Ohrimenko+ / Barnsley. Superfractals

Figure 3.55 Example of an IFS semigroup tiling of a picture. The domain of the picture is the complement of a Sierpinski triangle in the space X = .

while IFS#1 consists of the first and third transformations and IFS#3 consists of the second and fourth transformations.

E x e r c i s e 3.5.32 Find the IFSs used to make the middle and bottom picture tilings in Figure 3.56. The domains of these two orbital pictures are examples of reptiles, namely attractors of IFSs that can be used to tile R2; see for example [40] or simply type ‘fractal reptiles’ into your favourite internet search engine.

It is useful, for applications such as image compression, to think of an orbital picture of finite diversity as a kind of tiling that we call a panelling. In a panelling the ‘tiles’ are panels as illustrated in Figure 3.57, where we contrast picture tilings with panellings. An orbital picture of finite diversity is always a panelling, but may be a tiling only if the diversity is 1. When an orbital picture is a panelling, it can

Figure 3.57 Examples of panellings of orbital pictures. (i) A panelling of diversity 2. (ii), (iii) Panellings of diversity 1 that are also IFS semigroup tilings. (iv) A panelling of diversity 7. Can you think of a panelling of diversity 1 that is not an IFS semigroup tiling?

be constructed from the set of tiles obtained by applying the IFS semigroup to a finite set of condensation pictures, the elements of the space of limiting pictures. When | P0 | < ∞, the orbital picture is a panelling that consists of finitely many tiles, as illustrated in Figure 3.57(iii).

Figure 3.51 illustrates what appears to be an example of a panelling where | = ∞ and the diversity equals 1, yet this is not a picture tiling since some transforms of the square leaf-tile P0 overlap one another. We say ‘appears’ because we have not ruled out that some other IFS semigroup of transformations applied

to P0 could achieve the same orbital picture with no overlaps.

As in the case of IFS semigroup tilings of sets, pictures that are picture tilings can be represented with some efficiency. So how may we find P0 such that O(P0) is a picture tiling? One approach is to look for a set C such that O(C) is a tiling, as in Theorem 3.4.6, then choose P0 so that DP0 = C, again as in Theorem 3.4.6. Another approach is to look underneath O(P0), as will be discussed in the following subsection.

Underneath the orbital picture

Given an IFS semigroup S{ f1, f2,..., fN } and a condensation picture P0 we define an underneath picture to be a picture belonging to the sequence {Pn }∞n=1, where

P = fσ (n)(P0) fσ (n−1)(P0) · · · f1(P0) P0 = fσ (n)(P0) P − n n 1

and P0 := P0. Some examples of underneath pictures are shown in Figure 3.58.

Figure 3.58 The underneath pictures of the four orbital pictures in Figure 3.33. We have flipped the figure horizontally, so you can more easily imagine that you have turned Figure 3.33 upside down.

Notice that the underneath picture Pn is obtained from Pn−1 by putting fσ (n)(P0) on top of Pn−1. As a consequence the sequence of underneath pictures does not converge in general. Consider the case of a contractive IFS whose set attractor possesses a nonempty interior and let x be a point in this interior. Assume too that DP0 possesses a nonempty interior. Then for infinitely many values of n, say n = nk for k = 1, 2, . . . , it will occur that x belongs to the domain of fσ (nk )(P0). Furthermore, it can clearly occur that the sequence of colours { fσ (nk )(P0)(x)}∞k=1 does not converge. This is illustrated in the bottom row of underneath pictures in Figure 3.26. In this case the set attractor of the IFS is the leaf-shaped region, with a grainy yellow and green pattern, in the bottom right image. We actually computed many more pictures in this sequence of underneath pictures, and found that the parts of the pictures whose domains intersect the

attractor of the IFS seemed never to settle down; a restless sequence of beautiful textures was observed. We will show, in Section 4.8, that a model explanation for this effect, which we call the texture effect, lies with the ergodic theorem. It thus appears that a novel application of underneath pictures, and in particular of the ergodic theorem, is to the production of rich textures for computer graphics applications.

