The $ L ^{ 1 } $ -Potts Functional for Robust Jump-Sparse Reconstruction

We investigate the nonsmooth and nonconvex $L ^{ 1 }$ -Potts functional in discrete and continuous time. We show Γ-convergence of discrete $L ^{ 1 }$ -Potts functionals toward their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $L ^{ 1 }$ -Potts problem, we introduce an $O(n ^{ 2 }$ ) time and O(n) space algorithm to compute an exact minimizer. We apply $L ^{ 1 }$ -Potts minimization to the problem of recovering piecewise constant signals from noisy measurements ƒ. It turns out that the $L ^{ 1 }$ -Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $L ^{ 1 }$ -Potts functional. Furthermore, for strongly blurred signals and a known blurring operator, we derive an iterative reconstruction algorithm.