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The phase function is retrieved by the unknown-coefficient weighted product with (known values) of Zernike polynomial across the unit grid. Hence, coefficients can also be found by solving a linear system, for instance by matrix inversion.
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. These Zernike coefficients are linearly independent , thus individual aberration contributions to an overall wavefront may be isolated and quantified separately.
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
Among these the most important Zernike coefficients affecting visual quality are coma, spherical aberration, and trefoil. [6] Zernike polynomials are usually expressed in terms of polar coordinates (ρ,θ), where ρ is radial coordinate and θ is the angle.
The piston coefficient is typically expressed in wavelengths of light at a particular wavelength. Its main use is in curve-fitting wavefronts with Cartesian polynomials or Zernike polynomials . However, similar to a real engine piston moving up and down in its cylinder, optical piston values can be changed to bias the wavefront phase mean value ...
For example, Zernike polynomials are orthogonal on the unit disk. ... There are two popular examples: either the coefficients {} are matrices or : ...
Tilt quantifies the average slope in both the X and Y directions of a wavefront or phase profile across the pupil of an optical system. In conjunction with piston (the first Zernike polynomial term), X and Y tilt can be modeled using the second and third Zernike polynomials:
Defocus is modeled in Zernike polynomial format as (), where is the defocus coefficient in wavelengths of light. This corresponds to the parabola-shaped optical path difference between two spherical wavefronts that are tangent at their vertices and have different radii of curvature.