Map-colour Rendering of Ifs Fractals

In this paper a simple scheme of map colour is introduced for the colour rendering of IFS fractals. The way of incorporating it into Barnsley's random iteration and versions of Hutchinson's deterministic algorithm is detailed. Examples are presented showing an elegant stylised colouring of such fractals as the Barnsley Fern and the Sierpinski gasket that provide attractive motifs for visualization purposes. ITERATED FUNCTION SYSTEM FRACTALS The category of fractals we are concerned with is that of deterministic fractals mathematically specified as the 'attractor' of a set S of N contraction maps such as W i (i =0..N-1) : with S = {W 0 ,W 1 ,W 2 ,W 3 ,W 4 ,W 5 ,....W N-1 }. See {1][2][3][4][5]. For the case where the contraction maps are affine transformations, as per W       x y =       a b c d       x y +       e f with |ad -bc| < 1. where (x,y) is the co-ordinate pair of a pixel in the plane. The nomenclature was introduced by Barnsley et al [3][4] of referring to the set of transformations as an IFS [Iterated Function System], although in his recent book [5] Barnsley also uses the term IFS to refer to general sets of contraction maps. In this paper we describe formally a mapping scheme called map colour, and first give a global appreciation of how it serves to map subsets of an attractor. Then we present practical schemes for applying this colour rendering in conjunction with various algorithms for generating IFS fractals. MAP COLOUR SYSTEM GLOBAL PERSPECTIVE In this section a mathematical perspective of the IFS set colouring is outlined that depends on attributing a number colour, colour(r) to map W r from the IFS Set. Formally the attractor A(S) , is the (closure of the) set of points invariant under any finite composition of maps from S, This implies that W r [A(S) ]  A(S) So that  A(S) may be expressed as the union W 1 [A(S) ]W 2 [A(S) ]... W N [A(S) ] The simplest version of the fractal colouring scheme to be introduced involves associating a colour with each mapping, so that each subset of the attractor has a distinct colour: W r [A(S) ] has colour (r). Where the W r [A(S) ] are disjoint the colouring so induced is unique. In general, however, these sub-sets are not disjoint, and we specify that a given pixel is given a colour depending on a ranking of the possibly more than one colour that can be attributed to it. The above colouring scheme is here termed one-level colour. This is just the start of the colouring scheme. With more elaborate formalism, the attractor A(S) may be expressed as the union  r,s W r [W s [A(S) ]] To W r [W s [A(S) ]] we ascribe a two-level colour according to the formula colour(r,s) = N* colour(r) + colour(s)