Non-Convex Recovery from Phaseless Low-Resolution Blind Deconvolution Measurements using Noisy Masked Patterns

This paper addresses recovery of a kernel boldsymbolh∈ mathbbCn and a signal boldsymbolx∈ mathbbCn from the low-resolution phaseless measurements of their noisy circular convolution boldsymboly = left rvert boldsymbolFlo( boldsymbolxcircledast boldsymbolh) right rvert2 + boldsymbolη, where boldsymbolFlo∈ mathbbCm× n stands for a partial discrete Fourier transform (m<n), boldsymbolη models the noise, and lvert · rvert is the element-wise absolute value function. This problem is severely ill-posed because both the kernel and signal are unknown and, in addition, the measurements are phaseless, leading to many boldsymbolx-boldsymbolh pairs that correspond to the measurements. Therefore, to guarantee a stable recovery of boldsymbolx and boldsymbolh from boldsymboly, we assume that the kernel boldsymbolh and the signal boldsymbolx lie in known subspaces of dimensions k and s, respectively, such that mgg k+s. We solve this problem by proposing a blind deconvolution algorithm for phaseless super-resolution (BliPhaSu) to minimize a non-convex least-squares objective function. The method first estimates a low-resolution version of both signals through a spectral algorithm, which are then refined based upon a sequence of stochastic gradient iterations. We show that our BliPhaSu algorithm converges linearly to a pair of true signals on expectation under a proper initialization that is based on spectral method. Numerical results from experimental data demonstrate perfect recovery of both boldsymbolh and boldsymbolx using our method.