DOLPHIn - Dictionary Learning for Phase Retrieval

TL;DR

This paper introduces DOLPHIn, an algorithm that jointly learns a dictionary and reconstructs signals from phase retrieval measurements, leveraging sparsity to improve accuracy especially in noisy conditions.

Contribution

The work presents a novel method combining dictionary learning with phase retrieval, enabling better reconstruction by exploiting hidden sparsity without prior dictionary knowledge.

Findings

Significantly improved reconstruction quality in noisy phase retrieval scenarios.

Successful joint dictionary learning and signal reconstruction demonstrated through numerical experiments.

Provided a convergence proof for the proposed algorithm.

Abstract

We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal - possibly corrupted by noise - and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such "hidden" sparsity. Moreover, on the theoretical side, we provide a convergence result for our method.