Despite its ability to characterize 3-D environments, synthetic aperture radar (SAR) tomographic imaging, when applied to the characterization of targets concealed beneath forest canopies, may appear as an ill-conditioned estimation problem, with a complex mixture of numerous scattering mechanisms measured from a few different positions. Among the set of tomographic estimators that may be used to characterize such complex scattering environments, nonparametric tomographic techniques are more robust to focus on artifacts but limited in resolution and, hence, may fail to discriminate objects, whereas parametric ones provide better vertical resolution but cannot adequately handle continuously distributed volumetric scattering densities, characteristic of forest canopies. This letter addresses a new wavelet-based sparse tomographic estimation method for the 3-D imaging and discrimination of underfoliage objects that overcomes these limitations. The effectiveness of this new approach is demonstrated using L-band airborne tomographic SAR data acquired by the German Aerospace Center over Dornstetten, Germany.

SAR imaging of concealed targets beneath the canopies has to face a complex mixture of diverse scattering mechanisms. To characterize this complex scattering environment, nonparametric tomographic estimators are more robust to focusing artefacts but limited in resolution. Parametric tomographic estimators provide better vertical resolution but fail to adequately characterize continuously distributed volumetric scatterers such as forest canopies. To overcome these limitations, this paper addresses a new wavelet-based sparse estimation method for 3D imaging and characterization for underfoliage objects. The effectiveness of this new approach is demonstrated by using L-band Multi-Baseline PolInSAR Data over Dornstetten, Germany.

We consider a family of stochastic processes built from infinite sums of independent positive random functions on$\BBR_{+}$. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on$\BBR_{+}$. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic. Such processes bear some analogy with Lévy processes, but are more complex since their increments are neither stationary nor independent. In the work of Barral and Lévy Véhel, the Hausdorff multifractal spectrum of these processes was computed. We are interested here in their Large Deviation and Legendre multifractal spectra. These “statistical” spectra are seen to give, in this case, a richer information than the “geometrical” Hausdorff spectrum. In addition, our results provide a firm theoretical basis for the empirical discovery of the multifractal nature of TCP traffic.

This paper presents a Genetic Programming (GP) approach to synthesize estimators for the pointwise Hölder exponent in 2D signals. It is known that irregularities and singularities are the most salient and informative parts of a signal. Hence, explicitly measuring these variations can be important in various domains of signal processing. The pointwise Hölder exponent provides a characterization of these types of features. However, current methods for estimation cannot be considered to be optimal in any sense. Therefore, the goal of this work is to automatically synthesize operators that provide an estimation for the Hölderian regularity in a 2D signal. This goal is posed as an optimization problem in which we attempt to minimize the error between a prescribed regularity and the estimated regularity given by an image operator. The search for optimal estimators is then carried out using a GP algorithm. Experiments confirm that the GP-operators produce a good estimation of the Hölder exponent in images of multifractional Brownian motions. In fact, the evolved estimators significantly outperform a traditional method by as much as one order of magnitude. These results provide further empirical evidence that GP can solve difficult problems of applied mathematics.

We analyze the local regularity of RR traces from ECG through the computation of the so-called Hölder exponents. These exponents are at the basis of multifractal analysis, which has been shown to be relevant in the study of RR data. While multifractal analysis yields a global picture of the (statistical) distribution of regularity, we focus here on its time evolution. We show that this evolution is strongly correlated with the signal itself, a feature that seems to have remained unnoticed so far. We use this fact to build realistic synthetic RR traces.

We present a theoretical and experimental analysis of the influence of the local irregularity of the fitness function on the behavior of an (1+1)ES. Previous work on this subject suggests that the performance of an EA strongly depends on the irregularity of the fitness function. Several irregularity measures have been derived for discrete search spaces, in order to numerically characterize this type of difficulty for EA. These characterizations are mainly based on Holder exponents. Previous studies used however a global characterization of fitness regularity (the global Holder exponent), with experimental validations being conducted on test functions with uniform regularity. This work is extended here in two ways: Results are now stated for continuous search spaces, and pointwise instead of global irregularity is considered. In addition, we present a way to modify the genetic topology to accommodate for variable regularity: the mutation radius, which controls the size of the neighbourhood of a point, is allowed to vary according to the local irregularity of the fitness function. These results are explained through a simple theoretical analysis which gives a relation between the pointwise Holder exponent and the optimal mutation radius. Several questions connected to on-line measurements and usage of regularity in EAs are raised.

We consider the problem of interpolating a signal in Rdknown at a given resolution. In our approach, the signal is assumed to belong to a given (large) class of signals. This class is characterised by local regularity constraints, that can be described by a certain inter-scale behaviour of their wavelet coefficients. These constraints allow to predict the scale n coefficient from the ones at lower scales. We investigate some properties of this interpolation scheme. In particular, we give the Holder regularity of the refined signal, and present some asymptotic properties of the method. Finally, numerical experiments are presented. Both the theoretical and numerical results show that our regularity-based scheme allows to obtain good quality interpolated images

Interactive evolutionary algorithms (IEA) often suffer from what is called the "user bottleneck". In this paper, we propose and analyze a method to limit the user interactions, while still providing sufficient informations for the EA to converge. The method has been currently developed on a multifractal image denoising application: a multifractal denoising method is adapted to complex images, but depends on a set of parameters that are quite difficult to tune by hand. A simple IEA has been developed for this purpose in a previous work. We now experiment an approximation of the user judgment, via a "fitness map", that helps to reduce the number of user-interactions. The method is easily extensible to other interactive, or computationally expensive, evolutionary schemes.

