Ohrimenko+ / Barnsley. Superfractals

Figure 2.1 There are many types of transformation, not just mathematical ones. For example we have the following definition: ‘transformation n. 2 Zool. a change of form at metamorphosis, esp. of insects, amphibia, etc.’ (The Concise Oxford Dictionary, Clarendon Press, Oxford, 1990)

We formulate invariance properties of sets, measures and pictures under transformation in terms of fixed-point properties of transformations acting on appropriate spaces. This motivates us in Section 2.4 to discuss fixed-point theorems and to add to our collection a new metric space (P(X), dP ) whose elements are measures. These concepts will be used in Chapters 4 and 5 to construct fractal sets, measures and pictures.

But the central question which we need to answer and which we pursue in this chapter is how, specifically, do Mobius¨ and projective transformations deform space and, consequently, change or leave unaltered aspects of sets, pictures and measures? How do these transformations not only affect the locations of points within a picture but also, when they act upon a picture treated as a measure, alter contrast and brightness? In order to understand these questions better and so be able to model images with fractals, we will explore the geometry of Mobius¨ and projective transformations in a detailed and specific way.

Since transformations on real spaces relate to transformations on code spaces, we conclude this chapter with a section on transformations on code space, our ‘meristem’ or ‘formative tissue’. The relationship between transformations on code spaces and transformations on sets, pictures and measures is a key theme of this book.

Another theme of this chapter is that sets, measures and pictures founded in R2 may be complicated but even so can have invariance properties under geometrically simple transformations. Such invariances can in principle be used to reduce the amount of information needed to describe apparently complicated sets, measures

and pictures. Until the end of this chapter, code space is off the stage while we develop the theme of transformations. Then in Chapter 3 we start to combine the two themes.

Structure of this chapter

In Section 2.2 we define pictures and digital pictures and explain the action of transformations upon them. You might now like to glance ahead at Figures 2.2–2.6 to get a feel for the content of this section. We conclude Section 2.2 by illustrating the concept of a picture that is invariant under a transformation.

We start Section 2.3 by explaining in an intuitive and visual manner what a measure is. Again you might like to glance ahead, at Figure 2.10. Then we introduce fields and σ -algebras of subsets of a space X. Upon these we define and construct measures, and we introduce the space P(X) of normalized Borel measures upon a metric space (X, d). Then we explain how continuous transformations act on measures and give examples of transformations acting upon pictures of measures. We conclude this section by explaining what it means for a measure to be invariant under a transformation.

Then in Section 2.4 we consider fixed points. When does a transformation f : H(X) → H(X) possess a fixed point? We are interested because a fixed point of f is a set that is unchanged when f is applied to it. We introduce a metric dP on P(X). In the right circumstances (P(X), dP ) is a compact metric space, another remarkable example of inheritance. Contraction mappings on (P(X), dP ) possess unique fixed points, yielding measures unchanged by transformations. Also, since often the space P(X) will be linear and convex, the Schauder–Tychenoff fixedpoint theorem applies and ensures the existence of invariant measures in broad circumstances. We will need these ideas in the later chapters.

After Sections 2.2, 2.3 and 2.4 we will be in a position to discuss the actions of Mobius¨ and projective transformations on sets, measures and pictures founded on R2. Actually, the underlying space upon which Mobius¨ transformations act is the Riemann sphere, which is equivalent to R2 {∞} where ∞ is an additional point called ‘the point at infinity’. The underlying space upon which projective transformations act is RP2, which is equivalent to R2 {L∞} where L∞ is an additional straight line, the ‘line at infinity’. To explain these transformations we need first to understand in a geometrical way how linear transformations in R2 and R3 behave. This is considered in Section 2.5. Here we assume a basic knowledge of linear spaces and linear transformations but include a brief review of two-dimensional linear algebra as a reminder and as a way of introducing our notation. The main result that we need is that an invertible linear transformation in R3 can always be expressed as the composition of rescalings along three perpendicular directions, a possible reflection and a rotation. This is very useful!

We describe Mobius¨ transformations in Section 2.6. Mobius¨ transformations can be represented by linear transformations in C2 and so may be expressed with eight real parameters, which explains why they are efficient carriers of information. What intrigues us is that they map the set of straight lines and circles into itself while at the same time preserving angles, yet at the same time they can effect huge distortions. This really is remarkable: how can the nature of R2 {∞} be such that this is possible? Realization of the nature of Mobius¨ transformations was a key idea behind the discovery of non-euclidean geometry, which eluded geometers for nearly two thousand years.

