Computed Tomography: Algorithms, Insight, and Just Enough Theory

The front matter includes the title page, series page, copyright page, contributor page, TOC, preface, notation and getting started page.

First of all—what is tomography? The word comes from the Greek tomos, a section or slice, and graphos, to describe. In tomography we traditionally produce images of slices of an object—without actually slicing it! To see the inside, we need to obtain information from the outside. Typically this is done by measuring the transmission of waves or particles which more or less travel in straight lines but whose intensity is attenuated by the material through which they travel.

Apart from Chapter 4, which explains the physics underlying the X-ray CT scanner and how we arrive at measurements of the linear attenuation coefficient, the rest of the book is about computational methods for reconstructing an image of the object from these measurements. The schematic in Figure 2.1 illustrates this.

In this chapter, we summarize the notation and fundamental principles of linear algebra that are needed in the later chapters of the book. Readers who may wish to brush up their fundamentals of linear algebra are referred to, e.g., [14] (fundamentals) and [60], [105] (more advanced presentations).

This chapter sets the stage for the rest of the book by describing the basic physical mechanisms of common X-ray CT. The emphasis is on the physics needed to understand the generation of X-rays, the components of an X-ray scanner, and the measurement process, without going into too many details. We briefly discuss the basic statistical aspects of the measurements.

In this chapter, we introduce the Radon transform as the mathematical forward model of interest for tomographic reconstruction of 2D parallel-beam data. We start by defining and illustrating the Radon transform and describing some of its fundamental properties. We describe the connection to the Lambert–Beer law and give analytical expressions for the Radon transform of some simple geometric shapes. At the end of the chapter we briefly outline generalization to the 3D case.

We will now describe how we can invert the Radon transform, so that we can—in principle—reconstruct the image of the object from the measured data. Details of the implementation of analytic inversion methods are widely available, such as the self-contained treatment including necessary mathematical background in [43] and the more detailed treatment in [129], so we will not reproduce them here. The first part of the chapter gives an intuitive introduction to the filtered back-projection (FBP) method and the underlying Fourier slice theorem. Our emphasis in the second part of the chapter is on what we can learn from analytical theory about what constitutes sufficient data for a reconstruction, how unstable that reconstruction process is with respect to errors in the data, and how we can recognize valid data. Finally, we go on to briefly discuss the sufficiency of data and analytical reconstruction methods for the 3D case.

We have studied an efficient algorithm—FBP—for computing a parallel-beam CT reconstruction. And we saw that the reconstruction is sensitive to noise in the data. Some natural questions that now arise are as follows: How can we further study the Radon transform and its inverse, how can we describe its sensitivity to noise, and how can we possibly reduce the influence of the noise?

In this chapter, we investigate limited-data tomography. Limited or incomplete CT data occurs if some of the measured X-ray CT data is missing. Many tomography problems in medicine and industry involve limited data because one often cannot take data all around the object or is interested in only one small part of the object and will use only data through that part.

In this chapter, we describe how the tomography problem can be represented on a computer such that we—in later chapters—can perform numerical computations to analyze and solve the reconstruction problem. This process is called discretization, and it is applied to the object under study as well as to the forward- and back-projections. We focus on the 2D parallel-beam geometry and the Radon transform, but the ideas extend to other geometries and 3D problems in a straightforward way.

Using the discretization methods from the previous chapter, in Section 9.5.1 we obtained a linear algebra formulation for the CT reconstruction problem which takes the form A x = b. Note that in this form the image and the sinogram are represented by the vectors x and b, respectively. This and the next chapters take their basis in this linear algebra formulation. Readers who wish to brush up on their fundamentals of linear algebra are referred to Chapter 3; more details can be found, e.g., in [60] and [105].

We saw in Chapter 9 that when we discretize a CT reconstruction problem we arrive at a system of linear equations. In most of that chapter we used the notation f for the vector with all the pixel values, but in Section 9.5 we changed the notation for this vector to x in order to be consistent with the massive linear algebra literature. So our system of equations takes the form A x = b, where A is the system matrix, b is the right-hand side, and the vector x is the solution which represents the pixel values of the reconstruction we want to compute.

Inverse problems are mathematical problems that arise when our goal is to recover “interior” or “hidden” information from “outside”—or otherwise available—noisy data [3], [8], [67], [71]. Tomographic reconstruction is a classical example of an ill-posed inverse problem, and we have already seen that reconstructions are sensitive to measurement noise. Therefore, we cannot expect to compute satisfactory results by simply solving the system of linear equations—in the form of either a square system or a least-squares problem. In this chapter, we will introduce regularization techniques to incorporate prior information on the objects in order to stabilize the solutions with respect to measurement noise.

Many of the computational problems presented in the two previous chapters are naturally formulated as optimization problems. For example, MAP estimation and the various regularization problems inherently rely on optimization, and the algebraic iterative methods introduced in Chapter 11 may be viewed through an optimization lens as methods for minimizing a quadratic function. In this chapter, we show how an optimization perspective provides a natural and convenient way to generalize a number of basic reconstruction methods, e.g., to include regularization and/or constraints. The main focus is on the basic principles of first-order optimization methods (i.e., methods that make use of first-order partial derivatives of the objective function) and their application in CT. There is a wealth of more advanced optimization methods, but these are beyond the scope of this chapter.

Tomographic imaging is enjoying an explosion of growth as new measurement technology opens up novel tomographic modalities, with higher resolution in space and time, as well as adding extra dimensions with spectral, polarization, phase contrast, and diffraction measurement techniques. Energy-resolved detectors for X-rays, e.g., open up new methods that make use of scattered radiation, with some Compton tomography modalities resulting from integrals of volume images over surfaces instead of straight lines [174]. Problems in multistatic radar and seismic imaging result in weighted integrals over ellipses and ellipsoids [124]. While analytical methods are available for some generalizations of the Radon transform, practical problems typically have limited and noisy data. Once a discretization of the forward problem has been implemented generalizing the ideas in Chapter 9, regularized solutions can be obtained using the methods of Chapters 12 and 13.

The back matter includes bibliography, index, and back cover.