Mathematical framework to show the existence of attractor of partitioned iterative function systems

The technique of image compression using Iterative Function System (IFS) is known as fractal image compression. An extension of IFS theory is Partitioned or local Iterative Function System (PIFS) for coding the gray level images. Several techniques of PIFS based image compression have already been proposed by many researchers. The theory of PIFS appears to be different from the theory of IFS in the sense of application domain. The present article discusses some basic differences between IFS and PIFS and provides a separate mathematical formulation for the existence of attractor of partitioned IFS. In particular, it has been shown that the attractor exists and it is an approximation of the given target image. The experimental results have also been presented in support of the theory. The experimental results have been obtained by using a GA based PIFS technique proposed by Mitra et al [1]. key words : Image Compression, Iterative Fuction System (IFS), Partitioned Iterative Function System (PIFS), Attractor, Isometry.