Although the sequence {Pn }∞n=1 does not generally define a limiting picture it sometimes does. For example, the picture tiling in Figure 3.3 is of this type. In this case the domain of the condensation picture and the images of this domain under the IFS semigroup do not intersect the attractor of the IFS. The same situation occurs in the top right and bottom left images in Figure 3.58.

E x e r c i s e 3.5.34 Suppose that you are given a picture P and two transformations f1, f2 : R2 → R2, and you know that P represents the orbit of a picture P0 under the IFS semigroup S{ f1, f2}(R2). How would you find P0? Now suppose that you do not know f1 or f2. What can you say now? Suppose for example that you know that f1 and f2 are similitudes, but that is all. Can you design an algorithm, some sort of iterative procedure, to find f1 and f2?

There are clearly many pictures that we can associate with an orbit of pictures, in addition to the orbital picture and the underneath pictures. An interesting family of such pictures is provided by the tops semigroup generated by the infinite set of pictures {Pn = fσ (n)(P0) : n = 0, 1, 2, . . . }; see Section 3.2. It may be explored by a random iteration similar to that in Section 3.2 but using infinitely many pictures instead of finitely many. One may, for example, associate the probability

E x e r c i s e 3.5.35 Show that

pσ1 pσ2 . . . pσ|σ | = 1,

σ{1,2,··· ,N }

where the pσn are non-negative numbers such that p0 + p1 + · · · + pN = 1.

E x e r c i s e 3.5.36 As a special project choose a simple IFS semigroup and an interesting condensation picture P0 and explore the associated tops semigroup mentioned in the last paragraph above, using random iteration. How does the look and ‘feel’ of the pictures that you obtain change when the probabilities are altered?

The Henon transformation

Up to this point we have illustrated orbital pictures generated by IFS semigroups made of quite simple transformations, such as projective and Mobius¨

Figure 3.59 Some elements of an orbit of a picture P0 of a ‘Morning Glory’ flower, shown in the leftmost panel, under the IFS semigroup generated by the Henon transformation, Equation (3.5.19), with a = 1.0001 and c = 0.45. The lower left corner has coordinates (−1.5, −2.0) while the upper right corner has coordinates (1.7, 0.8). Where are the flowers going? See Figure 3.60.

transformations. Although most of the theory of orbital pictures is more generally applicable, we have emphasized pictures associated with contractive IFSs. So it is worthwhile here to consider briefly an example that involves a much more complicated mapping, namely the Henon transformation, which has been much studied from a dynamical systems point of view, [45]. Our goal is to emphasize the generality of the theory of orbital pictures and to illustrate that even with only one transformation there may be very complicated structure and hugely deformed picture tiles. In so doing we contact standard dynamical systems theory from the novel point of view of orbital pictures.

Orbits of dynamical systems, that is, of IFS semigroups generated by a single transformation, may lie on or be attracted to geometrically complicated structures called strange attractors, often by dint of a certain level of complication in the single underlying transformation. For example, consider the semigroup generated

where a and c are real numbers; for example a = 1.4 and c = 0.3. This transformation stretches and bends pictures upon which it acts, as illustrated in Figure 3.59. Figure 3.59 illustrates from left to right the pictures

P0, P0 fH enon (P0) and P0 fH enon (P0) fH◦2enon (P0),

where P0 is a picture of a ‘Morning Glory’ flower. It is seen that fH enon moves some pairs of points further apart while moving other pairs closer together. This behaviour contrasts with that in Figure 3.3, where the underlying transformation moves all pairs of points closer together. All orbits of points under the latter transformation converge to a single point. But some orbits of the Henon transformation are much more complicated: Figure 3.60 shows a plot of one million points of the orbit of the point (0.5, 0.5), outlining the structure of an associated attractor, defined below.

