and C I9 . The values provided by all these methods are deviated from the theoretical values due to the information lost in square to circular mapping. The performances of these methods are highly dependent on how many pixels are inside the unit disk and hence used for the Zernike moments computation. The same procedures are repeated for same artifical image using the proposed mapping technique. Eq. (36) is not suitable for computing the theoretical values of Zernike moments under this circumstance and hence a novel formula is proposed to compute them. The novel formula proposed specifically under this circumstance for computing the theoretical values of Zernike moments is given as p q s X X X s q n Z pq ¼ kp Bpqk w m n k¼q m¼0 n¼0 Z p1ffi Z p1ffi 2 2 xðk2mnÞ dx y ð2mþnÞ dy ð37Þ p1ffi p1ffi 2 2
Table 1 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = K, K = 1 using traditional mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 2 2 4 4 4 6 6 6 6 8 8 8 8 8 10 10 10 10 10 10
0 0 2 0 2 4 0 2 4 6 0 2 4 6 8 0 2 4 6 8 10
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.0345 0.1393 0 0.3614 0 0.0754 0.8257 0 0.0391 0 1.7764 0 0.0637 0 0.5101 3.7521 0 0.2868 0 1.2067 0
1.0345 0.0746 0 0.0179 0 0.0758 0.2105 0 0.0109 0 0.6119 0 0.1463 0 0.5095 0.9427 0 0.2644 0 0.6178 0
1.0345 0.1069 0 0.1898 0 0.0762 0.3101 0 0.0336 0 0.6003 0 0.0463 0 0.5087 1.4873 0 0.0108 0 0.9035 0
1.0345 0.1069 0 0.1879 0 0.0759 0.2824 0 0.0290 0 0.4144 0 0.0812 0 0.5093 0.6694 0 0.1809 0 0.9082 0
1.0345 0.1393 0 0.3634 0 0.0758 0.8554 0 0.0440 0 1.9796 0 0.0283 0 0.5095 4.6818 0 0.1265 0 1.2019 0
1.0345 0.1393 0 0.3614 0 0.0754 0.8257 0 0.0391 0 1.7757 0 0.0639 0 0.5101 3.7436 0 0.2893 0 1.2067 0
(Theo. = Theoretical values).
976
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
The results of EZM, ZOA and ESR with the proposed mapping technique are shown in Table 2. Note that the values provided by EZM are exactly match with the theoretical values computed using (37) while ZOA and ESR deviate from them except for C II9 . The deviation from theoretical values becomes more severe when higher order of Zernike moments are computed. Although the Zernike moment values computed using C II9 are very similar to the theoretical values, however it still not providing the exact Zernike moment values. (2) Artificial image II: For randomly generated small (4 · 4) artifical image, f(i,j) = F which is given as 9 8 16 2 3 13 > > > > > = < 5 11 10 8 > ð38Þ F¼ > 9 7 6 12 > > > > > ; : 4 14 15 1 The results are shown in Table 3. Once again only the values provided by EZM match the theoretical values while ZOA and ESR deviate severely from the theoretical values especially when the moment order increases. 7.2. Image reconstruction The comparative assessment between the approximated and exact Zernike moments in image reconstruction is performed through several numerical experiments using both the artificial and real images. (1) Random images: For the sake of fairness, randomly generated images are used in this experiment. The purpose of using randomly generated images is to minimize the effects of image content on the computed Zernike
moments. Initially, an artificial square image is generated randomly as f ði; jÞ ¼ randomðN ; N Þ;
0 6 f ði; jÞ 6 255; 8i; j
ð39Þ
The size of the image is determined by setting N = 64 and then is scaled to different sizes, i.e., 32, 16, 8. Zernike moments of these images are computed using the aforementioned methods up to maximum order, P = 40. The reconstruction error between the original and reconstructed images is measured using (31). Same procedures are repeated for 20 similar randomly generated images and their average error is shown in Fig. 9. It is clearly shown that for different image sizes, the exact Zernike moments consistently provide lesser reconstruction error if compared to the approximated Zernike moments (ZOA and ESR). The reconstruction error of exact Zernike moments is steadily decreasing while that of the approximated Zernike moments begin to increase after certain moment order. The reconstruction error is caused by the application of the truncated Zernike moments up to certain maximum order. The approximation error is deceasing as the increment of the moment order because more Zernike moments are included in reconstruction process. For the approximated Zernike moments, the reconstruction error is similar with the exact Zernike moments only when the moment order used is small due to small numerical error and sometimes can be ignored. However, as the moment order increases, the numerical error of the approximated Zernike moments becomes larger and hence causes the increment in reconstruction error. (2) Real images: Image reconstruction process is repeated for several grayscale real images. These real images are
Table 2 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = K, K = 1 using proposed mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 2 2 4 4 4 6 6 6 6 8 8 8 8 8 10 10 10 10 10 10
0 0 2 0 2 4 0 2 4 6 0 2 4 6 8 0 2 4 6 8 10
0.6366 0.6366 0 0.2122 0 0.2122 0.2122 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1273 0 0.1273 0 0.1273 0
0.6366 0.6366 0 0.2122 0 0.2122 0.2122 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1273 0 0.1273 0 0.1273 0
0.6366 0.6565 0 0.24402 0 0.2122 0.2215 0 0.2590 0 0.1491 0 0.1764 0 0.1269 0.1669 0 0.1765 0 0.2019 0
0.6366 0.6540 0 0.2401 0 0.2122 0.2199 0 0.2532 0 0.1451 0 0.1710 0 0.1269 0.1643 0 0.1677 0 0.1930 0
0.6366 0.6466 0 0.2284 0 0.2121 0.2150 0 0.2359 0 0.1332 0 0.1548 0 0.1268 0.1564 0 0.1414 0 0.1662 0
0.6366 0.6366 0 0.2119 0 0.2122 0.2143 0 0.2126 0 0.1316 0 0.1255 0 0.1269 0.1226 0 0.1378 0 0.1291 0
0.6366 0.6366 0 0.2122 0 0.2122 0.2121 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1275 0 0.1273 0 0.1273 0
(Theo. = Theoretical values).
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
977
Table 3 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = F using proposed mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 1 2 2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 10 10
0 1 0 2 1 3 0 2 4 1 3 5 0 2 4 6 1 3 5 7 0 2 4 6 8 1 3 5 7 9 0 2 4 6 8 10
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8038 0 1.8038 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0823 0 1.0823 0 1.0823 0.6171 2.4684i 0.0440 0.1758i 0.2836 + 1.1342i 0.4421 1.7683i 0.1525 0.6100i 1.0823 0 1.0823 0 1.0823 0
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8038 0 1.8038 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0823 0 1.0823 0 1.0823 0.6171 2.4684i 0.0440 0.1758i 0.2836 + 1.1342i 0.4421 1.7683i 0.1525 0.6100i 1.0823 0 1.0823 0 1.0823 0
5.4113 0 6.0877 0 0.3376 1.3505i 0.3376 1.3505i 2.7479 0 1.7967 0.4431 + 1.7725i 0.7069 + 2.8276i 0.2638 + 1.0551i 2.9963 0 3.4217 0 0.8001 + 3.2003i 0.1899 0.7596i 0.7807 3.123i 0.2093 + 0.8370i 3.0959 0 1.5725 0 1.0233 1.7076 6.8304i 0.3594 1.4375i 0.6363 + 2.5453i 0.8500 3.400i 0.1381 0.5523i 2.9857 0 6.6135 0 3.6239 0
5.4113 0 0.6540 0 0.6540 0 0.2401 0 0.2122 0.2401 0 0.2122 0.2199 0 0.2532 0 0.2199 0 0.2532 0 0.1451 0 0.1710 0 0.1269 0.1451 0 0.1710 0 0.1269 0.1643 0 0.1677 0 0.1930 0
5.4113 0 0.6466 0 0.6466 0 0.2284 0 0.2121 0.2284 0 0.2121 0.2150 0 0.2359 0 0.2150 0 0.2359 0 0.1332 0 0.1548 0 0.1268 0.1332 0 0.1548 0 0.1268 0.1564 0 0.1414 0 0.1662 0
5.4113 0 0.6366 0 0.6366 0 0.2119 0 0.2122 0.2119 0 0.2122 0.2143 0 0.2126 0 0.2143 0 0.2126 0 0.1316 0 0.1255 0 0.1269 0.1316 0 0.1255 0 0.1269 0.1226 0 0.1378 0 0.1291 0
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8027 0 1.8034 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0675 0 1.0780 0 1.0823 0.6157 2.4629i 0.0426 0.1703i 0.2841 + 1.1366i 0.4415 1.7659i 0.1525 0.6100i 1.1338 0 1.0760 0 1.0749 0
(Theo. = Theoretical values).
