Bayesian Inference with Error Variable Splitting and Sparsity Enforcing Priors for Linear Inverse Problems

Regularization and Bayesian inference based methods have been successfully applied for linear inverse problems. In these methods, often simple Gaussian or Poisson models for the forward model errors have been considered. In this work, we use variable splitting for the errors to model different sources of errors and their possible non-stationarity or impulsive nature using Student-t or other heavy tailed distributions. Also, as a prior model, a sparsity enforcing hierarchical model of Infinite Gaussian Mixture model is introduced. With these prior models, we obtain a complete Bayesian inference framework which can efficiently be implemented for any linear inverse problem. Interestingly, many recent regularization-based algorithms such as Alternating Direction Method of Multipliers (ADMM) as well as more classical Bayesian based methods such as Sparse Bayesian Learning (SBL) are obtained as particular cases. One advantage of the Bayesian approach is the possibility to estimate, jointly with the reconstruction, the hyper-parameters such as the regularization parameter, thus the capability of proposing unsupervised methods. Examples of implementation of the proposed method in different signal and image processing such as deconvolution in mass spectrometry, estimation of periodic components estimation in biological signals and computed tomography are mentioned and referenced.