Results 1  10
of
38
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
Abstract

Cited by 48 (10 self)
 Add to MetaCart
We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
The Object Instancing Paradigm for Linear Fractal Modeling
 IN PROC. OF GRAPHICS INTERFACE
, 1992
"... The recurrent iterated function system and the Lsystem are two powerful linear fractal models. The main drawback of recurrent iterated function systems is a difficulty in modeling whereas the main drawback of Lsystems is inefficient geometry specification. Iterative and recursive structures ext ..."
Abstract

Cited by 32 (4 self)
 Add to MetaCart
The recurrent iterated function system and the Lsystem are two powerful linear fractal models. The main drawback of recurrent iterated function systems is a difficulty in modeling whereas the main drawback of Lsystems is inefficient geometry specification. Iterative and recursive structures extend the object instancing paradigm, allowing it to model linear fractals. Instancing models render faster and are more intuitive to the computer graphics community. A preliminary section briefly introduces the object instancing paradigm and illustrates its ability to model linear fractals. Two main sections summarize recurrent iterated function systems and Lsystems, and provide methods with examples for converting such models to the object instancing paradigm. Finally, a short epilogue describes a particular use of color in the instancing paradigm and the conclusion outlines directions for further research.
Power domains and iterated function systems
 Information and Computation
, 1996
"... We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniquene ..."
Abstract

Cited by 30 (10 self)
 Add to MetaCart
We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domaintheoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution. For hyperbolic recurrent IFSs and Lipschitz maps, one can estimate the integral up to any threshold of accuracy.] 1996 Academic Press, Inc. 1.
Fast Hierarchical Codebook Search for Fractal Coding of Still Images
 EOS/SPIE Visual Communication and PACS for Medical Applications 93
, 1993
"... This paper presents a method for fast encoding of still images based on iterated function systems (IFSs). The major disadvantage of this coding approach, usually referred to as fractal coding, is the high computational effort of the encoding process compared to e.g. the JPEG algorithm [1]. This is m ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
This paper presents a method for fast encoding of still images based on iterated function systems (IFSs). The major disadvantage of this coding approach, usually referred to as fractal coding, is the high computational effort of the encoding process compared to e.g. the JPEG algorithm [1]. This is mainly due to the costly "full search" of the transform parameters within a fractal codebook. We therefore propose an hierarchical encoding scheme which is based upon a two level codebook search and a structural classification of its entries. By this way only a small subset of the codebook has to be considered, which increases encoding speed significantly. Refining the initial codebook and applying a second search even increases the reconstruction quality compared to the full search but with a fraction of its computational effort. 1.
On the Convergence of Fractal Transforms
, 1994
"... 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 nonlinear transformation in the finitedimensional vector space. Using spect ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
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 nonlinear transformation in the finitedimensional 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. 1.
Solving the Inverse Problem for Measures Using Iterated Function Systems: A New Approach
 Adv. Appl. Prob
, 1995
"... We present a systematic method of approximating, to an arbitrary accuracy, a probability measure ¯ on [0; 1] q ; q 1, with invariant measures for Iterated Function Systems by matching its moments. There are two novel features in our treatment: (1) An infinite number of fixed affine contraction ma ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
We present a systematic method of approximating, to an arbitrary accuracy, a probability measure ¯ on [0; 1] q ; q 1, with invariant measures for Iterated Function Systems by matching its moments. There are two novel features in our treatment: (1) An infinite number of fixed affine contraction maps on X; W = fw 1 ; w 2 ; : : :g, subject to an "fflcontractivity" condition, is employed. Thus, only an optimization over the associated probabilities p i is required. (2) We prove a Collage Theorem for Moments which reduces the moment matching problem to that of minimizing the "collage distance" between moment vectors. The minimization procedure is a standard quadratic programming problem in the p i which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0,1] are presented. AMS Subject Classifications: 28A, 41A, 58F 1. Introduction This paper is concerned with the approximation of probability measures on a compact metric space X ...
Signal Modeling With Iterated Function Systems
, 1993
"... this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
this memory requirement issue may become a factor, in which case the Random Iteration Algorithm could be adapted to overcome the shortcomings mentioned here with some simple checks on the path of the calculations. 2.3 Conclusion
Individual GP: an Alternative Viewpoint for the Resolution of Complex Problems.
"... An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Examples are presented ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
An unususal GP implementation is proposed, based on a more "economic" exploitation of the GP algorithm: the "individual" approach, where each individual of the population embodies a single function rather than a set of functions. The nal solution is then a set of individuals. Examples are presented where results are obtained more rapidly than with the conventional approach, where all individuals of the nal generation but one are discarded.
Adaptive Fractal Coding of Still Pictures
 In Proceedings of the International Picture Coding Symposium PCS’93
, 1993
"... This paper presents a method for lossy coding of still pictures using the theory of iterated function systems (IFS) which is a standard description for deterministic fractals. The coding concept is based on a blockwise fractal approximation of the original image by contraction mappings of itself usi ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
This paper presents a method for lossy coding of still pictures using the theory of iterated function systems (IFS) which is a standard description for deterministic fractals. The coding concept is based on a blockwise fractal approximation of the original image by contraction mappings of itself using affine transformations which are a special case of IFSs. Data reduction is achieved by exploiting the selfsimilarity within natural or artificial pictures. The proposed algorithm accounts for the local image characteristics by an adaptive quadtree segmentation. In contrast to existing fractal coding concepts an overall constant reconstruction quality is achieved. 1.
Fractal Image Compression and the Inverse Problem of Recurrent Iterated Function Systems
 Directions for Fractal Modeling in Computer Graphics. SIGGRAPH '94 Course Notes
, 1996
"... Fractal image compression currently relies on the partitioning of an image into both coarse #domain" segments and #ne #range" segments, and for each range element, determines the domain element that best transforms into the range element. Under normal circumstances, this algorithm produces a stru ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
Fractal image compression currently relies on the partitioning of an image into both coarse #domain" segments and #ne #range" segments, and for each range element, determines the domain element that best transforms into the range element. Under normal circumstances, this algorithm produces a structure equivalent to a recurrent iterated function system. This equivalence allows recent innovations to fractal image compression to be applied to the general inverse problem of recurrent iterated function systems. Additionally, the RIFS representation encodes bitmaps #bilevel images# better than current fractal image compression techniques. Keywords: bitmap, block coding, compression, fractals, imaging, recurrent iterated function system. 1 1 Introduction Fractal geometry provides a basis for modeling the in#nite detail found in nature #Mandelbrot, 1982#. Fractal methods are quite popular in the modeling of natural phenomena in computer graphics, ranging from random fractal models ...