Orthogonality and convergence of discrete Zernike polynomials

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.

Name thesis:degree_name

Mathematics

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Mathematics & Statistics

Year

2011

Author dc:creator

• Allen, Joseph

Contributors dc:contributor

• Embid, Pedro
• Pedro Embid
• Maria Cristina Pereyra
• Hugh Denham

• Orthogonalization methods
• Orthogonal polynomials--Asymptotic theory.

Language dc:language

English

Identifier

https://digitalrepository.unm.edu/math_etds/1

OAI identifier oai:identifier

oai:digitalrepository.unm.edu:math_etds-1000