M. F. Barnsley and L. Hurd. Fractal Image Compression. AK Peters, Wellsley, MA 02181, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....functions, the usual procedure in signal and image processing. The space L o ers some attractive simpli cations for fractal image compression. For example, the contractivity of the fractal transform T depends only upon the Lipschitz factors of the grey level maps i , unlike the case for L [9]. 1.2 IFS on L Spaces An IFS type method can easily be formulated over the space L (X) p 1 of Lebesgue integrable functions on X : X) ff : X R j jf(x)j dx 1g: 11) As before, let w i , 1 i N , denote a set of IFS contraction maps on X . Associated with each map w i ....

.... measure 0) then k Tu Tv k p C p k u v k p ; C p = c i K i ] 16) Note that C p C p K; K = max fK i g: 17) In the nonoverlapping case, with p = 1, k Tu Tv k 1 K k u v k 1 ; u; v 2 L This is the usual bound presented in the literature on fractal transforms [9]. In many treatments, both the IFS maps w i and the grey level maps i are assumed to be ane. In such cases, we refer to the IFSM (w; as an ane IFSM. In the case that X is one dimensional, e.g. X = 0; 1] the maps will have the form w i = s i a i ; i (t) i t i ; 1 i N; 19) ....

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M.F. Barnsley and L.P. Hurd, Fractal Image Compression, A.K. Peters, Wellesley, Mass. (1993).

....we want to reconstruct, compressed by means of the maps (w 1 ; w 2 ; wN ) A 2 K(X ) where (K(X ) h) is the complete metric space of compact subsets of X , with the Hausdor metrics. A is then the xed point of the IFS operator 2 W ( S N i=1 w i ( de ned on K(X ) see [2] and [3]) In order to decompress it, the algorithm which is tipically used is the socalled Chaos Game, that, roughly speaking, can be described by the following steps: x a starting point x 0 2 X ; choose a map w i with probability p i ; apply the map w i to x 0 , obtaining a new point x 1 = w ....

Barnsley M.F., Lyman P.H. (1992) Fractal Image Compression, AK Peters Ltd., Wellesley.

....publication of partitioned fractal interpolation functions, although it is a rather obvious extension of the ideas from fractal image compression to fractal interpolation functions. The PFIF is the FIF analogy of the partitioned iterated function system [7] and the local iterated function system [6]. 2.4 Fractal Interpolation Curves A parametric version of the curve can be constructed by assigning a value of t to each knot and using two functions x(t) and y(t) which interpolate appropriately. Hence, we are given a collection of input data points f(x i #y i #t i )g i=0 where (x i #y i ) ....

Michael F. Barnsley and Lyman P. Hurd. Fractal Image Compression. AK Peters, Wellesley, MA, 1993.

....1. INTRODUCTION JPEG and Vector Quantization have certainly been the most commonly used methods for image compression, followed more recently by wavelet methods. However, over the past few years, there has also been much interest and development in fractal based methods of image compression [3, 4, 7], accompanied by a number of significant developments [5] Most of these schemes are centered around the method of Iterated Function Sys tems [2] The common conception is that fractal based schemes exploit geometric self similarities that are inherent in images. However, amore realistic ....

M.F. Barnsley and L.P. Hurd, Fractal Image Com- pression, Wellesley, MA: A.K. Peters, 1993.

....provide adaptive solutions for problems involved in sub band coding of those images which may require di erent wavelets to achieve the best possible compression performance [4,71] 3.2. Fractal neural networks Fractally con gured neural networks [45,64,71] based on IFS (iterated function systems [3]) codes represent another example along the direction of developing existing image compression technology into neural networks. Its conventional counterpart involves representing images by fractals and each fractal is then represented by a so called IFS which consists of a group of a ne ....

....counterpart involves representing images by fractals and each fractal is then represented by a so called IFS which consists of a group of a ne transformations. To generate images from IFS, random iteration algorithm is the most typical technique associated with fractal based image decompression [3]. Hence, fractal based image compression features higher speed in decompression and lower speed in compression. By establishing one neurone per pixel, two traditional algorithms of generating images using IFSs are formulated into neural networks in which all the neurones are organized as a ....

M.F. Barnsley, L.P. Hurd, Fractal Image Compression, AK Peters Ltd, 1993, ISBN: 1-56881-000-8.