some benchmark images which are normally used in image processing field and are shown in Fig. 10. The average reconstruction error of these real images for different Zernike moments computation methods is shown in Fig. 11. The results validate the efficiency of the exact Zernike moments by outperforming all other approximation methods. All approximation methods facing the numerical instability problem after certain maximum order except for C II9 . The reconstruction error of C II9 is small and quite similar with the exact Zernike moments up to a quite large moment order. However, its deviation from exact Zernike moments becomes obvious after the moment order P = 35. This can be explained by the fact that the Zernike moments computed using C II9 are very similar to the exact Zernike moments up to a quite large moment order as shown in Tables 2 and 3. As the moment order increases, the deviation becomes larger and finally provides similar performance as other approximation methods.
7.3. Invariant properties Any proposed method either for efficient or fast computation of Zernike moments, must possess the ability to maintain the invariant properties of the Zernike moments. In order to show the robustness and feasibility of the proposed method, an experiment is performed to evaluate the invariant accuracy of the proposed method for scale and translation. The rotation invariant property of Zernike moments for the proposed method is not implemented since the Zernike moments are already rotation invariant. Since our proposed method in minimizing the approximation error of Zernike moments is based on the linear combination of geometric moments, the Zernike moment invariants can be obtained using the traditional indirect method based on the geometric moment invariants as shown in (40). The general formula of translation and scale invariants of Zernike moments of order p with repetition q, ~ pq , is given as in terms of M
978
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Fig. 9. Reconstruction error, for images with different sizes using EZM, ZOA and ESR.
Fig. 10. Greyscale real images used in the experiment with resolution of 256 · 256.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
979
Fig. 11. Average reconstruction error for images in Figure 10.
Table 4 Zernike moment values of the translated and scaled Chinese character images using the proposed methods. Images
Z pq ¼
Z20
Z22
Z40
Z42
Z44
6.3782e02
3.1891e02
3.4928e01
2.328501
5.8213e02
6.3771e02
3.1915e02
3.501101
2.329101
5.8232e02
6.3724e02
3.1962e02
3.502101
2.331401
5.8235e02
6.3809e02
3.1204e02
3.491601
2.324401
5.8260e02
6.3799e02
3.2099e02
3.504201
2.332801
5.8470e02
6.3846e02
3.2243e02
3.513201
2.354701
6.8667e02
p q s X X s q pþ1 X ~ k2mn;2mþn M Bpqk wn p k¼q m n m¼0 n¼0 ð40Þ
~ pq is the translation and scale exact geometric mowhere M ment invariants. Zernike moments of second and fourth orders which are computed from several translated and scaled images of Chinese character are shown in Table 4. The first three rows show the Chinese characters are translated, while rows 4–6 show the scaled images. Though the computed Zernike moment values for respective orders are not the same, but they are within the acceptable margin
of error. This is due to the image pixels lost during the randomly scaling process. 8. Concluding remarks The approximation error of the well known orthogonal Zernike moments for an image function can be decomposed into the geometrical and numerical errors. Efficient computation method is important in making the Zernike moments more reliable in real-world applications. The geometrical error is due to the necessary of the square to circular mapping during the computation of Zernike moments. A square to circular mapping technique which maps all the pixels of an image function inside the unit disk is used in
980
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
this paper. This mapping technique enables all the image pixels are included in the Zernike moments computation and hence prevents the information lost as normally occurs in the traditional mapping technique. The numerical error is caused by the needs of computing the double integral in the definition of Zernike moments and the use of the truncated approximation series of Zernike moments. The exact computation of Zernike moments is proposed to provide exact representation using Zernike moments and hence reduces the numerical error caused by the traditional zeroth order approximation and extended Simpson’s rules. The reconstruction error for exact Zernike moments is reducing with the increment of moment order while the reconstruction error for other approximated methods increases with the increment of moment order. The proposed method also not lacking in preserving the scale and translation invariant properties of Zernike moments. Acknowledgement The authors would like to extend their thanks to the anonymous reviewers for the valuable and constructive comments for making this manuscript more readable. References [1] M.R. Teague, Image analysis via the general theory of moments, J. Optical Soc. Am. 70 (1980) 920–930. [2] F. Zernike, Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode, Physical 7 (1934) 689–701. [3] S.O. Belkasim, M. Shridhar, M. Ahmadi, Pattern recognition with moment invariants: a comparative study and new results, Pattern Recogn. 24 (12) (1991) 1117–1138. [4] R. Mukundan, K.R. Ramakrishnan, Moment Functions in Image Analysis, World Scientific Publishing, Singapore, 1998. [5] M. Pawlak, On the reconstruction aspect of moment descriptors, IEEE Trans. Inf. Theory 38 (6) (1992) 1698–1708. [6] S.X. Liao, M. Pawlak, On image analysis by moments, IEEE Trans. Pattern Anal. Mach. Intell. 18 (3) (1996) 254–266. [7] C.H. Teh, R.T. Chin, On image analysis by the methods of moments, IEEE Trans. Pattern Anal. Mach. Intell. 10 (4) (1988) 496–512. [8] C.H. Teh, R.T. Chin, On image analysis by the methods of moments, in: Proceedings of Conference on Computer Vision and Pattern Recognition (CVPR’88), 1988, pp. 556–561. [9] C.-W. Chong, A formulation of a new class of continuous orthogonal moment invariants, and the analysis of their computational aspects, PhD thesis, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia, May 2003. [10] S.O. Belkasim, M. Shridhar, M. Ahamadi, Shape recognition using Zernike moments invariants, in: Proceedings of 23rd Annual Asilomar Conference on Signals Systems and Computers, 1989, pp. 167–171.
[11] S. Ghosal, R. Mehrotra, Zernike moment-based feature detectors, in: Proceedings of International IEEE Conference on Image Processing (ICIP-94), vol. 1, 1994, pp. 934–938. [12] C. Kan, M.D. Srinath, Invariant character recognition with Zernike and orthogonal Fourier–Mellin moments, Pattern Recogn. 35 (1) (2002) 143–154. [13] A. Khotanzad, J.H. Lu, Classification of invariant image representations using a neural network, IEEE Trans. Acoust., Speech, Signal Process. 38 (1990) 1028–1038. [14] L. Wang, G. Healey, Using Zernike moments for the illumination and geometry invariant classification of multispectral texture, IEEE Trans. Image Process. 7 (2) (1998) 196–203. [15] S. Liao, Image analysis by moments, PhD thesis, University of Manitoba, Winnipeg, Manitoba, Canada, 1993. [16] S. Ghosal, R. Mehrotra, Edge detection using orthogonal momentbased operators, in: Proceedings of 11th Image, Speech and Signal Analysis (IAPR) International Conference on Pattern Recognition, vol. III, 1992, pp. 413–416. [17] S. Ghosal, R. Mehrotra, Segmentation of range images: an orthogonal moment-based integrated approach, IEEE Trans. Robot. Autom. 9 (4) (1993) 385–399. [18] D.R. Iskander, M.J. Collins, B. Davis, Optimal modeling of corneal surfaces with Zernike polynomials, IEEE Trans. Biomed. Eng. 48 (1) (2001) 87–95. [19] D.R. Iskander, M.R. Morelande, M.J. Collins, B. Davis, Modelling of corneal surfaces with radial polynomials, IEEE Trans. Biomed. Eng. 49 (4) (2002) 320–328. [20] Y. Xin, S. Liao, M. Pawlak, Geometrically robust image watermarking via pseudo-Zernike moments, in: Proceedings of 2004 Canadian Conference on Electrical and Computer Engineering, vol. 2, 2004, pp. 939–942. [21] J. Haddadnia, M. Ahmadi, K. Raahemifar, An effective feature extraction method for face recognition, in: Proceedings of 2003 International Conference on Image Processing, vol. 3, 2003, pp. III– 917–920. [22] J. Haddadnia, M. Ahmadi, K. Faez, An feature extraction method with pseudo-Zernike moments in rbf neural network-based human face recognition system, EURASIP J. Appl. Signal Process. (2003) 890–901. [23] Y.-S. Kim, W.-Y. Kim, Content-based trademark retrieval system using visually salient features, in: Proceedings of 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1997, pp. 307–312. [24] Y.-H. Pang, Andrew B.-J. Teoh, David C.-L. Ngo, F.-S. Hiew, Palmprint verification with moments, J. WSCG 12 (2003) 1–3. [25] S.X. Liao, M. Pawlak, On the accuracy of Zernike moments for image analysis, IEEE Trans. Pattern Anal. Mach. Intell. 20 (12) (1998) 1358–1364. [26] C.W. Chong, P. Raveendran, R. Mukundan, A comparative analysis of algorithm for fast computation of Zernike moments, Pattern Recogn. 36 (3) (2003) 731–742. [27] A. Khotanzand, Y.H. Hong, Invariant image recognition by Zernike moments, IEEE Trans. Pattern Anal. Mach. Intell. 12 (5) (1990) 489– 497. [28] M. Abramowitz, I.A. Segun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, 1965.
