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
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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.
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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.
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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.
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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
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