[Paper Review] Phase Retrieval for Sparse Signals: Uniqueness Conditions

Juri Ranieri, Amina Chebira+2 more|arXiv (Cornell University)|Aug 14, 2013

Advanced X-ray Imaging Techniques28 references57 citations

TL;DR

This paper establishes uniqueness conditions for phase retrieval of sparse signals by linking the problem to the turnpike problem in combinatorics. It proves that when the autocorrelation function has no collisions, the solution is almost surely unique in one dimension and extends this to multi-dimensional signals, significantly improving prior uniqueness guarantees for sparse phase retrieval.

ABSTRACT

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?" In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.

Motivation & Objective

• To determine sufficient conditions under which the phase retrieval problem for sparse signals yields a unique solution.
• To bridge phase retrieval with the classical turnpike problem in combinatorics to derive new uniqueness criteria.
• To establish that the absence of collisions in the autocorrelation function ensures almost sure uniqueness in 1D sparse signals.
• To extend 1D uniqueness results to multi-dimensional signals by decomposing the problem into 1D subproblems.
• To provide a framework valid for both discrete and continuous domains, relying only on sparsity and support structure.

Proposed method

• The authors model the signal as sparse in the spatial domain, with non-zero entries confined to a known or estimable support.
• They establish a strong mathematical connection between phase retrieval and the turnpike problem, a classic problem in combinatorial reconstruction.
• They prove that collisions in the autocorrelation function can prevent unique recovery, thus identifying collision-free autocorrelation as a necessary condition for uniqueness.
• For 1D signals, they show that if the autocorrelation has no collisions, the solution is almost surely unique, leveraging probabilistic arguments on signal support.
• They extend the 1D result to multi-dimensional signals by reducing the problem to a set of 1D phase retrieval problems along coordinate directions.
• The method applies to both discrete and continuous domains, with the key condition being the absence of autocorrelation collisions and sparsity of the signal.

Experimental results

Research questions

• RQ1 Under what conditions is the phase retrieval problem uniquely solvable for sparse signals in one-dimensional domains?
• RQ2 How does the turnpike problem relate to the uniqueness of phase retrieval solutions?
• RQ3 Can the uniqueness condition derived for 1D signals be extended to multi-dimensional sparse signals?
• RQ4 What role do collisions in the autocorrelation function play in preventing unique recovery?
• RQ5 How does the probability of satisfying the uniqueness condition differ between discrete and continuous domains?

Key findings

• The solution to the phase retrieval problem is almost surely unique for 1D sparse signals when the autocorrelation function contains no collisions.
• The absence of autocorrelation collisions is both a necessary and sufficient condition for uniqueness in 1D sparse phase retrieval.
• The uniqueness condition derived for 1D signals extends to multi-dimensional signals by solving a set of 1D problems, ensuring uniqueness when the multi-dimensional autocorrelation has no collisions.
• The proposed uniqueness condition improves upon previously known results by providing a stronger and more general criterion for multi-dimensional phase retrieval.
• The framework applies uniformly to both discrete and continuous domains, with the only difference being the probability of satisfying the uniqueness condition due to domain structure.