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...

this paper we examine various methods to attack the inverse problem of function/measure approximation using generalized fractal transforms, which involve the use of IFS maps whose "range blocks" may overlap. As such, these results may not appear to be greatly effective in the problem of fractal image compression since overlapping blocks are viewed as redundant in the coding of an image. However, our philosophy has been to develop a systematic theory of approximation of functions, measures and distributions using a complete "basis set" of IFS maps. As in Paper I, we consider such target functions or images to be elements of an appropriate complete metric space (Y; d Y ). The underlying idea in fractal compression is the approximation, to some suitable accuracy, of a target y 2 Y

The most popular "fractal-based" algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) The "IFS component" of these methods is a set of N contraction maps w = fw 1 ; w 2 ; : : : ; wN g, w i : X ! X over a complete metric space (X; d), the "base space" representing the computer screen. Most discussions of these methods, both practical as well as theoretical in nature, assume that the sets w i (X) are nonoverlapping (or at least ignore any overlapping), i.e. that w \Gamma1 i (x) exists for only one value i 2 f1; 2; : : : Ng. As such, given an image function u, its so-called fractal transform (Tu)(x) at any point x 2 X is given by OE i (u(w \Ga...

Introduction This work has been done in collaboration with Bruno Forte, University of Verona, Italy. Much of the work to be outlined below was presented at the NATO ASI on Fractal Image Coding and Analysis held in Trondheim, Norway, July 8-17, 1995 and will appear as two papers in the Proceedings, edited by Yuval Fisher. Preliminary versions of these papers are available by anonymous ftp from links.uwaterloo.ca:/pub/Fractals/Papers/Waterloo/. The most popular fractal-based algorithms for both the representation as well as compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces, e.g. IFS with probabilities (IFSP), Iterated Fuzzy Set Systems (IFZS), Fractal Transforms (FT), the Bath Fractal Transform (BFT) and IFS with grey-level maps (IFSM). (FT and BFT are special cases of IFSM.) There are two major aspects of our work which are summarized below. First is the effort to formulat

. Most standard fractal image compression techniques rely on using an IFS operator directly on the image function. Sometimes, however, it is more convenient to work on a faithful representation of the image which, in certain applications, may be a transformed version of the image. For example, if an MRI image is scanned in as frequency data it may be more natural to work on the Fourier transform of the image rather than on the image itself. After a brief introduction to fractal transforms and classical fractal image compression, we discuss some generalities of IFS operators on transform spaces. We then illustrate with examples from Fourier, wavelet and Lebesgue transforms. We emphasize that the operations can be done completely in the transform domain. In some applications, e.g. measures, we may not even need or desire to return to the spatial domain. 1 Introduction In this paper we consider the construction of fractal, IFS-type (for Iterated Function Systems) operators on transforms ...

This chapter reviews the theoretical foundations and implementation issues of fractalbased image coding methods. The concepts of fractals, iterated function systems, and local iterated function systems are discussed and different implementations of compression of both still images and image sequences are reviewed. 1 INTRODUCTION Fractal-based image coding, which is sometimes called fractal image coding or attractor image coding, is a new method of image compression. In this method, similarities between different scales of the same image are used for compression. The method is rooted in the work of Mandelbrot, who introduced the concept of fractals and the fractal dimension. This chapter is organized as follows. Section 1 gives a brief review of the concept of fractals and its applications especially in image compression. In Section 2 the principles of iterated function systems (IFS) will be studied. IFS makes the basis of most fractal-based image compression methods. In Section 3, w...

Introduction In keeping with the philosophy of this workshop, the aim of this presentation is to provide an overview of the research done over the years at Waterloo on fractal-based methods of approximation and associated inverse problems. Near the end, some new and encouraging results regarding \fractal enhancement" will be presented. The paper concludes with thoughts and challenges on how the mathematical methods that underlie fractal image compression could be used in other areas of mathematics. Let us go back to rst principles for a moment in order to recall some of the early thinking behind fractal image compression (FIC). In fact, since the early work of Barnsley, Jacquin et al., there has been very little change in the basic idea of FIC. Most eorts have focussed on developing strategies to perform \collage coding" as eectively as possible { whether it be in the pixel or wavelet domain. This includes the the competition between employing the largest possible domain pools and

Abstract. Approximation of an image by the attractor evolved through iterations of a set of contractive maps is usually known as fractal image compression. The set of maps is called iterated function system (IFS). Several algorithms, with different motivations, have been suggested towards the solution of this problem. But, so far, the theory of IFS with probabilities, in the context of image compression, has not been explored much. In the present article we have proposed a new technique of fractal image compression using the theory of IFS and probabilities. In our proposed algorithm, we have used a multiscaling division of the given image up to a predetermined level or up to that level at which no further division is required. At each level, the maps and the corresponding probabilities are computed using the gray value information contained in that image level and in the image level higher to that level. A fine tuning of the algorithm is still to be done. But, the most interesting part of the proposed technique is its extreme fastness in image encoding. It can be looked upon as one of the solutions to the problem of huge computational cost for obtaining fractal code of images.

This work is devoted to the study of integration with respect to binomial measures. We develop interpolatory quadrature rules and study their properties. Local error estimates for these rules are derived in a general framework.