Stochastic Models for Sparse and Piecewise-Smooth Signals

M. Unser

Schloß Dagstuhl Seminar 11051, Sparse Representations and Efficient Sensing of Data (DS'11), Wadern, Federal Republic of Germany, January 30-February 4, 2011.

We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; this is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The non-standard aspect is that the models are driven by non-Gaussian noise (impulsive Poisson or alpha-stable) and that the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete distributional characterization of these processes. We also introduce signals that are the non-Gaussian (sparse) counterpart of fractional Brownian motion; they are non-stationary and have the same 1∕ω-type spectral signature. We prove that our generalized processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. Finally, we discuss implications for sampling and sparse signal recovery.

Slides of the presentation (PDF, 19.2 Mb)

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