We describe projective transformations in Section 2.7. Any projective transformation can expressed in terms of a linear transformation in three dimensions and can be represented using nine real parameters. When you view a picture on a flat plane, such as the screen of a modern television or movie screen, from two different positions, the relationship between the two images upon the retina of one eye will be provided by a projective transformation. Indeed the actual differences between the images, the distortions in going from one to the other, can be quite extreme. But the vision system compensates for such projective transformations. This observation motivated mathematicians of an earlier era to study projective geometry most intently, to discover what it is, mathematically, that is left unchanged by projective transformations. We recall some of their results in this section. But our goal in later chapters is to exploit these transformations by using finite collections of them to describe completely certain sets, pictures and measures. We discuss some transformations on code space in Section 2.8.

2.2Transformations of pictures

Definition of a picture

D e f i n i t i o n 2.2.1 We define a picture function P to be a function

P : DP R2 → C,

where C is a colour space and DP is called the domain of the picture. The value of P(x) gives the colour of the picture at the point x DP. We denote the space of all pictures with colour space C by = C.

Throughout this book we usually suppose that the colour space C is a subset of R3 such as

C = [0, ∞)3 R3, C = [0, 255]3 R3 or C = {0, 1, . . . , 255}3 R3.

When C R3 the components of a point c = (c1, c2, c3) C may be called the colour components, with c1 named the red component, c2 named the green component and c3 named the blue component.

But there are other possibilities, corresponding to different models for images: for example c1 might represent intensity, c2 saturation and c3 hue, with appropriate ranges of values. The points in the space C could be simply one dimensional, corresponding to intensities in greyscale pictures. Or they may have more than three dimensions. For example, in applications to the high-quality image printing industry a four-dimensional colour space is used, whose axes are cyan, magenta, yellow and black.

There are diverse possible choices for the domain DP of the picture function P; it may be a line segment, a curve, an open ball, a closed rectangle or any other subset of R2. It may represent the region that is yellow in a watercolour of a flower, the part of a piece of photographic paper on which a photo has been developed, the retina of your eye, the painted region of an artist’s canvas, the screen of a computer or a patch of vision in your mind’s eye.

In some cases we assume that

DP = = {(x, y) R2 : xL ≤ x ≤ xH , yL ≤ y ≤ yH }

where (xL , yL ) R2 is called the lower left corner and (xH , yH ) R2 is called the upper right corner of the (domain of the) picture. In the absence of other information, for mathematical purposes we take (xL , yL ) = (0, 0) and (xH , yH ) = (1, 1).

The domain of a picture function is an important part of its definition. The characteristic function of a subset S R2,

may be treated as a picture function that represents S. Another representation of S is provided by the picture function PS with domain S and constant value, say PS (x) = 1, for all x S. With the aid of χS or PS we can embed classical geometrical objects such as circles, lines or triangles in the space of picture functions.

We will usually refer to a picture function as a picture; our intention is that it should be clear from the context whether we mean a picture function or just a picture, as on the pages in this book. We refer to a picture as a geometer might refer to a triangle, meaning either a concrete image or the abstract mathematical entity.

Pictures are available to us in various forms. They may be defined explicitly, in much the same way as a parabola or a sphere is defined, by reference to mathematical algorithms and formulas, including for example the kinds of expressions produced by and interpreted by computer graphics software. They may be piecewise constant functions, such as digital pictures (see below), defined using arrays of data obtained from devices such as scanners and digital cameras that focus, sample, filter and interpolate real-world scenes. But for us they are always, in the end, mathematical entities.

We have defined pictures as having their domains in R2. But it is easy to see how this definition may be extended to pictures with domains in other fundamentally two-dimensional spaces such as a spherical shell or a projective plane.

Transformations of pictures

Let f : D f R2 → R2 be a one-to-one transformation and let P : DP R2 → C be a picture with DP D f . Then we define

f (P) : D f (P) R2 → R2

to be the picture P transformed by f , or equivalently, the transformation f applied to the picture P, where

D f (P) = f (DP)

and

f (P)(x) = P( f −1(x)) for all x D f (P).

We also denote f (P) by f ◦ P. Note that when f : R2 → R2 the picture f (P) is always well defined and f : → . Figure 2.2 shows the pictures produced when three different transformations f1, f2, f3 : R2 → R2 are applied to P, a picture of a fish.

E x e r c i s e 2.2.2 Why have we restricted transformations of pictures to being one-to-one?

