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) ....

[Article contains additional citation context not shown here]

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.

.... techniques PACT 1 Introduction Image coding methods that use wavelet transforms have been successful in providing high rates of compression while maintaining good image quality and have generated much interest in the scientific community as competitors to e.g. JPEG [42] and fractal image coding [3]. Even superior to these techniques are methods based on wavelet packet decompositions, which represent an adaptive generalization of multiresolution decompositions and comprise the entire family of subband coded tree decompositions. Despite of the relatively low complexity of the discrete wavelet ....

M.F. Barnsley and L.P. Hurd. Fractal image compression. AK Peters Ltd., Wellesley, Massachusetts, 1992.

....may differ in their essential requirements, they build upon a set of common ideas, and have some overlap in their usage. There are very efficient methods for compression of images available, such as JPEG (Pennebaker and Mitchell, 1993) Wavelet based (Froment and Mallat, 1992) and Fractal based (Barnsley and Hurd, 1993), which emphasize the bits per pixel rate. In interactive applications, however, if the intermediate steps of the decoding can be displayed quickly as meaningful images as is the case of image pyramids (Williams, 1983) for instance the output of the data as it becomes available can result in ....

....inefficient, especially if the fine coarse detail ratio is low. Many image representation schemes address this problem, most notably frequency domain codifications (Pennebaker and Mitchell, 1993) Froment and Mallat, 1992) quad tree based image models (Samet, 1984) and fractal image compression (Barnsley and Hurd, 1993). Most of these schemes use non homogeneous sampling, even though they are based on a regular rectangular grid. This originates aliasing artifacts, which are usually reduced by increasing the sampling rate and or band limiting the input image (Gomes and Velho, 1995) Approaches that do not use ....

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

....by using a fractal code which the texture mapper can exploit. The next section gives an introduction to the ane transform and its use in representing a fractal compressed image. This follows very similar lines to the development of the original fractal code by Barnsley and others from 1987 [6, 7]. Section 3 gives a review of the revolutionary addition to fractal coding developed initially be Jacquin in 1989 [8, 4, 5] An overview of the algorithm employed throughout these experiments is also given. Section 4 de nes and gives subjective and objective results on two important features of ....

....location. The Decoding Process Starting with an initial image, possibly just a plain grey image, the set of PIFS transforms are repeatedly applied until the attractor is closely reached. 3. 1 Range domain algorithm The general algorithm for all range domain fractal coding methods is given below [6, 4]. This creates the set W = S w i . The only parameter to this algorithm is the threshold value, T that gives an indication of the quality required. divide the image into a set of non overlapping ranges R i mark all ranges as uncovered while there exists an uncovered R i choose domain D i and ....

M. Barnsley and L. Hurd, Fractal Image Compression. AK Peters, Wellesley, Mass, 1993. ISBN: 1568810008.

....method for creating image e#ects based on patterns and textures. 1.1 Related Work Contraction mappings of the plane have been used for quite a while to obtain image compression. In this context the technique is called fractal image compression. The results are described in Barnsley and Hurd [2] and Barnsley and Jacquin [3] A very good explanation of the techniques can be found in Fisher [4] where you can find details of the main existing algorithms. This is the first work to use the powerful technique of contraction mapping encoding to obtain special e#ects with images. 1.2 ....

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

....partitions. Key Words: fractal image compression, combinatorial optimization, NP hard problems, local search, graph algorithms D R A F T August 28, 2000, 3:52pm D R A F T 2 RAOUF HAMZAOUI, DIETMAR SAUPE, AND MICHAEL HILLER 1. INTRODUCTION Fractal image compression was introduced by Barnsley [2] and Jacquin [15] Since then, many researchers improved the original approach in various ways [22] Unfortunately, the rate distortion results of the best fractal coders are still inferior to those of the state of the art in image compression [22] However, the potential of fractal image ....

....2. BACKGROUND In this section, we introduce terminology, provide a generic fractal coding scheme, and present previous work. 2.1. Terminology In fractal image compression, the code for an original image is given by a contractive mapping of a complete metric space of digital images (F ; d) [15, 2, 8]. The goal of the encoder is to nd a contractive mapping T of which the xed point f T is a good approximation to the original image. The decoder computes f T as the limit point of the sequence of iterates ff k g k 0 where f k 1 = T (f k ) and f 0 is an arbitrary starting image. Given a target ....

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

....environment like normal higher order functions[DGTJ95] 3 A Fractal Compression Algorithm in Haskell Because our main topic is parallelisation, we will only give a rough outline of the algorithm we used for our prototype. For a detailed introduction to fractal compression see [FBB 95, BH92] In fractal compression, a picture is regarded as an attractor of a partitioned iterated function system (PIFS) The task is to find an appropriate PIFS f for a given picture 2 p so that equation 1 holds. p = lim i 1 f i (p 0 ) 8p 0 2 Picture (1) A PIFS consists of a set of functions ....

