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

• Marc A. Berger
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 74)

## 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: ℝm: →ℝm, i=1,…, N; and it generates a discrete-time dynamical system (Xn) in ℝm according to
$$X_n = T_{\omega _N } X_{n - 1}$$
where (ωn) is an appropriately chosen sequence of indices ω ∈ {,…, N}. This sequence (ωn) 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 (Xn) 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
$$\mathcal{A} = \bigcup\limits_{{i = 1}}^{N} {{{T}_{i}}\mathcal{A}}$$

## Keywords

Image Compression Affine Transformation Iterate Function System Triangular Grid Eulerian Circuit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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