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Orthogonality and convergence of discrete Zernike polynomials


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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dc.contributor.author Allen, Joseph
dc.date.accessioned 2011-02-07T23:32:31Z
dc.date.available 2011-02-07T23:32:31Z
dc.date.issued 2011-02-07
dc.date.submitted December 2010
dc.identifier.uri http://hdl.handle.net/1928/12021
dc.description.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. en_US
dc.language.iso en_US en_US
dc.subject zernike en_US
dc.subject polynomial en_US
dc.subject condition number en_US
dc.subject gram matrix en_US
dc.subject orthogonal en_US
dc.subject optics en_US
dc.subject discrete en_US
dc.subject.lcsh Orthogonalization methods.
dc.subject.lcsh Orthogonal polynomials--Asymptotic theory.
dc.title Orthogonality and convergence of discrete Zernike polynomials en_US
dc.type Thesis en_US
dc.description.degree Applied Mathematics en_US
dc.description.level Masters en_US
dc.description.department University of New Mexico. Dept. of Mathematics and Statistics en_US
dc.description.advisor Embid, Pedro
dc.description.committee-member Embid, Pedro
dc.description.committee-member Pereyra, Cristina
dc.description.committee-member Denham, Hugh

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