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Approximation and Visualization of Sets Defined by Iterated Function Systems
Abstract and Figures
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.
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... If the implicit function underestimates the distance from any point to the object, then one can create a sequence that converges to the rst ray-object intersection . Such a distance function was devised for linear fractals in Hepting et al., 1990;Hepting, 1991 . Thus, linear fractals may be rendered using this distance estimate, though not in optimal time. ...
... In , the gradient of a distance estimate function provided a decent approximate surface normal. Such distance estimates exist for linear fractals Hepting et al., 1990;Hepting, 1991 but are relatively expensive to compute. ...
... on results derived inReuter, 1987;Hepting, 1991 for rendering 2-D linear fractals.The bounding volume hierarchy produced by the RIFS is more e cient when neighboring bounding volumes in the hierarchy do not intersect often. Ideally, the RIFS should have the open-set property with the initial bounding volume set to the closure of open set.Some attractors require overlapping constructions to be modeled e ciently. ...
Linear fractals are the fractal attractors of recurrent iterated function systems. Current computer graphics algorithms are not well suited for rendering such sets. This dissertation solves these problems by introducing new rendering methods designed specifically for linear fractals. The dissertation begins with a formal treatment of the iterated function system model, the recurrent iterated function system model, fractals in general, and specifically linear fractals. Interactive and automatic methods for modeling objects as linear fractals are also described. The new contributions to rendering linear fractals include: two methods for efficiently computing the intersection of a ray with a linear fractal, thereby solving the hidden surface problem; derivation of the bounds of the size of the projection of a pixel into object space, thereby eliminating the unnecessary computation of overly fine detail; two methods for computing an optimal initial bounding volume, thereby allowing arbitrary linear fractal models to be visualized; a method for shading the surface of a linear fractal, thereby producing orientation cues for surfaces lacking tangent planes; and a method for antialiased rasterization, thereby removing the jagged edges and twinkling pixels that often accompany fractal images and animations.
... The survey outlined the development of a relatively simple notion of contraction on average into substantial literature on iterated function systems. Indeed, our survey alone covers over 400 papers, several books, and a steady stream of PhD dissertations in Applied Probability and related areas of Mathematics: Giles (1990); Daryl (1991); ; Vines (1993); Doughty (1995); Stenflo (1998a) ...
We provide an overview of iterated function systems (IFS), where randomly chosen state-to-state maps are applied iteratively to a state. We aim to summarize the state of art and, where possible, identify fundamental challenges and opportunities for further research.
... This work began from specific efforts to create imagery to aid the study of numerical methods [115], dynamical systems [111], and fractals [109,116]. Fractals, in particular, have proven to be a prototypical example of the power of mathematical visualization. ...
This dissertation examines how the computer can aid the creative human endeavour which is data visualization. That computers now critically aid many fields is apparent, as is evidenced by the breadth of contemporary research on this topic.
Indeed, computers have contributed widely to the whole area of data comprehension, both in performing extensive computations and in producing visual representations of the results. Computers originally aided mathematicians who could both write the instructions necessary to direct the computer and interpret the resulting numbers. Even though modern computers include advanced graphical capabilities, many issues of access still remain: the users of data visualization software systems may not be experts in any computer-related
field, yet they want to see visual representations of their data which allow them insight into their problems. For example, today’s mathematicians who are generally expert in exploiting computational opportunities for experimentation may lack similar experience in opportunities for visual exploration.
Of particular concern is how a computer-aided visualization tool can be designed to support the user’s goal of obtaining insight. There are many visual representations for a given set of data, and different people may obtain insight from different visual representations. Selection of the “best” one for an individual can be exceedingly difficult, as the sheer number of possible representations may be staggering. Current software designs either recognize the possibility of overwhelming the individual and therefore employ some means of restricting the choices that the user is allowed to make, or the designs focus on providing only the raw materials necessary for constructing the representations, leaving the user unrestricted but potentially unaided in searching out the desired representation.
The novel approach presented in this dissertation adapts a genetic algorithm to provide a means for an individual to search alternative visual representations in a systematic and manageable way. Any visual representation is a combination of elements, each selected from a different component. This approach encourages the individual’s creativity without restricting available choices, and leaves the task of bookkeeping to the computer. A computer-aided visualization system which is driven by the unique preferences of each user has been developed. The efficacy of this system, cogito, is demonstrated through a software user study. From an initial specification of components and elements, the system provides a wide variety of visual representations. From within this range of available visual representations, the user pursues the goal of achieving insight by applying personal criteria for effectiveness to the iterative selection and evaluation of candidate representations.
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.
Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique 'attractor' which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design of fractal objects is made relatively simple and intuitive by the discovery of an important mathematical property relating the fractal sets to the IFS. The method also provides the possibility of solving the inverse problem. Given the geometry of an object, determine an IFS that will (approximately) generate that geometry. This paper presents the application of the theory of IFS to geometric modeling.
Introduces interactive computer graphics through a description of hardware and a simple graphics package. Geometrical transformations and 3-D viewing are covered, followed by discussion of design architecture and raster operations. Concludes with chapters on shading models and colour applications. -R.Harris
