The MFS can be defined on different  functions of the image intensity (output of various filters). We used three MFS vectors, which were defined on the  intensity, the  the  energy of edges, and the energy of the Laplacian.  Alternatively, the MFS can be defined on the  density function of the above quantities. The density computes the change of a quantity over scale.

Examples:

   image intensity                                                Gradient energy                                   Laplacian energy

Thus we obtain three vectors :

where dim(E_α) is the dimension of the set of image points with density α.

The MFS of the density of the intensity, the gradient energy, and the Laplacian energy of the grass texture above.

The most popular global statistical estimator is the histogram. It categorizes image points according to some criteria, for example the intensity value. For every set in the category it codes the number of points.

The MFS could be viewed as the histogram enhanced by an additional  multi-resolution analysis layer. Instead of just counting the number of pixels in a set, one computes the fractal dimension, which is obtained by counting the number of points under multiple resolutions ,and estimating the exponential changing ratio of the number of points with respect to the resolution. This multi-resolution analysis encodes information about the spatial distribution of the point set. Thus, the MFS can be viewed as a histogram which also codes geometric information, and which is invariant to geometric transformations.