Arnaud Jacquin. A fractal theory of iterated Markov operators with applications to digital image coding. PhD thesis, Georgia Institute of Technology, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., can be used to estimate the reconstruction error, which is used in the encoding process as the objective function to be minimized. Optimality of fractal transform coefficients, using the collage theorem, was investigated in [9] Most fractal algorithms, beginning with Jacquin s implementation [10], operate on a segmented image consisting of nonoverlapping square regions , called ranges, with . In order to avoid certain artifacts associated with square segmentation, alternative (triangular, hexagonal) segmentation schemes have been investigated [4] 7] 20] The block based segmentation, ....

A. E. Jacquin, "A fractal theory of iterated Markov operators with applications to digital image coding," Ph.D. dissertation, Georgia Inst. Technol., Atlanta, 1989.

....it uses a m 2 dimensional vector quantization approach. Similar ideas can be applied to volume compression, as is being done by Ning et al. Ning and Hesselink 92] Fractal Compression The theory behind fractal compression is non trivial, and some good reference could be found in Jacquin [Jacquin 89] and Barnsley et al. Barnsley et al. 88] In a nutshell, there is much resemblance between fractal compression and vector quantization. In both schemes, we search the code book and try to find and use one vector that best represents a given vector. However, they differ in at least two aspects. ....

....In both schemes, we search the code book and try to find and use one vector that best represents a given vector. However, they differ in at least two aspects. First, in fractal compression, sim ilarity is defined with respect to the results of a finite number of contractive , affine (see [Jacquin 89] or [Yang 00] for a detailed explanation) transforms on a given vector. Or to put it more precisely, for a given input vector, we will find a vector from the code book, which, under any contractive, affine transforms, will best approximate the given vector. Second, unlike in vector quantization, ....

A. Jacquin. A Fractal Theory of Iterated Markov Operators with Ap- plications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, August 1989.

....only a subset (i.e. sub sampling) of all the possible sub blocks are used during training. 4.2. 3 Fractal Compression Although the theory of fractal has been around for a while, its application on data compression just started about a decade ago, as being rst formalized and published in [Jac89] While the theory itself is intriguing (see [BDM 88] for a good reference) Fis95] provides a good tutorial on fractal image compression. Our interest here is mainly its applicability to volume data compression. Fortunately, the extension of fractal image compression to fractal volume ....

A. Jacquin. A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, August 1989.

....then equivalently, it is using a m 2 dimensional vector quantization approach. Similar idea can be applied to volume compression, as being done by Ning et al. in [NH92] Fractal Compression The theory behind fractal compression is non trivial, and some good reference could be found in Jacquin [Jac89] and Barnsley et al. BDM 88] In a nutshell, there is much resemblance between fractal compression and vector quantization. In both schemes, we search the code book and try to nd and use one vector that best represent a given vector. However, they di er in at least two aspects. First, in ....

....quantization. In both schemes, we search the code book and try to nd and use one vector that best represent a given vector. However, they di er in at least two aspects. First, in fractal compression, similarity is de ned with respect to results of a nite number of contractive , ane (see [Jac89] or [Yan00] for a detailed explanation) transforms on the given vector. Or to put it more precisely, for a given input vector, we will nd a vector from the code book, which, under any contractive, ane transforms, will best approximate the given vector. Second, unlike in vector quantization, in ....

A. Jacquin. A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, August 1989.

.... in contrast to the logical order of arguments. This paper summarizes the main results from our conference publications [27,28] The use of contractive transforms and their corresponding attractors for the compression of signals and images was proposed by Barnsley and Jacquin in the late 1980s [2,19]. Before the birth of fractal compression 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 ....

Jacquin, A. E., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, Doctoral Dissertation, Georgia Institute of Technology, August 1989.

....ahieve virtually lossless image compression at ratios of up to 100:1. The failure of Iterated Systems to publish the fractal transform led to a certain notoriety in the computer graphics community [11] but provided motivation for other workers. However, an algorithm described in 1990 by Jacquin [20, 21] is thought to be closely related to, if not identical to, the fractal transform. This algorithm is based on Iterated Transform Theory, an extension of IFS theory; some practical modifications have been suggested by Beaumont [10] The present work was conducted independently of Jacquin, as will be ....

