Multifractal analysis for multivariate data with application to remote sensing

Texture characterization is a central element in many image processing applications. Texture analysis can be embedded in the mathematical framework of multifractal analysis, enabling the study of the fluctuations in regularity of image intensity and providing practical tools for their assessment, the coefficients or wavelet leaders. Although successfully applied in various contexts, multi fractal analysis suffers at present from two major limitations. First, the accurate estimation of multifractal parameters for image texture remains a challenge, notably for small sample sizes. Second, multifractal analysis has so far been limited to the analysis of a single image, while the data available in applications are increasingly multivariate. The main goal of this thesis is to develop practical contributions to overcome these limitations. The first limitation is tackled by introducing a generic statistical model for the logarithm of wavelet leaders, parametrized by multifractal parameters of interest. This statistical model enables us to counterbalance the variability induced by small sample sizes and to embed the estimation in a Bayesian framework. This yields robust and accurate estimation procedures, effective both for small and large images. The multifractal analysis of multivariate images is then addressed by generalizing this Bayesian framework to hierarchical models able to account for the assumption that multifractal properties evolve smoothly in the dataset. This is achieved via the design of suitable priors relating the dynamical properties of the multifractal parameters of the different components composing the dataset. Different priors are investigated and compared in this thesis by means of numerical simulations conducted on synthetic multivariate multifractal images. This work is further completed by the investigation of the potential benefit of multifractal analysis and the proposed Bayesian methodology for remote sensing via the example of hyperspectral imaging

Bayesian Estimation of the Multifractality Parameter for Image Texture Using a Whittle Approximation

International audienceTexture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64 × 64 pixels

A Bayesian approach for the joint estimation of the multifractality parameter and integral scale based on the Whittle approximation

International audienceMultifractal analysis is a powerful tool used in signal processing. Multifractal models are essentially characterized by two parameters, the multifractality parameter c2 and the integral scale A (the time scale beyond which multifractal properties vanish). Yet, most applications concentrate on estimating c2 while the estimation of A is in general overlooked, despite the fact that A potentially conveys important information. Joint estimation of c2 and A is challenging due to the statistical nature of multifractal processes (i.e. the strong dependence and non-Gaussian nature), and has barely been considered. The present contribution addresses these limitations and proposes a Bayesian procedure for the joint estimation of (c2, A). Its originality resides, first, in the construction of a generic multivariate model for the statistics of wavelet leaders for multifractal multiplicative cascade processes, and second, in the use of a suitable Whittle approximation for the likelihood associated with the model. The resulting model enables Bayesian estimators for (c2, A) to also be computed for large sample size. Performance is assessed numerically for synthetic multifractal processes and illustrated for wind-tunnel turbulence data. The proposed procedure significantly improves estimation of c2 and yields, for the first time, reliable estimates for A

Estimation bayésienne locale du paramètre de multifractalité à l'aide d'un algorithme de Monte Carlo Hamiltonien

International audienceLa caractérisation de la texture d’une image peut être conduite via l’étude des fluctuations de la régularité locale de son amplitude dans le cadre théorique de l’analyse multifractale. Les images étant souvent composées de différentes textures, cette analyse doit être locale. Cet article s’attaque à ce problème en formulant un modèle bayésien par patch reposant sur un modèle semi-paramétrique récemment proposé pour la statistique du logarithme des coefficients dominants. Les estimateurs bayésiens sont obtenus via une procédure d’échantillonnage utilisant un algorithme de Monte-Carlo Hamiltonien. Les performances de ces estimateurs sont illustrées à l’aide de processus synthétiques

Analyse multifractale de données multivariées avec application à la télédétection

La caractérisation de texture est centrale dans de nombreuses applications liées au traitement d’images. L’analyse de textures peut être envisagée dans le cadre mathématique de l’analyse multifractale qui permet d’étudier les fluctuations de la régularité ponctuelle de l’amplitude d’une image et fournit les outils pratiques pour leur évaluation grâce aux coefficients d’ondelettes ou aux coefficients dominants. Bien que mise à profit dans de nombreuses applications, l’analyse multifractale souffre à présent de deux limitations majeures. Premièrement, l’estimation des paramètres multifractaux reste délicate, notamment pour les images de petites tailles. Deuxièmement, l’analyse multifractale a été jusqu’à présent uniquement considérée pour l’analyse univariée d’images, alors que les données à étudier sont de plus en plus multivariées. L’objectif principal de cette thèse est la mise au point de contributions pratiques permettant de pallier ces limitations. La première limitation est abordée en introduisant un modèle statistique générique pour le logarithme des coefficients dominants, paramétrisé par les paramètres multifractaux d’intérêt. Ce modèle statistique permet de contrebalancer la variabilité résultant de l’analyse d’images de petite taille et de formuler l’estimation dans un cadre bayésien. Cette approche aboutit à des procédures d’estimation robustes et efficaces, que ce soit pour des images de petites ou grandes tailles. Ensuite, l’analyse multifractale d’images multivariées est traitée en généralisant ce cadre bayésien à des modèles hiérarchiques capables de prendre en compte l’hypothèse d’une évolution lente des propriétés multifractales d’images multi-temporelles ou multi-bandes. Ceci est réalisé en définissant des lois a priori reliant les propriétés dynamiques des paramètres multifractaux des différents éléments composant le jeu de données. Différents types de lois a priori sont étudiés dans cette thèse au travers de simulations numériques conduites sur des images multifractales multivariées synthétiques. Ce travail est complété par une étude du potentiel apport de l’analyse multifractale et de la méthodologie bayésienne proposée pour la télédétection à travers l’exemple de l’imagerie hyperspectrale.Texture characterization is a central element in many image processing applications. Texture analysis can be embedded in the mathematical framework of multifractal analysis, enabling the study of the fluctuations in regularity of image intensity and providing practical tools for their assessment, the coefficients or wavelet leaders. Although successfully applied in various contexts, multi fractal analysis suffers at present from two major limitations. First, the accurate estimation of multifractal parameters for image texture remains a challenge, notably for small sample sizes. Second, multifractal analysis has so far been limited to the analysis of a single image, while the data available in applications are increasingly multivariate. The main goal of this thesis is to develop practical contributions to overcome these limitations. The first limitation is tackled by introducing a generic statistical model for the logarithm of wavelet leaders, parametrized by multifractal parameters of interest. This statistical model enables us to counterbalance the variability induced by small sample sizes and to embed the estimation in a Bayesian framework. This yields robust and accurate estimation procedures, effective both for small and large images. The multifractal analysis of multivariate images is then addressed by generalizing this Bayesian framework to hierarchical models able to account for the assumption that multifractal properties evolve smoothly in the dataset. This is achieved via the design of suitable priors relating the dynamical properties of the multifractal parameters of the different components composing the dataset. Different priors are investigated and compared in this thesis by means of numerical simulations conducted on synthetic multivariate multifractal images. This work is further completed by the investigation of the potential benefit of multifractal analysis and the proposed Bayesian methodology for remote sensing via the example of hyperspectral imaging

