Frames and Phase Retrieval

Date Issued:: 2019
Abstract/Description:: Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d ? 1 and d(d + 1)/2, inclusive. The complex Hilbert space case remains open.In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalizationof the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ? M such that |supp(x)| = k.Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices.Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recov-ering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.

Title:	Frames and Phase Retrieval.	32 views 14 downloads
Name(s):	Juste, Ted, Author Han, Deguang, Committee Chair Sun, Qiyu, Committee Member Dutkay, Dorin, Committee Member Wang, Dingbao, Committee Member University of Central Florida, Degree Grantor
Type of Resource:	text
Date Issued:	2019
Publisher:	University of Central Florida
Language(s):	English
Abstract/Description:	Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d ? 1 and d(d + 1)/2, inclusive. The complex Hilbert space case remains open.In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalizationof the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then \|supp(x)\| ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ? M such that \|supp(x)\| = k.Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices.Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recov-ering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.
Identifier:	CFE0007660 (IID), ucf:52503 (fedora)
Note(s):	2019-08-01 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information.
Subject(s):	frames -- phase retrieval -- operator-valued frames -- projective representation frames
Persistent Link to This Record:	http://purl.flvc.org/ucf/fd/CFE0007660
Restrictions on Access:	public 2019-08-15
Host Institution:	UCF

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