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Products of random matrices and computer image generation. (English) Zbl 0601.60066

Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 173-182 (1986).
[For the entire collection see Zbl 0581.00014.]
Following the discovery of the fractals by B. B. Mandelbrot [Fractal geometry of nature. (1982; Zbl 0504.28001)] J. Hutchinson [Fractals and self-similarity. Indiana Univ. Math. J. 30, 713-747 (1981)] gave a mathematical framework to study them. The authors of this paper observed that this method can be used for computer generation of pictures but further mathematical tools had to be developed. The paper summarizes these results. A Markov chain is defined by assigning probabilities to affine transformations in \(R^ d\) and properties of the stationary distribution are explored. Next, the inverse fractal problem is solved by presenting recursive formulas for the moments connected with the stationary distribution. Some open problems are also formulated.
Reviewer: A.Prekopa

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K99 Special processes
68T99 Artificial intelligence