Spatially regularized multifractal analysis for fMRI Data

International audienceScale-free dynamics is nowadays a massively used paradigm to model infraslow macroscopic brain activity. Mul-tifractal analysis is becoming the standard tool to characterize scale-free dynamics. It is commonly used on various modalities of neuroimaging data to evaluate whether arrhythmic fluctuations in ongoing or evoked brain activity are related to patholo-gies (Alzheimer, epilepsy) or task performance. The success of multifractal analysis in neurosciences remains however so far contrasted: While it lead to relevant findings on M/EEG data, less clear impact was shown when applied to fMRI data. This is mostly due to their poor time resolution and very short duration as well as to the fact that analysis remains performed voxelwise. To take advantage of the large amount of voxels recorded jointly in fMRI, the present contribution proposes the use of a recently introduced Bayesian formalism for multifractal analysis, that regularizes the estimation of the multifractality parameter of a given voxel using information from neighbor voxels. The benefits of this regularized multifractal analysis are illustrated by comparison against classical multifractal analysis on fMRI data collected on one subject, at rest and during a working memory task: Though not yet statistically significant, increased multifractality is observed in task-negative and task-positive networks, respectively

Multivariate Scale-free dynamics: Testing Fractal Connectivity

International audienceScale-free dynamics commonly appear in individual components of multivariate data. Yet, while the behavior of cross-components is crucial in modeling real-world multivariate data, their examination often suggests departures from exact multivariate self-similarity (also termed fractal connectivity). The present paper introduces a multivariate Gaussian stochastic process with Hadamard (i.e., entry-wise) self-similar scale-free dynamics, controlled by a matrix Hurst parameter H, that allows departures from fractal connectivity. The properties of its wavelet coefficients and wavelet spectrum are studied, enabling the estimation of H and of the fractal connectivity parameter. Furthermore, it permits the computation of closed-form confidence intervals for the estimates based on approximate (wavelet) covariances. Finally, these developments enable us to devise a test for fractal connectivity. Monte Carlo simulations are used to assess the accuracy of the proposed approximate confidence intervals and the performance of the fractal connectivity test

Bayesian estimation of the multifractality parameter for images via a closed-form Whittle likelihood

International audienceTexture analysis is central in many image processing problems. It can be conducted by studying the local regularity fluctuations of image amplitudes, and multifractal analysis provides a theoretical and practical framework for such a characterization. Yet, due to the non Gaussian nature and intricate dependence structure of multifractal models, accurate parameter estimation is challenging: standard estimators yield modest performance, and alternative (semi-)parametric estimators exhibit prohibitive computational cost for large images. This present contribution addresses these difficulties and proposes a Bayesian procedure for the estimation of the multifractality parameter c2 for images. It relies on a recently proposed semi-parametric model for the multivariate statistics of log-wavelet leaders and on a Whittle approximation that enables its numerical evaluation. The key result is a closed-form expression for the Whittle likelihood. Numerical simulations indicate the excellent performance of the method, significantly improving estimation performance over standard estimators and computational efficiency over previously proposed Bayesian estimators

Hyperspectral image analysis using multifractal attributes

International audienceThe increasing spatial resolution of hyperspectral remote sensors requires the development of new processing methods capable of combining both spectral and spatial information.In this article, we focus on the spatial component and propose the use of novel multifractal attributes, which extract spatial information in terms of the fluctuations of the local regularity of image amplitudes. The novelty of the proposed approach is twofold. First, unlike previous attempts, we study attributes that efficiently summarize multifractal information in a few parameters. Second, we make use of the most recent developments in multifractal analysis: wavelet leader multifractal formalism, the current theoretical and practical benchmark in multifractal analysis, and a novel Bayesian estimation procedure for one of the central multifractal parameters. Attributes provided by these state-of-the-art multifractal analysis procedures are studied on two sets of hyperspectral images. The experiments suggest that multifractal analysis can provide relevant spatial/textural attributes which can in turn be employed in tasks such as classification or segmentation