**1. Introduction**

The visible light being reflected by the majority of the objects which surround us, we can apprehend our environment only by the properties of the surface of the objects which com‐ pose it. To exceed this limit and to explore the intimacy of the matter, a dedicated imaging techniques and instruments, which used penetrating radiations like X-rays, neutrons, gam‐ ma, or certain electromagnetic or acoustic rays to explore internal structure, are developed. The tomography is one of these developed techniques that allow 2D and 3D interior object examination. By combination of a set of measures and thanks to computational and images reconstruction methods, it provides cartography of attenuation parameters characteristic of the radiation/object interaction, according to one or more transversal plans (slices). It thus makes possible to see on TV monitor the interior of bodies and objects, whereas before one had access either by pure imagination, by interpreting indirect measurements, or by cutting out the objects materially. In the case of the medical imaging, for example, this direct obser‐ vation requires a surgical operation. This formidable invention thus enables us to discover the interior of human bodies and different objects which surround us and their organization in space and time, without destroying them.

The tomography thus constitutes an instrument privileged to analyze and characterize mat‐ ter, that it is inert or alive, static or dynamic, of microscopic or macroscopic scale. While giv‐ ing access to the structure and the shape of the components, it makes possible the apprehension of the complexity of the objects studied. The computed tomography is a tech‐ nique of acquisition of digital images (projections). It generates a computer coding of a digi‐ tal representation of an area of interest through a patient, a structure or an object. The tomography thus provides a virtual representation of reality, in the intimacy of its composi‐ tion. This numerical coding then will facilitate the exploitation, the exchanges and the infor‐

© 2013 Kharfi; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Kharfi; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

mation storage associated. It becomes thus possible by appropriate processing to detect the presence of defects, to identify the internal structures and to study their form and their posi‐ tion, to quantify the variations of density, to model the internal components and to guide the instruments of intervention in medicine. Finally, the user will be also able to take benefit from a large variety of existent software and algorithms for the tomography digital images processing, analysis and visualization.

tion of a single layer. The superposition of such layers will constitute the 3D image or volume representing the studied object. By means of image processing tools, parts of the reconstructed

Mathematics and Physics of Computed Tomography (CT): Demonstrations and Practical Examples

An object O (x, y, z) is considered as a superposition of n layers of the same thickness along z axis, all located in planes parallel to the plane (x, y) and perpendicular to z (Fig. 1). Each layer represents a section in the object to be reconstructed. It is considered as a 2D function fn(x, y) that describes, for example, the distribution of linear attenuation coefficients as a function of the position or any other 2D function that can be measured and whose measurement signal is described by a full line. Any function maybe considered in tomography in condition to be limited and finite in a given region and equal to zero outside this region. The condition of a finite size is easily achievable for solid samples, liquid or gas contained in a box. The estab‐ lishment of condition is very difficult when it will be used for the tomography of electric or magnetic fields. The generalization of such function for the tomographic measurements is closely related to the determination of the nature of the interaction of the scanning beam (combshaped) with the object under examination. The purpose of tomography is the reconstruction of this 2D function, representing a layer or slice of the object, from the measured projections

3D volume can be extracted and analyzed separately from the rest of the data set.

y

ned along the angle θ and the transmitted intensity is stored in a system (t, s) of rotational coordinates.

x

t

**Figure 1.** The geometry of a studied object scanning in the {x,y,z} coordinates system. A layer in the plane (x, y) is scan‐

The layer is scanned (scanned) in an angle θ varying from 0 ° to 180 ° for the transmission tomography. The intensity of the transmitted beam is recorded like a translation function of of the position parameter t (Fig. 2). The transmitted intensity is given by Lambert's law given

> <sup>0</sup> ( , ) exp( ( , ) ) *path*

m

*Ixy I* = -

z

s

t

*x y ds* ò (1)

θ

ln(I/I0)

http://dx.doi.org/10.5772/52351

83

in a unique way.

x

by the following expression:

z

I0 I

A CT system gathers several technological components. Its development requires the partic‐ ipation of the end users - as the doctors, the physicists or the biologists - to specify the needs, of the engineers and researchers to develop the novel methods and, finally, of the in‐ dustrial teams to develop, produce and market these systems.

In this work, we are interested in the physics and mathematics related to the main phases of computed tomography, namely: the scanning or projection phase and the phase of 2D or 3D image reconstruction by various analytical methods such filtered back-projection method (FBP). In this context, all the mathematical equations and relations which seemed ambigu‐ ous or not clear are explained and mathematically demonstrated with some illustrative ex‐ amples. This chapter is organized in the form of a main body text with sub-sections presenting a brief explanation of most important concept or the detailed demonstration of any equation or expression, and this, each time that seemed necessary. The algebraic meth‐ ods for image reconstruction in tomography will be also outlined.

This chapter is intended as an introduction to Computed Tomography. It was written not only for those persons who have some familiarity with other imaging techniques such radia‐ tion transmission radiography but also for novices in the field of digital imaging. The chap‐ ter begins with some simple, yet fundamental, concepts regarding computed tomography and the physics and mathematics at the origin of CT. As one progresses through the chapter, more detail regarding the CT technique and methodology is meet. The reader should not be alarmed if his or her particular problem or preoccupation is not mentioned here. So many different CT developments have been achieved in the last thirty-three years that it would be difficult to describe them all in a single chapter. Some practical information are also present‐ ed that can be vital to obtaining good analytical results; it is sometimes difficult to find.

I hope that this introduction to the CT technique will provide useful information to those persons who are about to get involved with CT as well as present TC users and those with simply a curiosity about the technique.
