Self-Similar Models and How to Find Them: A Moment Theory Approach

Abstract

Self-similar modelling aims to capture how an object relates to itself, and can be done through fractal geometry. Creating a fractal that looks like a natural object is easy, nd- ing a fractal that models a given self-similar object is hard. The only known success of this inverse problem comes from fractal image compression (FIC). In this thesis we develop an alternative solution to the problem. Explicitly, we formulate a fractal moment approx- imation theory that provides more exibility in self-similar modelling to that present in FIC. This is done through placing a normalised measure on a self-similar set and using this measure's moments in its reconstruction. It is proven that the vector of these mo- ment values is the unique xed point of a linear operator on 1`1, the space of bounded sequences whose rst element is 1. A relationship between self-similar sets and measures is given through tools found in fractal tiling and dimension theory. An approximation theory is developed that gives a computationally feasible way to approximate (possibly non-self-similar) objects. These approximations are investigated in both theory and com- putation. Through our novel examples, this theory is extended to local fractal models that are frequent in application, and their relation to image compression is discussed.