In mathematics , pseudo-Zernike polynomials are well known and widely used in the analysis of optical systems. They are also widely used in image analysis as shape descriptors .
They are an orthogonal set of complex -valued polynomials
defined as
V
n
m
(
x
,
y
)
=
R
n
m
(
x
,
y
)
e
j
m
arctan
(
y
x
)
,
{\displaystyle V_{nm}(x,y)=R_{nm}(x,y)e^{jm\arctan({\frac {y}{x}})},}
where
x
2
+
y
2
≤
1
,
n
≥
0
,
|
m
|
≤
n
{\displaystyle x^{2}+y^{2}\leq 1,n\geq 0,|m|\leq n}
and orthogonality on the unit disk is given as
∫
0
2
π
∫
0
1
r
[
V
n
l
(
r
cos
θ
,
r
sin
θ
)
]
∗
×
V
m
k
(
r
cos
θ
,
r
sin
θ
)
d
r
d
θ
=
π
n
+
1
δ
m
n
δ
k
l
,
{\displaystyle \int _{0}^{2\pi }\int _{0}^{1}r[V_{nl}(r\cos \theta ,r\sin \theta )]^{*}\times V_{mk}(r\cos \theta ,r\sin \theta )\,dr\,d\theta ={\frac {\pi }{n+1}}\delta _{mn}\delta _{kl},}
where the star means complex conjugation, and
r
2
=
x
2
+
y
2
{\displaystyle r^{2}=x^{2}+y^{2}}
,
x
=
r
cos
θ
{\displaystyle x=r\cos \theta }
,
y
=
r
sin
θ
{\displaystyle y=r\sin \theta }
are the standard transformations between polar and Cartesian coordinates.
The radial polynomials
R
n
m
{\displaystyle R_{nm}}
are defined as[ 1]
R
n
m
(
r
)
=
∑
s
=
0
n
−
|
m
|
D
n
,
|
m
|
,
s
r
n
−
s
{\displaystyle R_{nm}(r)=\sum _{s=0}^{n-|m|}D_{n,|m|,s}\ r^{n-s}}
with integer coefficients
D
n
,
|
m
|
,
s
=
(
−
1
)
s
(
2
n
+
1
−
s
)
!
s
!
(
n
−
|
m
|
−
s
)
!
(
n
+
|
m
|
−
s
+
1
)
!
.
{\displaystyle D_{n,|m|,s}=(-1)^{s}{\frac {(2n+1-s)!}{s!(n-|m|-s)!(n+|m|-s+1)!}}.}
Examples are:
R
0
,
0
=
1
{\displaystyle R_{0,0}=1}
R
1
,
0
=
−
2
+
3
r
{\displaystyle R_{1,0}=-2+3r}
R
1
,
1
=
r
{\displaystyle R_{1,1}=r}
R
2
,
0
=
3
+
10
r
2
−
12
r
{\displaystyle R_{2,0}=3+10r^{2}-12r}
R
2
,
1
=
5
r
2
−
4
r
{\displaystyle R_{2,1}=5r^{2}-4r}
R
2
,
2
=
r
2
{\displaystyle R_{2,2}=r^{2}}
R
3
,
0
=
−
4
+
35
r
3
−
60
r
2
+
30
r
{\displaystyle R_{3,0}=-4+35r^{3}-60r^{2}+30r}
R
3
,
1
=
21
r
3
−
30
r
2
+
10
r
{\displaystyle R_{3,1}=21r^{3}-30r^{2}+10r}
R
3
,
2
=
7
r
3
−
6
r
2
{\displaystyle R_{3,2}=7r^{3}-6r^{2}}
R
3
,
3
=
r
3
{\displaystyle R_{3,3}=r^{3}}
R
4
,
0
=
5
+
126
r
4
−
280
r
3
+
210
r
2
−
60
r
{\displaystyle R_{4,0}=5+126r^{4}-280r^{3}+210r^{2}-60r}
R
4
,
1
=
84
r
4
−
168
r
3
+
105
r
2
−
20
r
{\displaystyle R_{4,1}=84r^{4}-168r^{3}+105r^{2}-20r}
R
4
,
2
=
36
r
4
−
56
r
3
+
21
r
2
{\displaystyle R_{4,2}=36r^{4}-56r^{3}+21r^{2}}
R
4
,
3
=
9
r
4
−
8
r
3
{\displaystyle R_{4,3}=9r^{4}-8r^{3}}
R
4
,
4
=
r
4
{\displaystyle R_{4,4}=r^{4}}
R
5
,
0
=
−
6
+
462
r
5
−
1260
r
4
+
1260
r
3
−
560
r
2
+
105
r
{\displaystyle R_{5,0}=-6+462r^{5}-1260r^{4}+1260r^{3}-560r^{2}+105r}
R
5
,
1
=
330
r
5
−
840
r
4
+
756
r
3
−
280
r
2
+
35
r
