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Sparse Stochastic Processes: A Statistical Framework for Compressed Sensing and Biomedical Image Reconstruction

M. Unser

Tutorial, Eleventh IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'14), Beijing, People's Republic of China, April 29-May 2, 2014.



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 tutorial focuses on 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 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 spline functions which is exploited to simplify the mathematics.

The proposed continuous-domain formalism lends itself naturally to the discretization of linear inverse problems. The reconstruction is formulated as a statistical estimation problem, which suggests some novel algorithms for biomedical image reconstruction, including magnetic resonance imaging and X-ray tomography. We present experiments where the proposed scheme outperforms the more traditional convex optimization techniques (in particular, total variation).

The lecture notes for the tutorial are available on the web at http://www.sparseprocesses.org


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