{"id":{"repo_id":"siu-theses","oai_identifier":"oai:opensiuc.lib.siu.edu:dissertations-1012"},"canonical_url":"https://search.dev.ndltd.org/etd/siu-theses/oai:opensiuc.lib.siu.edu:dissertations-1012","repository":{"repo_id":"siu-theses","name":"Southern Illinois University","base_url":"https://opensiuc.lib.siu.edu/do/oai/"},"display":{"title":"Reconstruction Of A Univariate Discrete Function From The Magnitude Of Its Fourier Transform","abstract":"In many branches of Physics and Engineering one comes across the problem of reconstructing a function $f$ using the Fourier transform $F$, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct $f$ when only the magnitude $|f|$ of the function and the magnitude $|F|$ of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct $f$ when only $|F|$ and some constraints on $f$, e.g., $f \\geq 0$, are available. In this paper we study the latter problem in a context where $f$ is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm.","abstract_html":"In many branches of Physics and Engineering one comes across the problem of reconstructing a function $f$ using the Fourier transform $F$, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct $f$ when only the magnitude $|f|$ of the function and the magnitude $|F|$ of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct $f$ when only $|F|$ and some constraints on $f$, e.g., $f \\geq 0$, are available. In this paper we study the latter problem in a context where $f$ is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm.","abstract_has_math":true,"creators":["Khurram, Alia"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"Campus Only Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kammler,David"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009-01-01T08:00:00Z","date_published":"2009-01-01T08:00:00Z","updated_at":"2026-07-24T04:33:23Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://opensiuc.lib.siu.edu/dissertations/12","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kammler,David"]},{"key":"dc:creator","label":"Author","values":["Khurram, Alia"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2009-11-04T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Only Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://opensiuc.lib.siu.edu/dissertations/12"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In many branches of Physics and Engineering one comes across the problem of reconstructing a function $f$ using the Fourier transform $F$, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct $f$ when only the magnitude $|f|$ of the function and the magnitude $|F|$ of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct $f$ when only $|F|$ and some constraints on $f$, e.g., $f \\geq 0$, are available. In this paper we study the latter problem in a context where $f$ is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm."]},{"key":"dc:title","label":"Title","values":["Reconstruction Of A Univariate Discrete Function From The Magnitude Of Its Fourier Transform"]}]}],"canonical_facts":{"dc:contributor":["Kammler,David"],"dc:creator":["Khurram, Alia"],"dc:date.available":["2009-11-04T08:00:00Z"],"dc:description.abstract":["In many branches of Physics and Engineering one comes across the problem of reconstructing a function $f$ using the Fourier transform $F$, when only partial information about the transform and the function is available. One of the most common examples is to reconstruct $f$ when only the magnitude $|f|$ of the function and the magnitude $|F|$ of the Fourier transform are known. This problem occurs in electron microscopy and wavefront sensing. Another problem which occurs in astronomy and crystallography is to reconstruct $f$ when only $|F|$ and some constraints on $f$, e.g., $f \\geq 0$, are available. In this paper we study the latter problem in a context where $f$ is univariate and discrete. We make use of Fienup's analysis and adapt the Gerchberg-Saxton algorithm to our problem. We devise ways to eliminate indeterminacy and we suggest ways to improve the rate of convergence of this algorithm."],"dc:identifier":["https://opensiuc.lib.siu.edu/dissertations/12"],"dc:title":["Reconstruction Of A Univariate Discrete Function From The Magnitude Of Its Fourier Transform"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Only Dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T04:33:23Z"}