....of a network s neurons as the functions or transforms of an IFS. As such, these neurons receive (X,Y) coordinates as input and return new ones in a recurrent manner. It has been suggested that coding fractals by Iterated Function Systems may be an e ective mechanism for compressing images [5]. As such this interpretation of network dynamics may form the basis of a highly e cient method for storing visual information and other related memories. A small sample of some of the fractals which a simple network of only four neurons can encode is shown in Figure 4.1. It is conceivable that ....

....iterative process, which is inherently not di erentiable. Second, the Hausdor distance e ectively uses only one point from each set. This point is not constant and its selection may lead to discontinuities. Our error function borrows principles from the Hausdor distance and the collage theorem [5]. The collage theorem provides the basis for most approaches to the fractal inverse problem or fractal image compression. It states that in order to nd an IFS for a given fractal attractor, it is necessary to nd a transformation which maps the attractor to itself. As such, our error function ....

M.F. Barnsley and L.P. Hurd. Fractal Image Compression. AK Peters, Wellesley, 1992.

....image coding based on quadtree partition is also proposed which can obtain a good trade off between compression ratio and image fidelity. 1. INTRODUCTION Since Jacquin introduced a practical block based fractal image coding scheme [1] in 1990, fractal coding has aroused a great deal of attention [2,3,4] as a new promising image compression technique. This method is based on IFS (iterated function system) which exploits the self affine similarity between different parts of the image. Through affine transformation one sub block of the original image can be approximated by another subblock in the ....

M. F. Barnsley and L. P. Hurd, Fractal Image Compression, AK Peter, Wellesley, 1992.

....is that the bitplane (and zerotree) algorithms are designed to approach the optimal PSNR reduction scheme of always reducing the currently largest coefficient error. As such, it can be quite difficult to compete with these algorithms on straight PSNR merits. On the other hand, it has been observed [6]that fractal methods are quite (visually) good at representing some types of image textures, and may result in images of better visual quality at the same PSNR. A similar result here would be interesting, even if PSNR was somewhat degraded. A common term for the number of bits a coder may ....

M. F. Barnsley and L. P. Hurd. Fractal Image Compression. AK Peters, Wellesley, MA, USA, 1993.

....is what we have done for these examples. Figure 6 demonstrates this application, using images of a set of maple trees and of a Dobag rug, respectively. An interesting area for future work is to choose the training pairs automatically for image compression, similar to fractal image compression [3]. 4.4 Texture transfer In texture transfer, we filter an image B so that it has the texture of a given example texture A # (Figure 7) Texture transfer is achieved by using the same texture for both A and A # . We can trade off the appearance between that of the unfiltered image B and that of ....

Michael F. Barnsley, Lyman P. Hurd, and Louisa F. Anson. Fractal Image Compression. A.K. Peters Ltd, 1993.

....search algorithms. 1. INTRODUCTION During the last decade, fractal geometry has captured increasing (1) E 1 N N n 1 I n I n IT B attention and interest. The application of fractal models to image compression has been promoted by Barnsley et al. [1]. Fractal image compression is based on the observation that all real world images are rich in affine redundancy. That is, under suitable affine transformations, larger blocks (domain blocks) of the image look like smaller blocks (range blocks) in the same image. The encoding process consists of ....

M. F. Barnsley, and L. P. Hurt, Fractal Image Compression, Wellesley, MA: A. K. Peters, 1993

....as a collection of self similarity properties. The better the collage fits the given image the higher the fidelity of the resulting decoded image. In this article we cannot explain any further details and variations of fractal image compression. For introductory texts or reviews see, for example, [2, 4, 5, 10]. For a bibliographic survey of the field of fractal image compression see our paper [13] JPEG can be termed symmetric in the sense that the encoding and decoding phases require about the same number of operations. On the contrary, fractal image compression allows fast decoding but suffers from ....

Barnsley, M., Hurd, L., Fractal Image Compression, AK Peters, Wellesley, 1993.

....optimization is required for each pair of range and codebook block in order to determine the optimal luminance transformation and the resulting approximation error. Since codebooks consist of many thousands of blocks, the straightforward implementation of fractal image compression by brute force [1] su#ers from long encoding times.Therefore, much e#ort has been undertaken to find a variety of ways to speed up the process; for a survey see [22] Most techniques are lossy acceleration schemes in the sense that they sacrifice image reconstruction quality for the sake of speedup. Forexample, an ....

