Sparse stochastic processes

Sparse stochastic processes are continuous-domain processes that admit a parsimonious representation in some matched wavelet-like basis. Such models are relevant for image compression, compressed sensing, and, more generally, for the derivation of statistical algorithms for solving ill-posed inverse problems.

This course introduces an extended family of sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. It presents the mathematical tools for their characterization. The two leading threads of the exposition are

• the statistical property of infinite divisibility, which induces two distinct types of behavior—Gaussian vs. sparse—at the exclusion of any other;
  
• the structural link between linear stochastic processes and spline functions, which is exploited to simplify the mathematical analysis.

The core of the course is devoted to the investigation of sparse processes, including the complete description of their transform-domain statistics. The final part develops signal-processing techniques that are based on these models. This leads to a reinterpretation of popular sparsity-promoting processing schemes—such as total-variation denoising, LASSO, and wavelet shrinkage—as MAP estimators for specific types of sparse processes. It also suggests alternative Bayesian recovery procedures that minimize the estimation error. The framework is illustrated with the reconstruction of biomedical images (deconvolution microscopy, MRI, X-ray tomography) from noisy and/or incomplete data.