The MFS for texture

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.

Relationship of the MFS to the histogram

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.