On the computational aspects of Zernike moments Chong-Yaw Wee, Raveendran Paramesran
*
Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Lembah Pantai, 50603 Kuala Lumpur, Malaysia Received 18 June 2005; received in revised form 30 June 2006; accepted 12 July 2006
Abstract The set of Zernike moments belongs to the class of continuous orthogonal moments which is defined over a unit disk in polar coordinate system. The approximation error of Zernike moments limits its applications in real discrete-space images. The approximation error of Zernike moments is divided into geometrical and numerical errors. In this paper, the geometrical and numerical errors of Zernike moments are explored and methods are proposed to minimize them. The geometrical error is minimized by mapping all the pixels of discrete image inside the unit disk. The numerical error is eliminated using the proposed exact Zernike moments where the Zernike polynomials are integrated mathematically over the corresponding intervals of the image pixels. The proposed methods also overcome the numerical instability problem for high order Zernike moments. Experimental results prove the superiority and reliability of the proposed methods in providing better image representation and reconstruction capabilities. The proposed methods are also not lacking in preserving the scale and translation invariant properties of Zernike moments. 2006 Elsevier B.V. All rights reserved. Keywords: Zernike moments; Approximation error; Geometrical error; Numerical error; Square-to-circular mapping; Exact Zernike moments
1. Introduction The set of orthogonal Zernike moments was first introduced for image analysis by Teague [1]. It is a set of complex orthogonal functions with a simple rotational invariant property which forms a complete orthogonal basis over the class of square integrable functions named as Zernike polynomials which is defined over the unit circle [2]. Although it is computationally complex if compared to other moment functions such as geometric and Legendre moments, Zernike moments had been proven to be superior in terms of their feature representation capability [3,4], image reconstruction capability [5,6] and low noise sensitivity [7,8]. Besides that, the orthogonal property also enables the separation of the individual contributions of each order moment to the reconstruction process. This will make the *
Corresponding author. Tel.: +60 3 7967 5253; fax: +60 3 7967 5316. E-mail addresses: [email protected] (C.-Y. Wee), [email protected] (R. Paramesran). 0262-8856/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2006.07.010
image reconstruction or inverse transform process of Zernike moments much more easier to be performed if compared to other moment functions. Zernike moment is rotational invariant by nature and the translation and scale invariants of Zernike moment are achieved easily through the normalization of its orthogonal polynomials [9]. These invariant properties enable the Zernike moments to form a robust set of image feature representation. Due to these superiorities, the set of Zernike moments is widely applied in image analysis includes invariant pattern or object recognition [10,11,12,13,14], image reconstruction [5,15], edge detection [16], image segmentation [17], optimal corneal surfaces modeling [18,19], watermarking [20], face recognition [21,22], content-based retrieval [23] and palmprint verification [24]. The definition of Zernike moments has a form of mapping the discrete-space image function which is normally in square or rectangular shape onto Zernike polynomials over a unit disk. The set of Zernike polynomials needs to be approximated by sampling at fixed interval when it is applied to a discrete-space image. Hence, these
968
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
approximate values of Zernike moments introduce the approximation error which in somehow degraded its information representation ability. Not like Legendre moments, the approximation error of Zernike moments can be divided into two categories, namely the geometrical error and numerical error [25]. The geometrical error is caused by the projection of a square discrete image onto the unit disk of Zernike polynomials. In the traditional approach, the pixel where its center falls outside the border of unit disk is not included in Zernike moments computation [25]. This approach will not introduce any error only when the input image is also circular in shape. For square image, information at the four corners of the image will be excluded in computation and hence information lost occurs. However, this geometrical error generally decreases as the moment order used is increased [15]. The numerical error is caused by the needs of calculating the double integral over a fixed sampling intervals. The traditional approach for solving the double integral is based on the zeroth order approximation. Another approach proposed by Liao and Pawlak in [25] to minimize this error is based on the extended version of Simpson’s rules (multipledimension cubature rules). As opposed to the geometrical error, this numerical error is increasing as the moment order used is increased [15]. In this paper, the computational aspect of Zernike moments in terms of error analysis is carried out in details. The inherent limitation in precision of computing the Zernike moments due to geometric nature of a circular domain is concerned. In order to minimize this geometrical error, a square to circular mapping technique is used where all pixels of a discrete square image are mapped inside the unit disk of Zernike polynomials and hence included in moment computation. This approach is very similar to the method proposed by Chong and et al. [9,26] excepts the way in computing the transformed image coordinate. This transformed image coordinate is important for the computation of exact Zernike moments through simple mathematical integration. The numerical error of Zernike moments is solved by mathematically integrating the Zernike polynomials over the corresponding intervals of the image pixels. This method will provide the theoretical exact values of the double integral of Zernike polynomials only under the aforementioned square to circular mapping method. If these exact Zernike moments are computed with traditional mapping approach, the moment values computed are not exact anymore due to information lost. The set of Zernike moments can be expressed as the linear combinations of geometric moments [7]. Hence, it can be implied that by first computing the exact values of the set of monomials and then combining them linearly, we can formulate the set of exact Zernike moments. The effectiveness and advantages of the proposed methods are assessed through several numerical experiments by
means of the moment computation, image reconstruction, numerical instability and invariant property. The organization of the rest of this paper is given as follow. The theoretical overview of Zernike moments is provided in Section 2. Section 3 provides the error analysis of Zernike moments when applied to a two-dimensional discrete image. The square to circular mapping method used for minimizing the geometrical error is discussed in detail in Section 4. Section 5 gives the derivation of the exact Zernike moments. Section 6 provides the experimental validation of the proposed mapping technique in minimizing the geometrical error. Section 7 provides the experimental validation of the proposed exact Zernike moments in eliminating the numerical error. Section 8 concludes the study. 2. Zernike moments The two-dimensional Zernike moments of order p with repetition q of an image intensity function f(x, y), are defined as Z Z pþ1 Z pq ¼ V pq ðx; yÞf ðx; yÞ dx dy ð1Þ p x2 þy 2 61
The pth order Zernike polynomials are defined as ð2Þ V pq ðx; yÞ ¼ Rpq ðrÞejqh ; r 2 ½1; 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y 2 is the length of the vector from the origin to the pixel (x, y), and h = tan1(y/x) is the angle between the vector r and the principle x-axis. The Zernike real-valued radial polynomials are given by Rpq ðrÞ ¼
ðpqÞ=2 X
ð1Þ
s¼0
ðp sÞ!
