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 @math pixels.
Multifractal analysis has matured into a widely used signal and image processing tool. Due to the statistical nature of multifractal processes (strongly non-Gaussian and intricate dependence) the accurate estimation of multifractal parameters is very challenging in situations where the sample size is small (notably including a range of biomedical applications) and currently available estimators need to be improved. To overcome such limitations, the present contribution proposes a Bayesian estimation procedure for the multifractality (or intermittence) parameter. Its originality is threefold: First, the use of wavelet leaders, a recently introduced multiresolution quantity that has been shown to yield significant benefits for multifractal analysis; Second, the construction of a simple yet generic semi-parametric model for the marginals and covariance structure of wavelet leaders for the large class of multiplicative cascade based multifractal processes; Third, the construction of original Bayesian estimators associated with the model and the constraints imposed by multifractal theory. Performance are numerically assessed and illustrated for synthetic multifractal processes for a range of multifractal parameter values. The proposed procedure yields significantly improved estimation performance for small sample sizes. We present an approximated maximum likelihood method for the multifractal random walk processes of [E. , Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the R computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.