Ohrimenko+ / Barnsley. Superfractals

Figure 5.23 Example of a 2-variable fractal top rendered using colour-stealing. The picture from which the colours were stolen is shown inset. The attractors of the two IFSs that comprise the superIFS are illustrated at top left and top right.

Another example of fractal sets belonging to a 2-variable superfractal is illustrated in Figure 5.24. In this case M = 2 and the projective IFSs used are those in Tables 5.4 and 5.5. One of the goals of this example is to illustrate how closely similar images can be produced, with ‘random’ variations, so the two IFSs were chosen to be quite similar. Let us refer to images such as those in the bottom row of Figure 5.24 as ‘ti-trees’. Then each transformation maps approximately the unit square := {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}, in which each ti-tree lies, into itself. Both f21(x, y) and f22(x, y) map ti-trees to lower right branches of ti-trees. Both f11(x, y) and f12(x, y) map a ti-tree to a ti-tree minus the lower right branch. In this case, in place of computing successive pairs of sets we actually computed successive pairs of pictures (P1,k , P2,k ), for k = 1, 2, . . . , according to

(P1,k+1, P2,k+1) = f ak (P1,k , P2,k ),

where ak was chosen equal to a A with probability pa independently of all other choices, in the usual manner. Here P1,1 represents a green filled rectangle with

rounded corners and P2,1 is similar to P1,1 but purple. This illustrates how we may generalize the concept of random orbits of 1-variable pictures to V -variable pictures for V ≥ 1. It also illustrates the ‘texture effect’, discussed in Section 5.11; see for example Figure 5.20.

5.13 V -variable pictures with stolen colours, and V -variable orbital pictures

We assume here that the functions that comprise the IFSs in the superIFS are all one-to-one and thus invertible on their ranges. This assumption is always needed when we apply transformations to pictures.

The transformations f a : H(X)V → H(X)V for a A may be used to define, consistently with Equation (4.12.1),

fTOPa : {1,2,...,N } (X)V → {1,2,...,N } (X)V .

Let P = (P1, P2, . . . , PV ) {1,2,...,N } (X)V and let Dv denote the domain of Pv : Dv → {1,2,...,N } for v = 1, 2, . . . , V . P is a vector of picture functions whose colour values are points in code space. Then we define

f a (P1, P2, . . . , PV )

for each (P1, P2, . . . , PV ) {1,2,...,N } (X)V . To show what this notation means let us consider the picture function ((Nm2−1 + 1) f1m2 (Pv2,1 )). We start at the right

of the expression: f1m2 (Pv2,1 ) is the function whose domain is f1m2 (Dv2,1 ) and

whose value at x f1m2 (Dv2,1 ) is Pv2,1 (( f1m2 )−1(x)). This value is an element of the code space {1,2,...,N }, namely an infinite string of symbols from the alphabet

{1, 2, . . . , N }. Then the value of the picture function ((Nm2−1 + 1) f1m2 (Pv2,1 )) is obtained by putting the symbol for the number Nm2−1 + 1 at the beginning of this string. See also the discussion at the start of Section 5.9.

Notice that the domain of the function f a (P) is f a (D), where D =

(D1, D2, . . . , DV ).

Table 5.4 A projective IFS code. This is used in Figure 5.24

Table 5.5 A projective IFS code. This is used in Figure 5.24

T h e o r e m 5.13.1 Let the V -variable IFS F(V ) = {H(X)V ; f a , Pa , a A} be given, as above. Let FTOPa : {1,2,...,N } (X)V → {1,2,...,N } (X)V be defined as in Equation (5.13.1). If P1, P2 {1,2,...,N } (X)V have the same domain, D XV ,

P r o o f The proof is completely mechanical and follows the same lines as the proof of Theorem 4.12.1.

Theorem 5.13.1 tells us much less about the case V > 1 than Theorem 5.9.1 tells us about the case V = 1. But it does tell us just enough to be useful: that the ‘colours’ of any sequence of pictures which we obtain via the chaos game converge and that the sequence of pictures which is obtained depends asymptotically only on the domains of the initial pictures and not on their ‘colours’, namely their code space values. Of course, we also know that the domains of the pictures converge to V -variable fractal sets belonging to the appropriate superfractal.

We are certainly able to compute sequences of pictures of V -variable fractal sets, rendered using colour-stealing, that have a reasonable level of stability. So they clearly have applications to the creation of synthetic content and textures.

Figure 5.24 Some elements of a sequence of images that are converging towards 2-variable fractals. Convergence of the silhouttes, to within the numerical resolution, has occurred in the lower left and centre images. Note the subtle but real differences between the silhouettes of these two sets. A variant of the texture effect can also be seen: the purple points appear to dance forever on the green ti-trees, while the ti-trees dance forever on the superfractal.

