{"id":{"repo_id":"unm","oai_identifier":"oai:digitalrepository.unm.edu:math_etds-1000"},"canonical_url":"https://search.dev.ndltd.org/etd/unm/oai:digitalrepository.unm.edu:math_etds-1000","repository":{"repo_id":"unm","name":"University of New Mexico","base_url":"https://digitalrepository.unm.edu/do/oai/"},"display":{"title":"Orthogonality and convergence of discrete Zernike polynomials","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Allen, Joseph"],"institution":null,"degree_name":"Mathematics","degree_level":"Masters","degree_discipline":"Mathematics & Statistics","degree_department":null,"school":null,"contributors":["Embid, Pedro","Pedro Embid","Maria Cristina Pereyra","Hugh Denham"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-02-07T08:00:00Z","date_published":"2011-02-07T08:00:00Z","updated_at":"2026-07-24T05:27:37Z","subjects":["Orthogonalization methods","Orthogonal polynomials--Asymptotic theory."],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalrepository.unm.edu/math_etds/1"],"render_values":[{"text":"https://digitalrepository.unm.edu/math_etds/1","href":"https://digitalrepository.unm.edu/math_etds/1","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1928/12021","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Embid, Pedro","Pedro Embid","Maria Cristina Pereyra","Hugh Denham"]},{"key":"dc:creator","label":"Author","values":["Allen, Joseph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics & Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters","Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Orthogonalization methods","Orthogonal polynomials--Asymptotic theory."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/1928/12021","https://digitalrepository.unm.edu/math_etds/1"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Orthogonality and convergence of discrete Zernike polynomials"]}]}],"canonical_facts":{"dc:contributor":["Embid, Pedro","Pedro Embid","Maria Cristina Pereyra","Hugh Denham"],"dc:creator":["Allen, Joseph"],"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 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."],"dc:identifier":["http://hdl.handle.net/1928/12021","https://digitalrepository.unm.edu/math_etds/1"],"dc:language":["English"],"dc:subject":["Orthogonalization methods","Orthogonal polynomials--Asymptotic theory."],"dc:title":["Orthogonality and convergence of discrete Zernike polynomials"],"thesis:degree_discipline":["Mathematics & Statistics"],"thesis:degree_level":["Masters","Thesis"],"thesis:degree_name":["Mathematics"]},"updated_at":"2026-07-24T05:27:37Z"}