A hierarchical sparsity-smoothness Bayesian model for ℓ0 + ℓ1 + ℓ2 regularization

Sparse signal/image recovery is a challenging topic that has captured a great interest during the last decades. To address the ill-posedness of the related inverse problem, regularization is often essential by using appropriate priors that promote the sparsity of the target signal/image. In this context, ℓ₀ + ℓ₁ regularization has been widely investigated. In this paper, we introduce a new prior accounting simultaneously for both sparsity and smoothness of restored signals. We use a Bernoulli-generalized Gauss-Laplace distribution to perform ℓ₀ + ℓ₁ + ℓ₂ regularization in a Bayesian framework. Our results show the potential of the proposed approach especially in restoring the non-zero coefficients of the signal/image of interest.