“Generalized fractal transforms, contraction maps and associated inverse problems”
In the spirit of iterated function systems (IFS), we (”Waterloo Fractal Analysis and Coding Project”) have been interested in ”generalized fractal transforms” (GFTs) over various spaces. A GFT, T, defined on a complete metric space (X,d) acts on an element x in (X,d) in the following ”fractal-like” way: It first makes N modified copies of x (e.g., spatially-contracted, range-transformed and translated), then combines these ”fractal components”, x(i), (in a manner appropriate to the space X) to produce a new element y = T x.
Under certain conditions, the operator T is contractive on X which, from Banach’s Fixed Point Theorem, implies the existence of a unique fixed point p = Tp . From the action of T, p is ”self-similar” in the sense that it is a combination of modified copies of itself.
In the 1980’s, fractal image coding naturally led to the formulation of GFTs on various function spaces. More recently, we have looked at GFTs on multifunctions, as well as function- and measure- valued mappings, both of which have applications in image processing, as will be shown.