Fractal Top for Contractive and Non-Contractive Transformations

Fractal Top for Contractive and Non-Contractive Transformations

Sarika Jain (IT Department, Amity School of Engineering & Technology, Amity University, Noida, India), S. L. Singh (Department of Mathematics, Gurukula Kangri Vishwavidalaya, Haridwar, Uttarakhand, India) and S. N. Mishra (Department of Mathematics, Walter Sisulu University, Mthatha, South Africa)
Copyright: © 2012 |Pages: 17
DOI: 10.4018/ijalr.2012100104
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Abstract

Barnsley (2006) introduced the notion of a fractal top, which is an addressing function for the set attractor of an Iterated Function System (IFS). A fractal top is analogous to a set attractor as it is the fixed point of a contractive transformation. However, the definition of IFS is extended so that it works on the colour component as well as the spatial part of a picture. They can be used to colour-render pictures produced by fractal top and stealing colours from a natural picture. Barnsley has used the one-step feed- back process to compute the fractal top. In this paper, the authors introduce a two-step feedback process to compute fractal top for contractive and non-contractive transformations.
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1. Introduction

Points in a space, such as the real interval [0, 1] and the euclidean plane R2, are organized by means of a system of addresses or coordinates. For example, points in R2 may be addressed by ordered pairs of decimal expansions. Addresses are themselves members of certain types of spaces called code spaces (see, for instance, Devaney (1986, 1992), Jacquin (1992) and Parry (1966)).

A code space is a significant set, which consists of uncountable infinite points. Its importance lies in the fact that it can be embedded in a very small interval. The power of code space in the study of fractals is significant. It is used to give coordinates to the points of a self-similar set. The associated theory has potential applications in biological sciences, relating the biology and human anatomy with the topology and code space in mathematical sciences. For details regarding code space and its association with fractal generation, one may refer to Devaney (1986, 1992), Jacquin (1992) and Parry (1966).

It has a noticeable role in generating fractal tops of an Iterated Function System (IFS). Iterated Function Systems are basis of developing IFS Fractals. The IFS has the ability to create realistic images by encoding a complex natural scene. For a detailed description and applications of fractals, one may refer to Barnsley (1993, 2002, 2005, 2006, 2009), Barnsley and Barnsley (2004), Barnsley and Demko (1985), Barrallo and Jones (1999), Devaney (1986, 1992), Encarnacao et al. (1992), Hutchinson (1981), Jacquin (1992), Lapidus and Frankenhuysen (2004), Mandelbrot (1982) and Peitgen, Jürgens and Saupe (2004).

The study of fractal top is initiated by Barnsley (2005, 2006). It is an addressing function for a set attractor of an IFS, which assigns a unique largest address to each point of the set attractor. Until now, set and measure attractors were the only known fixed points of an IFS. However, it has recently been shown in Barnsley (2006, p. 352) that fractal top is the unique largest fixed point of a contractive transformation associated with certain code space.

Fractal top may be obtained by deterministic algorithms or by a variant of chaos game. This new operator not only works on the pixel co-ordinates of the picture but on the colour component as well. It defines a picture that is invariant under an IFS consisting of contractive transformations which appears to have theoretical as well as practical importance in mathematical sciences and simulation of natural objects.

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