Stochastic Models for Sparse and Piecewise-Smooth Signals

Abstract

Abstract-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 -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.

Generalized total variation denoising via augmented Lagrangian cycle spinning with Haar wavelets

“…We consider a 1-D signal that follows the innovation model in [10] and generate a compound Poisson process (piecewiseconstant signal) for which TV regularization is particularly well tailored. We show in Figure 1 the plots of the TV objective function for a signal of length 4000, generated with Prob(x = 0) = 0.9 and a Gaussian amplitude distribution of variance 1.…”

Bayesian denoising of generalized poisson processes with finite rate of innovation

“…1 It is shown in [2] that, for symmetric probability distribution of c k s (pc) and for rapidly decaying test functions ϕ(x), we have…”

“…The generalized Poisson processes characterized in [2] are the stochastic counterparts of the signals with Finite Rate of Innovation (FRI) [3]. The latter family includes nonuniform splines and piecewise-polynomial functions as particular cases.…”

“…of [1,2], the linear operators acting on the Dirac stream (innovations) play a role similar to the continuous-domain prefilter in the FRI setup.…”

Left-inverses of fractional Laplacian and sparse stochastic processes

“…We One of the primary application of the p-integrable Riesz potentials is the construction of generalized random processes by suitable functional integration of white noise [12][13][14]. These processes are defined by the stochastic partial differential equation (1.3), the motivation being that the solution should essentially display the same invariance properties as the defining operator (fractional Laplacian).…”