....is then regarded as a transformation on the whole image; the codes for all range blocks together give an overlapping IFS. The unified transformation W is redefined so that, if x 2 B, then W Gamma1 (x) w Gamma1 B (x) where wB is the particular transformation for range block B. Jacquin [20] shows that such a transformation W has an attractor which is close to the original image. Our methods and ITT coding represent two contrasting approaches to blockcoding an image: we compute many SASs of minimal order; Jacquin computes one SAS of maximal order. However, both methods regard ....

Jacquin, A. E.: A fractal theory of iterated Markov operators with applications to digital image coding, Ph.D thesis, Georgia Tech (1989)

....Abnormalities Analysis Extraction Classification Encoding Quadtree FAR Feature Generation Filter #2 Filter #N . List Partition decoding. The topic of fractal image compression has received considerable attention since the publication of the first practical scheme by Jacquin in [4]. In this paper, we present a rather superficial discussion of this topic that serves our purposes. The interested reader is referred to [5] for a more detailed treatment of the subject matter. Fractal compression, and fractal encoding exploit the property of self similarity of fractal objects. ....

Jacquin, A., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, Ph.D. Thesis, Georgia Institute of Technology, August 1989.

....ON IMAGE PROCESSING, VOL. XX, NO. Y, MONTH 1999 102 I. Introduction IN fractal image compression an image is modeled as the unique fixed point of a contractive operator on the space of images. This type of image representation was first proposed by Barnsley and Sloan [1] 2] and Jacquin [3], 4] who devised the first practical fractal coder. Fractal coding has since been a topic of active research because it has opened up a refreshing new view to image compression. It leads to visually pleasing results at high compression ratios, and it provides resolution independent image ....

A. E. Jacquin, A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD thesis, Georgia Institute of Technology, August 1989.

....the coder is drastically increased. It may be kept low by preprocessing using an edge detection scheme to select the pertinent measure. The reconstruction algorithm remains the same. 1 INTRODUCTION Automatic fractal coding algorithms on still images were first introduced by Barnsley and Jacquin [1] and improved by others, see [2] for example. The main idea is to model one block (range block) of an image with an affine transformation of another one (domain block) taken from the image itself. The matching domain block and transformation minimize the mean square error between the range block ....

....the range block and the transformed domain block. However, the coded image is often a blurred, due to the choice of the distortion measure. The mean square error possesses no edge preserving property and does not correspond to the human perception [3, 4] One solution, proposed by Jacquin in [1] is block classification, with respect to geometric characteristics: shade, midrange, or edge blocks. Only the class which the range block belongs to is searched. Edges are better coded and the computation time is reduced. Fisher proposes another direction of research in [2] detection and ....

A. E. Jacquin, A fractal theory of iterated Markov operators with applications to digital image coding. PhD thesis, Georgia Institute of Technology, 1989.

....images and blocks because it is not efficient to send a new codebook for every image or every block in an image. Fractal based methods are typically based on the work by Barnsley [1] 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 ....

....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 ....

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A. E. Jacquin, A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, Aug. 1989.

....corresponds to the number of iterations of the traditional iterative algorithm. Keywords: Fractals; Image coding; Neural networks 1 Introduction Fractal image compression, based on the theory of contractive transformations, began with the work of Michael Barnsley [1, 2] and Arnauld Jacquin [3, 4]. Since then a large amount of work on the topics has been undertaken which has given the fractal image coding the power to become a serious competitor of the established compression techniques. Neural networks (NN) were suggested for image decoding in fractal image coding schemes in [5] where two ....

.... the pixel averaging operator avg and then 8 isometries of the square ( 0 identity, 1 ; 2 reflections about mid horisontal and mid vertical axes, 3 ; 4 reflections about both diagonals, and 5 ; 6 ; 7 rotations through 90 ffi , 180 ffi , 270 ffi ) have been applied on it [3, 6]. The resulting pool C of the size 8m is called a codebook pool. Next we can consider each range block as a vector R 2 R p where p is the number of pixels in the range block R. The encoding problem for the range block R (using a codebook block C) is then the least squares problem min x2R 2 ....

Jacquin, A. E.: A fractal theory of iterated Markov operators with applications to digital image coding, PhD thesis, Georgia Institute of Technology, Atlanta, August 1989.

....thus reduce the complexity of the search problem. 1. INTRODUCTION Fractal image coding has gained immense attraction by image coding researchers. In just a few years it found its way into end user products such as Microsoft s Encarta or as a Netscape plug in by Iterated Systems. The background [1, 2, 3] is well known and subsequently reconsidered briefly as a reference for the remaining sections. Starting with an arbitrary image f 0 , according to Banach s Fixed Point Theorem the iterative application of contractive transforms converges to a unique fixed point fF . We adopt the notation of ....