This work presents an approach for signal/image denoising in a semi-parametric frame. Our model is a wavelet-based one, which essentially assumes a minimal local regularity. This assumption translates into constraints on the multifractal spectrum of the signals. Such constraints are in turn used in a Bayesian framework to estimate the wavelet coefficients of the original signal from the noisy ones. Our scheme is well adapted to the processing of irregular signals, such as (multi-)fractal ones, and is potentially useful for the processing of e.g. turbulence, bio-medical or seismic data.

We present an approach for image denoising based on the analysis of the local Holder regularity. The method takes the point of view that denoising may be performed by increasing the Holder regularity at each point. Under the assumption that the noise is additive and white, our procedure is asymptotically minimax, provided the original signal belongs to a ball in some Besov space. Such a scheme is well adapted to the case where the image to be recovered is itself very irregular, e.g. nowhere differentiable with rapidly varying local regularity. The method is implemented through a wavelet analysis. We show an application to SAR image denoising where this technique yields good results compared to other algorithms.

previous studies have shown that TCP traffic displays strong multifractal scaling. However a physical explanation of why such a behavior occurs is still elusive. We propose a cascade model that is based on the retransmission and congestion avoidance mechanisms of TCP. At the same time, it relates to the physical, tree-like organization of networks. This model allows one to relate the most salient multifractal features with basic traffic parameters as the round trip time (RTT) and the loss probability. Numerical experiments confirm that such a parsimonious model is able to give a satisfactory explanation for a number of features pertaining to multifractality, including the range of scales where it is observed. We believe these results open the way to a more profound understanding of the small time scale properties of TCP traffic.

Multifractional Brownian motion (mBm) was introduced to overcome certain limitations of the classical fractional Brownian motion (fBm). The major difference between the two processes is that, contrarily to fBm, the almost sure Holder exponent of mBm is allowed to vary along the trajectory, a useful feature when one needs to model processes whose regularity evolves in time, such as Internet traffic or images. Various properties of mBm have been studied in the literature, related to its dimensions or the statistical estimation of its pointwise Holder regularity. However, the covariance structure of mBm has not been investigated so far. We present in this work an explicit formula for this covariance. Since mBm is a zero mean Gaussian process, this provides a full characterization of its stochastic properties. We report on some applications, including the synthesis problem and the long term structure: in particular, we show that the increments of mBm exhibit long range dependence under general conditions.

We present a method for digital image watermarking based on the modification of certain subsets of the wavelet packet decomposition. These subsets are determined both from a secret key and an image dependent procedure that chooses a best basis from an energy criterion. The mark is set by imposing a parity constraint at each level of the decomposition. We elaborate on the choice of some of the parameters of the model, showing how they can be tuned so as to obtain good resistance to attacks. Examples are illustrated to assess the validity of our approach.

We present a deception analysis for Holder functions. Our approach uses a decomposition on the Haar basis, which reflects in a natural way the Holder structure of the function. This allows the relation of the deception, the Holder exponent, and some parameters of the genetic algorithms (GAs). These results prove that deception is connected to the irregularity of the fitness function and shed a new light on the schema theory. In addition, this analysis may assist in understanding the influence of some of the parameters on the performance of a GA.

We propose a multifractal approach to the problem of change detection in image sequences, such as registrated remotely sensed images of the same scene or sequences of medical images. We show that the multifractal analysis of images-based on a modelisation of the two-dimensional signal as measure-can be of great help if we want to detect changes without any a priori knowledge of the objects to be extracted. We first present a simple change detection method based on the classical multifractal analysis of images w.r.t. the Lebesgue measure. We then describe an improved method based on the analysis of images w.r.t. a reference measure, which in this case is the first image of the sequence. We finally show some results on real data.

Image analysis using texture and multifractal paradigms is addressed. Multifractal theory and its application to image description are discussed, and it is shown that this approach allows the discrete signal to be worked on directly. A system for texture classification that is based on a learning scheme and does not make use of any a priori model is introduced. Image segmentation is then considered, and the notion of mixed classes, which allows accurate detection of texture boundaries on complex images, is introduced.>

The extension of the iterated function system (IFS) theory dealing with probability functions instead of numbers is applied to texture analysis. The results allow encoding and reconstruction of textured images, and hence the compression of data to tackle the problem of texture segmentation in a rigorous manner.>

Two techniques are applied for segmentation of different states of one texture (e.g. deformations of a homogeneous texture): fractal geometry, which deals with the analysis of complex irregular shapes which cannot be described by the classical Euclidean geometry, and integral geometry, which treats sets globally and makes it possible to introduce robust measures. The author focuses on the study of two parameters, lacunarity and Favard length, and proves a theoretical link between them. As an application, the author achieves with an excellent accuracy automatic classification of lung diseases on SPECT images. Classical techniques tried on those images given poor results.>

We present some new techniques for shape approximation with fractals, using Iterated Function System, a powerful method which allows good control on the resulting fractal. The main point discussed here can be stated as follows : given a grey level image A, find a few number of functions and associated probabilities that approximately generate A. Two directions have been explored : the first uses a gradient method, thus it was necessary to define a smooth error function ; the second one is based upon the ideas of simulated annealing. We then generalize the methods to a broader class of functions, and present some results.

The use of fractals to synthesize complex objects is of current interest in the computer graphics community. A powerful way to compute fractals is the use of IFS (iterated function system) which is a set of contractions with associated probabilities which characterize the fractal. This theory, developed by M. Barnsley and al., can produce very complicated objects. We present a method to solve the inverse problem for these globally constructed fractals : given a set A (attractor), find an IFS that will approximately generate A. We use an optimisation method to minimize a distance between A and the current set L. Several distances have been tested and an algorithm has been implemented which gives good results. A test image is presented.