The successful use of multifractal analysis in applications crucially relies on accurate procedures for assessing and comparing the parameters of multifractal models. We will show how parameters associated with the multifractal spectrum can be estimated in practice in a robust manner, based on wavelet expansions of finite resolution time series or images. We will begin with revisiting the computation of estimates through log-log plot regressions and discuss the advantages, limitations and pitfalls of this approach. Then we will show how certain multifractal parameters can be assessed within a Bayesian model for the logarithm of wavelet leaders, with the practical benefit of stabilizing estimates. Finally, motivated by the fact that in an increasing number of applications, the acquired data are naturally multivariate (i.e., they consist of a collection of spatially/temporally/spectrally organized time series or images), we will show how the associated collections of multifractal parameters can be estimated jointly within appropriate hierarchical Bayesian models, enabling their regularization through the use of suitable priors. Several illustrations will be provided involving real-world multivariate time series and images from biomedical and remote sensing applications.


The slides of the course with supplementary material (appendices) can be downloaded here.

3D Animations for estimation for c2 for multivariate images

  • Estimation for a sequence of synthetic multifractal images (slide 89): download (.mp4 format).
  • Estimation for a real-world hyperspectral image cube (slide 99): download (.mp4 format).

MATLAB toolbox
Bayesian univariate and multivariate models and estimators for (c1,c2)

This MATLAB toolbox collects the different estimators for the multifractal parameters c1 and c2 for univariate and multivariate signals and images.

  • The standard estimator for c1 and c2 based on a simple linear regression.
  • The Bayesian estimator IG using the univariate (data augmented) model.
  • The multivariate Bayesian estimators using the joint gamma Markov random field prior for c2.
  • The multivariate Bayesian estimators using the joint SAR prior for c1.

Download: You can download the MATLAB files here.

If you use the code in your work, please cite the following references:
[JI.12 .bib]
"Bayesian Estimation of the Multifractality Parameter for Image Texture Using a Whittle Approximation"
[JI.26 .bib]
"Multifractal analysis of multivariate images using gamma Markov random field priors"

Demo Files

You will find several demo files that explain and illustrate the use of the different estimators for different data scenarios.

MATLAB Tutorials
Bayesian univariate and multivariate models and estimators for (c1,c2)

These tutorials illustrate the basic use of the different estimators implemented in the toolbox, for different data scenarios, and discuss further topics beyond "off the shelf" use, such as the possibility to change the integral scale.