A. E. Jacquin, A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD thesis, Georgia Institute of Technology, 1989.

....Barnsley originally proposed the idea to use deterministic fractal geometry to obtain a compressed representation of digital images. Some years Preprint submitted to Elsevier Preprint 1 February 1998 later, one of his students devised the first algorithm capable to partially achieve that goal [10]. The idea of fractal coding is to represent the signal, or 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 ....

....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 approximated by a ....

Jacquin, A. E. A fractal theory of iterated Markov operators with applications to digital image coding. PhD thesis, Georgia Tech, 1989.

....the values of pixels. The algorithm is compared with the traditional iterative decoding algorithm. Keywords: Fractals; Image coding 1 Introduction Fractal image compression, based on the theory of contractive transformations, began with the work of Michael Barnsley [1, 2] and Arnauld Jacquin [3, 4]. Since then a large amount of work on the topics has been undertaken which has given the fractal image coding the power to become a serious competitor of the established compression techniques. Traditionally, a fractal image block coding scheme is based on the use of domain blocks which are ....

....In this case square domain blocks of twice the size of range blocks, distributed uniformly over the image I, have been used. Each domain block D i 2 D has been scaled down to the size of a range block using a pixel averaging operator and then 8 isometries of the square have been applied to it [3, 5]. The resulting pool C of the size 8m is called a codebook pool. Next we can consider each range block as a vector R 2 R p where p is the number of pixels in the range block R. The encoding problem for the range block R is then the least squares problem min x2R 2 kR Gamma Axk (1) where A is ....

JACQUIN, A. E.: A fractal theory of iterated Markov operators with applications to digital image coding, PhD thesis, Georgia Institute of Technology, Atlanta, August 1989. 9

....the images were encoded by interactive computer programs. This resulted in codes for images which were extremely compact in size, but their decoded images had very low quality [12] This was until the work by Arnaud E. Jacquin (a student of Barnsley) who automated this method for the first time [67, 68, 69, 70]. The code generated by Jacquin s program for an image was not as compact as before, but the compression ratio and the quality of Fractal Based Image and Video Coding 271 the decoded images looked promising. The work by Jacquin provided a platform for others to continue this line of research. ....

....[38] In this method, edge points were detected by wavelet transform and the dilation parameter is controlled by the fractal dimension. Fractal Coding of Contours One of the first applications of the theory of iterated function systems proposed by Barnsley and Jacquin was in contour coding [23, 67]. A similar method was later used for this purpose by Jacobs et al. 65] Jacquin s Method as a Second Generation Method Fractal coding methods based on Jacquin s method basically use redundancies in an image at different scales, i.e. they use the fact that different parts of the image at ....

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A. E. Jacquin. A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, Aug. 1989.

....and propose two new techniques. Keywords: Image compression, classification, clustering, nearest neighbors, time complexity, fractals, Hilbert curve. 1 Introduction Fractal image compression began with the visionary conception of M. Barnsley in 1987 and the ground breaking work of A. Jacquin [1, 2] in 1989. Since then a lot of work on the topic has been undertaken which has given fractal image compression the power to become a serious competitor of the established compression techniques. Fractal image compression is explained in the books [3, 4] and a recent survey of the rapidly growing ....

....the D. Saupe and R. Hamzaoui 5 method extends to the general case allowing p 1, provided that certain modifications are made. Essentially, this amounts to considering the transformed domains OE(D i ) see (2.3) in place of the original domains. 3.1. 1 Jacquin s approach In his original work [1, 2] Jacquin used a classification scheme coming from a study of Ramamurthi and Gersho [17] The domain blocks are classified according to their perceptual geometric features. Only three major types of blocks are differentiated: shade blocks, edge blocks, and midrange blocks. In shade blocks the image ....

Jacquin, A. E., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD Thesis, Georgia Institute of Technology, August 1989.

....of a picture of a face one should see tiny little distorted copies of the face everywhere. This seemed not only unnatural but also technically infeasible. Then, in 1989, Arnaud Jacquin, one of the graduate students of Barnsley, realized a first automatic fractal encoding system in his dissertation [Jacq89c], leaving behind the rigid thinking in terms of global IFS mappings. This broke the ice for a new direction of research in image coding. 1.1 The fractal goldrush The basic new idea in Jacquin s approach was very simple. An image should not be thought of as a collage of copies of the entire image, ....

