Image Compression with Iterated Function Systems, Finite Automata and Zerotrees: Grand Unification

Abstract

Fractal image compression, Culik's image compression and zerotree prediction coding of wavelet image decomposition coefficients succeed only because typical images being compressed possess a significant degree of self-similarity. Besides the common concept, these methods turn out to be even more tightly related, to the point of algorithmical reducibility of one technique to another. The goal of the present paper is to demonstrate these relations. The paper offers a plain-term interpretation of Culik's image compression, in regular image processing terms, without resorting to finite state machines and similar lofty language. The interpretation is shown to be algorithmically related to an IFS fractal image compression method: an IFS can be exactly transformed into Culik's image code. Using this transformation, we will prove that in a self-similar (part of an) image any zero wavelet coefficient is the root of a zerotree, or its branch. The paper discusses the zerotree coding of (wavelet/projection) coefficients as a common predictor/corrector, applied vertically through different layers of a multiresolutional decomposition, rather than within the same view. This interpretation leads to an insight into the evolution of image compression techniques: from a causal single-layer prediction, to non-causal same-view predictions (wavelet decomposition among others) and to a causal cross-layer prediction (zero-trees, Culik's method).