Graph-theoretical analysis of the fractal transform
09 May 1995-Vol. 4, pp 2559-2562
TL;DR: An algorithm to analyze the structure of the dependence graph is presented and it is shown that the affine transformations for some range blocks can be optimized based on the domain blocks extracted from the reconstructed rather than the original image.
Abstract: A part of a fractal code is an assignment of a domain block to every range block. The assignment is used to construct the dependence graph of a fractal code. The vertices of the graph represent the range blocks. Two vertices z and y are connected by a directed edge from y to x if the range block y is overlapped, fully or partially, by the domain block assigned to the range block x. An algorithm to analyze the structure of the dependence graph is presented. The exposed structure of the graph can be used for three different purposes. The first one is convergence analysis: the affine transformations linking domain and range blocks can be classified into those that affect convergence and those that do not. The second one is decoding time reduction: certain range blocks can be reconstructed in a non-iterative way. The third one is improving upon collage coding: the affine transformations for some range blocks can be optimized based on the domain blocks extracted from the reconstructed rather than the original image.
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Proceedings Article•
31 Jul 1997
TL;DR: This paper has chosen the similarity to a particular variant of vector quantization as the most direct approach to fractal image compression and surveys some of the advanced concepts such as fast decoding, hybrid methods, and adaptive partitionings.
Abstract: Fractal image compression is a new technique for encoding images compactly. It builds on local self-similarities within images. Image blocks are seen as rescaled and intensity transformed approximate copies of blocks found elsewhere in the image. This yields a self-referential description of image data, which --- when decoded --- shows a typical fractal structure. This paper provides an elementary introduction to this compression technique. We have chosen the similarity to a particular variant of vector quantization as the most direct approach to fractal image compression. We discuss the hierarchical quadtree scheme and vital complexity reduction methods. Furthermore, we survey some of the advanced concepts such as fast decoding, hybrid methods, and adaptive partitionings. We conclude with a list of relevant WEB resources including complete public domain C implementations of the method and a comprehensive list of up-to-date references.
66 citations
TL;DR: A novel algorithm for fractal video sequence coding, based on the circular prediction mapping and the noncontractive interframe mapping is proposed, which can effectively exploit the temporal correlation in real image sequences.
Abstract: We propose a novel algorithm for fractal video sequence coding, based on the circular prediction mapping and the noncontractive interframe mapping. The proposed algorithm can effectively exploit the temporal correlation in real image sequences, since each range block is approximated by the domain block in the adjacent frame, which is of the same size as the range block. The computer simulation results demonstrate that the proposed algorithm provides very promising performance at low bit rate, ranging from 40-250 kbyte/s.
39 citations
TL;DR: The stereo fractal video coding is proposed which matches the macroblock with two reference frames in left and right view results in increasing compression ratio and reducing bit rate when transmitting compressed stereo data.
Abstract: Based on the classical fractal video compression method, an improved object-based stereo video compression scheme with Shape-Adaptive DCT is proposed in this paper. Firstly, we use more effective macroblock partition scheme instead of classical quadtree partition scheme; thus reducing the block searching strategy. The stereo fractal video coding is proposed which matches the macroblock with two reference frames in left and right view results in increasing compression ratio and reducing bit rate when transmitting compressed stereo data. The stereo codec combines the Motion Compensation Prediction (MCP) and Disparity Compensation Prediction (DCP). Fractal coding is adopted and each object is encoded independently by a prior video segmentation alpha plane, which is defined exactly as in MPEG-4. The testing results with the nature monocular and stereo video sequences provide promising performances at low bit rate coding. We believe it will be a powerful and efficient technique for the object-based monocular and stereo video sequences coding.
25 citations
13 Sep 1995
TL;DR: Why effective non-linear transformations are not easy to find and a model based on conformai mappings in the geometric domain that are a natural extension of the affine model are proposed.
Abstract: Most recent advances in fractal image coding have been concentrating on better adaptive coding algorithms, on extending the variety of the blocks and on search strategies to reduce the encoding time. Very little has been done to challenge the linear model of the fractal transformations used so far in practical applications. In this paper we explain why effective non-linear transformations are not easy to find and propose a model based on conformai mappings in the geometric domain that are a natural extension of the affine model. Our compression results show improvements over the linear model and support the hope that a deeper understanding of the notion of self-similarity would further advance fractal image coding.
23 citations
TL;DR: An algorithm is proposed that starts from an initial mapping obtained by collage coding and iteratively provides a sequence of contractive mappings whose fixed points monotonically approach the original image.
21 citations
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References
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Book•
01 Jan 1990TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher:
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects.
In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage.
As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds.
Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.
21,651 citations
01 Oct 1993
TL;DR: An approach to image coding based on a fractal theory of iterated contractive transformations defined piecewise is described, and the design of a system for the encoding of monochrome digital images at rates below 1 b/pixel is described.
Abstract: An approach to image coding based on a fractal theory of iterated contractive transformations defined piecewise is described. The main characteristics of this approach are that it relies on the assumption that image redundancy can be efficiently captured and exploited through piecewise self-transformability on a block-wise basis, and it approximates an original image by a fractal image, obtained from a finite number of iterations of an image transformation called a fractal code. This approach is referred to as fractal block coding. The general coding-decoding system is based on the construction, for an image to be encoded, of a fractal code-a contractive image transformation for which the original image is an approximate fixed point-which, when applied iteratively on any initial image of the decoder, produces a sequence of images which converges to a fractal approximation of the original. The design of a system for the encoding of monochrome digital images at rates below 1 b/pixel is described. Ideas and extensions from the work of other researchers are presented. >
559 citations
Book•
01 Dec 1994TL;DR: This paper begins by presenting a simple explanation of the main ideas in fractal image compression followed by a brief discussion of the current state of the art along with some results comparing fractal encoding, JPEG, and a wavelet scheme.
Abstract: This paper begins by presenting a simple explanation of the main ideas in fractal image compression. It then presents a brief discussion of the current state of the art along with some results comparing fractal encoding, JPEG, and a wavelet scheme. The conclusion contains references to many of the latest theoretical and implementation results.
305 citations
27 Apr 1993
TL;DR: It is shown how the problem of decoder convergence speed in an attractor (fractal-based) image compression system can be sidestepped by introducing a modification providing an iteration-free decoder.
Abstract: The problem of decoder convergence speed in an attractor (fractal-based) image compression system is considered. The general principles of such a system are described in the context of discrete linear spaces. A problem with previous implementations is that the decoding rests on an iterative procedure whose complexity is image-dependent. It is shown how this problem can be sidestepped by introducing a modification providing an iteration-free decoder. The coding results are not affected by this modification. The new decoding algorithm requires only one multiplication and three additions per pixel. The algorithm is suitable for applications where fast display of images is essential, such as retrieval of images from compressed databases. >
51 citations
19 Apr 1994
TL;DR: This paper reports on investigations concerning the convergence of fractal transforms for signal modelling by using spectral theory to derive a contractivity measure derived from the eigenvalues of a special linear operator.
Abstract: 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. >
26 citations