Fractal Approximation of Sets (Barnsley's algorithm)

We can conveniently work in the unit square $S=[0,1] \times [0,1] \subset {{\rm I\!R}}^2$ $S=[0,1] \times [0,1] \subset {{\rm I\!R}}^2$ for the space X. The IFS under consideration is appropriately scaled to fit into S.

In practice, we are dealing with approximations of fractals rather than with fractals themselves. We work not in the space X but in its pixel representation $\tilde X$ and, therefore, we deal not with $({\cal H}(X),h)$but with $({\cal H}(\tilde X),h)$ and, correspondingly, with $\tilde F = (\tilde f_1, \dots, \tilde f_q)$ and $\tilde S$. In what follows under S, F and f we mean $\tilde S$, $\tilde F$and $\tilde f$, respectively.

One way to build a fractal [4], specified by IFS, looks as follows. We take an initial $A_0 \in {\cal H}(X)$ and define

$$A_{n+1}=F(A_n) \equiv \bigcup_{i=1}^q f_i(A_n), \hspace*{10mm} n=0, \dots, \infty.$$ (5)

The sequence $\{A_n\}_{n=0}^\infty$ converges to A_F in the Hausdorff metric. We do N iterations of F. When the number N of iterations becomes great enough, we have an approximation

$$F^N(A_0) \approx A_F.$$ (6)

Barnsley's deterministic algorithm [2] for building a fractal implies the calculation of each $f_k \in F$ for every point $s_i \in S$, $i \in \{1, \dots, \vert S\vert\}$ for every iteration. When we wish to rebuild the fractal, we have to repeat the whole above procedure.