E x e r c i s e 2.2.3 Where in the real world do we see interesting transformations of pictures? Some sources are mirrors, uneven glass, the distortions produced by water in a fish tank or by hot rising air, and the reflections in shiny metal surfaces such as the surface of a ball bearing. Name some other sources.

Invariant sets and pictures

A set S X is said to be invariant under a transformation f : X → X iff

f −1(S) = S.

We will refer to such a set S as an invariant set of the transformation f . Note that this implies

f (S) = S,

but the converse is not true unless f is one-to-one.

Figure 2.2 A picture function P representing a fish, and three different Mobius¨ transformations of it, f1 ◦ P, f2 ◦ P and f3 ◦ P.

Similarly, a picture P is said to be invariant under a one-to-one transformation f : R2 → R2 iff

f (P) = P.

We will refer to such a picture P as an invariant picture of the transformation f .

For example, Figure 2.3 shows a picture that is invariant under the transformation defined by f (x, y) = (−x, y) and Figure 2.4 shows a picture that is invariant under the transformation f (x, y) = (x, −y). Figure 2.5 illustrates a set and a picture that are invariant under the same transformation Rθ : R2 → R2, a clockwise rotation through θ = 36◦.

Figure 2.5 Both pictures here are invariant under the rotation transformation R36◦ . The left-hand picture also represents an invariant set.

There are many instances of sets and pictures that are invariant under transformations. In graphic design and art the transformations under which a picture is invariant may be referred to as its symmetries. Wallpaper pictures, pictures of flowers and architectural motifs may be invariant under translational and/or rotational transformations.

As a more complicated example, Figure 2.6 illustrates a set S R2 that is invariant under the Mobius¨ transformation (Section 2.6) M = Mρ ◦ Rθ ◦ Mρ , where Mρ : C → C is defined by

for values of ρ > 1. This transformation obeys Mρ (0) = 0 and Mρ (1) = 1. The

a rotation through angle θ about the origin.

An even more complicated example of an invariant picture is illustrated in

Elaborate sets and pictures that are invariant under simple transformations, those whose formulas may be described explicitly in a succinct manner involving less than say sixteen free parameters, can be produced in various ways. In Chapter 3 we show how new pictures generated by groups of simple transformations, such as Mobius¨ transformations and projective transformations, acting on a given picture

may be used to define invariant pictures. Such families of transformations may be produced by autonomous differential equations that model physical systems. Indeed, phase portraits associated with autonomous systems in two dimensions can be thought of as invariant sets for appropriate transformations.

E x e r c i s e 2.2.4 Show that if g : R2 → R2 is invertible and if the picture P is invariant under the rotation Rθ then the picture P := g(P) is invariant under the transformation g := g ◦ Rθ ◦ g−1.

E x e r c i s e 2.2.5 Define a transformation f : {0,1,2} → {0,1,2} by the expression f (σ1σ2σ3 · · · ) = 2σ1σ2σ3 · · · for all σ = σ1σ2σ3 · · · {0,1,2}. Find an invariant set for f .

Digital pictures

Let W, H N = {1, 2, 3, . . . }. Suppose that C is a discrete space, such as {0, 1, . . . , 255}3, and that the picture function P : R2 → C is constant on each rectangular region in a W × H array of rectangular regions w,h , each of the same width and height,

( w,h ) := { w,h : w = 1, 2, . . . , W ; h = 1, 2, . . . , H }

such that

= w,h

and

w,h ∩ w ,h = whenever (w, h) =(w , h ).

Then P is called a digital picture.

We will suppose that the array of rectangles ( w,h ) is organized similarly to the elements of a matrix but flipped and transposed, so that 1,1 is in the lower left corner of and W,H is in the upper right corner of , as illustrated in Figure 2.8. Each rectangle may be open, closed or partly open and partly closed, as indicated. We may write w,W,hH to denote w,h more precisely. Exercise 1.9.5 provides a canonical set of choices for w,W,hH .

The picture Pw,h : w,h R2 → C, where Pw,h is the restriction of the digital picture P to the rectangle w,h , is called a pixel function or, simply, a pixel. The constant value of Pw,h (x) C for x w,h is called the colour of the pixel Pw,h .

We will denote a typical digital image, as described here, by PW ×H and a typical pixel as Pw,h or more specifically as Pw,W,hH . We call W the width of the digital image and H the height of the digital image, in pixel units. We refer to min{W, H } as the resolution of the digital picture PW ×H .

Let f : R2 → R2 and let PW ×H be a digital picture. Then in general f ◦ PW ×H is not a digital picture. So the set of digital images is not mapped into itself under