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Concept
Zernike polynomials
Summary
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging.
There are even and odd Zernike polynomials. The even Zernike polynomials are defined as
(even function over the azimuthal angle ), and the odd Zernike polynomials are defined as
(odd function over the azimuthal angle ) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), is the azimuthal angle, ρ is the radial distance , and are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. . The radial polynomials are defined as
for an even number of n − m, while it is 0 for an odd number of n − m. A special value is
Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:
A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:
for n − m even.
The factor in the radial polynomial may be expanded in a Bernstein basis of for even or times a function of for odd in the range . The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:
Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds a matrix elements.
To enumerate the rows and columns of these matrices by a single index, a conventional mapping of the two indices n and l to a single index j has been introduced by Noll. The table of this association starts as follows .
The rule is the following.
The even Zernike polynomials Z (with even azimuthal parts , where as is a positive number) obtain even indices j.
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