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The Escape Buffer: Efficient Computation of Escape Time for Linear Fractals
Abstract and Figures
The study of linear fractals has gained a great deal from the study of quadratic fractals, despite important differences. Methods for classifying points in the complement of a fractal shape were originally developed for quadratic fractals, to provide insight into their underlying dynamics. These methods were later modified for use with linear fracta ls. This paper reconsiders one such classification, called escape time, and presents a new algorithm for its computation that is significantly faster and conceptually simp ler. Previous methods worked backwards, by mapping pixels into classified regions, whereas the new forward algorithm uses an "escape buffer" to mapping classified regions onto pixels. The efficiency of the escape buffer is justified by a careful ana lysis of its performance on linear fractals with various properties.
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... [10][11][12] While fractal patterns are very complex, only a small amount of information is 1 needed to generate them, e.g., in the IFS, only information about a finite number of contractive mappings is needed. In the literature, there are many methods of generating fractal patterns, including deterministic algorithm and random iteration algorithm, 2 the escape time algorithm, 13 and even the IFS ray tracing algorithm. 14 In 2000, Frame and Cogevina introduced a method called circle inversion fractals 15 that was based on circle inversion transformation, a well-known concept in geometry. ...
In the paper, we generalized the idea of circle inversion to star-shaped sets and used the generalized inversion to replace the circle inversion transformation in the algorithm for the generation of the circle inversion fractals. In this way, we obtained the star-shaped set inversion fractals. The examples that we have presented show that we were able to obtain very diverse fractal patterns by using the proposed extension and that these patterns are different from those obtained with the circle inversion method. Moreover, because circles are star-shaped sets, the proposed generalization allows us to deform the circle inversion fractals in a very easy and intuitive way.
... The deterministic algorithm is a brute force algorithm, the random iteration algorithm requires sometimes millions of points to be evaluated before pixels are covered with enough points to obtain a detailed and stable image of µ, the escape time algorithm also involves a lot of computation. Several other algorithms have been developed [3,5,4,9,10] that are more efficient. For example, it is possible to limit the number of calculated points that fall into any given pixel. ...
A fast method to generate fractal imagery is presented. Iterated Function Systems (IFS) are based on repeatedly copying transformed images. We show that this can be directly translated into standard graphics operations: Each image is generated by texture mapping and blending copies of the previous image. Animations of dy-namic IFS codes are generated at fifty frames per second on a notebook PC, using commodity graphics hardware. Various extensions and variations are presented.
... On the other hand, escape-time is a visualization tool used to understand the dynamics of fractal representations. It has many forms and has been used by several algorithms, for instance, Prusinkiewic, et al. [29] adopted the discrete and continuous escapetime classification to larger class of linear fractals, especially those represented by language-restricted iterated function systems, Daryl et al. [11] modifies escape-time computation for generating linear fractals, Hussein et al. [16], [17] simulate escape-time fractal image synthesis using PN, in [18] extended escape-time algorithm for representing 3-D linear fractals, and in [19] generalized the original escape-time algorithm of quadratic order into higher order mapping. The object of our paper is to make escape-time more computationally effective by manipulating PN reachability problem [14] in order to generate unlimited class of set of patterns as well as fractal patterns based only on the concept of iteration principle of escape-time process without using any iteration function on the complex plane which is a timeconsuming and chaos process. ...
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This paper proposes a method to integrate basic origami processes into fractal concepts. The starting point is an area of polygonal shape. It is folded until one or more smaller copies result. The crease pattern is scaled to the reduced copies, and then the process repeats. This is equivalent to maintaining the original crease pattern if the reduced objects are magnified. As will become clear in the paper, the fact that the folded object is enlarged implies that functions are used that are the inverse of contracting functions, as is familiar in inverse techniques for iterated function systems. The method is generalized by allowing cuts before folding. With a well-chosen colorization method, the images obtained reveal complex structures suggestive of depth and light.
The objective of this paper is to explain the machinery of the Chaos Game used in the context of various types of iterated function system specifications. Although the discussion is kept in the framework of a mathematical proof, in contrast to the previous work, only the fundamentals of probability are employed. Moreover, unlike the known approaches, the way of reasoning presented in this paper is the same for all the discussed variants of the Chaos Game. As a result, the logical background of the algorithm can be followed by anyone who is interested in fractals and computer graphics.
This monograph was written to be used in a course of the Summer School in Computer Graphics at IMPA in 1993. Since then it has heen used to present a tuturial at SIBGRAPI 94 (Brazilian Simposium on Computer Graphics and Image Processing). It covers the area of modeling with implicit objects from the beginning, addressing both theoretical and implementation issues.
