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
where formula_2 and orthogonality on the unit disk is given as
where the star means complex conjugation, and
formula_4, formula_5, formula_6
are the standard transformations between polar and Cartesian coordinates.
The radial polynomials formula_7 are defined as
formula_8
with integer coefficients
Examples.
Examples are:
formula_10
formula_11
formula_12
formula_13
formula_14
formula_15
formula_16
formula_17
formula_18
formula_19
formula_20
formula_21
formula_22
formula_23
formula_24
formula_25
formula_26
formula_27
formula_28
formula_29
formula_30
Moments.
The pseudo-Zernike Moments (PZM) of order formula_31 and repetition formula_32 are defined as
where formula_34, and formula_32 takes on positive and negative integer
values subject to formula_36.
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.