# From Fractal Image Compression to Fractal-based Methods in Mathematics

Wednesday, January 17, 2001 - 4:00pm - 5:00pm

Keller 3-180

Edward Vrscay (University of Waterloo)

I shall briefly review some mathematical approaches involving fractal transforms and the inverse problem (work with B. Forte) that were inspired by earlier work in fractal image coding. These include (i) the construction of IFS-type operators over various spaces, (ii) the formulation of appropriate inverse problems using contraction maps and (iii) the solution of such inverse problems. (A very recent development in (i) is IFS over vector-valued measures which, among other things, can be used to compute line integrals of smooth vector fields over fractal curve attractors of IFS. This is work with Franklin Mendivil, who will describe the method in more detail in a later talk.)

Fractal-wavelet transforms, an IFS-type operation on wavelet coefficient trees, have the potential to draw from the best of both worlds: 1) the multiresolution framework of wavelets and 2) the scaling property of fractals. There is still hope that the duality of FW transforms can be used to develop both theory and practical coding from the two directions: (i) that IFS theory/methods can be enhanced using the enormous amount of results in wavelet analysis and (ii) that practical wavelet coding methods can be enhanced using fractal-type methods. Some encouraging results have been found and will be reviewed here. There is still a great deal of work to be done.

Finally, IFS-type inverse problems are centered around the idea of approximating a target by the fixed point of a contraction mapping. The inverse problem is usually reformulated in terms of the collage theorem - finding an operator that sends the target as close to itself as possible. This naturally leads to the question: Where else in (applied) mathematics can we possibly formulate and use such inverse problems involving contraction mappings? An obvious place to start was the initial value problem in ODEs, using the Picard integral operator: Given a trajectory or set of trajectories, find the best vector field (work with Herb Kunze, who will describe the method in more detail in a later talk). Indeed, such parameter identification problems have been of great interest in a number of areas, e.g. kinetics and population dynamics, and the collage method rigorously justifies the various heuristic solutions/algorithms that have been devised over the years. This indeed serves as an inspiration to look at other fields of mathematics and science, either fractal or nonfractal in nature.

Fractal-wavelet transforms, an IFS-type operation on wavelet coefficient trees, have the potential to draw from the best of both worlds: 1) the multiresolution framework of wavelets and 2) the scaling property of fractals. There is still hope that the duality of FW transforms can be used to develop both theory and practical coding from the two directions: (i) that IFS theory/methods can be enhanced using the enormous amount of results in wavelet analysis and (ii) that practical wavelet coding methods can be enhanced using fractal-type methods. Some encouraging results have been found and will be reviewed here. There is still a great deal of work to be done.

Finally, IFS-type inverse problems are centered around the idea of approximating a target by the fixed point of a contraction mapping. The inverse problem is usually reformulated in terms of the collage theorem - finding an operator that sends the target as close to itself as possible. This naturally leads to the question: Where else in (applied) mathematics can we possibly formulate and use such inverse problems involving contraction mappings? An obvious place to start was the initial value problem in ODEs, using the Picard integral operator: Given a trajectory or set of trajectories, find the best vector field (work with Herb Kunze, who will describe the method in more detail in a later talk). Indeed, such parameter identification problems have been of great interest in a number of areas, e.g. kinetics and population dynamics, and the collage method rigorously justifies the various heuristic solutions/algorithms that have been devised over the years. This indeed serves as an inspiration to look at other fields of mathematics and science, either fractal or nonfractal in nature.