The Radon Transform and Local Tomography

Alexander G. Ramm, Alex I. Katsevich

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9780849394928
• eBook ISBN: 9781000151770
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Over the past decade, the field of image processing has made tremendous advances. One type of image processing that is currently of particular interest is "tomographic imaging," a technique for computing the density function of a body, or discontinuity surfaces of this function. Today, tomography is widely used, and has applications in such fields as medicine, engineering, physics, geophysics, and security. The Radon Transform and Local Tomography clearly explains the theoretical, computational, and practical aspects of applied tomography. It includes sufficient background information to make it essentially self-contained for most readers.

CHAPTER 1

INTRODUCTION

1.1. Brief description of new results and the aims of the book

The Radon transform of a function f(x), x ∈ ℝⁿ, is defined to be

$Rf≔\hat{f}(\alpha ,p)≔{\displaystyle \underset{l_{\alpha p}}{\int }f(x)ds},$

(1.1.1)

where α ∈ Sⁿ⁻¹, Sⁿ⁻¹ is the unit sphere in ℝⁿ, p ∈ ℝ, l_αp ≔ {x: α · x = p} is a plane, and ds is the Lebesgue measure on this plane. It is assumed that f(x) is integrable over any plane. A natural question to ask is: how can one recover f(x) knowing $\hat{f}(\alpha ,p)$? This question, in the case n = 2, was posed and solved by Radon in 1917 [Ra]. However, according to [De2, p.3], Uhlenbeck (1925) wrote that Lorentz, the famous Dutch physicist, the author of the theory of electrons and the Nobel prize winner in physics for 1902 (jointly with P. Zeeman), knew the inversion formula for the Radon transform already around the turn of the century. In the mathematical literature the Radon transform was used in the books [J3] and [GS] for construction of fundamental solutions to partial differential equations with constant coefficients. Inversion of the Radon transform was one of the first problems of integral geometry, which is a branch of mathematics dealing with the recovery of functions knowing their integrals over a family of manifolds [LRS].

Radon’s paper was not used for a long time. The rapid development of applications of the Radon transform started in the early 1970s. It is not possible to mention all the relevant works in medicine, astronomy, optics, physics, geophysics, and other areas. A vast bibliography is given in [De2], [Nat3]. The most well known applications are in computed tomography (X-ray transmission tomography, emission tomography, and ultrasound tomography). The Nobel prize in physiology and medicine was awarded in 1979 to A. Cormack and G. Hounsfield for their work on applications of tomography to medical diagnostics.

The Radon transform is a particular case of the more general X_m-ray transform, which is defined as follows:

$X_{m}f={\displaystyle \underset{{ℝ}^n\cap M_m}{\int }fds},$

(1.1.2)

where M_m runs through the set of m-dimensional affine manifolds, 1 ≤ m ≤ n − 1, and ds is the m-dimensional Lebesgue measure on ℝⁿ ∩ M_m. If m = 1, we obtain X-ray transform – the integrals of f over straight lines:

$Xf≔{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x+\alpha t)dt≔g(x,\alpha )},$

(1.1.3)

where α ∈ Sⁿ⁻¹ and x ∈ ℝⁿ runs through a subset of ℝⁿ. For example, this subset may be a curve, a surface or other manifold in ℝⁿ. If m = n − 1, we obtain the Radon transform (1.1.1), that is X_n−1 f = Rf.

Let us consider the practically important case n = 2. In this case the Radon transform and X-ray transform coincide and, basically, one is given the integrals of a three-dimensional object f along all lines located on a fixed plane through the object. Such data can be collected observing attenuation of X-rays passing through the object (see Section 1.2.1, p. 6). The problem is: given the line integral data, recover f on the plane. This gives a two-dimensional slice of f. Stacking many two-dimensional slices, if necessary, one can recover the three-dimensional object. Unfortunately, the conventional two-dimensional reconstruction is not local: to compute f at a point x one needs to know the integrals of f along all lines on the plane intersecting the support of f, even along the lines far removed from x. Moreover, in practice, it might be impossible to collect the complete data set: for example, if the object is too big. Suppose now that one is interested in the recovery of f not for all x ∈ supp f, but for x only in some subset U ⊂ supp f. The subset U will be call...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Preface
• Table of Contents
• Chapter 1. Introduction
• Chapter 2. Properties of the Radon transform and inversion formulas
• Chapter 3. Range Theorems and reconstruction algorithms
• Chapter 4. Singularities of the Radon transform
• Chapter 5. Local Tomography
• Chapter 6. Pseudolocal Tomography
• Chapter 7. Geometrical tomography
• Chapter 8. Inversion of incomplete tomographic data
• Chapter 9. Inversion of cone-beam data
• Chapter 10. Radon transform of distributions
• Chapter 11. Abel-type integral equation
• Chapter 12. Multidimensional algorithm for finding discontinuities of signals from noisy discrete data
• Chapter 13. Test of randomness and its applications
• Chapter 14. Auxiliary Results
• Research Problems
• Bibliographical notes
• References
• List of notations
• Index