Zernike polynomials
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In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after Frits Zernike, they play an important role in geometrical optics.
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[edit] Definitions
There are even and odd Zernike polynomials. The even ones are defined as
and the odd ones as
where m and n are nonnegative integers with 
, 
is the azimuthal angle in radians, and 
is the normalized radial distance. The radial polynomials 
are defined as
for n − m even, and are identically 0 for n − m odd.
For m = 0, the even definition is used which reduces to 
.
[edit] Applications
In optometry and ophthalmology the Zernike polynomials are used to describe aberrations of the cornea or lens from an ideal spherical shape, which result in refraction errors.
They are commonly used in adaptive optics where they can be used to effectively cancel out atmospheric distortion. Obvious applications for this are IR or visual astronomy, and spy satellites. For example, one of the zernike terms (for 
) is called 'de-focus'.[1] By coupling the output from this term to a control system, an automatic focus can be implemented.
Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of diffraction and aberrations.
Zernike polynomials are widely used as basis functions of image moments.
[edit] References
- Born and Wolf, "Principles of Optics", Oxford: Pergamon, 1970
- Eric W. Weisstein et al., "Zernike Polynomial", at MathWorld.
