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.

Once we select the graph model of a network, various algorithms can be used to efficiently design and analyze a network architecture. Some of the most fundamental algorithms among them are finding trees in a graph with minimum cost (where cost is defined appropriately) or finding a minimum spanning tree, visiting nodes of a tree in a specific order, finding connected components of a graph, finding the shortest paths from a node to another node, from a node to all nodes, and from all nodes to all nodes in a distributed or centralized fashion, and assigning flows on various links for a given traffic matrix.

In the following we describe some useful graph algorithms that are important in network design. Recall that N represents the number of nodes and M represents the number of links in the graph.

Shortest-path routing

Shortest-path routing, as the name suggests, finds a path of the shortest length in the network from a source to a destination. This path may be computed statically for the given graph regardless of the resources being used (or assuming that all resources are available to set up that path). In that case, if at a given moment all resources on that path are in use then the request to set a path between the given pair is blocked. On the other hand, the path may be computed for the graph of available resources. This will be a reduced graph that is obtained after removing all the links and the nodes that may be busy at the time of computing from the original graph.

Technological advances in semiconductor products have essentially been the primary driver for the growth of networking that led to improvements and simplification in the long-distance communication infrastructure in the twentieth century. Two major networks of networks, the public switched telephone network (PSTN) and the Internet and Internet II, exist today. The PSTN, a low-delay, fixed-bandwidth network of networks based on the circuit switching principle, provides a very high quality of service (QoS) for large-scale, advanced voice services. The Internet provides very flexible data services such as e-mail and access to the World Wide Web. Packet-switched internet protocol (IP) networks are replacing the electronic-switched, connection-oriented networks of the past century. For example, the Internet is primarily based on packet switching. It is a variable-delay, variable-bandwidth network that provides no guarantee on the quality of service in its initial phase. However, the Internet traffic volume has grown considerably over the last decade. Data traffic now exceeds voice traffic. Various methods have evolved to provide high levels of QoS on packet networks – particularly for voice and other real-time services. Further advances in the area of telecommunications over the last half a century have enabled the communication networks to see the light. Over the 1980s and 1990s, research into optical fibers and their applications in networking revolutionized the communications industry. Current telecommunication transmission lines employ light signals to carry data over guided channels, called optical fibers. The transmission of signals that travel at the speed of light is not new and has been in existence in the form of radio broadcasts for several decades.