Generalized fractal transforms and self-similarity : recent results and applications

Most practical as well as theoretical works in image processing and mathematical imaging consider images as real-valued functions, u: X → R<inf>g</inf>, where X denotes the base space or pixel space over which the images are defined and R<inf>g</inf> ⊂ R is a suitable greyscale space. A variety of function spaces F(X) may be considered depending on the application. Fractal image coding seeks to approximate an image function as a union of spatially-contracted and greyscale-modified copies of subsets of itself, i.e., u ≈ Tu, where T is the so-called Generalized Fractal Transform (GFT) operator. The aim of this paper is to show some recent developments of the theory of generalized fractal transforms and how they can be used for the purpose of image analysis (compression, denoising). This includes the formulation of fractal transforms over various spaces of multifunctions, i.e., set-valued and measure-valued functions. The latter may be useful in nonlocal image processing.