s
sÞ!ðpjqj sÞ! s!ðpþjqj 2 2
rp2s
ð3Þ
where p jqj is even, 0 6 jqj 6 p and p P 0. By letting s fi (p k)/2, Zernike polynomials in (2) can be represented as V pq ðx; yÞ ¼
p X
Bpqk rk ejqh
ð4Þ
k¼q
with the polynomial coefficient, Bpqk is defined as pk
Bpqk ¼
ð1Þ 2 ðpþk Þ! 2 Þ!ðkþq Þ!ðkq Þ! ðpk 2 2 2
ð5Þ
Hence, the definition of Zernike moments in (1) can be rewritten in polar form as Z Z p pþ1 1 p X Z pq ¼ Bpqk rk ejqh f ðr; hÞr dr dh ð6Þ p 1 p k¼q with dx dy = rdr dh and p 6 h 6 p. Graphs for the first four orders of radial polynomials, Rpq(r) are shown in Fig. 1.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
with Zpq is computed over the same unit disk and kp = (p + 1)dA/p is the normalizing constant. dA is the elemental area of the normalized square image in discrete form when the square image is mapped over the unit disk of Zernike polynomials. This elemental area provides different values with different square to circular mapping techniques. Therefore, proper mapping technique plays an crucial role in minimizing the geometrical error. If only Zernike moments of order 6Pmax are given, then the function f(x, y) can be approximated by a continuous function in a truncated series as
Orthogonal Radial Polynomials, Rpq(r) R00
1 0.8
R40
Radial Polynomials Values
0.6
R11
0.4
R22 R33
0.2
R44
0 R20 R31
R42
969
f^ ðx; yÞ ¼
p P max X X
kp Z pq V pq ðx; yÞ
ð12Þ
p¼0 q¼0
0
0.1
0.2
0.3
0.4
0.5 r
0.6
0.7
0.8
0.9
1
Fig. 1. Graphs for the first four orders of radial polynomials, Rpq(r).
The set of Zernike polynomials, Vpq(r, h) forms a complete orthogonal set on the interval [1, 1] as Z 1 Z p V pq ðr; hÞ½V p0 q0 ðr; hÞ ; r dr dh 1 p ð7Þ p d 0 d 0 p ¼ p 0 ; q ¼ q0 ; pþ1 pp qq ¼ 0 otherwise and its radial polynomials also satisfy the orthogonality relation as ( Z 1 1 dpp0 p ¼ p0 ; ð8Þ Rpq ðrÞ½Rp0 q ðrÞ ; r dr ¼ 2ðpþ1Þ 0 otherwise 1 where dij is the Kronecker delta. The set of Zernike moments possesses the inherit rotational invariant property. The invariant features are obtained by taking the magnitude values of Zernike moments since these values are remain identical to those image functions before and after rotation. The set of Zernike moments of an image which is rotated by a angle is given as Z rpq
¼ Z pq e
jqa
ð9Þ
where Z rpq are the Zernike moments of rotated image and Zpq are the Zernike moment of original image. The rotation invariant Zernike moments are extracted by considering only their magnitude values as jZ rpq j ¼ jZ pq ejqa j ¼ jZ pq j
ð10Þ
Only magnitude of Zernike moments with q P 0 are concerned since Z p;q ¼ Z pq and jZp,qj = jZp,qj [27]. The piece-wise continuous and bounded image intensity function f(x, y) can be expressed as an infinite series of Zernike polynomials expansion over the unit disk as f ðx; yÞ ¼
p 1 X X p¼0 q¼0
kp Z pq V pq ðx; yÞ
ð11Þ
3. Approximation error of Zernike moments The aforementioned favorable properties of the Zernike moments are valid as long as one uses a truly analog image function. Since most of the computer images are discrete in format and square or rectangle in shape, there is an inherent error when computing Zpq due to the circular nature and the approximation of the double integral term in Zernike polynomials. For a general two-dimensional image, this approximation error, Epq between the continuous original Zpq and discrete approximated Z~ pq can be decomposed into geometrical and numerical errors. The geometrical error is caused by the square to circular mapping procedure while the numerical error is caused by the discrete approximation of double integral [25]: ðnÞ Epq ¼ Z pq Z~ pq ¼ EðgÞ pq þ E pq
ð13Þ
ðnÞ where EðgÞ pq denotes the geometrical error and E pq denotes the numerical error. The details of the approximation error is explored in following subsections.
3.1. Geometrical error Since the set of Zernike polynomials is defined in terms of polar coordinate (r, h) with jrj 6 1 as given in (2), then the computation of Zernike moments requires a linear mapping process to map the image coordinates (i, j) to a suitable unit circular domain (r, h) 2 R2. The general form of mapping technique is given as xi ¼ c1 i þ c2 ;
y j ¼ c1 j þ c2
ð14Þ
where the coefficients c1 and c2 are given as c1 ¼
2h ; N 1
c2 ¼ h
ð15Þ
with 0 6 h 6 1 and h 2 R. The value of h is determined based on the applied square to circular mapping technique. The traditional mapping approach is showed in Fig. 2. This approach maps the (N · N) square image plane onto a unit disk by first taking the center of the image as the origin.
970
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
pþ1 Z~ pq ¼ p
N 1 X N 1 X i¼0
f ðxi ; y j ÞV pq ðxi ; y j ÞDxi Dy j
ð17Þ
j¼0
This approximation of double integral term causes error in the computed Zernike moments and the error can be determined as ~ EðnÞ pq ¼ Z pq Z pq ¼ Fig. 2. The traditional square to circular mapping technique.
The pixel coordinates are then normalized over the range of the unit disk as shown in Fig. 2(b). In this traditional approach, the h value is equal to the radius of unit disk; i.e., h = 1. Hence, the values of c1 and c2 are obtained as 2/(N 1) and 1, respectively. Therefore, the elemental area of this projection, dA, which is the ratio of the area of unit disk to the total number of pixels in image plane is given as p/N2. Unfortunately, this traditional approach introduces the geometrical error during Zernike moments computation. The pixels which their center fall outside the border of the unit disk are ignored during the moment computation. All the information carried by these pixels will be lost and the condition becomes worse if these pixels contain important information of the image. The lost of these information will degrade the image representation capability and also the quality of reconstructed image. 3.2. Numerical error ðnÞ The numerical error, Epq is caused by the needs of computing the double integral term in the definition of Zernike moments as shown in (1) and (6). Let the image coordinates of a discrete-space (N · N) square image, f(x, y) is defined as a discrete set of points (xi, yj). Then, the set of Zernike moments in (1) can be rewritten as Z xi þDx2 i N 1 X N 1 pþ1 X f ðxi ; y j Þ Z pq ¼ Dx p i¼0 j¼0 xi 2 i
Z
yj þ
Dy j 2
Dy yj 2j
V pq ðx; yÞ dx dy
ð16Þ
where (Dxi = xi+1 xi) and (Dyj = yj+1 yj) are the intervals between two successive image pixels along x and y Cartesian axis, respectively. In the traditional approach, the double integral term is often evaluated using the zeroth order approximation method where the values of Zernike polynomials are assumed to be constant over the intervals ½xi Dx2 i ; xi þ Dx2 i Dy Dy and ½y j 2 j ; y j þ 2 j , and the value of each interval is obtained by sampling the Zernike polynomials at the central point of these intervals. Hence, the set of approximated Zernike moments computed using the zeroth order approximation order is given as
N 1 X N 1 pþ1 X f ðxi ; y j Þ p i¼0 j¼0 "Z Dxi Z # Dy j xi þ 2 yj þ 2 V pq ðx; yÞ dx dy V pq ðxi ; y j ÞDxi Dy j Dy xi
Dxi 2
yj
j
2
ð18Þ It should be noted that this error increases as the number of sampling points decreases and increases further if the order of moments is increased [15]. The increment of numerical error is caused by the sampling approximation of Zernike radial polynomials. The Zernike radial polynomials are oscillating like sine and cosine functions within the interval [1, 1] as shown in Fig. 1. As the moment order increases, the shape of polynomial will oscillate at higher spatial-frequency. Since the moment kernels of the approximated Zernike moments are approximated by sampling the Zernike polynomials, the moment kernels are susceptible to information loss if it is undersampled. The number of sampling points is determined by the size of the image, (N · N). Hence, if the moment order of p < N is considered, then the sampling of the Zernike polynomials becomes insufficient and the resulting moment kernels are undersampled. Consequently, the set of Zernike moments computed based on these undersampled Zernike polynomials cannot get rid from the approximation error. In order to minimize the error caused by this double integral approximation, Liao and et al. had proposed several cubature formulas based on the extended Simpson’s rules which use a two-dimensional approximation approach for the double integral term [6,15,25]. This approach successfully improves the image reconstruction performance. However, the computed moment values is still not exact and the reconstruction error is increasing after certain moment order. For certain images, high order Zernike moments are required to represent the finest information of the images. However, numerical instability is arisen when the moment order used is over certain limit due to the involvement of factorial terms in polynomial definition. The numerical instability causes the computed moment values deviate from the actual values and hence causes tremendous degradation in the reconstructed images. In order to overcome this problem, Liao et al. had proposed a modified set of Zernike moments to increase the maximum moment order before the occurrence of numerical instability. This modified set of Zernike moments is computed by restricting the radius as r = x2 + y2 < 1. This
