The generalized phase retrieval problem over compact groups
The classical phase retrieval problem involves estimating a signal from its Fourier magnitudes (power spectrum) by leveraging prior information about the desired signal. This paper extends the problem to compact groups, addressing the recovery of a set of matrices from their Gram matrices. In this broader context, the missing phases in Fourier space are replaced by missing unitary or orthogonal matrices arising from the action of a compact group on a finite-dimensional vector space. This generalization is driven by applications in multi-reference alignment and single-particle cryo-electron microscopy, a pivotal technology in structural biology. We define the generalized phase retrieval problem over compact groups and explore its underlying algebraic structure. We survey recent results on the uniqueness of solutions, focusing on the significant class of semialgebraic priors. Furthermore, we present a family of algorithms inspired by classical phase retrieval techniques. Finally, we propose a conjecture on the stability of the problem based on bi-Lipschitz analysis, supported by numerical experiments.
Code (0)
등록된 구현이 없습니다.
Tasks
RetrievalMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
The stability of generalized phase retrieval problem over compact groups
The generalized phase retrieval problem over compact groups aims to recover a set of matrices, representing an unknown signal, from their associated Gram matrices, leveraging prior structural knowledge about the signal. …
RetrievalSpectral Method for Multiplexed Phase Retrieval and Application in Optical Imaging in Complex Media
We introduce a generalized version of phase retrieval called multiplexed phase retrieval. We want to recover the phase of amplitude-only measurements from linear combinations of them. This corresponds to the case in whic…
RetrievalApplication of the Non-Hermitian Singular Spectrum Analysis to the exponential retrieval problem
We present a new approach to solve the exponential retrieval problem. We derive a stable technique, based on the singular value decomposition (SVD) of lag-covariance and crosscovariance matrices consisting of covariance …
RetrievalTime SeriesTime Series AnalysisUniqueness of phase retrieval from offset linear canonical transform
The classical phase retrieval refers to the recovery of an unknown signal from its Fourier magnitudes, which is widely used in fields such as quantum mechanics, signal processing, optics, etc. The offset linear canonical…
RetrievalExtended Successive Convex Approximation for Phase Retrieval with Dictionary Learning
Phase retrieval aims at reconstructing unknown signals from magnitude measurements of linear mixtures. In this paper, we consider the phase retrieval with dictionary learning problem, which includes an additional prior i…
Dictionary LearningRetrieval