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} 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 }
The radial polynomials R n m {\displaystyle R_{nm}} [ 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} l {\displaystyle l}
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 } l {\displaystyle l} 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" . 
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