Inverse and approximation problem for two-dimensional fractal sets

Abstract

The geometry of fractals is rich enough that they have extensively been used to model natural phenomena and images. 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. In this paper, we consider two distinct problems: an inverse problem and an approximation problem. The inverse problem involves finding the IFS parameters of a signal that is exactly generated via an IFS. We make use of the wavelet transform and of the image moments to solve the inverse problem. The approximation problem involves finding a fractal IFS-generated image whose moments match, either exactly or in a mean squared error sense, a range of moments of the original image. The approximating measures are generated by an IFS model of a special form and provide a general basis for the approximation of arbitrary images. Experimental results verifying our approach will be presented.