IFS Algorithms for Wavelet Transforms, Curves and Surfaces, and Image Compression

Abstract

This report concerns research topics discussed while the author was at the Institute Mauro Picone (June 25–28, 1990), under the sponsorship of IAC-CNR. These topics involve applications of affine iterated function systems (IFS) [1]. An affine IFS consists of affine transformations T _i: ℝ^m: →ℝ^m, i=1,…, N; and it generates a discrete-time dynamical system (X_n) in ℝ^m according to

$$ X_n = T_{\omega _N } X_{n - 1} $$

where (ω_n) is an appropriately chosen sequence of indices ω ∈ {,…, N}. This sequence (ω_n) is said to drive the dynamics. In most IFS applications it is an i.i.d. sequence

$$\begin{array}{*{20}{c}} {\mathbb{P}(\omega = i) = {{p}_{i}} > 0,} & {i = 1, \ldots ,N} \\ \end{array}$$

where the p _i’s are pre-assigned weights. When the transformations T _i are strictly contractive, the IFS process (X_n) is a recurrent Markov chain, and its orbit is dense in the attractor A for the IFS, with probability one. The attractor is the unique non-empty compact set satisfying

$$\mathcal{A} = \bigcup\limits_{{i = 1}}^{N} {{{T}_{i}}\mathcal{A}}$$