LOCALLY STABLE PHASE RETRIEVAL

Abstract

A frame (xj )j∈J for a Hilbert space H is said to do phase retrieval if, for all distinct vectors x, y ∈ H the magnitude of the frame coefficients (|⟨x, xj ⟩|)j∈J and (|⟨y, xj ⟩|)j∈J distinguish x from y (up to a unimodular scalar). The problem of phase retrieval shows up in many different settings as speech recognition [BR], coherent diffraction imaging [MCKS], X-ray crystallography [T], and transmission electron microscopy [K]. In this dissertation, we present some results about frame and phase retrieval. Chapter 1 contains some background about frames and phase retrieval. In Chapter 2, we consider the weaker condition where the magnitude of the frame coefficients distinguishes x from every vector y in a small neighborhood of x (up to a unimodular scalar). We prove that some essential theorems for phase retrieval hold for this local condition, whereas some theorems are completely different. We also prove that when considering the stability of phase retrieval, the worst stability inequality is always witnessed at orthogonal vectors. This allows for much simpler calculations when considering optimization problems for phase retrieval.