Phase retrieval algorithms with random masks

Peng-Wen Chen

Department of Applied Mathematics

National Chung Hsing University

pengwen@nchu.edu.tw

Phase retrieval aims to recover one unknown vector from its magnitude measurements, e.g., coherent diffractive imaging, where phase information is missing. The recovery of phase information can be formulated as one minimization problem subject to a non convex high-dimensional torus set. In theory, uniqueness of solutions can be obtained under random masks. The introduction of random masks actually breaks the symmetry of Fourier matrices and creates spectral gap for the local convergence of many phase retrieval algorithms, including alternative projection methods and Fourier Douglas-Rachford algorithms. The spectral gap is related to the local convergence rate.On the other hand, these alternative algorithms still could fail to generate the global solution effectively. To alleviate the stagnation of possible local solutions, we propose one null vector method as an initialization method for phase retrieval algorithms. The method is motivated by the following observation: Gaussian random vectors in high dimensional space are always nearly orthogonal to each other. According to magnitude data, we can construct one sub-matrix assembled from the sensing vectors nearly orthogonal to the unknown vector. One candidate for the initialization vector is given by the singular vector of the sub-matrix corresponding to the least singular value. Thanks to isometric Fourier matrices, this vector coincides with the dominant singular vector of the complement sub-matrix. Empirical studies (non-ptychography and ptychography) indicate that its incredible closeness to the unknown vector, compared with other existing methods. In this talk, we present one nonasymptotic error bound in the case of random complex Gaussian matrices, which sheds some light on its superior performance in the Fourier coherent diffractive case with random masks.

Keyword: random masks, phase retrieval, null vector method