Pseudo-Zernike polynomials

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_{nm}(x,y) = R_{nm}(x,y)e^{jm\arctan(\frac{y}{x})}, $$

where $$x^2+y^2\leq 1, n\geq 0, |m|\leq n$$ and orthogonality on the unit disk is given as



\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 $$ r^2 = x^2+y^2$$, $$x=r\cos\theta$$, $$y=r\sin\theta $$ are the standard transformations between polar and Cartesian coordinates.

The radial polynomials $$R_{nm}$$ are defined as

$$ R_{nm}(r) = \sum_{s=0}^{n-|m|}D_{n,|m|,s}\ r^{n-s} $$

with integer coefficients



D_{n,|m|,s} = (-1)^s\frac{(2n+1-s)!}{s!(n-|m|-s)!(n+|m|-s+1)!}. $$

Examples
Examples are:

$$R_{0,0} = 1$$

$$R_{1,0} = -2+3 r$$

$$R_{1,1} = r$$

$$R_{2,0} = 3+10 r^2-12 r$$

$$R_{2,1} = 5 r^2-4 r$$

$$R_{2,2} = r^2$$

$$R_{3,0} = -4+35 r^3-60 r^2+30 r$$

$$R_{3,1} = 21 r^3-30 r^2+10 r$$

$$R_{3,2} = 7 r^3-6 r^2$$

$$R_{3,3} = r^3$$

$$R_{4,0} = 5+126 r^4-280 r^3+210 r^2-60 r$$

$$R_{4,1} = 84 r^4-168 r^3+105 r^2-20 r$$

$$R_{4,2} = 36 r^4-56 r^3+21 r^2$$

$$R_{4,3} = 9 r^4-8 r^3$$

$$R_{4,4} = r^4$$

$$R_{5,0} = -6+462 r^5-1260 r^4+1260 r^3-560 r^2+105 r$$

$$R_{5,1} = 330 r^5-840 r^4+756 r^3-280 r^2+35 r$$

$$R_{5,2} = 165 r^5-360 r^4+252 r^3-56 r^2$$

$$R_{5,3} = 55 r^5-90 r^4+36 r^3$$

$$R_{5,4} = 11 r^5-10 r^4$$

$$R_{5,5} = r^5$$

Moments
The pseudo-Zernike Moments (PZM) of order $$n$$ and repetition $$l$$ are defined as



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 $$n = 0, \ldots \infty$$, and $$l$$ takes on positive and negative integer values subject to $$|l|\leq n$$.

The image function can be reconstructed by expansion of the pseudo-Zernike coefficients on the unit disk as



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.