Sébastien Combrexelle

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

Texture 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 framework for the multifractal analysis of images using data augmentation and a whittle approximation

Texture analysis is an image processing task that can be conducted using the mathematical framework of multifractal analysis to study the regularity fluctuations of image intensity and the practical tools for their assessment, such as (wavelet) leaders. A recently introduced statistical model for leaders enables the Bayesian estimation of multifractal parameters. It significantly improves performance over standard (linear regression based) estimation. However, the computational cost induced by the associated nonstandard posterior distributions limits its application. The present work proposes an alternative Bayesian model for multifractal analysis that leads to more efficient algorithms. It relies on three original contributions: A novel generative model for the Fourier coefficients of log-leaders; an appropriate reparametrization for handling its inherent constraints; a data-augmented Bayesian model yielding standard conditional posterior distributions that can be sampled exactly. Numerical simulations using synthetic multifractal images demonstrate the excellent performance of the proposed algorithm, both in terms of estimation quality and computational cost.

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

Texture 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.

Multivariate Hadamard self-similarity: Testing fractal connectivity

While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters.

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

Multifractal 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.

Bayesian estimation for the local assessment of the multifractality parameter of multivariate time series

Multifractal analysis (MF) is a widely used signal processing tool that enables the study of scale invariance models. Classical MF assumes homogeneous MF properties, which cannot always be guaranteed in practice. Yet, the local estimation of MF parameters has barely been considered due to the challenging statistical nature of MF processes (non-Gaussian, intricate dependence), requiring large sample sizes. This present work addresses this limitation and proposes a Bayesian estimator for local MF parameters of multivariate time series. The proposed Bayesian model builds on a recently introduced statistical model for leaders (i.e., specific multiresolution quantities designed for MF analysis purposes) that enabled the Bayesian estimation of MF parameters and extends it to multivariate non-overlapping time windows. It is formulated using spatially smoothing gamma Markov random field priors that counteract the large statistical variability of estimates for short time windows. Numerical simulations demonstrate that the proposed algorithm significantly outperforms current state-of-the-art estimators.

Hyperspectral image analysis using multifractal attributes

The 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.

Multifractal Analysis of Multivariate Images Using Gamma Markov Random Field Priors

Texture characterization of natural images using the mathematical framework of multifractal analysis (MFA) enables the study of the fluctuations in the regularity of image intensity. Although successfully applied in various contexts, the use of MFA has so far been limited to the independent analysis of a single image, while the data available in applications are increasingly multivariate. This paper addresses this limitation and proposes a joint Bayesian model and associated estimation procedure for multifractal parameters of multivariate images. It builds on a recently introduced generic statistical model that enabled the Bayesian estimation of multifractal parameters for a single image and relies on the following original key contributions: First, we develop a novel Fourier domain statistical model for a single image that permits the use of a likelihood that is separable in the multifractal parameters via data augmentation. Second, a joint Bayesian model for multivariate images is formulated in which prior models based on gamma Markov random fields encode the assumption of the smooth evolution of multifractal parameters between the image components. The design of the likelihood and of conjugate prior models is such that exploitation of the conjugacy between the likelihood and prior models enables an efficient estimation procedure that can handle a large number of data components. Numerical simulations conducted using sequences of multifractal images demonstrate that the proposed procedure significantly outperforms previous univariate benchmark formulations at a competitive computational cost.

Multivariate scale-free dynamics: Testing fractal connectivity

Scale-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.

A Bayesian approach for the multifractal analysis of spatio-temporal data

Multifractal (MF) analysis enables the theoretical study of scale invariance models and their practical assessment via wavelet leaders. Yet, the accurate estimation of MF parameters remains a challenging task. For a range of applications, notably biomedical, the performance can potentially be improved by taking advantage of the multivariate nature of data. However, this has barely been considered in the context of MF analysis. This paper proposes a Bayesian model that enables the joint estimation of MF parameters for multivariate time series. It builds on a recently introduced statistical model for leaders and is formulated using a 3D gamma Markov random field joint prior for the MF parameters of the voxels of spatio-temporal data, represented as a multivariate time series, that counteracts the statistical variability induced by small sample size. Numerical simulations indicate that the proposed Bayesian estimator significantly outperforms current state-of-the-art algorithms.