DOLPHIn—Dictionary Learning for Phase Retrieval

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.

Denoising Poisson phaseless measurements via orthogonal dictionary learning

“…Remark II.2. Tillmann, Eldar, and Mairal [36] proposed the following dictionary learning based model to denoising Gaussian noised measurements for real-valued images min u∈R n ,D,α…”

Undersampled Sparse Phase Retrieval via Majorization–Minimization

“…Besides these, the authors in [24], [25] have shown the advantage of learning an overcomplete dictionary to sparsely represent a signal in an image denoising application. Recently, the dictionary learning techniques [26], [27] have also been exploited to solve the phase retrieval problem [23]. Inspired by these sparse coding ideas, we propose a simple and efficient algorithm in this section to solve the undersampled phase retrieval problem for signals that are not sparse in the standard basis.…”

“…The second, more general, setting considers cases where the original signals are not sparse in the standard basis (or any other bases). We share the same idea of applying the sparse coding techniques to the phase retrieval problem [23], inspired by the fact that a lot of image and video signals can be sparsely approximated by a linear combination of a few columns in a dictionary [24]- [27]. Recently the authors in [23] have shown encouraging results of exploiting sparse coding for the oversampled phase retrieval problem (DOLPHIn algorithm).…”

“…We share the same idea of applying the sparse coding techniques to the phase retrieval problem [23], inspired by the fact that a lot of image and video signals can be sparsely approximated by a linear combination of a few columns in a dictionary [24]- [27]. Recently the authors in [23] have shown encouraging results of exploiting sparse coding for the oversampled phase retrieval problem (DOLPHIn algorithm). In this paper, we propose efficient algorithms to recover the unknown signals with high accuracy for the undersampled phase retrieval problem by jointly designing the dictionary and the sparse codes.…”

Online convolutional dictionary learning for multimodal imaging

“…Starting from early work by Olshausen and Field [4,5], dictionary learning has become a standard tool for various tasks in image processing [6][7][8][9][10]. Our approach builds upon two prior lines of research, one on convolutional sparse representations [11][12][13] and one on online dictionary learning [14][15][16].…”