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
971
approach successfully increases the maximum moment order from around 25th to around 50th order [15]. However, same numerical instability problem occurs when more than 50th order of Zernike moments are computed. 4. Geometrical error minimization In order to minimize the geometrical error and provide the essential conditions for the exact Zernike moments computation, a square to circular mapping technique which is very similar to the technique proposed by Chong et al. [9,26] is used in this paper. In this mapping technique, the image projection operation used is same with the technique in [26] but the operation of computing the transformed coordinate is slightly different. Both technique map the square image into the unit disk of Zernike polynomials as shown in Fig. 3. By using this technique, the problem of information lost in traditional approach is overcome and hence the geometrical error is minimized. As shown in Fig. 3, the entire square image is bounded inside the unit disk. This approach ensures that there is no pixel loss during moment computation and thus all information of the square image can be preserved during moment computation. The transformed coordinate (xi, yj) of this mapping technique is computed as below. Firstly,pthe ffiffiffi h value of this mapping technique is obtained as 1= 2. The coefficients c1ffiffiffiand c2 are determined p pffiffiffi based on (15) and are equal to 2=ðN 1Þ and 1= 2, respectively. The new elemental area is derived from the ratio between the area of normalized square image to the total pixels of the image and is given as dA = 2/N2. For the technique in [26], the transformed coordinate is computed using the traditional formula in (14); i.e., the computed transformed coordinate is located at the edges of the transformed image pixel as shown in Fig. 4. However in our technique, the computed transformed coordinate is located at the center of the transformed image pixel as shown in Fig. 5. This is to ensure that the mapping technique used can provide the essential conditions for the exact Zernike moment computation through the simple mathematical integration. Hence, the area of the computed
a
Fig. 4. The transformed coordinate and intervals computed in [26].
Fig. 5. The transformed coordinate and intervals computed in this paper.
intervals, Dxi and Dyj are different in location for both techniques. In order to make sure that the center of each pixel is used in moment computation, the value of ‘0.5’ is included in computing every (xi, yj). Hence, the general formula used to compute the transformed coordinate (xi, yj) in (14) is modified as xi ¼ c1 ði þ 0:5Þ þ c2 ;
where the coefficients c1 and c2 are given as pffiffiffi 1 2 ; c2 ¼ pffiffiffi c1 ¼ N 2
N-1
ð19Þ
ð20Þ
Nevertheless, this technique is not a linear transformation because a pixel in the unit disk represents more than one pixel of a square image. It causes the area of square image has been scaled down by 50% after mapping inside the unit disk. Furthermore, this approach has lower reconstruction power and requires higher order of Zernike moments for image reconstruction if compared to the aforementioned traditional approach. This is because the reconstructed coarse information of the image through the low order
b 0
y j ¼ c1 ðj þ 0:5Þ þ c2
y
i 1
h
x
N-1 j
Fig. 3. The square to circular mapping approach where whole square image is mapped inside a unit disk.
972
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Zernike moments is degraded by the scaling process. Nevertheless, this approach performs well in reconstructing the fine information using the high order Zernike moments. It also increases the maximum moment order before the occurrence of numerical instability besides maintaining all the information carried by the square image.
Notice that Hp(xi) and Hq(yj) are independent of the image and can always be precomputed, stored and recalled later to avoid repetitive computation. Based on (25) and (26), (22) can be rewritten as N 1 X N 1 X f ðxi ; y j ÞH p ðxi ÞH q ðy j Þ ð27Þ M pq ¼ i¼0
5. Numerical error minimization In the previous section, we have discussed that the numerical error is due to the zeroth order approximation of the double integral term as shown in (18). This problem can be solved by mathematically integrating the Zernike polynomials over the corresponding intervals of the image pixels. By using this method, the computed Zernike moments are exactly equal to the theoretical values. The exact computation of Zernike moments is derived based on the exact geometric moments and the relationship between geometric moments and Zernike moments. 5.1. Exact computation of geometric moments The geometric moments of order (p + q) for (N · N) square image f(x, y) are defined within the interval of [1, 1] · [1, 1] as Z 1 Z 1 M pq ¼ xp y q f ðx; yÞ dx dy ð21Þ 1
1
If the image is defined only for a discrete set of points (xi, yj), then (21) can be rewritten as Z xi þDx2 i Z y j þDy2 j N 1 X N 1 X M pq ¼ f ðxi ; y j Þ xp y q dx dy Dy i¼0
¼
xi
j¼0
N 1 X N 1 X i¼0
Dxi 2
yj
j
2
f ðxi ; y j Þhp ðxi Þhq ðy j Þ
ð22Þ
j¼0
with hp(xi) and hq(yj) are given as Z xi þDx2 i xp dx hp ðxi Þ ¼ xi
Dxi 2
ð23Þ
hq ðy j Þ ¼
Z
yj þ
Dy j 2
Dy j yj 2
y q dy
ð24Þ
The exact computation of hp(xi) and hq(yj) based on simple mathematical integration rule are given as pþ1 xi þDx2 i Z xi þDx2 i x p x dx ¼ ¼ H p ðxi Þ ð25Þ Dx p þ 1 xi Dxi xi 2 i 2
and Z
yjþ
yj
Dy j 2
Dy j 2
y qþ1 y dy ¼ qþ1 q
Note that there is no approximation involved in (27), so the moments computed using this method is named as the exact geometric moments. 5.2. Exact computation of Zernike moments Since different sets of polynomials up to the same order define the same subspace, any complete set of moments up to a given order can be obtained from any other set of moments up to same order, at least in theory [7,8]. Hence, most of the moment functions include Zernike moments can be computed exactly or approximately from the simple geometric moments. The set of Zernike moments can be computed using the geometric moments up to the same order as [7] p q s X X s q pþ1 X Z pq ¼ Bpqk wn M k2mn;2mþn p k¼q m n m¼0 n¼0 ð28Þ where i q > 0; w¼ þi q 6 0
y j þDy2 j yj
Dy j 2
¼ H q ðy j Þ
ð26Þ
ð29Þ
with pffiffiffiffiffiffiffi 1 s ¼ ðk qÞ; and i ¼ 1 2 Based on (28) and (27), the set of Zernike moments can be computed exactly as p q s X X s q pþ1 X Z pq ¼ Bpqk wn p k¼q m n m¼0 n¼0 ð30Þ N 1 X N 1 X f ðxi ; y j ÞH k2mn ðxi ÞH 2mþn ðy j Þ i¼0
and
j¼0
j¼0
Similar to the exact geometric moments computation, there is no approximation involved in computing the double integral in exact Zernike moments computation. By using Eq. (30), the numerical error occurs in Zernike moment computation is eliminated because it provides the exact Zernike moment values. 6. Experimental study on geometrical error In this section, the proposed square to circular mapping technique is compared with the traditional mapping technique. The performance of both techniques are evaluated through the image reconstruction of several grayscale images. The testing images used are five 30 · 30 grayscale Chinese character images as shown in Fig. 6.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
973
Fig. 6. Grayscale Chinese character images used in computer simulation. From left to right are C1, C2, C3, C4 and C5.
