Introdução à compressão fractal de imagens através de sistemas de funções iteradas

Abstract

The study object of this work is the fractal image compression method through systems of iterated functions. This technique consists of describing, through affine transformations, fractals that have a special characteristic: self-similarity. To understand this method of compression, we make a brief explanation about fractal geometry, start a study on linear transformations and define affine transformations in the plane. Then, we focus on the concepts of Metric Spaces necessary for understanding Banach’s Fixed Point Theorem, which is the key for the application of systems of iterated functions in the construction of self-similar fractals. We present the Hausdorff distance, as it is used in the compression of real images that have little or no similarity and, finally, we show the application in practice by building two very important fractals: the Sierpinski Triangle and the Sierpinski Carpet.