  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.
