A multiresolution algorithm to generate images of generalized fuzzy fractal attractors

Abstract

We provide a new algorithm to generate images of the generalized fuzzy fractal attractors described in [Str17]. We also provide some important results on the approximation of fractal operators to discrete subspaces with application to discrete versions of the deterministic algorithm for fractal image generation in the cases of IFS recovering the classical images from Barnsley et al., Fuzzy IFS from [Vrs92] and GIFS's from [Str16].

Fuzzy-set approach to invariant idempotent measures

“…It bases on the idea on approximating IFSs by discrete ones, defined on appropriately dense grids. See [COS20] and [COS21].…”

“…The picture of µ S produced by the algorithm DiscreteIFSIdempMeasureDraw(S) do not show much graphically because its density is equal to −∞ outside of the Cantor ser. However, the correspondent fuzzy set, that is, u S = Θ(µ S ), for θ(t) = 1.1 t , t ≤ 0, given on the left side of Figure 2, is indistinguishable from the right one, which was produced by the algorithm FuzzyIFSDraw(S) (see [COS21]), for the associated fuzzy IFS with admissible system of grey level maps r 1 (t) = θ(q 1 + θ −1 (t)) = t and r 2 (t) = θ(q 2 + θ −1 (t)) = 1.1 −1+ ln t ln(1.1) , as predicted by Theorem 5.4. As both approaches are equivalent, starting with the computation of u S by using the algorithm FuzzyIFSDraw(S), with a discretization with 1000 points and 15 iterations, we obtain again µ S (ϕ) = −0.999000 for µ S = Θ −1 (u S ).…”

“…In [OS17] we considered a fuzzy version of GIFSs, and in [COS21] we introduced the discrete algorithm for generating images of fuzzy GIFSs attractors. It is easy to rewrite algorithms DiscreteIFSIdempMeasureDraw(S) and DeterminIFSIdempMeasureDraw(S) for max plus normalized GIFSs (whose definition is analogous to the definition of max plus normalized IFSs, and is clear from the context).…”

“…Example 8.2. This example came from [COS21,Example 11.6]. The approximation of the discrete idempotent invariant measure, through DiscreteGIFSIdempMeasureDraw(S), is presented in Figure 6.…”

Making the computation of approximations of invariant measures and its attractors for IFS and GIFS, through the deterministic algorithm, tractable

“…As can be seen in [4,5,7] and [6,8] the discrete algorithms exhibit a good performance, but require a lot of technical detail to its implementation. On the other hand deterministic algorithms are easy to describe and implement, in its naive version, but they are impractical computationally.…”