This paper is devoted to the characterization of an extended family of continuous-time autoregressive moving average (CARMA) processes that are solutions of stochastic differential equations driven by white Lévy innovations. These are completely specified by: 1) a set of poles and zeros that fixes their correlation structure and 2) a canonical infinitely divisible probability distribution that controls their degree of sparsity (with the Gaussian model corresponding to the least sparse scenario). The generalized CARMA processes are either stationary or nonstationary, depending on the location of the poles in the complex plane. The most basic nonstationary representatives (with a single pole at the origin) are the Lévy processes, which are the non-Gaussian counterparts of Brownian motion. We focus on the general analog-to-discrete conversion problem and introduce a novel spline-based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes. We also rely on the concept of generalized increment process, which suppresses all long range dependencies, to specify an equivalent discrete-domain innovation model. A crucial ingredient is the existence of a minimally supported function associated with the whitening operator L; this B-spline, which is fundamental to our formulation, appears in most of our formulas, both at the level of the correlation and the characteristic function. We make use of these discrete-domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous-domain model.

We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson, or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern, which is determined by the type of noise excitation. The latter is fully specified by a Lévy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some Lp bound for p ≥ 1. We present a novel operator-based method that yields an explicit characterization of all Lévy-driven processes that are solutions of constant-coefficient stochastic differential equations. When the underlying system is stable, we recover the family of stationary continuous-time autoregressive moving average processes (CARMA), including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Lévy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.

In this paper, we give a general characterization of regularization functionals for vector field reconstruction, based on the requirement that the said functionals satisfy certain geometric invariance properties with respect to transformations of the coordinate system. In preparation for our general result, we also address some commonalities of invariant regularization in scalar and vector settings, and give a complete account of invariant regularization for scalar fields, before focusing on their main points of difference, which lead to a distinct class of regularization operators in the vector case. Finally, as an illustration of potential, we formulate and compare quadratic (L(2)) and total-variation-type (L(1)) regularized denoising of vector fields in the proposed framework.

In this note we introduce a vector generalization of fractional Brownian motion. Our definition takes into account directional properties of vector fields-such as divergence, rotational behaviour, and interactions with coordinate transformations-that have no counterpart in the scalar setting. Apart from the Hurst exponent which dictates the scale-dependent structure of the field, additional parameters of the new model control the balance between solenoidal and irrotational behaviour. This level of versatility makes these random fields potentially interesting candidates for the stochastic modelling of physical phenomena in various fields of application such as fluid dynamics, field theory, and medical image processing.

We provide an extension of the L2-spline pyramid (Unser et al., 1993) using polyharmonic splines. We analytically prove that the corresponding error pyramid behaves exactly as a multi-scale Laplace operator. We use the multiresolution properties of polyharmonic splines to derive an efficient, non-separable filterbank implementation. Finally, we illustrate the potentials of our pyramid by performing an estimation of the parameters of multivariate fractal processes.

In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBm-like processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.

When producing a mosaic of multiple multi-spectral images one needs to harmonize the colours so that the tone transition is smooth from one image to the other. Given two images Imaand Imb, a transform T is sought to map Imbto an image that is harmonious in multi-spectral appearance to Ima. We give the above problem of tonal harmonization an analytical framework, in which both ideal and practical solutions of the problem are studied. Using a physically motivated image formation model, we prove that a perfect tonal harmonizing operator cannot in general be found, but that whenever such an operator exists it is linear. In the latter case, finding the optimal harmonizing transformation can be cast as a linear programme (LP), which is a type of problem that can be efficiently solved using known techniques. Finally, strong empirical evidence is provided for the efficacy of the proposed solution.

This letter introduces a new representation of discrete signals based on the mathematical notions of functionals and continuous dual spaces. A new and more general sampling theorem is also suggested. Next, the problems of interpolating and resampling discrete signals are addressed; and a general solution using functional interpolation-which is applicable to many different settings-is proposed. Families of resampling filters dubbed de Boor-Ron filters that use de Boor-Ron interpolation are introduced, and their numerical realization is discussed. Some applications of this research are suggested

Multi-dimensional datasets containing local averages of a function arise in many applications such as processing of CCD captures and medical images. Motivated by this fact we introduce multi-dimensional average-interpolating refinement on arbitrary lattices in arbitrary dimensions. Our refinement algorithm results in smooth scaling functions of compact support. This method forms a basis for multi-dimensional multi-resolution analysis and subdivision on datasets obtained by locally averaging a smooth function. As an example, we present two-dimensional polynomial average-interpolating subdivision on the quincunx lattice and show that the resulting scaling functions are highly regular in the sense of Sobolev.