Bayesian-driven criterion to automatically select the regularization parameter in the ℓ1-Potts model

This contribution focuses, within the ℓ₁-Potts model, on the automated estimation of the regularization parameter balancing the ℓ₁ data fidelity term and the TVℓ₀ penalization. Variational approaches based on total variation gained considerable interest to solve piecewise constant denoising problems thanks to their deterministic setting and low computational cost. However, the quality of the achieved solution strongly depends on the tuning of the regularization parameter. While recent works have tailored various hierarchical Bayesian procedures to additionally estimate the regularization parameter for Gaussian noise, less attention has been granted to Laplacian noise, of interested in numerous applications. This contribution promotes a fast and parameter-free denoising procedure for piecewise constant signals corrupted by Laplacian noise, that includes automated selection of the regularization parameter. It relies on the minimization of a Bayesian-driven criterion whose similarities with the ℓ₁-Potts model permit to derive a computationally efficient algorithm.