Bayesian estimation of the multifractality parameter for image texture using a Whittle approximation

TL;DR

This paper introduces a Bayesian method with a Whittle approximation for accurately estimating the multifractality parameter in image textures, especially effective for small images, improving over existing methods.

Contribution

It develops a novel semi-parametric Bayesian model and a Whittle approximation technique for better multifractal parameter estimation in images.

Findings

Significantly improves estimation accuracy over benchmarks.

Effective for small image patches as small as 64x64 pixels.

Enhances discrimination between different multifractal process models.

Abstract

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 two-dimensional (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 semi-parametric 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 64x64 pixels.