Aurélia Fraysse

How smooth is almost every function in a Sobolev space

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Holder regularity are fractal sets, and we determine their Hausdorff dimension.

GENERIC VALIDITY OF THE MULTIFRACTAL FORMALISM

The multifractal formalism is a conjecture which gives the spectrum of singularities of a signal using numerically computable quantities. We prove its generic validity by showing that almost every function in a given function space is multifractal and satisfies the multifractal formalism.

On the Uniform Quantization of a Class of Sparse Sources

We consider the uniform scalar quantization of a class of mixed distributed memoryless sources, namely sources having a Bernoulli Generalized Gaussian (BGG) distribution. Both for low and high resolutions, asymptotic expressions of the distortion for a pth-order moment error measure, and close approximations of the entropy are provided for these sources. Operational rate-distortion functions at high bit rate and their slope factors at low bit rate are derived. The dependence of these results on p and the distribution parameters as well as the relation to the Shannon optimal rate-distortion bound are then discussed. The application of these results to transform coding in two simple cases is finally highlighted.

A Measure-Theoretic Variational Bayesian Algorithm for Large Dimensional Problems

In this paper we provide an algorithm adapted to variational Bayesian approximation. The main contribution is to transpose a classical iterative algorithm of optimization in the metric space of measures involved in Bayesian methodology. Once given the convergence properties of this algorithm, we consider its application to large dimensional inverse problems, especially for unsupervised reconstruction. The interest of our method is enhanced by its application to large dimensional linear inverse problems involving sparse objects. Finally, we provide simulation results. First we show the good numerical performances of our method compared to classical ones on a small example. Then we deal with a large dimensional dictionary learning problem.

A gradient-like variational Bayesian algorithm

In this paper we provide a new algorithm allowing to solve a variational Bayesian issue which can be seen as a functional optimization problem. The main contribution of this paper is to transpose a classical iterative algorithm of optimization in the metric space of probability densities involved in the Bayesian methodology. Another important part is the application of our algorithm to a class of linear inverse problems where estimated quantities are assumed to be sparse. Finally, we compare performances of our method with classical ones on a tomographic problem. Preliminary results on a small dimensional example show that our new algorithm is faster than the classical approaches for the same quality of reconstruction.

Efficient Variational Bayesian Approximation Method Based on Subspace Optimization

Variational Bayesian approximations have been widely used in fully Bayesian inference for approximating an intractable posterior distribution by a separable one. Nevertheless, the classical variational Bayesian approximation (VBA) method suffers from slow convergence to the approximate solution when tackling large dimensional problems. To address this problem, we propose in this paper a more efficient VBA method. Actually, variational Bayesian issue can be seen as a functional optimization problem. The proposed method is based on the adaptation of subspace optimization methods in Hilbert spaces to the involved function space, in order to solve this optimization problem in an iterative way. The aim is to determine an optimal direction at each iteration in order to get a more efficient method. We highlight the efficiency of our new VBA method and demonstrate its application to image processing by considering an ill-posed linear inverse problem using a total variation prior. Comparisons with state of the art variational Bayesian methods through a numerical example show a notable improvement in computation time.

A Bit Allocation Method for Sparse Source Coding

In this paper, we develop an efficient bit allocation strategy for subband-based image coding systems. More specifically, our objective is to design a new optimization algorithm based on a rate-distortion optimality criterion. To this end, we consider the uniform scalar quantization of a class of mixed distributed sources following a Bernoulli-generalized Gaussian distribution. This model appears to be particularly well-adapted for image data, which have a sparse representation in a wavelet basis. In this paper, we propose new approximations of the entropy and the distortion functions using piecewise affine and exponential forms, respectively. Because of these approximations, bit allocation is reformulated as a convex optimization problem. Solving the resulting problem allows us to derive the optimal quantization step for each subband. Experimental results show the benefits that can be drawn from the proposed bit allocation method in a typical transform-based coding application.

Rate-distortion results for Generalized Gaussian distributions

In this paper, we provide operational rate-distortion results for memoryless generalized Gaussian sources. Close approximations of the entropy are provided for these sources, after a uniform scalar quantization at low/high resolution. Asymptotic expressions of the distortion for an arbitrary p-th order error measure are also given. The resulting approximations at low/high bitrate of the operational rate-distortion function are thus compared with the Shannon optimal bound showing the overall good performance of uniform quantization rules.

Accurate rate-distortion approximation for sparse Bernoulli-Generalized Gaussian models

The objective of this paper is to study rate-distortion properties of a quantized Bernoulli-Generalized Gaussian source. Such source model has been found to be well-adapted for signals having a sparse representation in a transformed domain. We provide here accurate approximations of the entropy and the distortion functions evaluated through a p-th order error measure. These theoretical results are then validated experimentally. Finally, the benefit that can be drawn from the proposed approximations in bit allocation problems is illustrated for a wavelet-based compression scheme.

Floating Costa Scheme with fractal structure for information embedding

Nowadays, multimedia data protection widely uses data hiding technology like Digital Watermarking or steganography. Among the large offer of watermarking techniques several ones are now based on the side information scheme proposed by Costa. These techniques mainly use scalar quantization for embedding a given information. Such solutions have good properties in terms of robustness and capacity but has lack of statistical transparency. In this paper we propose a new watermarking scheme based on a floating quantizer with a fractal structure. The aim of this study is to get rid of the unwanted fluctuations in the probability density function (PDF) of the watermarked signal, caused by the use of a scalar quantizer in the Scalar Costa Scheme approach (SCS). We present here the structure of a Fractal based Costa Scheme (FCS) and detail its performances. Afterwards, we compare the performances of our FCS proposed method with those obtained with the reference SCS. We show that we are able to reduce the statistical distortions obtained with this proposed scheme and keep good robustness properties.