New techniques in computer graphics are playing an increasingly important role in the digital-content creation industry, as evidenced by the succession of successes of computer-generated films. Part of the appeal of such films is the artistic quality of the graphics. It appears that V -variable fractals can provide, efficiently, new types of rendered digital imagery, significantly extending standard IFS graphics, as discussed for example in [27] and [8]. Figures 5.25–5.27 illustrate a few rendered 2-variable fractal sets to hint at the diversity of possibilities.

V -variable fractal sets may have applications to biological modelling. This theme is illustrated in Figure 5.28, which is a smaller version of Figure 0.5 in the Introduction. The top twelve pictures illustrate elements of a random sequence of 2-variable fractal sets belonging to a superfractal of fern-like sets associated with the two IFSs given in Tables 5.6 and 5.7. After approximately 130 iterations one of the IFSs was changed in a subtle way, see Figure 5.29, and the picture from which colours were stolen was switched.

We can imagine that two types of fern were growing close together a long time ago. They are distinguished by the amount of tilt in the main frond after the initial

Table 5.7 Another projective IFS code. This too is used in Figure 5.28

pair of fronds has been produced. This tilting is a function of the meristem and is supposed, in this story, to be activated by a gene which is switched on and off at each successive generation, up the fern and along the fronds, by factors that relate to the fern type. We can further imagine that the two types of fern interact by sharing their DNA in a 2-variable manner, resulting in numerous attempted new types of fern, each of which switches on and off the tilting mechanism at each level in the successive meristems according to its own 2-variable pattern. Further sharing of DNA maintains 2-variability. The resulting types strive for longer-term existence by being bountiful with their own spores. Sadly, none of them survives to tell the tale. But a random mutation of the gene led to a new sequence of trials. Perhaps a better informed botanist can tell a real story like this one?

5.14V -variable fractal interpolation

The technique of fractal interpolation, see for example [6], [11], [66] and [74], has many applications, including the modelling of speech signals, altitude maps in geophysics and stock-market indices. A simple version of this technique adapted to the V -variable setting is as follows. Let a set of real interpolation points

−∞ < x0 < x1 < · · · < xL < ∞

and a set of data

xl , ylm R2 for l = 0, 1, 2, . . . , L and m = 1, 2, . . . , M

be given, such that

y0m = y0 and yLm = yL

Figure 5.28 The top twelve pictures belong to a random sequence of 2-variable sets, starting from the set at the top left. After about 130 iterations, one of the IFSs in the superIFS was altered in a subtle manner and the picture from which colours were stolen was changed. The bottom four pictures show elements of the continuing random orbit, now moving onto a new superfractal. Perhaps this new species will be more successful?

are independent of m {1, 2, . . . , M}. We may or may not also require that ylm is independent of m, and say equal to yl , for all l {1, 2, . . . , L − 1} and m {1, 2, . . . , M}. We will suppose that L ≥ 2 and that M ≥ 1.

It is desired to find a superfractal of continuous functions fσ : [x0, xL ] → R such that fσ (x0) = y0 and fσ (xL ) = yL for all σ , where is an appropriate code space. It may also be desired, when the values ylm are independent of m for all l {1, 2, . . . , L} and all m {1, 2, . . . , M}, that fσ (xl ) = yl for all

Figure 5.29 Pictures of the attractors of the three IFSs which were used in connection with Figure 5.28. We may think of these attactors as representing three fern phenotypes.

l {1, 2, . . . , L − 1} and all σ . Furthermore, we require that Gσ = {(x, y) R2 : y = fσ (x)} is a V-variable fractal set for all σ , possibly with a specified Hausdorff dimension.

and dlm (−1, 1), for m {1, 2, . . . , M} and l {1, 2, . . . , L}. Then the superfractal A(V ) associated with the corresponding IFS F(V ) is a set of graphs of

functions with the desired properties and may be explored by means of the chaos game in the usual manner. The superfractal and the corresponding set of functions { fσ : σ } depend on the free parameters {dlm : m = 1, 2, . . . , M, l = 1, 2, . . . , L}, which may be used to control the distribution of fractal dimensions of the graphs or the Holder¨ exponents of the functions. Some 1-variable fractal interpolation functions are illustrated in Figure 5.8.

In the case M = 1 this situation reduces to standard affine fractal interpolation, as described in [9]. The Hausdorff dimension D = dimH G of the only graph G belonging to the superfractal is either equal to 1 or, in non-degenerate cases, to the