Author

Joseph Allen

Publication Date

2-7-2011

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 coe\ufb03cients 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.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Advisor

Embid, Pedro

First Committee Member (Chair)

Embid, Pedro

Second Committee Member

Pereyra, Cristina

Third Committee Member

Denham, Hugh

Language

English

Keywords

Orthogonalization methods, Orthogonal polynomials--Asymptotic theory.

Document Type

Thesis

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