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The L1-Potts Functional for Robust Jump-Sparse Reconstruction

A. Weinmann, M. Storath, L. Demaret

SIAM Journal on Numerical Analysis, vol. 53, no. 1, pp. 644-673, 2015.

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 ƒ. 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.

AUTHOR="Weinmann, A. and Storath, M. and Demaret, L.",
TITLE="The $L^{1}$-{P}otts Functional for Robust Jump-Sparse
JOURNAL="{SIAM} Journal on Numerical Analysis",

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