Statistics of Medical Imaging

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X-Ray CT Physics a n d Math ematics

2.1 Introduction

The word tomography consists of two Greek words tomos (slice ) and graphein

(draw). Historically, the term tomography has referred to a technique of X-

ray photography in which only o ne plane of the internal struc ture inside the

object is photographed in sha pe focus. Linear and transaxial tomography are

two examples.

In the linear tomogr aphy as shown in Figur e 2.1, an X-ray source and a

photographic plate (which is parallel to the cr oss section of an object) are

placed on two sides of the object. By moving the X-ray source at a fixed

sp eed parallel to the cross section in one direction, and moving the plate at

an appropriate speed in the opposite direction, a point in the cross section

(denoted by •) is always projected onto the same point in the plate, but

the point above or below the plate is projected onto different points in the

plate. Thus, on the photographic plate the cross section stands out while the

remainder of the object is blurred.

In transaxial tomography as shown in Figure 2.2, an object sits in a turn-

table in an upright position. The photographic plate is on a ho rizontal ta-

ble next to the object. X-rays are directed obliquely through the object

and projected onto the photog raphic plate. Both the object and the pho-

tographic plate are rotating in the same direction and at the same speed.

Only those points actua lly on the focal section of the object remain in sharp

fo cus throughout a rota tion. Points that are above or below the focal section

are blurred.

Nowadays, tomogr aphy refers to the cross-sectional imaging of an object

from either transmission or reflection data collected by illuminating the object

from many different directions.

Computed tomography (abbrev iated as CT ) is differe nt from conventional

tomography in that the images of the cross sections in CT are not influenced

by the objects outside those sections.

In X-ray tra nsmission CT (abbreviated as X-ray CT), a single X-ray sourc e

is contained in a tube and the detector devic e consists of an array of X-ray

detectors. The X-ray tube and the collimator are on one side and the detector

device a nd the data acquisition unit are on the other side of the object. Both

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8 Statistics of Medical Imaging

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x-ray source

photographic plate

cross section

t

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t

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t

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t

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FIGURE 2.1

Linear tomography: t

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and t

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denote the time ins tants and −→ represents the

moving direction.

the object and this apparatus are statio nary, and the apparatus is ro tating

around the object. An image of a slice of the object (which is determined by

the X-ray beams from the sour ce to the detectors) can be created [1–4, 7–9,

18, 37, 53].

X-ray CT technology has improved dramatically over the past 50 years and

now CT scanners are an important and routinely use d diagnostic imaging

instrument. From the first X-ray CT scanner

∗

to the modern 3D cone beam

scanner, there may not be a clear definition of the generatio ns of X-ray CT.

However, based on X- ray generation, detection, data acquisition, and image

reconstruction, the milestones in its research and development may include,

but are not limited to, (1) the parallel beams (a tr anslation-rotatio n mode),

(2) the narrow divergent beams (a translation-rotation mode), (3) the wide

divergent be ams (a rotation mode), (4) the wide divergent beams with the

closed detector ring (a rotation mode), (5) electron beams, (6) beams on a

spiral or a helical path, (7) the cone beam geo metry, as well as the micro-CT

and PET-CT combined scanner, etc.

X-ray CT is closely related to a co mputational approach often known as

image reconstruction from projections. As stated in [4], “Image reconstruction

from projections is the process of producing an image of a two-dimensional

distribution (usually of some physical property) from estimation of its line in-

tegrals along a finite number of lines of known locations.” In fact, X-ray CT

is an ex ample of the application of this approach. O ver the past 50 years, the

∗

It utilized 180 views, 160 parallel projections i n each view, algebraic reconstruction

method, and took 2.5 hours to reconstruct a 64 × 64 slice image.