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Orthogonality and convergence of discrete Zernike polynomials

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Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/12021

Orthogonality and convergence of discrete Zernike polynomials

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Title: Orthogonality and convergence of discrete Zernike polynomials
Author: Allen, Joseph
Advisor(s): Embid, Pedro
Committee Member(s): Embid, Pedro
Pereyra, Cristina
Denham, Hugh
Department: University of New Mexico. Dept. of Mathematics and Statistics
Subject(s): zernike
polynomial
condition number
gram matrix
orthogonal
optics
discrete
LC Subject(s): Orthogonalization methods.
Orthogonal polynomials--Asymptotic theory.
Degree Level: Masters
Abstract: The Zernike polynomials are an infinite set of orthogonal polynomials over the unit disk, which are rotationally invariant. They are frequently utilized in optics, opthal- mology, and image recognition, among many other applications, to describe spherical aberrations and image features. Discretizing the continuous polynomials, however, introduces errors that corrupt the orthogonality. Minimizing these errors requires numerical considerations which have not been addressed. This work examines the orthonormal polynomials visually with the Gram matrix and computationally with the rank and condition number. The convergence of the Fourier-Zernike coefficients and the Fourier-Zernike series are also examined using various measures of error. The orthogonality and convergence are studied over six grid types and resolutions, polynomial truncation order, and function smoothness. The analysis concludes with design criteria for computing an accurate analysis with the discrete Zernike polynomials.
Graduation Date: December 2010
URI: http://hdl.handle.net/1928/12021

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