We propose a new and effective image modeling dual technique which
is capable of simulating both texture image synthesis and non-texture
images like fractals. The technique uses the algebraic approach of graph
grammars theory as a new simulation tool for both texture and
non-texture image synthesis via its graph production, derivation and
double-pushout construction. Validation of our approach is given by
discussion and an illustration of some experimental results. An
investigation of the relationships between the generated patterns and
their corresponding graph grammars is also discussed
Linear fractal models such as the iterated function system (IFS),
recurrent IFS (RIFS), and Lindenmeyer system (L-system) concisely
describe complex objects using self-reference. These models hold much
promise in computer graphics as geometric representations of detail. The
use of fractals for image compression sacrifices its fractal origins in
the search for optimal coding gain. This article rebuilds the
relationship between the fields of fractal image compression (FIC) and
fractal geometry to better facilitate the sharing of new results. Many
geometric representations exist for smooth shapes, and each has certain
benefits and drawbacks. Computer-aided geometric design has produced
many algorithms to convert a given curve or surface description into the
most appropriate geometric representation for a given task. Likewise,
there are several models for linear fractal shapes and several methods
for converting between the representations. The representation used by
FIC has been called partitioned IFS or local IFS. This article describes
a method for converting FIC's partitioned/local IFS to fractal
geometry's RIFS. This conversion algorithm allows FIC to represent any
input shape as a linear fractal and permits algorithms developed for
linear fractals to be applied to a wider variety of shapes. When every
domain element can be expressed as the union of range elements, FIC
produces a structure that is equivalent to a RIFS, enabling it to
automatically model arbitrary shapes
An iterated function system (IFS) is defined to be a set of contractive affine transformations. When iterated, these transformations define a closed set, called the attractor of an IFS, which has fractal characteristics. Fractals of any sort are currently a topic of great popular appeal, largely due to the exciting images to which they lend
themselves. Iterated function systems represent one of the newest sources of fractal images. Research to date has focused on exploiting IFS techniques for the generation of fractals and for use in modelling applications. Both areas of this research are well suited to computer graphics, and this thesis examine the IFS techniques from a
computer graphics perspective.
As a source of fractals, iterated function systems have some relationship to other methods of fractal generation. In particular, the relationship between IFS attractors and Julia sets will be examined throughout the thesis. Many insights can be gained from the previous work done by Peitgen, Richter and Saupe [32, 33] both in terms of methods for the generation of the fractal sets and methods for their visualization. The differences between the linear transformations which compose an IFS and the quadratic polynomials which define Julia sets are significant, but not moreso than their similarities.
This thesis deals with the related questions of approximation and visualization. The method of constructing the approximating set of points is dependent upon the visualization method in use. Methods have been developed both to visualize the attractor and its complement. The two techniques used to examine the complement set are based on the distance and escape-time functions. The modelling power of standard IFS techniques is limited in that they cannot be used to model any object which is not strictly self-affine. To combat this, methods
for controlling transformation application are examined which allow objects without
strict self-affinity to be modelled.
As part of this research, an extensible software system was developed to allow
experimentation with the various concepts discussed. A description of that system is
included in Chapter 6.
The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that i) it relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis, and ii) it approximates an original image by a fractal image. The author refers to the approach as fractal block coding. The coding-decoding system is based on the construction, for an original image to encode, of a specific image transformation--a fractal code--which, when iterated on any initial image, produces a sequence of images that converges to a fractal approximation of the original. It is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0.5-1.0 b/pixel. The fractal block coder has performance comparable to state-of-the-art vector quantizers.
Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In section I, a series of computer generated and rendered random fractal shapes provide a visual introduction to the concepts of fractal geometry. These complex images, with details on all scales, are the result of the simplest rules of fractal geometry. Their success as forgeries of the natural world has played an important role in the rapid establishment of fractal geometry as a new scientific discipline and exciting graphic technique. Section II presents a brief mathematical characterization of these forgeries, as variations on Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. Finally, section III will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms.
Implicit surface techniques provide useful tools for modeling and rendering smooth surfaces. Deriving implicit formulations for fractal representations extends the scope of implicit surface techniques to rough surfaces. Linear fractals modeled by recurrent iterated function systems may be defined implicitly using either geometric distance or escape time. Random fractals modeled using Perlin's noise function are already defined implicitly when described as "hypertexture."Deriving new implicit formulae is only the first step. Unlike their smooth counterparts, rough implicit surfaces require special rendering techniques that do not rely on continuous differentiation of the defining function. Preliminary experiments applying blending operations to rough surfaces have succeeded in an initial attempt to overcome current challenges in natural modeling. The grafting of a stem onto the base of a linear fractal leaf continuously blends smooth detail into rough detail. The blend of two textured cylinders interpolates geometric bark across branching points in a tree.
The goal of this chapter, which is a continuation of Section 3.3, is to demonstrate how genuine mathematical research experiments open a door to a seemingly inexhaustible new reservoir of fantastic shapes and images. Their aesthetic appeal stems from structures which at the same time are beyond imagination and yet look extremely realistic. Being the result of well defined and mostly very simple mathematical processes — which depend on a few parameters — animations of evolution or metamorphosis of fantastic forms are easy to obtain.
Five approximations of fractal sets A are compared. The first three were described by Hutchinson, Barnsley-Demko and Williams, respectively. Two new other approximations are still efficient in all circumstances. For any algorithm, error estimates in the approximations are found. Numerical comparisons are also done through classical examples of attractors in the plane.
The problem of indefinite integration is one of the easiest problems of mathematics to describe: given a function f(x), find a function g(x) such that g´(x) =f(x)