Reconstruction Algorithms for Computed Tomography

Claas Bontus; Thomas Köhler    Philips Research Europe–Hamburg, Sector Medical Imaging Systems, Röntgenstrasse 24–26, D-22?335 Hamburg, Germany

I INTRODUCTION

Computed tomography (CT) is of vital importance for medical diagnosis, intervention planning, and treatment evaluation. CT yields three-dimensional (3D) tomographic images of large parts of the human body within a few seconds. The rather wide bore allows use of CT scans for obese patients and claustrophobia is rarely an issue. At the same time, the current pace of technological progress is unprecedented. Two major advances are important. First, X-ray detectors are larger, allowing scanning larger volumes in shorter times. Second, rotation speeds of the tube-detector system are steadily increasing.

Although the technological progress is of great benefit for patients and medical staff, the engineering challenges are great. X-ray tubes must be able to withstand the centrifugal forces increasing with the rotation speed. These tubes also must be able to illuminate detectors of increasing sizes. Integration times of the detector elements become shorter. The amount of data measured per time unit increases with increasing rotation speed and detector sizes. All these data must be transmitted and processed in an appropriate amount of time.

Data processing is separated into different steps: preprocessing, reconstruction and image processing. Preprocessing covers correction methods to treat disturbing effects and system imperfections, such as beam hardening, cross-talk, and afterglow. For example, beam hardening is a nonlinear effect resulting from the fact that X-rays experience weaker absorption with increasing photon energy. Image-processing methods cover all algorithms that are applied to the image data to assist radiologists in image analysis. Examples for image-processing methods are tools that automatically segment different organs, as well as computer-aided diagnosis (CAD) tools.

This chapter covers the field of CT reconstruction, i.e., the mathematics to obtain the image data from the measurements. In other words, reconstruction deals with an inverse problem. Reconstruction algorithms must fulfill certain criteria to be useful in practice. They must be numerically stable and yield images of sufficient quality with respect to artefacts, resolution, and signal to noise ratio (SNR). Furthermore, reconstruction algorithms should be sufficiently fast so that images can be obtained in an adequate amount of time.

Historically, most CT scanners were so-called single-row scanners. These scanners yield two-dimensional (2D) images of cross-sectional slices. The mathematics for single-row scanners is well understood. Modern cone-beam CT scanners, in which the detector consists of many rows, require completely new reconstruction algorithms. The first reconstruction algorithms proposed for cone-beam scanners were extensions of 2D methods. These algorithms, which are of approximative nature, yield good image quality as long as the number of detector rows does not become too large. Once the number of rows becomes large, more effort is necessary and the underlying mathematics must be analyzed to develop more sophisticated algorithms.

In 2002 Alexander Katsevich (2002) published the first reconstruction algorithm for helical CT, which is mathematically exact, of the filtered backprojection (FBP) type, and in which the filtering is shift invariant. In particular, the exactness guarantees that remaining artefacts originate from the discrete reformulation of the inversion formulas. This discretization is required by the discrete sampling of the projection data. FBP, in combination with the shift invariance of the underlying filter, ensures that the algorithm can be implemented in an efficient way. Katsevich's work resulted in many publications analyzing the underlying mathematics or extending the algorithm to different settings.

The chapter is devoted to reconstruction algorithms of the FBP type that are derived from an exact method. The underlying mathematics is discussed on a broad basis. Nevertheless, more effort is placed on descriptive discussions than on waterproof mathematical analyses. Section II provides a short summary of the principles. The measurement process as well as different kinds of detectors and trajectories are discussed. Section III covers the field of reconstruction in several subsections, beginning with the Radon transform in Section III.A. As it turns out, the mathematics associated with the discussed algorithms can be separated into two parts. While Sections III.B and III.C deal with the formulas that are independent of the particular system settings, the subsequent sections specify the parameters for different types of acquisitions. The mathematics of the circle and line trajectory (CLT) can be understood readily. This trajectory, therefore, serves as the first example, covered in Section III.D. The Katsevich algorithm for a so-called Pi acquisition is covered in Section III.E, while Section III.F deals with the so-called EnPiT algorithm. The EnPiT algorithm can be considered an extension of the Katsevich method useful for more freely choosing the patient table feed. Cardiac CT is one of the most important applications for modern CT scanners. The CEnPiT algorithm, which was developed for cardiac cone-beam CT based on a helical acquisition, is briefly summarized in Section III.G.

II PRINCIPLES OF COMPUTED TOMOGRAPHY

A X-Ray Attenuation

Figure 1 shows a modern CT scanner. Within such a CT system a tube emits X-rays in the direction of a detector. The X-rays penetrate a certain part of the patient's body before they enter the detector. During the scan the tube-detector system (termed as the gantry) rotates around the patient (see Figure 2) so that projections from a large number of directions are taken. The patient lies on a patient table, that can be moved during the scan. In this manner, a relative motion of the patient with respect to the rotation plane can be realized. In practice, this is used, for example, to obtain a relative trajectory in which the X-ray tube moves on a helix around the patient.

Consider the X-ray tube itself. Let us denote the flux density of X-ray photons with energy E emitted into a certain direction by 0(E). Typical units for the photon flux density are I]=1/ssrkeV; that is (E) is the number of photons emitted within each second per solid angle and per photon energy. Some X-ray photons are scattered or absorbed when penetrating the patient. The number of photons, that pass through the patient (not scattered and not absorbed) is given by the Beer–Lambert law:

(E)=I0(E)exp(−∫dℓμ(x(ℓ),E)).

Here, (ℓ) parameterizes the path along which the X-rays travel. The scalar function (x) are the so-called absorption coefficients. Function (x) is also known as the object function. Certainly, the values of μ are different for different kinds of tissue.

From Eq. (1) we easily obtain

dℓμ(x(ℓ),E)=ln(I0(E)/I(E)).

In other words, a monochromatic CT system measures line integrals. Unfortunately, X-rays emitted from today's tubes have a wide spectrum and the detectors average over the photon energies. Therefore, some preprocessing steps are necessary before the measurements can approximately be interpreted as line integrals.

B Parameterization of the Measurements

The following sections assume that the preprocessing steps yield data, which can be interpreted as line integrals in the sense of Eq. (2) with =E0, where 0 is some mean energy. We choose a coordinate system, which is fixed to the patient. At a particular time the X-ray source will be at a point y with respect to this coordinate system. The line integrals associated with the measurements at this point in time all originate from y. Terms source position and focal spot are used synonymously throughout this chapter and correspond to vectors y.

Each line integral can be parameterized as

(y,θ)=∫0∞dℓμ(y+ℓθ),

where θ is a unit vector pointing into the direction of the line integral. The different values of θ are determined by the locations of the different detector elements.

If the patient table is fixed during the scan (see Figure 2), the source moves on a circular trajectory with respect to the coordinate system defined above. This trajectory can...