The reconstruction error between the original image and the reconstructed image is used as the measurement of the reconstruction capability for the traditional and proposed mapping techniques. The reconstruction error, e is determined as ( ) 2 N 1 X N 1 X ½f ði; jÞ f^ ði; jÞ e¼ ð31Þ ½f ði; jÞ2 i¼0 j¼0 where f(i, j) and f^ ði; jÞ are the original and reconstructed images, respectively. Average reconstruction error for five Chinese character images with different maximum order of Zernike moments are shown in Fig. 7. From Fig. 7, it is very clear that the proposed square to circular mapping technique outperforms the traditional mapping technique. The average reconstruction error of traditional technique up to 45th order is decreasing from 0.85 to 0.40. The minimum reconstruction error that can be achieved using the traditional technique is about 0.40 at 45th order. The large reconstruction error when using the traditional technique is caused by the geometrical error where the pixels which their center located outside the unit disk are excluded in Zernike moment computation. All the information carried by these pixels is lost after reconstruction. These excluded pixels are about 21.5% of the overall pixels of a (N · N) image. The reduction of reconstruction error using the proposed technique is significant. The average reconstruction
error is reduced to as low as 0.05 at 45th order. The significant reduction is due to the inclusion of all image pixels in the Zernike moment computation. The effect of geometrical error is reduced to its minimum level by using the proposed mapping technique. This experimental result also proves that the geometrical error dominates the reconstruction error due to the inherit property of Zernike polynomials which are defined over a unit disk. The increment of moment order used causes the decrement in reconstruction error (geometrical error) which is congruence with the findings in [15]. The reconstructed images for five testing images with different maximum order using the traditional and proposed techniques are shown in Fig. 8. For traditional technique, the reconstructed images facing the problem of missing information at their four corners. The intensity values after reconstruction also smaller as shown by the lighter illumination of the reconstructed characters. These problems are not faced by the proposed technique. Therefore, the reconstruction error for proposed technique is much smaller and better reconstruction capability is shown. 7. Experimental study on numerical error In this section, computer simulations are performed to validate the proposed theoretical framework in providing the exact Zernike moments and hence minimizing the numerical error. The reconstruction power of the proposed method also been explored. 7.1. Moment computation
Average Reconstruction Error vs. Moment Order 1 Proposed Traditional
The set of Zernike moments is computed from simple artificial images using several Zernike moment computation methods with two mapping approaches, i.e., the traditional and proposed approaches. The computation methods used included the proposed exact Zernike moments (EZM), zeroth order approximation (ZOA) and several extended Simpson’s rules (ESR). The ESR used are two five-dimensional cubature formulas and two nine dimensional cubature formulas. The five-dimensional cubature formulas for first type, C I5 and second type, C II5 were proposed by Liao and et al. and are given as [25]
0.9
Average Reconstruction Error
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
45
50
Moment Order
Fig. 7. Average reconstruction error for five Chinese character images with different maximum order of Zernike moments using the traditional and proposed mapping methods.
1 C I5 ðV Þ ¼ f8V ð0; 0Þ þ V ð0; 1Þ þ V ð1; 0Þ þ V ð0; 1Þ 3 þ V ð1; 0Þg ð32Þ 4 C II5 ðV Þ ¼ fV ð0; 0Þ þ V ð0; 0:5Þ þ V ð0:5; 0Þ þ ð0; 0:5Þ 3 þ V ð0:5; 0Þg ð33Þ
974
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Fig. 8. The reconstructed images using traditional and proposed mapping approaches with different maximum order. (Trad. = Traditional approach, Prop. = Proposed approach.)
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
The nine-dimensional cubature formulas of first type, C I9 and second type, C II9 are given as [28] C I9 ðV Þ ¼
1 f16V ð0; 0Þ þ 4½V ð0; 1Þ þ V ð1; 0Þ þ V ð0; 1Þ 36 þV ð1; 0Þ þ ½V ð1; 1Þ þ V ð1; 1Þ þ V ð1; 1Þ
þV ð1; 1Þg ð34Þ n pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 C II9 ðV Þ ¼ 64V ð0; 0Þ þ 40½V ð0; 0:6Þ þ V ð 0:6; 0Þ 324 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi þV ð0; 0:6Þ þ V ð 0:6; 0Þ þ 25½V ð 0:6; 0:6Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi þV ð 0:6; 0:6Þ þ V ð 0:6; 0:6Þ þ V ð 0:6; pffiffiffiffiffiffiffi o 0:6Þ ð35Þ (1) Artificial image I: The artificial image used is a (8 · 8) image with all its pixel intensity values are constant, i.e., f(i, j) = K, K = constant, "(i, j). Small artificial image is used in order for the approximation error to be observed readily by directly compared with the theoretical values. In traditional mapping technique, it is evident that all Zernike moments should provide the value of zero, except Z00. This can be shown mathematically by performing the Zernike moments computation through the following equation p q s X X X s q Z pq ¼kp Bpqk wn m n k¼q m¼0 n¼0 " # pffiffiffiffiffiffiffi Z 1 Z 1x2 ðk2mnÞ ð2mþnÞ x dy dx pffiffiffiffiffiffiffi y K ¼ 0
1x2
1
if p ¼ q ¼ 0;
ð36Þ
otherwise:
975
Providing that if (k 2m n) and/or (2m + n) is odd, then (36) will give zero values. Therefore, only the even order of Zernike moments are taken into the consideration in this case. For case K = 1, only the zeroth order Zernike moment gives the nonzero value, i.e., Z00 = 1 while others give zero values. The results for exact Zernike moments (EZM, Zpq (30)), the zeroth order approximation (ZOA, Z~ pq (17)) and several extended Simpson’s rules (ESR, C I5 (32), C II5 (33), C I9 (34), C II9 (35)) are shown in Table 1. The values given by the EZM are quite far away from the theoretical values calculated using (36). The C II5 of ESR provides closest values and followed by ZOA, C I5 , EZM, C II9 and C I9 . The values provided by all these methods are deviated from the theoretical values due to the information lost in square to circular mapping. The performances of these methods are highly dependent on how many pixels are inside the unit disk and hence used for the Zernike moments computation. The same procedures are repeated for same artifical image using the proposed mapping technique. Eq. (36) is not suitable for computing the theoretical values of Zernike moments under this circumstance and hence a novel formula is proposed to compute them. The novel formula proposed specifically under this circumstance for computing the theoretical values of Zernike moments is given as p q s X X X s q n Z pq ¼ kp Bpqk w m n k¼q m¼0 n¼0 Z p1ffi Z p1ffi 2 2 xðk2mnÞ dx y ð2mþnÞ dy ð37Þ p1ffi p1ffi 2 2
Table 1 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = K, K = 1 using traditional mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 2 2 4 4 4 6 6 6 6 8 8 8 8 8 10 10 10 10 10 10
0 0 2 0 2 4 0 2 4 6 0 2 4 6 8 0 2 4 6 8 10
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1.0345 0.1393 0 0.3614 0 0.0754 0.8257 0 0.0391 0 1.7764 0 0.0637 0 0.5101 3.7521 0 0.2868 0 1.2067 0
1.0345 0.0746 0 0.0179 0 0.0758 0.2105 0 0.0109 0 0.6119 0 0.1463 0 0.5095 0.9427 0 0.2644 0 0.6178 0
1.0345 0.1069 0 0.1898 0 0.0762 0.3101 0 0.0336 0 0.6003 0 0.0463 0 0.5087 1.4873 0 0.0108 0 0.9035 0
1.0345 0.1069 0 0.1879 0 0.0759 0.2824 0 0.0290 0 0.4144 0 0.0812 0 0.5093 0.6694 0 0.1809 0 0.9082 0
1.0345 0.1393 0 0.3634 0 0.0758 0.8554 0 0.0440 0 1.9796 0 0.0283 0 0.5095 4.6818 0 0.1265 0 1.2019 0
1.0345 0.1393 0 0.3614 0 0.0754 0.8257 0 0.0391 0 1.7757 0 0.0639 0 0.5101 3.7436 0 0.2893 0 1.2067 0
(Theo. = Theoretical values).
