The L1-Potts Functional for Robust Jump-Sparse Reconstruction

We investigate the nonsmooth and nonconvex L1-Potts functional in discrete and continuous time. We show Γ-convergence of discrete L1-Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete L1-Potts problem, we introduce an O(n2) time and O(n) space algorithm to compute an exact minimizer. We apply L1-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements f. It turns out that the L1-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the L1-Potts functional. Furthermore, for strongly blurred signals and a known blurring operator, we derive an iterative reconstruction algorithm.