Phase Retrieval - A Deconvolution Perspective

Phase retrieval finds applications in various optical imaging modalities such as X-ray crystallography, holography, frequency-domain optical-coherence tomography, etc. The sensors used in optical imaging can measure only the magnitudes of incoming wavefronts and the phase information is not measured directly. This necessitates developing appropriate phase retrieval algorithms to reconstruct the object as phase contains most of the structural information. The phase retrieval problem naturally arises in the Fourier imaging context, where the measurement is the Fourier magnitude/intensity spectrum. Reconstruction from the Fourier intensity results in the autocorrelation and not the signal. We therefore address the equivalent problem of signal retrieval from the autocorrelation. Since the signal autocorrelation can be expressed as a convolution of the signal with its flipped version, we propose to solve the phase retrieval problem within a deconvolution framework. We consider a non-convex cost in two vector variables, the signal and its flipped version. An alternating minimization (Alt. Min.) strategy is employed to arrive at an optimal estimate of the signal, given the autocorrelation. Due to non-convexity of the cost function, the accuracy of the estimation is critically dependent on the initialization. We establish that the Alt. Min. iterates ensure that the cost is nonincreasing. For the specific case of causal, delta-dominant signals, the proposed framework results in exact reconstruction with an all zero-phase initialization. We shall also consider the effect of random initialization on the estimation accuracy. © 2018 APSIPA organization