A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy

Details A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy

2003-12-09

arxiv.org/abs/math/0312187v1

Fractals, 13, No. 2, 111-146 (2005)

Mathematics

Probability

51 pages, 31 figures

Scientific paper

We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" : at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example $\mathbb{R}^{2}$, and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

Affiliated with

Also associated with

No associations

Rating

A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Fractal Valued Random Iteration Algorithm and Fractal Hierarchy will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-41512