chapter 2, Preliminaries

from: Computed Tomography, Second Edition: Principles, Design, Artifacts, and Recent Advances, Second

Author(s): Jiang Hsieh

Published: 26 October 2009

Chapter DOI: 10.1117/3.817303.ch2

Chapter Page Count: 31 pages

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Front Matter

1. Introduction

2. Preliminaries

3. Image Reconstruction

4. Image Presentation

5. Key Performance Parameters of the CT Scanner

6. Major Components of the CT Scanner

7. Image Artifacts: Appearances, Causes, and Corrections

8. Computer Simulation and Analysis

9. Helical or Spiral CT

10. Multislice CT

11. X-ray Radiation and Dose-Reduction Techniques

12. Advanced CT Applications

Back Matter

Preview

Chapter Contents

2.1 Mathematics Fundamentals
2.1.1 Fourier transform and convolution
2.1.2 Random variables
2.1.3 Linear algebra
2.2 Fundamentals of X-ray Physics
2.2.1 Production of x rays
2.2.2 Interaction of x rays with matter
2.3 Measurement of Line Integrals and Data Conditioning
2.4 Sampling Geometry and Sinogram
2.5 Problems
References

Excerpt

This chapter covers two major topics: the fundamentals of the mathematics and the fundamentals of the physics involved in x-ray physics. The objective is to provide a general review of the important mathematics tools and background knowledge of x-ray physics that are used throughout the book. The chapter serves mainly as a refresher. Readers who have not been previously exposed to these topics will find a list of recommended reference materials at the end of the chapter so that detailed studies can be performed.

Although these topics can be integrated into other chapters when they are encountered, many of the topics appear multiple times and at different locations. The goal is to provide a convenient and quick reference point to the readers by consolidating these subjects in a separate chapter.

2.1 Mathematics Fundamentals

2.1.1 Fourier transform and convolution

The one-dimensional (1D) Fourier transform of a function f(x) is defined as

where j = √−1. In this equation, e^−j2πux = cos2πux − j sin 2πux. Substituting this expression into Eq. (2.1), the 1D Fourier transform can also be expressed as