Phase Retrieval from Linear Canonical Transforms. (1st November 2022)

• Record Type: Journal Article
• Authors: Chen, Yang; Qu, Na
• Is Part Of: Numerical functional analysis and optimization. Volume 43:Number 15(2022)
• Page Start: 1760
• Page End: 1777
• Publication Date: 2022-11-01
• Subjects: Chirped Hermite functions -- conjugate phase retrieval -- linear canonical transform -- phase retrieval -- vectorial phase retrieval; 42A05 -- 94A08 -- 94A12 -- 94A20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2022.2132511
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The classical phase retrieval problem aims to recover an unknown function from the Fourier magnitudes. The linear canonical transform C A has a more generalized form of the well-known (fractional) Fourier transform and a wide range of engineering applications such as optics and quantum mechanism. In this paper, we consider the linear canonic phase retrieval problem of determining a function from the magnitudes of the linear canonic transforms. We show that a compactly supported function f can be determined, up to a global phase, from the magnitudes | C A f |, A ∈ Γ, of multiple linear canonic transforms, where Γ is a class of real unimodular matrices. It generalizes the results of phase retrieval from multiple fractional Fourier transforms. On the other hand, we show that a compactly supported function f can be determined, up to a global phase, from the interference linear canonic magnitudes | C A f |, | C A Q f |, | C A ( f + Q f ) | and | C A ( f + i Q f ) |, where Q f ( t ) = t f ( t ) . Moreover, if the ambiguity of conjugate reflection is taken into account, the compactly supported function f can be determined, up to a rotation and conjugate reflection, from the linear canonic magnitudes | C A f | and | C A Q f | .