A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY

By Michael F. Barnsley, John E. Hutchinson, Orjan Stenflo. 2003

62 citations recorded by the source corpus.

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 "variabil- ity": 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 R2, and the transformations are compu- tationally 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.

Subfields: Artificial Intelligence, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f2632b56cefb99d795126b12b0d048120cec50b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Orjan Stenflo
2. Michael F. Barnsley
3. John E. Hutchinson

In Condensed Matter Physics

1. Orjan Stenflo
2. Michael F. Barnsley
3. John E. Hutchinson
4. Marco A. M. Guaraco

In Mathematical Physics

1. Orjan Stenflo
2. Michael F. Barnsley
3. John E. Hutchinson

Citations in the corpus

• V -variable fractals: Fractals with partial self similarity ✩ (2008, methodology): “The following theorem includes and strengthens Theorems 15‐24 from [ BHS05 ].”
• A survey of average contractive iterated function systems (2012, background): “[6], [7] and [8] for more on the theory of V-variable fractals.”
• Developments in fractal geometry (2013, unknown): “Another topic not discussed is V -variable fractals and superfractals [16,26– 28].”