2016/01/27 by Rima Alaifari, Alaifari, Rima, Ingrid Daubechies +5
Computer Science · Engineering · Physics and Astronomy · #42C15 #49N45 #94A12 #94A20 #Advanced X-ray Imaging Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Image Processing Techniques and Applications #Optical measurement and interference techniques
paper · pdf · doi:10.48550/arxiv.1601.07579
openalex publication_date 2016/01/27 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
In this paper we consider the following problem of phase retrieval: Given a\ncollection of real-valued band-limited functions\n \ψ\λ \λ\∈ \Λ\⊂ L2(\ℝd) that\nconstitutes a semi-discrete frame, we ask whether any real-valued function f\n\∈ L2(\ℝd) can be uniquely recovered from its unsigned convolutions\n |f \∗ \ψ_\λ| \λ \∈ \Λ.\n We find that under some mild assumptions on the semi-discrete frame and if\nf has exponential decay at \∞, it suffices to know |f \∗\n\ψ_\λ| on suitably fine lattices to uniquely determine f (up to a\nglobal sign factor).\n We further establish a local stability property of our reconstruction\nproblem. Finally, for two concrete examples of a (discrete) frame of\nL2(\ℝd), d=1,2, we show that through sufficient oversampling one\nobtains a frame such that any real-valued function with exponential decay can\nbe uniquely recovered from its unsigned frame coefficients.\n