Regularization dimension

The general idea is to define a fractal index for irregular functions based on the behavior of the lengths of less and less regularized versions of their graphs.

The regularized versions are obtained by convoluting the function with a smooth kernel dilated by a scale parameter a. When a tends to 0, the kernel tends to the Dirac distribution and the regularized versions to the original graph. See the mathematical definition.

A "fractal" graph will typically exhibit a power law for the regularized lengths as a function of a, with exponent -d. In this case, the regularization dimension is defined to be 1+d. In general, one uses as is customary lower and upper limits to obtain well defined dimensions. The basic properties of the regularization dimension are similar to those of classical fractal dimensions, such as the box dimension.

However, it has a number of advantages over other dimensions in the frame of signal and image processing. Indeed, thanks to its adaptive and analytical definition, it usually leads to estimations methods which are more precise than is the case for the box dimension (and of course the Hausdorff dimension). It also has interesting statistical properties. Indeed, if the data are corrupted with an additive white noise, one can easily distinguish, in the computation of the regularization dimension, the contribution of the noise and the one of the noise-free signal. This allows to obtain a robust estimator of the regularization dimensions of noisy signals. This applies to 1D as well as to 2D data.

For more details on this and other aspects of the regularization dimension, see the paper A Regularization Approach to Fractional Dimension Estimation.

Since its introduction, regularization dimension has been used by various teams worldwide in many different applications, including the characterization of certain stochastic processes, statistical estimation, the study of mammographies or galactograms for breast carcinomas detection, ECG analysis for the study of ventricular arrhythmia, encephalitis diagnosis from EEG, human skin analysis, discrimination between the nature of radioactive contaminations, analysis of porous media textures, well-logs data analysis, agro-alimentary image analysis, road profile analysis, remote sensing, mechanical systems assessment, and the analysis of video games.

For a sample of these works, look here.