Inverse Problems with Fourier-Domain Measurements and gTV Regularization: uniqueness and reconstruction algorithm
Meeting • 22 September 2020AbstractWe study the super-resolution problem of recovering a sparse periodic continuous-domain function from its low-frequency information. We provide a new analysis of constrained optimization problems with total-variation (TV) regularization over Radon measures. In particular, we demonstrate that the solution is not necessarily unique, and we identify a general sufficient condition for uniqueness, expressed in terms of the Fourier-domain measurements. We then apply this result to prove that when the TV regularization includes a derivative operator of any order (generalized TV), the solution is always unique and is a periodic spline. We propose an adaptation of the sliding Frank-Wolfe algorithm for spline reconstruction with generalized TV regularization in a noisy setting.