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 where
+
y
≤ 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 where the star means complex conjugation, and
2
=
x
2
{\displaystyle r^{2}=x^{2}+y^{2}}
x = r cos θ
y = r sin θ
are the standard transformations between polar and Cartesian coordinates.
The radial polynomials
{\displaystyle R_{nm}}
are defined as[1]
{\displaystyle R_{nm}(r)=\sum _{s=0}^{n-|m|}D_{n,|m|,s}\ r^{n-s}}
with integer coefficients Examples are:
The pseudo-Zernike Moments (PZM) of order
and repetition
are defined as where
takes on positive and negative integer values subject to
The image function can be reconstructed by expansion of the pseudo-Zernike coefficients on the unit disk as 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.