{\displaystyle R_{5,1}=330r^{5}-840r^{4}+756r^{3}-280r^{2}+35r}
R
5
,
2
=
165
r
5
−
360
r
4
+
252
r
3
−
56
r
2
{\displaystyle R_{5,2}=165r^{5}-360r^{4}+252r^{3}-56r^{2}}
R
5
,
3
=
55
r
5
−
90
r
4
+
36
r
3
{\displaystyle R_{5,3}=55r^{5}-90r^{4}+36r^{3}}
R
5
,
4
=
11
r
5
−
10
r
4
{\displaystyle R_{5,4}=11r^{5}-10r^{4}}
R
5
,
5
=
r
5
{\displaystyle R_{5,5}=r^{5}}
The pseudo-Zernike Moments (PZM) of order
n
{\displaystyle n}
and repetition
l
{\displaystyle l}
are defined as
A
n
l
=
n
+
1
π
∫
0
2
π
∫
0
1
[
V
n
l
(
r
cos
θ
,
r
sin
θ
)
]
∗
f
(
r
cos
θ
,
r
sin
θ
)
r
d
r
d
θ
,
{\displaystyle A_{nl}={\frac {n+1}{\pi }}\int _{0}^{2\pi }\int _{0}^{1}[V_{nl}(r\cos \theta ,r\sin \theta )]^{*}f(r\cos \theta ,r\sin \theta )r\,dr\,d\theta ,}
where
n
=
0
,
…
∞
{\displaystyle n=0,\ldots \infty }
, and
l
{\displaystyle l}
takes on positive and negative integer
values subject to
|
l
|
≤
n
{\displaystyle |l|\leq n}
.
The image function can be reconstructed by expansion of the pseudo-Zernike coefficients on the unit disk as
f
(
x
,
y
)
=
∑
n
=
0
∞
∑
l
=
−
n
+
n
A
n
l
V
n
l
(
x
,
y
)
.
{\displaystyle f(x,y)=\sum _{n=0}^{\infty }\sum _{l=-n}^{+n}A_{nl}V_{nl}(x,y).}
Pseudo-Zernike moments are derived from conventional Zernike moments and shown
to be more robust and less sensitive to image noise than the Zernike moments.[ 1]
^ a b Teh, C.-H.; Chin, R. (1988). "On image analysis by the methods of moments". IEEE Transactions on Pattern Analysis and Machine Intelligence . 10 (4): 496–513. doi :10.1109/34.3913 .
Belkasim, S.; Ahmadi, M.; Shridhar, M. (1996). "Efficient algorithm for the fast computation of zernike moments". Journal of the Franklin Institute . 333 (4): 577–581. doi :10.1016/0016-0032(96)00017-8 .
Haddadnia, J.; Ahmadi, M.; Faez, K. (2003). "An efficient feature extraction method with pseudo-zernike moment in rbf neural network-based human face recognition system" . EURASIP Journal on Applied Signal Processing . 2003 (9): 890–901. Bibcode :2003EJASP2003..146H . doi :10.1155/S1110865703305128 .
T.-W. Lin; Y.-F. Chou (2003). A comparative study of zernike moments . Proceedings of the IEEE/WIC International Conference on Web Intelligence. pp. 516–519. doi :10.1109/WI.2003.1241255 . ISBN 0-7695-1932-6 .
Chong, C.-W.; Raveendran, P.; Mukundan, R. (2003). "The scale invariants of pseudo-Zernike moments" (PDF) . Pattern Anal. Applic . 6 (3): 176–184. doi :10.1007/s10044-002-0183-5 .
Chong, Chee-Way; Mukundan, R.; Raveendran, P. (2003). "An Efficient Algorithm for Fast Computation of Pseudo-Zernike Moments" (PDF) . Int. J. Pattern Recogn. Artif. Int . 17 (6): 1011–1023. doi :10.1142/S0218001403002769 . hdl :10092/448 .
Shutler, Jamie (1992). "Complex Zernike Moments" .