Image Reconstruction in Radiology

J. A. Parker

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351090261
• Edition: 1
• Topic: Derecho
• Subtopic: Ciencias forenses

This one-of-a-kind resource provides a very readable description of the methods used for image reconstruction in magnetic resonance imaging, X-ray computed tomography, and single photon emission computed tomography. The goal of this fascinating work is to provide radiologists with a practical introduction to mathematical methods so that they may better understand the potentials and limitations of the images used to make diagnoses. Presented in four parts, this state-of-the-art text covers (1) an introduction to the models used in reconstruction, (2) an explanation of the Fourier transform, (3) a brief description of filtering, and (4) the application of these methods to reconstruction. In order to provide a better understanding of the reconstruction process, this comprehensive volume draws analogies between several different reconstruction methods. This informative reference is an absolute must for all radiology residents, as well as graduate students and professionals in the fields of physics, nuclear medicine, and computer-assisted tomography.

I. System Models

Chapter 1

Introduction

The primary goal of this book is to explain the mathematical methods used for image reconstruction. Image reconstruction has become increasingly important in Radiology as computer performance has made image processing practical for day-to-day operation. It is likely that these methods will become even more important in the future.

The presentation of mathematical methods in this book is most similar to that in an area called signal processing in electrical engineering. In electrical engineering many of the signals of interest are one dimensional. Although images are two-dimensional signals, almost all of the one-dimensional signal processing methods can be directly extended to two dimensions. Consequently, the first three parts of the book will deal mostly with one-dimensional signals. In the last part of the book these methods will be applied to the task of image reconstruction.

Signal processing methods find a wider application in Radiology than image reconstruction. For example, these methods are very useful for understanding tracer kinetics and for understanding nuclear magnetic resonance spectroscopy. Hence, many of the examples in the first three parts of the book will come from nonimaging applications.

I. Structure of the Book

This book is divided into four parts — System Models, Transformations, Filtering, and Reconstruction. The object of the first three parts of the book is to provide a background for Part IV which explains image reconstruction. However, the methods described in the first three parts are useful in themselves for understanding several aspects of radiology. Throughout the book, an attempt will be made to show how the mathematical methods apply to radiological problems.

We assume that the reader has taken a college calculus course. However, since some of the mathematics may be only dimly remembered, Chapter 2 provides a brief review of the calculus which will be used in the rest of the book. Other mathematical background material, necessary to understand reconstruction, is described in greater detail — complex numbers in Chapter 6, vectors and matrices in Chapters 8 and 9, and statistics in Chapters 10, 11, and 12.

A. Part I. System Models

Part I, System Models, deals with the various models which can be used to describe a system. The concept of a system is central to this book. Anytime one signal depends upon another signal, the relationship can be modeled as a system. The system is said to transform one signal, the input signal, into the other signal, the output signal. For example, the set of attenuation coefficients in a cross section of the body could be an input signal. The system could be a computed tomographic scanner. The output signal would then be the computed tomographic image.

Part I will explain the concept of a system by developing several different models of systems. The major models are convolution, simultaneous linear equations, and stochastic processes (noisy signals). In addition, differential equations are briefly mentioned in order to explain the idea of resonant signals.

Convolution is a method of describing simple linear systems. Convolution and Fourier transforms come from the field called linear systems theory. Chapters 3, 4, 5 and 6 deal with linear systems theory. Simultaneous linear equations provide a second method of describing linear systems. Simultaneous linear equations come from the field called linear algebra. Chapters 8 and 9 deal with linear algebra. We shall learn that a major feature of both of these models is that they assume that a system is linear.

For both convolution and simultaneous linear equations, we shall learn that there are special signals, called eigenfunctions, which do not change form as they pass through a system. These signals are described in Chapter 5. The concept of eigenfunction is often described in more complicated mathematical texts. We shall try to show that the basic concept of an eigenfunction is quite simple and will help in understanding how systems effect signals. Furthermore, in Part II, we shall find that eigenfunctions provide insights into the operation of the Fourier transform.

Complex numbers greatly simplify the description of linear systems and of the Fourier transform. Most readers either will have never heard of complex numbers or will have only a superficial acquaintance with complex numbers. In Chapter 6 we shall try to familiarize the reader with complex numbers. Although learning about complex numbers requires some mental anguish, the reward will be that much of the rest of the book will be considerably simplified.

Both convolution and simultaneous linear equations need to be combined with the idea of measurement noise in order to be useful for radiological applications. Chapter 10 introduces some basic statistical methods. Chapter 11 introduces statistical signals (stochastic processes). In Part IV, stochastic processes will be used to model both noisy measurements and reconstructed images.

Chapter 12 introduces the idea of estimation from a set of noisy measurements. Part III will combine the concept of a signals with estimation. Estimation i...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• The Author
• Dedication
• Table of Contents
• I. SYSTEM MODELS
• II. TRANSFORMATIONS
• III. FILTERING
• IV. RECONSTRUCTION
• Appendix: Symbols
• INDEX