976
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
The results of EZM, ZOA and ESR with the proposed mapping technique are shown in Table 2. Note that the values provided by EZM are exactly match with the theoretical values computed using (37) while ZOA and ESR deviate from them except for C II9 . The deviation from theoretical values becomes more severe when higher order of Zernike moments are computed. Although the Zernike moment values computed using C II9 are very similar to the theoretical values, however it still not providing the exact Zernike moment values. (2) Artificial image II: For randomly generated small (4 · 4) artifical image, f(i,j) = F which is given as 9 8 16 2 3 13 > > > > > = < 5 11 10 8 > ð38Þ F¼ > 9 7 6 12 > > > > > ; : 4 14 15 1 The results are shown in Table 3. Once again only the values provided by EZM match the theoretical values while ZOA and ESR deviate severely from the theoretical values especially when the moment order increases. 7.2. Image reconstruction The comparative assessment between the approximated and exact Zernike moments in image reconstruction is performed through several numerical experiments using both the artificial and real images. (1) Random images: For the sake of fairness, randomly generated images are used in this experiment. The purpose of using randomly generated images is to minimize the effects of image content on the computed Zernike
moments. Initially, an artificial square image is generated randomly as f ði; jÞ ¼ randomðN ; N Þ;
0 6 f ði; jÞ 6 255; 8i; j
ð39Þ
The size of the image is determined by setting N = 64 and then is scaled to different sizes, i.e., 32, 16, 8. Zernike moments of these images are computed using the aforementioned methods up to maximum order, P = 40. The reconstruction error between the original and reconstructed images is measured using (31). Same procedures are repeated for 20 similar randomly generated images and their average error is shown in Fig. 9. It is clearly shown that for different image sizes, the exact Zernike moments consistently provide lesser reconstruction error if compared to the approximated Zernike moments (ZOA and ESR). The reconstruction error of exact Zernike moments is steadily decreasing while that of the approximated Zernike moments begin to increase after certain moment order. The reconstruction error is caused by the application of the truncated Zernike moments up to certain maximum order. The approximation error is deceasing as the increment of the moment order because more Zernike moments are included in reconstruction process. For the approximated Zernike moments, the reconstruction error is similar with the exact Zernike moments only when the moment order used is small due to small numerical error and sometimes can be ignored. However, as the moment order increases, the numerical error of the approximated Zernike moments becomes larger and hence causes the increment in reconstruction error. (2) Real images: Image reconstruction process is repeated for several grayscale real images. These real images are
Table 2 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = K, K = 1 using proposed mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 2 2 4 4 4 6 6 6 6 8 8 8 8 8 10 10 10 10 10 10
0 0 2 0 2 4 0 2 4 6 0 2 4 6 8 0 2 4 6 8 10
0.6366 0.6366 0 0.2122 0 0.2122 0.2122 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1273 0 0.1273 0 0.1273 0
0.6366 0.6366 0 0.2122 0 0.2122 0.2122 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1273 0 0.1273 0 0.1273 0
0.6366 0.6565 0 0.24402 0 0.2122 0.2215 0 0.2590 0 0.1491 0 0.1764 0 0.1269 0.1669 0 0.1765 0 0.2019 0
0.6366 0.6540 0 0.2401 0 0.2122 0.2199 0 0.2532 0 0.1451 0 0.1710 0 0.1269 0.1643 0 0.1677 0 0.1930 0
0.6366 0.6466 0 0.2284 0 0.2121 0.2150 0 0.2359 0 0.1332 0 0.1548 0 0.1268 0.1564 0 0.1414 0 0.1662 0
0.6366 0.6366 0 0.2119 0 0.2122 0.2143 0 0.2126 0 0.1316 0 0.1255 0 0.1269 0.1226 0 0.1378 0 0.1291 0
0.6366 0.6366 0 0.2122 0 0.2122 0.2121 0 0.2122 0 0.1273 0 0.1273 0 0.1273 0.1275 0 0.1273 0 0.1273 0
(Theo. = Theoretical values).
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
977
Table 3 Comparison of EZM (Zpq), ZOA ðZ~ pq Þ and ESR ðC I5 ; C II5 ; C I9 ; C II9 Þ with the theoretical values for f(i, j) = F using proposed mapping technique p
q
Theo.
Zpq
Z~ pq
C I5
C II5
C I9
C II9
0 1 2 2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 10 10
0 1 0 2 1 3 0 2 4 1 3 5 0 2 4 6 1 3 5 7 0 2 4 6 8 1 3 5 7 9 0 2 4 6 8 10
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8038 0 1.8038 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0823 0 1.0823 0 1.0823 0.6171 2.4684i 0.0440 0.1758i 0.2836 + 1.1342i 0.4421 1.7683i 0.1525 0.6100i 1.0823 0 1.0823 0 1.0823 0
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8038 0 1.8038 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0823 0 1.0823 0 1.0823 0.6171 2.4684i 0.0440 0.1758i 0.2836 + 1.1342i 0.4421 1.7683i 0.1525 0.6100i 1.0823 0 1.0823 0 1.0823 0
5.4113 0 6.0877 0 0.3376 1.3505i 0.3376 1.3505i 2.7479 0 1.7967 0.4431 + 1.7725i 0.7069 + 2.8276i 0.2638 + 1.0551i 2.9963 0 3.4217 0 0.8001 + 3.2003i 0.1899 0.7596i 0.7807 3.123i 0.2093 + 0.8370i 3.0959 0 1.5725 0 1.0233 1.7076 6.8304i 0.3594 1.4375i 0.6363 + 2.5453i 0.8500 3.400i 0.1381 0.5523i 2.9857 0 6.6135 0 3.6239 0
5.4113 0 0.6540 0 0.6540 0 0.2401 0 0.2122 0.2401 0 0.2122 0.2199 0 0.2532 0 0.2199 0 0.2532 0 0.1451 0 0.1710 0 0.1269 0.1451 0 0.1710 0 0.1269 0.1643 0 0.1677 0 0.1930 0
5.4113 0 0.6466 0 0.6466 0 0.2284 0 0.2121 0.2284 0 0.2121 0.2150 0 0.2359 0 0.2150 0 0.2359 0 0.1332 0 0.1548 0 0.1268 0.1332 0 0.1548 0 0.1268 0.1564 0 0.1414 0 0.1662 0
5.4113 0 0.6366 0 0.6366 0 0.2119 0 0.2122 0.2119 0 0.2122 0.2143 0 0.2126 0 0.2143 0 0.2126 0 0.1316 0 0.1255 0 0.1269 0.1316 0 0.1255 0 0.1269 0.1226 0 0.1378 0 0.1291 0
5.4113 0 5.4113 0 0.3376 1.3505i 0.3376 1.3505i 1.8038 0 1.8038 0.2321 + 0.9285i 0.4959 + 1.9835i 0.2638 + 1.0551i 1.8027 0 1.8034 0 0.5495 + 2.1980i 0.1328 0.5311i 0.4669 1.8675i 0.2154 + 0.8616i 1.0675 0 1.0780 0 1.0823 0.6157 2.4629i 0.0426 0.1703i 0.2841 + 1.1366i 0.4415 1.7659i 0.1525 0.6100i 1.1338 0 1.0760 0 1.0749 0
(Theo. = Theoretical values).
some benchmark images which are normally used in image processing field and are shown in Fig. 10. The average reconstruction error of these real images for different Zernike moments computation methods is shown in Fig. 11. The results validate the efficiency of the exact Zernike moments by outperforming all other approximation methods. All approximation methods facing the numerical instability problem after certain maximum order except for C II9 . The reconstruction error of C II9 is small and quite similar with the exact Zernike moments up to a quite large moment order. However, its deviation from exact Zernike moments becomes obvious after the moment order P = 35. This can be explained by the fact that the Zernike moments computed using C II9 are very similar to the exact Zernike moments up to a quite large moment order as shown in Tables 2 and 3. As the moment order increases, the deviation becomes larger and finally provides similar performance as other approximation methods.