M. Barnsley and L. Hurd. Fractal Image Compression. AK Peters, Ltd., Wellesly, Ma, December 1992.

....X for which u(x 0 ) 1. Fractal Functions [22] which also includes Fractal Interpolation Functions [1, 5] the space of k times continuously differentiable functions C k (X) IFSM [16] L p (X; the space of p integrable functions with respect to a measure , 1 p 1. Fractal Transforms [7, 12, 13] are a special case of IFSM as are Fractal Functions. The Bath Fractal Transform [23, 24] is an IFSM with place dependent grey level maps. IFSP [19, 3] M(X) the set of probability measures on B(X) the oe algebra of Borel subsets of X. 2 IFSD [17] the linear space D 0 (X) of distributions ....

.... C p d p (u; v) C p = N X k=1 jJ k jK p k # 1=p : 28) Note that C p C p K; K = max 1kN K k : 29) In the nonoverlapping case, with p = 1, k Tu Gamma Tv k1 K k u Gamma v k1 ; 8u; v 2 L 1 (X; m) 30) This is the usual bound presented in the literature on fractal transforms [7, 13]. In applications, we shall be using affine IFSM, i.e. w k 2 Aff 1 (X) and OE k (t) ff k t fi k ; t 2 R; 1 k N: 31) If the associated operator T is contractive on L p (X; then its fixed point u satisfies the equation u(x) N X k=1 [ff k k (x) fi k k (x) 32) where i ....

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

....of the training set. We proves the efficiency of the proposed approach both in terms of PSNR bit rate and computation time. 1. INTRODUCTION Fractal image compression using self similarity has recently drawn considerable attention since the Iterated Function System (IFS) was proposed by Barnsley [1]. The first automated fractal coding algorithm based on Local Iterated Function Systems (LIFS) was developed by Jacquin [2] This approach using self affine transformations leads to very encouraging results at low bit rates [3] 4] The fractal encoding process is very time consuming. This limits ....

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

....between an image and a spatially averaged copy, while wavelets demonstrate the scale invariance of edges in an image. We shall now cover the background of both of these areas of research. 2 Fractal Block Coding Fractal block coding is based on the ground breaking work of Barnsley [1] 2] [3] and was developed to a usable state by Jacquin[11] The basic concept underlying this technique is that for each image, there exists a block wise transform upon the image that will leave the image unchanged. The roots of fractal block coding lie in the mathematical world of metric spaces and, in ....

....d) together with a finite set of contraction mappings fw n g; w n : X X with contractivity factors s n ; n = 1; 2; N . The notation for this IFS is fX;w n ; n = 1; 2; Ng and its contractivity factor is s = maxfs n ; n = 1; 2; Ng. Then, according to Barnsley s IFS Theorem[3], the transformation W : H(X) H(X) defined by W (B) N [ n=1 w n (B) 8B 2 H(X) is a contraction mapping with contractivity factor s. In particular, W has a unique fixed point, A 2 H(X) given by A = lim n 1 W n (B) Here, H(X) is the Hausdorff space of X, that is the set of all ....

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

....well suited for use with highly irregular image partitions for which most traditional (lossy) acceleration schemes lose a large part of their e#ciency. For large ranges our approach outperforms other currently known lossless acceleration methods. 1 Introduction Fractal image compression [1,8,11] exploits redundancy given by self similarities within an image. The image to be coded is partitioned into a set of image blocks (called ranges in fractal coding parlance) For each range one searches for another part of the image called a domain that gives a good approximation to the range when ....

....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. For example, ....

[Article contains additional citation context not shown here]

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

....fractal compression of digital images has attracted much attention. It is based on the mathematical theory of iterated function systems (IFS) developed by Hutchinson [1] and Barnsley [2] The application of IFS theory to image compression was proposed originally by Jacquin [3] and Barnsley [4]. Since their seminal work, many variants of the IFS method have been reported in the literature [5] The IFS theory and its applications to image compression together with recent research results are collected in the book edited by Y. Fisher [6] The basic idea of the fractal image compression ....

M. Barnsley, L. Hurd, `Fractal Image Compression', Jones and Bartlett, 1992.

....as well. Finally, an algorithm using the ideas of both fractal and transform methods will be presented. Chapter 2 Signal Compression 2.1 Introduction This chapter presents a brief overview of the basic principles used in signal compression. The interested reader can find more details in [13] [2] and [19] Signal compression is achieved by two complementary means: lossy and lossless compression. Lossy compression essentially consists of quantizing a signal with coarser precision than the one at which it was originally digitized. The new coarser quantization step is chosen based on human ....

