Phase Retrieval from Linear Canonical Transforms
Yang Chen, Na Qu
Abstract
The classical phase retrieval problem aims to recover an unknown function from the Fourier magnitudes. The linear canonical transform CA 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 |CAf|,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 |CAf|,|CAQf|,|CA(f+Qf)| and |CA(f+iQf)|, where Qf(t)=tf(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 |CAf| and |CAQf|.