The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white L 'evy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized L 'evy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.
Several behaviors can be observed within this extended family of random processes. For instance, self-similar Gaussian processes exhibit fractal behaviors. In one dimension, they include the fractional Brownian motion @cite_32 and its higher-order extensions @cite_10 . In higher dimensions, our framework covers the family of fractional Gaussian fields @cite_21 @cite_38 @cite_36 and finite-variance self-similar fields that appear to converge to fractional Gaussian fields at large scales @cite_16 . The self-similarity property is also compatible with the family of @math -stable processes @cite_37 which have the particularity of having unbounded variances or second-order moments (when non-Gaussian). More generally, every process considered in our framework is part of the L 'evy family, including Laplace processes @cite_42 and Student's processes @cite_14 . Upon varying the operator @math , one recovers L 'evy processes @cite_1 , CARMA processes @cite_30 @cite_15 , and their multivariate generalizations @cite_23 @cite_47 . Unlike those examples, the compound-Poisson processes, although members of the L 'evy family, are piecewise-constant and have a finite rate of innovation (FRI) in the sense of @cite_26 . For a signal, being FRI means that a finite quantity of information is sufficient to reconstruct it over a bounded domain.
@cite_30 @cite_38 @cite_37 @cite_14 @cite_47 @cite_26 @cite_36 @cite_21 @cite_42 @cite_32 @cite_1 @cite_23 @cite_15 @cite_16 @cite_10
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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. It is well documented that natural images are compressible in wavelet bases and tend to exhibit fractal properties. In this paper, we investigate statistical models that mimic these behaviors. We then use our models to make predictions on the statistics of the wavelet coefficients. Following an innovation modeling approach, we identify a general class of finite-variance self-similar sparse random processes. We first prove that spatially dilated versions of self-similar sparse processes are asymptotically Gaussian as the dilation factor increases. Based on this fundamental result, we show that the coarse-scale wavelet coefficients of these processes are also asymptotically Gaussian, provided the wavelet has enough vanishing moments. Moreover, we quantify the degree of Gaussianity by deriving the theoretical evolution of the kurtosis of the wavelet coefficients across scales. Finally, we apply our analysis to one- and two-dimensional signals, including natural images, and show that the wavelet coefficients ... In a companion paper (see Self-Similarity: Part I-Splines and Operators), we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order gamma. We used these operators to specify regularization functionals for a series of Tikhonov-like least-squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)-based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically self-similar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel'fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) p(BH(t)|Y), where Y= BH(k)+n[k] kisinZ are the noisy samples of the fBm BH(t) with Hurst exponent H. We find that the conditional mean is a fractional spline of degree 2H, which proves that this class of functions is indeed optimal for the estimation of fractal-like processes. The result also yields the optimal [minimum mean-square error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.