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

A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY