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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 Xray 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 phaseretrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the ddimensional real Hilbert space case, exact phaseretrievable 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 phaseretrievability by studying maximal phaseretrievable subspaces with respect to a given frame. These maximal PRsubspaces 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 phaseretrievable frames. In the basis case, we prove that if M is a kdimensional PRsubspace then supp(x) ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a kdimensional PRsubspace is maximal if and only if there exists a vector x ? M such that supp(x) = k.Chapter 5 is devoted to investigating phaseretrievable operatorvalued frames. We obtain some characterizations of phaseretrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vectorvalued 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 recovering of the phaseretrieval 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 phaseretrieval with Gabor (or STFT) measurements.
Title:  Frames and Phase Retrieval. 
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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 Xray 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 phaseretrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the ddimensional real Hilbert space case, exact phaseretrievable 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 phaseretrievability by studying maximal phaseretrievable subspaces with respect to a given frame. These maximal PRsubspaces 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 phaseretrievable frames. In the basis case, we prove that if M is a kdimensional PRsubspace then supp(x) ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a kdimensional PRsubspace is maximal if and only if there exists a vector x ? M such that supp(x) = k.Chapter 5 is devoted to investigating phaseretrievable operatorvalued frames. We obtain some characterizations of phaseretrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vectorvalued 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 recovering of the phaseretrieval 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 phaseretrieval with Gabor (or STFT) measurements.  
Identifier:  CFE0007660 (IID), ucf:52503 (fedora)  
Note(s): 
20190801 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  frames  phase retrieval  operatorvalued frames  projective representation frames  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0007660  
Restrictions on Access:  public 20190815  
Host Institution:  UCF 