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

Abstract

This paper addresses recovery of a kernel h ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and a signal x ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> from the low-resolution phaseless measurements of their noisy circular convolution y = |F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lo</inf> (x ⊛ h)| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + η, where F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lo</inf> ∈ ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m×n</sup> 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.