Bayesian Estimation of the Parameters of the Joint Multifractal Spectrum of Signals and Images

Abstract

Multifractal analysis has become a reference tool for signal and image processing. Grounded in the quantification of local regularity fluctuations, it has proven useful in an increasing range of applications, yet so far involving only univariate data (scalar valued time series or single channel images). Recently the theoretical ground for multivariate multifractal analysis has been devised, showing potential for quantifying transient higher-order dependence beyond linear correlation among collections of data. However, the accurate estimation of the parameters associated with a multivariate multifractal model remains challenging, severely limiting their actual use in applications. The main goal of this thesis is to propose and study practical contributions on multivariate multifractal analysis of signals and images. Specifically, the proposed approach relies on a novel and original joint Gaussian model for the logarithm of wavelet leaders and leverages on a Whittle-based likelihood approximation and data augmentation for the matrix-valued parameters of interest. This careful design enables efficient estimation procedures to be constructed for two relevant choices of priors using Bayesian inference. Algorithms based on Monte Carlo Markov Chain and Expectation Maximization strategies are designed and used to approximate the Bayesian estimators. Monte Carlo simulations, conducted on synthetic multivariate signals and images with various sample sizes, numbers of components and multifractal parameter settings, demonstrate significant performance improvements over the state of the art. In addition, theoretical lower bounds on the variance of the estimators are designed to study their asymptotic behavior. Finally, the relevance of the proposed multivariate multifractal estimation framework is shown for two real-world data examples, drowsiness detection from multichannel physiological signals and potential remote sensing applications in multispectral satellite imagery.