Introduction

In this diaper we introduce the theory and some applications of superfractals. Superfractals are families of sometimes beautiful fractal objects which can be explored by means of the chaos game (see Figure 5.1) and which span the gap between fully ‘random’ fractal objects and deterministic fractal objects. Our presentation is via elementary examples and theory together with brief descriptions of natural feasible extensions. This chapter depends heavily on the earlier material. You may grasp intuitively the key ideas of superfractals and ‘2-variability’ by studying the experiment described in Section 5.2. But be careful notto miss subtleties such as those that enable the construction of superfractals whose elements are vast collections of homeomorphic pictures, as for example those illustrated in Figures 5.13 and 5.17.

A superfractal (see Figure 5.2) is associated with a single underlying hyperbolic IFS. It has its own underlying logical structure, called the ‘V-variability’ of the superfractal, for some V ∈{1, 2,…}, which enables us to sample the superfractal by means of the chaos game and produce generalized fractal objects such as fractal sets, pictures, measures and so on, one after another. The property of V-variability enables us to ‘dance on the superfractal’, sometimes producing wondrous objects in splendid succession.

Introduction

Any picture may be conceived as a mathematical object, lying on part of the euclidean plane, each point having its own colour. Then it is a strange and wonderful entity. It is mysterious, for you probably cannot see it. And worse, you cannot even describe it in the type of language with which you normally talk about objects you can see; at least, not without making a lot of assumptions. But we want to be able to see, to describe and to make pictures on paper of fractals and other mathematical objects that we feel ought to be capable of representation as pictures. We want to make mathematical models for real-world images, biological entities such as leaves and many other types of data. To be able to do this we need certain parts of the language of mathematics, related to set theory, metric spaces and topology.

Code space There is a remarkable set, called a code space, which consists of an uncountable infinity of points and which can be embedded in the tiniest real interval. A code space can be reorganized in an endless variety of amazing geometrical, topological, ways, to form sets that look like leaves, ferns, cells, flowers and so on. For this reason we think of a code space as being somehow protoplasmic, plastic, impressionable and capable of diverse re-expressions, like the meristem of a plant; see Figure 1.1. This idea is a theme of this chapter and of the whole book.

This book is about tomography, which is a way to see what is inside an object without opening it up. If you are intrigued with this idea, then, no matter what your background, you will find that at least some portion of this book will provide interesting reading. If this idea is not intriguing, then I would recommend some other publication for your reading pleasure.

The unifying idea of tomography is the Radon transform, which is introduced in an informal and graphic way in chapter 1. Reading chapter 1 will give you a good idea of the precise meaning of tomography. Reading chapter 2 will give you a very good idea of the meaning of tomography and if you read the last few chapters you will have a really good understanding of this idea. However, some of the later chapters will only be accessible to specialists.

I tried to write this book with two main ideas in mind. I wanted it to appeal to the broadest possible group of readers and I wanted it to be as comprehensive as possible. Therefore, chapter 1 has almost no mathematics in it – at least it does not require the reader to have any background beyond a good course in secondary school mathematics. CT (computerized tomography) scanners are used for medical diagnosis and produce detailed pictures of the human anatomy without opening up the patient. The dedicated reader will learn, in a very graphic way, how a CT scanner works.

Introduction

There are two main ways of generalizing the Radon transform. One can add a weight to the hyperplane integral or one can integrate over subvarieties that are not linear. The first idea leads to the generalized Radon transform in ℝn that we study in section 5.4. An example of a generalized Radon transform is the attenuated Radon transform. This transform is of practical importance because it models single-photon emission computed tomography (SPECT). We investigate the attenuated Radon transform in section 5.3 along with the exponential Radon transform that is also used in SPECT. In this section we also mention positron emission tomography (PET), although this version of tomography reduces to the standard Radon transform.

Another way of generalizing the Radon transform is to integrate over more general submanifolds of ℝn than hyperplanes and k planes. More generally, we can integrate over submanifolds of a manifold X that is more general than ℝn. For example, Funk in his 1916 paper [184] showed that an even function on a sphere is completely determined by its integrals over great circles. This was a year before Radon's paper [508] appeared. The transform that sends a function on the unit sphere to its integrals over great circles is now called the Funk transform or the spherical Radon transform. A major advance in studying Radon transforms over nonlinear varieties was created in 1964 by Helgason [267] and we introduce this idea in section 5.5.

The divergent beam transform, a relative of the Radon transform is introduced and studied in section 5.2.

Introduction

Let us define a k plane to be any translation of a k-dimensional subspace of ℝn. Therefore, a k plane has the form η + x, where η is a k-dimensional subspace and x ∈ ℝn. Note that a hyperplane is therefore an (n – 1) plane.

The Radon transform can be generalized so that the integration is performed on k planes instead of hyperplanes. The related transform is called a k-dimensional Radon transform or a k-plane transform. Some authors use the term Radon–John transform. We use the terms synonymously, and in this chapter we develop the theory of these transforms.

The main part of this chapter begins in section 3.3 with an investigation of the set of all k-dimensional linear subspaces of ℝn. This set is called the Grassmannian and is denoted by Gk,n. Grassmannians are not only sets, but they are also manifolds and measure spaces. We do not require the manifold structure, but we do need to know how to define a suitable measure on Grassmannians. This is done by introducing homogeneous spaces and Haar measure.

Once we have Grassmannians, it is easy to describe the set of all k planes and integration on k planes. This leads to the definition of the k-plane transform and its adjoint. We study the basic properties of the k-plane transform in sections 3.4 and 3.5.

An inversion formula for the k-plane transform is of great interest. We provide four main approaches to the inversion of the k-plane transform.

Introduction

The purpose of this chapter is to give an informal introduction to the subject of tomography. There are very few mathematical requirements for this chapter, so readers who are not specialists in the field, indeed who are not mathematicians or scientists, should find this material accessible and interesting. Specialists will find a graphic and intuitive presentation of the Radon transform and its approximate inversion.

Tomography is concerned with solving problems such as the following. Suppose that we are given an object but can only see its surface. Could we determine the nature of the object without cutting it open? In 1917 an Austrian mathematician named Johann Radon showed that this could be done provided the total density of every line through the object were known. We can think of the density of an object at a specific point as the amount of material comprising the object at that point. The total density along a line is the sum of the individual densities or amounts of material.

In 1895 Wilhelm Roengten discovered x-rays, a property of which is their determining of the total density of an object along their line of travel. For this reason, mathematicians call the total density an x-ray projection. It is immaterial whether the x-ray projection was obtained via x-rays or by some other method; we still call the resulting total density an x-ray projection.