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.
Definition 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
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 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}
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