....the process; for a survey see [22] Most techniques are lossy acceleration schemes in the sense that they sacrifice image reconstruction quality for the sake of speedup. Forexample, an acceleration method that only 1 We restrictoursel es to conventional fractal coding based on thecol23 theorem [1,8]; we do not consider direct attractor optimization here. 2 considers a subset of the canonical codebooks results in a speedup by choosing an acceptable but suboptimal codebook blockand, therefore, loses some of the possible coding quality since the canonical codebooks are superior to smaller ....

M.Barnsl5 , L. Hurd, Fractal Image Compression, Academic Press, San Diego, 1988

....Experimental results show that the new scheme can achieve nearly optimal partition of wavelet subtree with substantially computational reduction as compared with Davis scheme. 1. INTRODUCTION Recently, G. M. Davis [8] and H. Krupnik et al. 9] independently generalized the fractal coding [4 6] from spatial domain to the wavelet domain [1 3] Davis coined a new term wavelet subtree for representing the hierarchical data structure of an image decomposed in wavelet pyramid. The wavelet subtree consists of the wavelet coefficients that has the same spatial location but with different ....

M. F. Barnsley and L. P. Hurd, Fractal Image Compression, AK peters, Wellesley, Massachusetts, 1993.

.... has been a very active area of research in fractal geometry [73, 8, 29, 83, 84, 48, 9] and has found applications in diverse areas such as mathematical finance, signal processing, computer graphics, image compression, learning automata, neural nets, statistical physics and real number computation [11, 12, 7, 10, 22, 83, 84, 15, 13, 44]. A simple example of an IFS can be constructed for the decimal representation of real numbers in [0, 1] Let f i : x ## x i 10 : 0, 1] # [0, 1] with i # 0, 1, 2, 9 . Then the decimal representation of any real number in [0, 1] can be expressed by the IFS [0, 1] f 0 , ....

.... of the above IFS results for the so called recurrent IFS, i.e. an IFS which is equipped with a stochastic matrix rather than just a probability vector, and also for the so called vector recurrent IFS [34] which is the basis of Barnsley s software for fractal image compression using measures [10]. The domain theoretic framework for IFS, as we have indicated in Section 4 and in this section, has the unifying feature that several aspects of the theory of IFS, namely (i) the proof of existence and uniqueness of the attractor of a weakly hyperbolic IFS and that of the invariant measure of a ....

M. F. Barnsley and L. P. Hurd, Fractal image compression, AK Peters, Ltd, 1993.

....and without technical applications in mind, Williams [29] and Hutchinson [18] had published mathematical studies of compositions of contractions and iterated function system. During the last 10 years about 400 papers were published in the eld of fractal compression, as well as four books [4,10,20,11]. Several studies have attempted to nd attractor functions f that are better approximations to a target f than the collage attractors f p c . Indeed, these studies have typically employed the collage attractor f p c as a starting point. For example, Barthel [5] and then Lu [20] ....

Barnsley, M. F., Hurd, L., Fractal Image Compression, AK Peters, Wellesley, 1993.

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M. F. Barnsley and L. Hurd. Fractal Image Compression. AK Peters, Wellsley, MA 02181, 1993.

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M. F. Barnsley and L. P. Hurd, Fractal Image Compression. Massachusetts: AK Press, 1(res

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M. F. Barnsley and L. Hurd, Fractal Image Compression. AK Peters, Wellesley, 1993.

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Barnsley, M. and Hurd, L. #1993#. Fractal Image Compression. AKPeters.

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A399, London, pp. 243~75. Barnsley, M & Hurd, LP 1993, Fractal Image Compression, AK Peters. Ltd.

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M.F. Barnsley and L.P. Hurd. Fractal Image Compression. AK Peters Ltd., Wellesley, 1993.

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Barnsley M.F., Lyman P.H. (1992) Fractal Image Compression, AK Peters Ltd., Wellesley.

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Michael F. Barnsley and Lyman P. Hurd. Fractal Image Compression. AK Peters, Wellesley, MA, 1993.

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Barnsley M.F., Lyman P.H. (1992) Fractal Image Compression, AK Peters Ltd., Wellesley. 21

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Michael Barnsley, Lyman Hurd. Fractal Image Compression. AK Peters Ltd, 1993.

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