Inverse problem for two-dimensional fractal sets using the wavelet transform and the moment method

Fractal geometry has provided statistical and deterministic models for classes of signals and images that represent many natural phenomena and objects. Iterated function systems (IFS) theory provides a convenient way to describe and classify deterministic fractals in the form of a recursive definition. As a result, it is conceivable to develop image representation schemes based on the IFS parameters that correspond to a given fractal image. The authors propose the use of the wavelet transform and of the moment method for the solution of the inverse problem of recovering IFS parameters from fractal images. The redundancy of a fractal with respect to scale variation is mirrored by its wavelet decomposition, thus providing a method to estimate the scaling parameters for a class of IFSs modeling the image. Displacement parameters and probabilities are then found using the moment method. Experimental results verifying the approach are presented