Julien Fageot

Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Splines come in a variety of flavors that can be characterized in terms of some differential operator

On the Besov regularity of periodic Lévy noises

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Wavelet Statistics of Sparse and Self-Similar Images ∗

. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, on...

Compressibility of symmetric-α-stable processes

In this paper, we study the Besov regularity of Levy white noises on the d-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Levy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-α-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical σ-field of the space of generalized functions. These results pave the way to the characterization of the n-term wavelet approximation properties of stochastic processes.

Interpretation of continuous-time autoregressive processes as random exponential splines

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 ...

Representer Theorems for Sparsity-Promoting

Within a deterministic framework, it is well known that n-term wavelet approximation rates of functions can be deduced from their Besov regularity. We use this principle to determine approximation rates for symmetric-α-stable (SαS) stochastic processes. First, we characterize the Besov regularity of SαS processes. Then the n-term approximation rates follow. To capture the local smoothness behavior, we consider sparse processes defined on the circle that are solutions of stochastic differential equations.

\ell _{1}

We consider the class of continuous-time autoregressive (CAR) processes driven by (possibly non-Gaussian) Lévy white noises. When the excitation is an impulsive noise, also known as compound Poisson noise, the associated CAR process is a random non-uniform exponential spline. Therefore, Poisson-type processes are relatively easy to understand in the sense that they have a finite rate of innovation. We show in this paper that any CAR process is the limit in distribution of a sequence of CAR processes driven by impulsive noises. Hence, we provide a new interpretation of general CAR processes as limits of random exponential splines. We illustrate our result with simulations.

Regularization

We present a theoretical analysis and comparison of the effect of l1 versus l2 regularization for the resolution of ill-posed linear inverse and/or compressed sensing problems. Our formulation covers the most general setting where the solution is specified as the minimizer of a convex cost functional. We derive a series of representer theorems that give the generic form of the solution depending on the type of regularization. We start with the analysis of the problem in finite dimensions and then extend our results to the infinite-dimensional spaces l2(Z) and l1(Z). We also consider the use of linear transformations in the form of dictionaries or regularization operators. In particular, we show that the l2 solution is forced to live in a predefined subspace that is intrinsically smooth and tied to the measurement operator. The l1 solution, on the other hand, is formed by adaptively selecting a subset of atoms in a dictionary that is specified by the regularization operator. Beside the proof that l1 solutions are intrinsically sparse, the main outcome of our investigation is that the use of l1 regularization is much more favorable for injecting prior knowledge: it results in a functional form that is independent of the system matrix, while this is not so in the l2 scenario.

Hermite Snakes With Control of Tangents

We introduce a new model of parametric contours defined in a continuous fashion. Our curve model relies on Hermite spline interpolation and can easily generate curves with sharp discontinuities; it also grants direct access to the tangent at each location. With these two features, the Hermite snake distinguishes itself from classical spline-snake models and allows one to address certain bioimaging problems in a more efficient way. More precisely, the Hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image. Using the formalism of spline theory, the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties. Finally, the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our Hermite snake model, and its usefulness is illustrated on real biological images acquired using brightfield, phase-contrast, differential-interference-contrast, and scanning-electron microscopy.

Gaussian versus Sparse Stochastic Processes

Although our work lies in the field of random processes, this thesis was originally motivated by signal processing applications, mainly the stochastic modeling of sparse signals. We develop a mathematical study of the innovation model, under which a signal is described as a random process s that can be linearly and deterministically transformed into a white noise. The noise represents the unpredictable part of the signal, called its innovation, and is part of the family of Levy white noises, which includes both Gaussian and Poisson noises. In mathematical terms, s satisfies the equation Ls=w where L is a differential operator and w a Levy noise. The problem is therefore to study the solution of a stochastic differential equation driven by a Levy noise. Gaussian models usually fail to reproduce the empirical sparsity observed in real-world signals. By contrast, Levy models offer a wide range of random processes going from typically non-sparse (Gaussian) to very sparse ones (Poisson), and with many sparse signals standing between these two extremes. Our contributions can be divided in four parts. First, the cornerstone of our work is the theory of generalized random processes. Within this framework, all the considered random processes are seen as random tempered generalized functions and can be observed through smooth and rapidly decaying windows. This allows us to define the solutions of Ls=w, called generalized Levy processes, in the most general setting. Then, we identify two limit phenomenons: the approximation of generalized Levy processes by their Poisson counterparts, and the asymptotic behavior of generalized Levy processes at coarse and fine scales. In the third part, we study the localization of Levy noise in notorious function spaces (Holder, Sobolev, Besov). As an application, characterize the local smoothness and the asymptotic growth rate of the Levy noise. Finally, we quantify the local compressibility of the generalized Levy processes, understood as a measure of the decreasing rate of their approximation error in an appropriate basis. From this last result, we provide a theoretical justification of the ability of the innovation model to represent sparse signals. The guiding principle of our research is the duality between the local and asymptotic properties of generalized Levy processes. In particular, we highlight the relevant quantities, called the local and asymptotic indices, that allow quantifying the local regularity, the asymptotic growth rate, the limit behavior at coarse and fine scales, and the level of compressibility of the solutions of generalized Levy processes.