Fractal-Based Image Compression

摘要

Iterated Function Systems (IFS) offers a method of describing complicated digitalfiles with a small set of functions exhibiting fractal properties. In coding an image, first cover it with contractive affine transformations of itself and then save the coefficients of the transformations. Decoding is performed by generating a dynamical system whose attractor is suitably close to the original image. The amount of distortion is dependent on the quality of the initial covering. This paper will describe the mathematics of IFS, the coding and decoding of a digital image with IFS, error analysis of IFS compression, and comparison to other compression techniques. Mandelbrot coined the term fractal to describe sets that demonstrate detail under any arbitrary magnification. Mandelbrot described the properties of sets with non-integer dimension, fractal sets, and produced many examples from nature: coastlines, clouds, brownian motion, trees, etc... Human-made mechanisms also display fractal behavior: the stock market, bouncing balls, dripping faucets, etc... Mandelbrot, made it clear that in nature fractal action has a limit while pure mathematical fractals have structure at all scales, but both have in common statistical invariance under transformation of a scale. An IFS generates a set with fractal properties. (kr)