Biomedical Imaging Group

Sparse Processes

The lecture notes for the tutorial are available on the web at:

http://www.sparseprocesses.org

Lecture 1: Introduction to sparse stochastic processes

Lecture 2: Applications in signal processing and imaging

Lecture 3: Wavelets and independent component analysis

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.

The tutorial will introduce the participants to the 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. We shall provide a complete functional characterization of these processes and highlight some of their properties.

The two leading threads that underlie the exposition are:

- 1) the statistical property of infinite divisibility, which induces two distinct types of behavior-Gaussian vs. sparse-at the exclusion of any other;
- 2) the structural link between linear stochastic processes and splines.

The formalism lends itself to the derivation of the transform-domain statistics of these processes and to the identification of "optimal" (ICA-like) representations. We also show that these models are applicable to the derivation of statistical algorithms for solving ill-posed inverse problems, including compressed sensing. The proposed formulation 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, but it also suggests alternative Bayesian recovery procedures that minimize the estimation error.

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