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Another application of the Zernike polynomials is found in the Extended Nijboer–Zernike theory of diffraction and aberrations. Zernike polynomials are widely used as basis functions of image moments .
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.
The theory of aberrated point spread functions close to the optimum focal plane was studied by Zernike and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike's circle polynomials that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have ...
"An efficient feature extraction method with pseudo-zernike moment in rbf neural network-based human face recognition system". EURASIP Journal on Applied Signal Processing. 2003 (9): 890–901. Bibcode:2003EJASP2003..146H. doi: 10.1155/S1110865703305128. T.-W. Lin; Y.-F. Chou (2003). A comparative study of zernike moments. Proceedings of the ...
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:
Several orthogonal polynomials, including Jacobi polynomials P (α,β) n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, Zernike polynomials can be written in terms of hypergeometric functions using (, + + +; +;) =!
The 9-year-old was reluctant to speak and told officials that their parents "don't want them talking about trouble" and to "keep it in the family,” CBS News reported.
There are even and odd Zernike polynomials. The even Zernike polynomials are defined as (,) = and the odd Zernike polynomials as (,) = (), where m and n are nonnegative integers with , Φ is the azimuthal angle in radians, and ρ is the normalized radial distance. The radial polynomials have no azimuthal dependence, and are defined as