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23
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 ..."
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Cited by 32 (4 self)
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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.
A Schur–Newton method for the matrix pth root and its inverse
 SIAM J. Matrix Anal. Appl
"... And by contacting: ..."
A fast algorithm for Julia sets of hyperbolic rational functions
 Proc. of CCA 2004, in ENTCS, vol 120
, 2005
"... Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of nontrivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, beco ..."
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Cited by 16 (0 self)
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Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of nontrivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers. In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z ↦ → z 2 + c, where c  < 1/4, in [RW03] and an earlier result in [Zho98] on the recursiveness of the Julia sets of hyperbolic polynomials. The algorithm given in this paper computes Julia sets locally in time O(k · M(k)) (where M(k) denotes the time needed to multiply two kbit numbers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide a set of disks of spherical diameter 2 −k so that the union of these disks has Hausdorff distance at most 2 −k+2. This allows to give reliable pictures of Julia sets to arbitrary precision. Key words: Julia Sets, Computational Complexity. 1
Valuations of Languages, with Applications to Fractal Geometry
, 1994
"... Valuations  morphisms from (\Sigma ; \Delta; ) to ((0; 1); \Delta; 1)  are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [28, 41, 11, 9, 10, 42]. This paper shows that valuations are useful not only within the theory of codes, but also when dealing wit ..."
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Cited by 8 (4 self)
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Valuations  morphisms from (\Sigma ; \Delta; ) to ((0; 1); \Delta; 1)  are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [28, 41, 11, 9, 10, 42]. This paper shows that valuations are useful not only within the theory of codes, but also when dealing with ambiguity, especially in contextfree grammars, or for defining outer measures on the space of omegawords which are of some importance to the theory of fractals. These connections yield new formulae to determine the Hausdorff dimension of fractal sets (especially in Euclidean spaces) defined via formal languages. The class of fractals describable with contextfree languages strictly includes that of MRFSfractals introduced in [58, 18]. Some of the results of this paper also appear as part of the author's PhD thesis [36] and in [30, 31].
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 ..."
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Cited by 7 (3 self)
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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 ...
Infinite Iterated Function Systems
, 1994
"... : We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the wellknown case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff d ..."
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Cited by 7 (3 self)
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: We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the wellknown case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded nonempty sets not describable by IIFS. Keywords: Fractal geometry, iterated function systems, complete metric spaces, Baire space, Hausdorff measure, Hausdorff dimension, selfsimilarity. AMS classification: 28A80, 54E50, 54E52, 28A78, 54F45. 1. Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper [14], gained more and more interest. Several books on this topic are available [3, 7, 5, 18, 19] which have become popular even amo...
Memory across eyemovements: 1/f dynamic in visual search
 Nonlinear Dynamics, Psychology, & Life Sciences
, 2002
"... The ubiquity of apparently random behavior in visual search (e.g., Horowitz & Wolfe, 1998) has led to our proposal that the human oculomotor system has subtle deterministic properties that underlie its complex behavior. We report the results of one subject’s performance in a challenging search task ..."
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Cited by 6 (1 self)
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The ubiquity of apparently random behavior in visual search (e.g., Horowitz & Wolfe, 1998) has led to our proposal that the human oculomotor system has subtle deterministic properties that underlie its complex behavior. We report the results of one subject’s performance in a challenging search task in which 10,215 fixations were accumulated. A number of statistical and spectral tests revealed both fractal and 1/f structure. First, scaling properties emerged in differences across eye positions and their relative dispersion (SD/M)—both decreasing over time. Fractal microstructure also emerged in an iterated function systems test and delay plot. Power spectra obtained from the Fourier analysis of fixations produced brown (1/f 2) noise and the spectra of differences across eye positions showed 1/f (pink) noise. Thus, while the sequence of absolute eye positions resembles a random walk, the differences in fixations reflect a longerterm dynamic of 1/f pink noise. These results suggest that memory across eyemovements may serve to facilitate our ability to select out useful information from the environment. The 1/f patterns in relative eye positions together with models of complex systems (e.g., Bak, Tang & Wiesenfeld, 1987) suggest that our oculomotor system may produce a complex and selforganizing search pattern providing maximum coverage with minimal effort. KEY WORDS: visual search; eyemovements; attention; Fourier analysis; pink noise; selforganized criticality. A classic problem in the study of perception centers on our ability to perceive a stable world despite the dynamic nature of the retinal image.
PérezJiménez: Fractals and P systems
 In Proceedings of Fourth Brainstorming Week on Membrane Computing
"... Summary. In this paper we show that the massive parallelism, the synchronous application of the rules, and the discrete nature of their computation, among other features, lead us to consider P systems as natural tools for dealing with fractals. Several examples of fractals encoded by P systems are p ..."
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Cited by 2 (1 self)
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Summary. In this paper we show that the massive parallelism, the synchronous application of the rules, and the discrete nature of their computation, among other features, lead us to consider P systems as natural tools for dealing with fractals. Several examples of fractals encoded by P systems are presented and we wonder about using P systems as a new tool for representing and simulating the fractal nature of tumors. 1
The Most Unreliable Technique in the World to compute pi
 Workshop at the IIIrd Summer School on Advanced Functional Programming
, 1998
"... This paper is an atypical exercice in lazy functional coding, written for fun and instruction. It can be read and understood by anybody who understands the programming language Haskell. We show how to implement the BaileyBorweinPlouffe formula for π in a corecursive, incremental way which prod ..."
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This paper is an atypical exercice in lazy functional coding, written for fun and instruction. It can be read and understood by anybody who understands the programming language Haskell. We show how to implement the BaileyBorweinPlouffe formula for π in a corecursive, incremental way which produces the digits 3, 1, 4, 1, 5, 9. . . until the memory exhaustion. This is not a way to proceed if somebody needs many digits! Our coding strategy is perverse and dangerous, and it provably breaks down. It is based on the arithmetics over the domain of infinite sequences of digits representing proper fractions expanded in an integer base. We show how to manipulate: add, multiply by an integer, etc. such sequences from the left to the right ad in nitum, which obviously cannot work in all cases because of ambiguities. Some deep philosophical consequences are discussed in the conclusions.