BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Sparse Modeling and the Resolution of Inverse Problems in Bimedical Imaging

M. Unser

Invited lecture, 2015 HIT International Summer School on Pure and Applied Mathematics (ISSPAM'15), Harbin, People's Republic of China, July 6-August 9, 2015.

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 is devoted to the study of the broad 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 that underly 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 mathematics.

The concepts are illustrated with the derivation of algorithms for the recovery of sparse signals, with applications to biomedical image reconstruction. In particular, this leads to a Bayesian 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. The formulation also suggests alternative recovery procedures that minimize the estimation error.

The course is targeted to an audience of graduate students and researchers with an interest in signal/image processing, compressed sensing, approximation theory, machine learning, or statistics.For more details, including table of content, see http://www.sparseprocesses.org/. 

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