Stochastic Models for Sparse and Piecewise-Smooth Signals

• Published in: IEEE transactions on signal processing Vol. 59; no. 3; pp. 989 - 1006
• Main Authors: Unser, M, Tafti, P D
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Compounds; Detection, estimation, filtering, equalization, prediction; Differential equations; Equivalence; Exact sciences and technology; Fractals; Gaussian; Information, signal and communications theory; innovation models; Miscellaneous; Noise; Noise reduction; non-Gaussian statistics; Operators; Poisson processes; Signal and communications theory; Signal processing; Signal, noise; sparsity; Spline; splines; stochastic differential equations; Stochastic models; Stochastic processes; Stochasticity; Studies; Technological innovation; Telecommunications and information theory; wavelet transform; White noise; Random signal; Stochastic model; Differential equation; innovation models; wavelet transform; Fractals; sparsity; Boundary condition; Stochastic process; Poisson process; Spline; Spectral signature; Innovation; Gaussian noise; non-Gaussian statistics; Wavelet base; A posteriori estimation; White noise; Sparse representation; Stationary process; stochastic processes; Stochastic equation; Scale invariance; Specification; stochastic differential equations; splines; Poisson processes; Brownian motion; Pulse noise; Gaussian process; Signal processing
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2010.2091638

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; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finite-rate-of-innovation signals within Gelfand's framework of generalized stochastic processes. We then focus on the class of scale-invariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, spline-type signals that are piecewise-smooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the same 1/ω-type spectral signature. We prove that the generalized Poisson processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TV-denoising algorithm.