7.3. Invariant properties Any proposed method either for efficient or fast computation of Zernike moments, must possess the ability to maintain the invariant properties of the Zernike moments. In order to show the robustness and feasibility of the proposed method, an experiment is performed to evaluate the invariant accuracy of the proposed method for scale and translation. The rotation invariant property of Zernike moments for the proposed method is not implemented since the Zernike moments are already rotation invariant. Since our proposed method in minimizing the approximation error of Zernike moments is based on the linear combination of geometric moments, the Zernike moment invariants can be obtained using the traditional indirect method based on the geometric moment invariants as shown in (40). The general formula of translation and scale invariants of Zernike moments of order p with repetition q, ~ pq , is given as in terms of M
978
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
Fig. 9. Reconstruction error, for images with different sizes using EZM, ZOA and ESR.
Fig. 10. Greyscale real images used in the experiment with resolution of 256 · 256.
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
979
Fig. 11. Average reconstruction error for images in Figure 10.
Table 4 Zernike moment values of the translated and scaled Chinese character images using the proposed methods. Images
Z pq ¼
Z20
Z22
Z40
Z42
Z44
6.3782e02
3.1891e02
3.4928e01
2.328501
5.8213e02
6.3771e02
3.1915e02
3.501101
2.329101
5.8232e02
6.3724e02
3.1962e02
3.502101
2.331401
5.8235e02
6.3809e02
3.1204e02
3.491601
2.324401
5.8260e02
6.3799e02
3.2099e02
3.504201
2.332801
5.8470e02
6.3846e02
3.2243e02
3.513201
2.354701
6.8667e02
p q s X X s q pþ1 X ~ k2mn;2mþn M Bpqk wn p k¼q m n m¼0 n¼0 ð40Þ
~ pq is the translation and scale exact geometric mowhere M ment invariants. Zernike moments of second and fourth orders which are computed from several translated and scaled images of Chinese character are shown in Table 4. The first three rows show the Chinese characters are translated, while rows 4–6 show the scaled images. Though the computed Zernike moment values for respective orders are not the same, but they are within the acceptable margin
of error. This is due to the image pixels lost during the randomly scaling process. 8. Concluding remarks The approximation error of the well known orthogonal Zernike moments for an image function can be decomposed into the geometrical and numerical errors. Efficient computation method is important in making the Zernike moments more reliable in real-world applications. The geometrical error is due to the necessary of the square to circular mapping during the computation of Zernike moments. A square to circular mapping technique which maps all the pixels of an image function inside the unit disk is used in
980
C.-Y. Wee, R. Paramesran / Image and Vision Computing 25 (2007) 967–980
this paper. This mapping technique enables all the image pixels are included in the Zernike moments computation and hence prevents the information lost as normally occurs in the traditional mapping technique. The numerical error is caused by the needs of computing the double integral in the definition of Zernike moments and the use of the truncated approximation series of Zernike moments. The exact computation of Zernike moments is proposed to provide exact representation using Zernike moments and hence reduces the numerical error caused by the traditional zeroth order approximation and extended Simpson’s rules. The reconstruction error for exact Zernike moments is reducing with the increment of moment order while the reconstruction error for other approximated methods increases with the increment of moment order. The proposed method also not lacking in preserving the scale and translation invariant properties of Zernike moments. Acknowledgement The authors would like to extend their thanks to the anonymous reviewers for the valuable and constructive comments for making this manuscript more readable. References [1] M.R. Teague, Image analysis via the general theory of moments, J. Optical Soc. Am. 70 (1980) 920–930. [2] F. Zernike, Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode, Physical 7 (1934) 689–701. [3] S.O. Belkasim, M. Shridhar, M. Ahmadi, Pattern recognition with moment invariants: a comparative study and new results, Pattern Recogn. 24 (12) (1991) 1117–1138. [4] R. Mukundan, K.R. Ramakrishnan, Moment Functions in Image Analysis, World Scientific Publishing, Singapore, 1998. [5] M. Pawlak, On the reconstruction aspect of moment descriptors, IEEE Trans. Inf. Theory 38 (6) (1992) 1698–1708. [6] S.X. Liao, M. Pawlak, On image analysis by moments, IEEE Trans. Pattern Anal. Mach. Intell. 18 (3) (1996) 254–266. [7] C.H. Teh, R.T. Chin, On image analysis by the methods of moments, IEEE Trans. Pattern Anal. Mach. Intell. 10 (4) (1988) 496–512. [8] C.H. Teh, R.T. Chin, On image analysis by the methods of moments, in: Proceedings of Conference on Computer Vision and Pattern Recognition (CVPR’88), 1988, pp. 556–561. [9] C.-W. Chong, A formulation of a new class of continuous orthogonal moment invariants, and the analysis of their computational aspects, PhD thesis, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia, May 2003. [10] S.O. Belkasim, M. Shridhar, M. Ahamadi, Shape recognition using Zernike moments invariants, in: Proceedings of 23rd Annual Asilomar Conference on Signals Systems and Computers, 1989, pp. 167–171.
[11] S. Ghosal, R. Mehrotra, Zernike moment-based feature detectors, in: Proceedings of International IEEE Conference on Image Processing (ICIP-94), vol. 1, 1994, pp. 934–938. [12] C. Kan, M.D. Srinath, Invariant character recognition with Zernike and orthogonal Fourier–Mellin moments, Pattern Recogn. 35 (1) (2002) 143–154. [13] A. Khotanzad, J.H. Lu, Classification of invariant image representations using a neural network, IEEE Trans. Acoust., Speech, Signal Process. 38 (1990) 1028–1038. [14] L. Wang, G. Healey, Using Zernike moments for the illumination and geometry invariant classification of multispectral texture, IEEE Trans. Image Process. 7 (2) (1998) 196–203. [15] S. Liao, Image analysis by moments, PhD thesis, University of Manitoba, Winnipeg, Manitoba, Canada, 1993. [16] S. Ghosal, R. Mehrotra, Edge detection using orthogonal momentbased operators, in: Proceedings of 11th Image, Speech and Signal Analysis (IAPR) International Conference on Pattern Recognition, vol. III, 1992, pp. 413–416. [17] S. Ghosal, R. Mehrotra, Segmentation of range images: an orthogonal moment-based integrated approach, IEEE Trans. Robot. Autom. 9 (4) (1993) 385–399. [18] D.R. Iskander, M.J. Collins, B. Davis, Optimal modeling of corneal surfaces with Zernike polynomials, IEEE Trans. Biomed. Eng. 48 (1) (2001) 87–95. [19] D.R. Iskander, M.R. Morelande, M.J. Collins, B. Davis, Modelling of corneal surfaces with radial polynomials, IEEE Trans. Biomed. Eng. 49 (4) (2002) 320–328. [20] Y. Xin, S. Liao, M. Pawlak, Geometrically robust image watermarking via pseudo-Zernike moments, in: Proceedings of 2004 Canadian Conference on Electrical and Computer Engineering, vol. 2, 2004, pp. 939–942. [21] J. Haddadnia, M. Ahmadi, K. Raahemifar, An effective feature extraction method for face recognition, in: Proceedings of 2003 International Conference on Image Processing, vol. 3, 2003, pp. III– 917–920. [22] J. Haddadnia, M. Ahmadi, K. Faez, An feature extraction method with pseudo-Zernike moments in rbf neural network-based human face recognition system, EURASIP J. Appl. Signal Process. (2003) 890–901. [23] Y.-S. Kim, W.-Y. Kim, Content-based trademark retrieval system using visually salient features, in: Proceedings of 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1997, pp. 307–312. [24] Y.-H. Pang, Andrew B.-J. Teoh, David C.-L. Ngo, F.-S. Hiew, Palmprint verification with moments, J. WSCG 12 (2003) 1–3. [25] S.X. Liao, M. Pawlak, On the accuracy of Zernike moments for image analysis, IEEE Trans. Pattern Anal. Mach. Intell. 20 (12) (1998) 1358–1364. [26] C.W. Chong, P. Raveendran, R. Mukundan, A comparative analysis of algorithm for fast computation of Zernike moments, Pattern Recogn. 36 (3) (2003) 731–742. [27] A. Khotanzand, Y.H. Hong, Invariant image recognition by Zernike moments, IEEE Trans. Pattern Anal. Mach. Intell. 12 (5) (1990) 489– 497. [28] M. Abramowitz, I.A. Segun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, 1965.