Non-Convex Recovery from Phaseless Low-Resolution Blind Deconvolution Measurements using Noisy Masked Patterns
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This paper addresses recovery of a kernel h∈ℂ^n and a signal x∈ℂ^n from the low-resolution phaseless measurements of their noisy circular convolution y = |F_lo( x⊛h) |^2 + η, where F_lo∈ℂ^m× n stands for a partial discrete Fourier transform (m<n), η models the noise, and |·| 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 x-h pairs that correspond to the measurements. Therefore, to guarantee a stable recovery of x and h from y, we assume that the kernel h and the signal x lie in known subspaces of dimensions k and s, respectively, such that m≫ 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 h and x using our method.
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