Fractal Image Coding as Cross-Scale Wavelet Coefficient Prediction

Abstract

Fractal image compression techniques, introduced by Barnsley and Jacquin [3], are the product of the study of iterated function systems (IFS)[2]. These techniques involve an approach to compression quite different from standard transform coder-based methods. Transform coders model images in a very simple fashion, namely, as vectors drawn from a wide-sense stationary random process. They store images as quantized transform coefficients. Fractal block coders, as described by Jacquin, assume that “image redundancy can be efficiently exploited through selftransformability on a blockwise basis” [16]. They store images as contraction maps of which the images are approximate fixed points. Images are decoded via iterative application of these maps. In this chapter we examine the connection between the fractal block coders as introduced by Jacquin [16] and transform coders. We show that fractal coding is a form of wavelet subtree quantization. The basis used by the Jacquin-style block coders is the Haar basis. Our wavelet based analysis framework leads to improved understanding of the behavior of fractal schemes. We describe a simple generalization of fractal block coders that yields a substantial improvement in performance, giving results comparable to the best reported for fractal schemes. Finally, our wavelet framework reveals some of the limitations of current fractal coders. The term “fractal image coding” is defined rather loosely and has been used to describe a wide variety of algorithms. Throughout this chapter, when we discuss fractal image coding, we will be referring to the block-based coders of the form described in [16] and [11].