Why does fractal image compression work? What is the implicit image model underlying fractal block coding? How can we characterize the types of images for which fractal block coders will work well? These are the central issues we address. We introduce a new waveletbased framework for analyzing block-based fractal compression schemes. Within this framework we are able to draw upon insights from the well-established transform coder paradigm in order to address the issue of why fractal block coders work. We show that fractal block coders of the form introduced by Jacquin[1] are a Haar wavelet subtree quantization scheme. We examine a generalization of this scheme to smooth wavelets with additional vanishing moments. The performance of our generalized coder is comparable to the best results in the literature for a Jacquin-style coding scheme. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergen...

The redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images, but it has not been fully recognized and exploited for general images. Recently, fractal block coders have exploited the self-similarity among blocks in images. In this work we devise an image coder in which the causal similarity among blocks of different subbands in a multiresolution decomposition of the image is exploited. In a pyramid subband decomposition, the image is decomposed into a set of subbands which are localized in scale, orientation and space. The proposed coding scheme consists of predicting blocks in one subimage from blocks in lower resolution subbands with the same orientation. Although our prediction maps are of the same kind of those used in fractal block coders, which are based on an iterative mapping scheme, our coding technique does not impose any contractivity constraint on the block maps. This makes the decoding procedure very simple and...

This paper is concerned with function approximation and image representation using a new formulation of Iterated Function Systems (IFS) over the general function spaces L p (X; ¯): An N-map IFS with grey level maps (IFSM), to be denoted as (w; \Phi), is a set w of N contraction maps w i : X ! X over a compact metric space (X; d) (the "base space") with an associated set \Phi of maps OE i : R ! R. Associated with each IFSM is an operator T which, under certain conditions, may be contractive with unique fixed point u 2 L p (X; ¯). A rigorous solution to the following inverse problem is provided: Given a target v 2 L p (X; ¯) and an ffl ? 0, find an IFSM whose attractor satisfies k u \Gamma v k p ! ffl. An algorithm for the construction of IFSM approximations of arbitary accuracy to a target set in L 2 (X; ¯), where X ae R D and ¯ = m (D) (Lebesgue measure), is also given. The IFSM formulation can easily be generalized to include the "local IFSM" (LIFSM) which considers the...

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: In fractal image compression the encoding step is computationally expensive. A large number of sequential searches through a list of domains (portions of the image) are carried out while trying to find a best match for another image portion. We show that this step can be replaced by multi-dimensional nearest neighbor search which runs in logarithmic time instead of linear time required for the common sequential search. Keywords: Image compression, multi-dimensional search, nearest neighbors, fractals. 1 Introduction With the ever increasing demand for images, sound, video sequences, computer animations and volume visualization, data compression remains a critical issue regarding the cost of data storage and transmission times. While JPEG currently provides the industry standard for still image compression there is ongoing research in alternative methods. Fractal image compression is one of them. JPEG can be termed symmetric in the sense that the encoding and decoding phases require ...

Fractal image compression was one of the earliest compression schemes to take advantage of image redundancy in scale. The theory of iterated function systems motivates a broad class of fractal schemes but does not give much guidance for implementation. Fractal compression schemes do not fit into the standard transform coder paradigm and have proven difficult to analyze. We introduce a wavelet-based framework for analyzing fractal block coders which simplifies these schemes considerably. Using this framework we find that fractal block coders are Haar wavelet subtree quantization schemes, and we thereby place fractal schemes in the context of conventional transform coders. We show that the central mechanism of fractal schemes is an extrapolation of fine-scale Haar wavelet coefficients from coarse-scale coefficients. We use this insight to derive a wavelet-based analog of fractal compression, the self-quantization of subtrees (SQS) scheme. We obtain a simple SQS decoder convergence proof and a fast SQS decoding algorithm which simplify and generalize existing fractal compression results. We describe an adaptive SQS compression scheme which outperforms the best fractal schemes in the literature by roughly 1 dB in PSNR across a broad range of compression ratios and which has performance comparable to some of the best conventional wavelet subtree quantization schemes.

A video coding method is proposed which is based upon fractal block coding. The method utilizes a novel three-dimensional partitioning of input frames for which a number of efficient block-matching search methods can be used, and permits spatio-temporal splitting of the input blocks to improve overall-encoding quality. After describing the basic fractal block coding algorithm, the details of the proposed three-dimensional algorithm are presented along with encoding and decoding results from two standard video test sequences, representative of videoconferencing data. These results indicate that average compression rates ranging from 40 to 77 can be obtained with subjective reconstruction quality of video-conferencing quality. The results also indicate that, in order to meet the compression rates required for very low bit rate coding, it is necessary to employ additional techniques such as entropy encoding of the fractal transformation coefficients. 18 October 1994 11:04 Page 2 of 33 V...

. Since the conception of fractal image compression by Michael F. Barnsley around 1987 the research literature on this topic has experienced a rapid growth, which is hard to keep track of. This paper contains a brief description of the major advances in the field. The corresponding relevant papers are presented in the form of the original (slightly edited) abstracts. Also included is a comprehensive bibliography, the largest published on this topic to this date. 1 Introduction While JPEG is becoming the industry standard for image compression technology there is ongoing research in alternative methods. Currently there are at least two exciting new developments: wavelet based methods and fractal image compression. This article is intended to provide the reader with an overview and a resource of the research on the latter. We attempt to put the work into a historical perspective and aim at providing the most comprehensive and up-to-date list of references in the field, which has truly b...

This paper reports on investigations concerning the convergence of fractal transforms for signal modelling. Convergence is essential for the functionality of fractal based coding schemes. The coding process is described as non-linear transformation in the finite-dimensional vector space. Using spectral theory, a necessary and sufficient condition for the contractivity is derived from the eigenvalues of a special linear operator. In the same way some constraints for the choice of the encoding parameters are deduced which are less strict than those imposed so far. The proposed contractivity measure can be calculated directly from the transformation parameters during the encoding process. For complex encoding schemes the calculation of the eigenvalues may be infeasible. For those cases a contractivity criterion derived from the norm of the operator is suggested. 1.

In this paper we present a method to encode a single image by finding an Iterated Function System (IFS) that describes an approximation to the image we want to compress by using Genetic Algorithms (GA). The search was restricted to IFS with a fixed number of maps, and a fixed contractivity factor, like in Barnsley 's brute force method. To improve the convergence for this problem, a modified mating operator was used in the genetic algorithm. By doing this, the time needed to get an IFS is reduced by about 30% compared with Barnsley's brute force method if similar image quality is desired, but more to the point, unlike with other algorithms, using a GA we can vary the time the compression will last as a function of the error we tolerate by varying parameters such as population size and number of generations we allow the algorithm to proceed. The algorithm was extended to deal with image sequences by using the population that has evolved for an image as the initial population for the nex...