....We cannot hope to find an expression for the joint probability distribution of the gray level of each pixel of an image. We can nevertheless find out some of its properties by seeking transformations that leave this probability distribution invariant. The content of this section is inspired from [2] and [7] We can represent a gray scale picture as a function g 2 L 2 (D) where D is a bounded subset of R 2 . This function simply gives the luminosity of each point of a picture supported on D. The L 2 space is chosen because we expect luminosity to be bounded in some way and because we will ....

[Article contains additional citation context not shown here]

Michael Barnsley. Fractal Image Compression. A. K. Peters, Wellesley, Massachusetts, 1993.

....quantization [30] fractal compression 3 (a) b) c) d) Figure 1.1: Examples of binary document images of the types usually found in a document image database. These are images from the University of Washington document database [32] cropped (automatically) to the main body of the text. 4 [12], pattern matching and substitution [10, 64] and other approaches) depends on the types of images being compressed, and on their texture and content characteristics. Algorithms often do well on some classes of images and not so well on others. Since document images differ significantly from scene ....

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

....log 2 1 p i When the log 2 1 p i are integers, we can code each s i using l i = log 2 1 p i bits with a Huffman coding scheme. Otherwise, a more complicated scheme called arithmetic coding can be used so that the average coding length l average is as close to the entropy bound as possible [1]. Huffman and arithmetic coding are both often called entropy coding techniques. When we analyse the efficiency of a compression method theoretically, we can use the entropy of its output message without actually encoding the message. It can be verified that the entropy defined by Eq. 1) has ....

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

....complexity, when compared to full 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

....referenced the more recent or easily accessible work. In addition to the proceedings [14] 15] of the 1995 NATO conference on the subject, of which many of the papers are referenced in this review, there are currently three books devoted entirely to this subject. The book by Barnsley and Hurd [7], the rst on the subject, reveals relatively little practical detail. The book edited by Fisher [3] contains two introductory chapters and a collection of signi cant work by a number of authors, while the recent book by Lu [16] combines introductory material with an in depth discussion of many ....

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

....who proposed to use fractal properties of natural images for image compression. Based on Barnsley s work, Jacquin [2, 3] developed an algorithm for automatic compression of images. The work of Barnsley and Jacquin on Iterated Function Systems (IFS) and Recurrent Iterated Function Systems (RIFS) [1, 2, 4, 5] has made a basis for development of a series of fractal based methods by other researchers, for compression of both still images [6, 7, 8, 9, 10, 11, 12] and image sequences [13, 14, 15] The essence of most fractal based coding methods is to approximate a range block by a linear combination of ....

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

.... pulse C 0 ( 0 x ; 0 y ) ffi ( 0 x Gamma s 0 x ; 0 y Gamma s 0 y ) 5) at the logarithmized location identical to the scaling factors, which are given by s x = e s 0 x ; s y = e s 0 y : 6) Figure 1 shows the detected scalefactors of a Sierpinski triangle , cf. [9], with selfsimilarities at horizontal as well as vertical scaling levels being arbitrary powers of 2 and demonstrates the potential of this method for fractal analysis. Using a transformation to circular coordinates with logarithmized radial axis instead, the method allows the displacement ....

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

....a general encryption algorithm. Thus, instead of the whole encoded image, only key parameters of the IFS code need to be encrypted. 2. A REVIEW OF FRACTAL IMAGE CODING The basic idea about fractal image coding is to represent the image by a set of transforms associated with an iterated process [2]. The goal is to assure that this process converge towards a fixed point called attractor that is an approximation of the original image. The compression gain provided by this technique is due to the compact encoding of the set of transforms that represent the image data. In order to formalize ....

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

....be considered as a special case of our model. In our case, however, we are not interested in classification, but in discovering scale affine autosimilarity. The ability of the LRAAM to represent similar patterns by similar pointers can be exploited to identify a fractal approximation of the image [1, 2, 11, 8]. CONCLUSIONS The LRAAM model seems ideal to code a pyramid representing an image with scale affine redundancy. The hidden representations developed by the LRAAM seem to capture redundancies present in the image at different scales. The LRAAM can be used either to compress the pyramid or to ....

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

....better, the function that one wants to approximate, solely by the relations that are present between affinely transformed parts of the signal and the signal itself. Through the removal of of self affine redundancy , one hopes to obtain a more compact representation than the original one. Barnsley [4], Jacquin [10 12] and Fisher [9] presented different methods for looking for the similarities present in digital images. For simplicity of implementation the search for similarity was performed only between blocks in which the image was preventively decomposed. The brightness of a block was being ....

Barnsley, M. F., and Hurd, L. P. Fractal Image Compression. AK Peters Ltd., 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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M.F. Barnsley and L.P. Hurd, Fractal Image Compression, AK Peters, Ltd (1992).

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

No context found.

Barnsley F. M., Hurd P. L., "Fractal Image Compression," AK Peters, Ltd., Massachusetts, 1993.

No context found.

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

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