Jacquin, A. E., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD Thesis, Georgia Institute of Technology, August 1989.

....and fast search algorithm is designed, based on orthogonal range searching and avoiding the curse of dimensionality problem of classical best match search methods. 1 Introduction 1. 1 Fractal Image Compression The fractal compression method has been implemented for the first time in 1989 ([12]) and extensively described since then in many different publications (e.g. 9] In fractal image compression, we try to find a mapping W in the image space so that the fixed point of this mapping exists, is unique, and is as close as possible to the image I we want to encode. W is itself ....

....is very slow, and the compression of a single image may take hours. The simplest improvement is the use of classification schemes : dividing the domain pool in a number of classes, and comparing only the domains falling in the same class as the range. Classification schemes are numerous (e.g. [12], 9] but none of them decreases the complexity order of the search. There are anyway some methods that achieve this goal, often with a slight quality loss (see e.g. 1] The structure of the remainder of this paper is as follows. In section 1.2, we review the idea of feature vectors, and ....

Jacquin, A., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD. thesis, Georgia Institute of Technology, August 1989.

....and avoiding the curse of dimensionality problem of classical best match searching methods. The application of the technique to vector quantization is also discussed. 1 Introduction 1. 1 Fractal Image Compression The fractal compression method has been implemented for the first time in 1989 ([12]) and extensively described since in many different publications (e.g. 7] In fractal image compression, we try to find a mapping W in the images space so that the fixed point of this mapping exists, is unique, and is as close as possible to the image I we want to encode. W is itself composed of ....

....is very slow, and the compression of a single image may take hours. The most simple improvement is the use of classification schemes : divide the domain pool in a number of classes, and compare only the domains falling in the same class than the range. Classification schemes are numerous (e.g. [12], 7] but none of them decreases the complexity order of the search. There are anyway a lot of methods that achieve this goal, often with a slight quality loss (see e.g. 1] 1.2 A Fast General Method Using Feature Vectors A good method to solve the problem of finding the best domain block for ....

Jacquin, A., A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding, PhD. thesis, Georgia Institute of Technology, August 1989.

....than what is typical of fractal coding; so an implementation on parallel architecture represents its natural solution. 1. INTRODUCTION Fractal image coding, based on the theory of Partitioned Iterated Function Systems, is emerging as a promising compression technique since its proposal in 1989 [1]. Already today it provides performances in terms of compression quality not worse than JPEG if it is particularly considered that nearly all proposed schemes of PIFS based coders are lacking in final stages of entropic compression (for example Huffman or arithmetic coding) which are instead ....

A. Jacquin, "A fractal theory of iterated Markov operators with applications to digital images coding", PhD Thesis, Georgia Institute of Technology, August 1989.

....there are two common ways of formalizing a monochrome image, namely, as a measure describing the ink distribution, and as a brightness function. The former one is used in the paradigm of fractal image synthesis and compression, the Iterated Function System (IFS) 1, 9] and in its extensions [2, 7]. The latter one is found in several recent compression strategies [5, 8, 4] one of which is based on Partitioned Iterated Function Systems (PIFSs) In [3] this concept is introduced from a practical point of view in order to meet the need for IFSs in which each function is only iterated on some ....

....8, 4] one of which is based on Partitioned Iterated Function Systems (PIFSs) In [3] this concept is introduced from a practical point of view in order to meet the need for IFSs in which each function is only iterated on some part of the image instead of the entire image. For the same reason, in [7] an analogue one is considered using the measure formalization of an image. However, the precise form proposed is not theoretically founded, nor is it clear in [3] what assumptions are implicitly made in using it. That is what we intend to investigate in this paper, by making the transition from ....

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Arnaud Jacquin. A fractal theory of iterated Markov operators with applications to digital image coding. PhD thesis, Georgia Institute of Technology, 1989.

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Arnaud Jacquin. A fractal theory of iterated Markov operators with applications to digital image coding. PhD thesis, Georgia Institute of Technology, 1989.

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A. Jacquin, A fractal theory of iterated markov operators with applications to digital image coding (PhD thesis, Georgia Institute of Technology, 1989).

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A. Jacquin, "A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding," Ph.D. dissertation, Georgia Tech. Atlanta, 1989.

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A. E. Jacquin, A Fractal Theory of Iterated Markov Operators with Applications to Digital Image Coding. PhD thesis, Georgia Institute of Technology, Atlanta, GA, Aug. 1989.

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