# Phase Retrieval for Finitely-Supported Complex Measures via the Fourier Transform

dc.contributor.committeeMember | Bodmann, Bernhard G. | |

dc.contributor.committeeMember | Mixon, Dustin G. | |

dc.contributor.committeeMember | Labate, Demetrio | |

dc.contributor.committeeMember | Kalantar, Mehrdad | |

dc.creator | Abouserie, Ahmed | |

dc.creator.orcid | 0000-0002-1436-7394 | |

dc.date.accessioned | 2022-12-29T02:10:43Z | |

dc.date.created | May 2022 | |

dc.date.issued | 2022-05-06 | |

dc.date.updated | 2022-12-29T02:10:44Z | |

dc.description.abstract | We study the recovery of a finitely-supported complex measure $\mu=\sum_{j=1}^{s}c_{j}\delta_{t_{j}}$ from the magnitudes of linear measurements. The distribution $\mu$ is completely determined by the amplitude vector $c\in\mathbb{C}^{s}$ and the support set $\{t_{1},t_{2},\dots,t_{s}\}\subset [0,\Lambda]$, where $\Lambda>0$ is assumed to be known. We show that by using magnitudes of point evaluations of the Fourier transform $\widehat{\mu}$ of $\mu$ at $\{v_{1},v_{2},\dots,v_{n}\}\subset[-\Omega,\Omega]$, along with magnitudes of differences of modulated point evaluations of $\widehat{\mu}$, we can construct injective maps over the space of all such measures of support length at most $s$. We follow a measurement design by Alexeev et al. \cite{alexeev_bandeira_fickus_mixon_2014} whereby point evaluations of $|\widehat{\mu}|^{2}$ are encoded as vertices of a graph, and edges of this graph correspond to interference measurements. In particular, if $\Lambda\Omega<s$, $d\geq 3$ and there is a Ramanujan graph, $\Gamma$, that is $d$-regular on $n>\frac{6(1+\sfrac{6}{\ln{(\sfrac{s}{\Lambda\Omega})}})s}{1-\sfrac{2\sqrt{d-1}}{d}}$ vertices, then a set of $M=(d+1)n$ magnitude measurements associated with $\Gamma$ is sufficient for identifying $\mu$ up to an overall unimodular multiplicative constant. Under some additional assumptions, we provide two recovery algorithms. The first algorithm is based on phase propagation and the Prony method. We show that the reconstruction problem can be reduced to applying linear inverses and finding roots of a polynomial in the case of exact measurements for almost every signal of the above form. In the second algorithm, at the cost of introducing a truncation error, we follow the technique presented by Cand\`es and Fernandez-Granda in \cite{cand'es_fernandez-granda_2013} to show that the solution to a total-variation norm minimization problem defined by the given intensity measurements yields an approximation of $\mu$. We give explicit error bounds for recovery using this method depending on the number of given samples, and discuss the effect of noise in this approach. | |

dc.description.department | Mathematics, Department of | |

dc.format.digitalOrigin | born digital | |

dc.format.mimetype | application/pdf | |

dc.identifier.uri | https://hdl.handle.net/10657/13145 | |

dc.language.iso | eng | |

dc.rights | The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s). | |

dc.subject | Phase retrieval | |

dc.subject | Linear Algebra | |

dc.subject | Fourier Analysis | |

dc.title | Phase Retrieval for Finitely-Supported Complex Measures via the Fourier Transform | |

dc.type.dcmi | Text | |

dc.type.genre | Thesis | |

dcterms.accessRights | The full text of this item is not available at this time because the student has placed this item under an embargo for a period of time. The Libraries are not authorized to provide a copy of this work during the embargo period. | |

local.embargo.lift | 2024-05-01 | |

local.embargo.terms | 2024-05-01 | |

thesis.degree.college | College of Natural Sciences and Mathematics | |

thesis.degree.department | Mathematics, Department of | |

thesis.degree.discipline | Mathematics | |

thesis.degree.grantor | University of Houston | |

thesis.degree.level | Doctoral | |

thesis.degree.name | Doctor of Philosophy |

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