Figure 3.61 Three underneath pictures associated with the IFS semigroup generated by fHenon−1 when a = 1.4 and c = 0.3 in Equation (3.5.19). The successive flowers in the orbit of the condensation picture

are being drawn towards a repeller of fHenon.

T h e o r e m 3.5.40 Let A be an attractor of an invertible dynamical system f : X → X and let V be a neighbourhood of A such that f (V ) V . Let P0 be a condensation picture, with compact domain DP0 such that DP0 V \ A. Let P be the orbital picture generated by the IFS semigroup S{ f }(X) acting on P0. Then P is either a picture tiling or has diversity 2.

is layered.

It now follows from Theorem 3.4.6, wherein we take C = DP0 , that DP0 is a semigroup tiling of P = O(DP0 ) iff DP0 ∩ f (DP0 ) = . Hence, by Definition 3.5.31, O(P0) is a picture tiling iff DP0 ∩ f (DP0 ) = .

So, suppose that O(P0) is not a picture tiling; then DP0 ∩ f (DP0 ) = . Consequently, C0 = D f (P0)\DP0 must be nonempty and hence Q1 is a panel distinct from P0. The orbital picture generated by Q1 is a tiling since it is layered and, as can be readily checked, DQ1 ∩ f (DQ1 ) = . Moreover, O(Q1) and P0 are disjoint pictures and O(P0) = P0 O(Q1). Hence, the space of limiting pictures contains exactly two distinct elements, P0 and f −1(Q1).

Applications of orbital pictures

Here we speculate briefly on possible applications of orbital pictures.

Orbital pictures have obvious applications to computer graphics. Indeed, many standard pictures of fractals, such as Julia sets surrounded by ribbons of colour,

278 Semigroups on sets, measures and pictures

and tree-like sets where each branch looks like a small copy of the trunk, may be interpreted as orbital pictures. From computational experiments it appears that some orbital pictures vary smoothly in appearance when the condensation picture and the IFS are varied, so it is clearly possible to design attractive-looking pictures using orbital pictures. To make a fresh-looking advertisement on the internet one might use an IFS whose set attractor is in the shape of a corporate logo with a condensation set that is a picture of a brand of the company and then adjust parameters to animate the resulting orbital picture. Orbital pictures may also be used to generate intricate textures and patterns that may be wrapped around wireframe models to fill in backgrounds in synthetic imagery. Some related ideas are explored in [95].

In some cases it may be possible to decompose a picture approximately into a tops union of orbital pictures. In turn the condensation pictures may themselves be approximated by orbital pictures in the same manner. If such a recursive decomposition is possible, with some stability, then a new type of method for image approximation and compression would result, distinct from block-based fractal image compression, as described in [53], [12] and [38] for example.

The code space of an orbital picture and the diversity or growth rate of diversity of an orbital picture are parameters that may be applied to the problem of classifying real-world pictures and textures. Quantities which one might associate with realworld pictures such as photographs and which are based on these types of ideas would be invariant under homeomorphism. Such quantities would be of a character altogether different from those based on fractal dimension, which are invariant under transformations that provide equivalent metrics but are not robust against more ferocious transformations.

For example, one may wish to compare pictures of leaves of different plants. The boundaries of the leaves could have different experimental fractal dimensions yet the pictures might be well described by equivalent orbital pictures. In such a case one might define an empirical diversity and use it to classify and compare the leaves.

The applications of orbital pictures to biological modelling may be considered as refinements of approaches, already used with some success, based on the ideas of Lindemeyer; see for example [54] and [80]. It is in the non-commutative interaction, via the tops union, of the images of the condensation picture under the semigroup that enriches the approach via orbital pictures; this interaction reminds me of the way in which, in the expression of a genetic code, some genes become active only in certain circumstances.

For biological modelling applications it is interesting to apply what we call orbit-stealing. Let two related IFS semigroups S{ f1, f2,... f N } (X) and S{ f 1, f 2,... f N } (X) and two pictures P0 and Q0 (X) be given. Use P0 to construct P(P0) and, in