Abstract

We treat the phase retrieval (PR) problem of reconstructing the interest signal from its Fourier magnitude. Since the Fourier phase information is lost, the problem is ill-posed. Several techniques have been used to address this problem by utilizing various priors such as non-negative, support, and Fourier magnitude constraints. Recent methods exploiting sparsity are developed to improve the reconstruction quality. However, the previous algorithms of utilizing sparsity prior suffer from either the low reconstruction quality at low oversampled factors or being sensitive to noise. To address these issues, we propose a framework that exploits sparsity of the signal in the translation invariant Haar pyramid (TIHP) tight frame. Based on this sparsity prior, we formulate the sparse representation regularization term and incorporate it into the PR optimization problem. We propose the alternating iterative algorithm for solving the corresponding non-convex problem by dividing the problem into several subproblems. We give the optimal solution to each subproblem, and experimental simulations under the noise-free and noisy scenario indicate that our proposed algorithm can obtain a better reconstruction quality compared to the conventional alternative projection methods, even outperform the recent sparsity-based algorithms in terms of reconstruction quality.

Highlights

  • In science and engineering fields, such as crystallography, neutron radiography, astronomy, signal processing, and optical imaging [1, 2], it is difficult to design sophisticated measuring setups to allow direct recording of the phase, which carries the critical structural information of the test object or signal [1]

  • Our contributions can be summarized as follows: 1. We propose a sparse representation regularization term based on the translation invariant Haar pyramid (TIHP) tight frame for phase retrieval

  • We propose a framework of utilizing TIHP tight frame, and experimental results indicate its efficiency for natural images

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Summary

Introduction

In science and engineering fields, such as crystallography, neutron radiography, astronomy, signal processing, and optical imaging [1, 2], it is difficult to design sophisticated measuring setups to allow direct recording of the phase, which carries the critical structural information of the test object or signal [1]. In the image PR field, the image regularization, such as l1 regularization [13, 14], is focused by researchers They often formulate the non-convex l1 minimization problem and solve the problem by alternating directions method of multipliers (ADMM) [15], which can obtain a suboptimal solution to the non-convex problem. We combine the sparse representation regularization term with the data consistency term and object constraint term of utilizing the indicator function to formulate a new phase retrieval problem. The sparse representation regularization term of utilizing TIHP tight frame is helpful to retrieve the missing phase as well as recover the image at low oversampled factors. Experimental results indicate that our proposed algorithm can obtain better reconstruction quality compared with the alternative projection algorithms of utilizing the same sparsity prior.

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