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One popular set of basis functions are the Zernike polynomials.
Zernike polynomials are widely used as basis functions of image moments.
For example, Zernike polynomials are orthogonal on the unit disk.
There are even and odd Zernike polynomials.
The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations.
The first few Zernike polynomials are:
Circular wavefront profiles associated with aberrations may be mathematically modeled using Zernike polynomials.
Applications often involve linear algebra, where integrals over products of Zernike polynomials and some other factor build the matrix elements.
Its main use is in curve-fitting wavefronts with Cartesian polynomials or Zernike polynomials.
In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses, in order to achieve desired system performance.
It is very common to compare an arbitrary DM to an ideal device that can perfectly reproduce wavefront modes in the form of Zernike polynomials.
As in Fourier synthesis using sines and cosines, a wavefront may be perfectly represented by a sufficiently large number of higher-order Zernike polynomials.
The Hankel transform of Zernike polynomials are essentially Bessel Functions (Noll 1976):
A complex, aberrated wavefront profile may be curve-fitted with Zernike polynomials to yield a set of fitting coefficients that individually represent different types of aberrations.
In optometry and ophthalmology Zernike polynomials are used to describe aberrations of the cornea or lens from an ideal spherical shape, which result in refraction errors.
Since Zernike polynomials are orthogonal to each other, Zernike moments can represent properties of an image with no redundancy or overlap of information between the moments.
Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of diffraction and aberrations.
The Shack-Hartmann AO system allows for corrections of the wavefront's aberrations caused by scintillation (degraded seeing), to higher Zernike polynomials.
In conjunction with piston (the first Zernike polynomial term), X and Y tilt can be modeled using the second and third Zernike polynomials:
Fast algorithms to calculate the forward and inverse Zernike transform use symmetry properties of trigonometric functions, separability of radial and azimuthal parts of Zernike polynomials, and their rotational symmetries.
Because there is no limit to the number of terms that may be used by Zernike polynomials, vision scientists use the first 15 polynomials, based on the fact that they are enough to obtain a highly accurate description of the most common aberrations found in human eye.
However, wavefronts with very steep gradients or very high spatial frequency structure, such as produced by propagation through atmospheric turbulence or aerodynamic flowfields, are not well modeled by Zernike polynomials, which tend to low-pass filter fine spatial definition in the wavefront.