Iterated Function Systems with Place-Dependent Probabilities and the Inverse Problem of Measure Approximation Using Moments

Abstract

The study of iterated function systems has close ties with the subject of fractal-based analysis. One important application is the approximation of a target object by the fixed point of a contractive iterated function system. In recent decades, substantial evidence has been put forth suggesting that images (as the mathematical object) are amenable to compression by these fractal-based techniques. With images as our eventual goal, we present research on the 1-dimensional case- the reconstruction of a data set based on a smaller subset of data. Formally posed here as the inverse problem, a myriad of possible solution methods exist already in literature. We explore and improve further a generalization in method that entails denotation of the target object as a measure and matching the moments of this measure by optimizing over free parameters in the moments of the invariant measure resulting from the action of an iterated function system with associated place dependent probabilities. The data then required to store an approximation to the target measure is only that of the parameters for the iterated function system and the probabilities. Our generalization allows for these associated probabilities to be place-dependent, with the effect of reducing the approximation error. Necessarily this technique introduces complications in calculating the moments of the invariant measure, but we exhibit an effective means of circumventing the problem.