13
Principles of Computed
Tomography
13.1 INTRODUCTION AND OVERVIEW
Before applying any image processing technique described in the previous chapters
to analyze a biomedical image, one needs to create an image. This is often done
using computational techniques specialized to exploit the physical laws governing
the imaging system as well as the tissue to be imaged. Despite the differences in the
physical laws and principles of imaging modalities such as MRI, x-ray CT, ultrasound, and PET, surprisingly, the core computational techniques used to create an
image in all these modalities are more or less the same. These computational techniques are often referred to as “computed tomography” or CT. We start this section
with the description of the main concepts of CT and its importance in biomedical
image processing.
While mathematical representation of CT (as will be discussed in detail) is rather
complex, the concept of CT is very much simple and intuitional. Simply put, CT is
a process in which the contents of a black box are estimated and visualized based
on the reading and measurements made on the surface or around the box. In other
words, in CT, one needs to know the contents of a box without opening it. This
simple definition explains the importance of CT in biomedical sciences. In medical
diagnosis, physicians need to “see” the inside of a body as a two-dimensional (2-D)
or three-dimensional (3-D) image noninvasively, i.e., without having to cut the skin,
organ, or tissue. The popularity and widespread use of all the existing imaging
systems in the recent decades witness to the importance of CT.
From the simple definition of CT given earlier, one might think that this is not
a feasible task and no math can handle it. In order to see if this task might be possible even without any math, let us make a quick journey to our younger age. Piggy
banks are not only popular play objects among children but also the best examples
to witness to the feasibility of CT. Children are often curious to know how full or
empty their piggy banks are, i.e., they try to estimate the contents of their black
box (piggy bank) at least roughly. In order to do so, they shake (i.e., “stimulate”)
their piggy bank and get two types of feedback: tactile and audio information.
Based on the way the shacked piggy bank “touch” and “sound,” children decide
not only whether it is full or not but also whether the contents are mainly notes or
coins. This CT process happens in the brain apparently without any math as most
of these children do not even know how to add numbers!
The reader might think that the examples of CT are limited to early medical imaging,
but CT problems are encountered in many areas of science such as nondestructive tests
249
Principles of Computed
Tomography
13.1 INTRODUCTION AND OVERVIEW
Before applying any image processing technique described in the previous chapters
to analyze a biomedical image, one needs to create an image. This is often done
using computational techniques specialized to exploit the physical laws governing
the imaging system as well as the tissue to be imaged. Despite the differences in the
physical laws and principles of imaging modalities such as MRI, x-ray CT, ultrasound, and PET, surprisingly, the core computational techniques used to create an
image in all these modalities are more or less the same. These computational techniques are often referred to as “computed tomography” or CT. We start this section
with the description of the main concepts of CT and its importance in biomedical
image processing.
While mathematical representation of CT (as will be discussed in detail) is rather
complex, the concept of CT is very much simple and intuitional. Simply put, CT is
a process in which the contents of a black box are estimated and visualized based
on the reading and measurements made on the surface or around the box. In other
words, in CT, one needs to know the contents of a box without opening it. This
simple definition explains the importance of CT in biomedical sciences. In medical
diagnosis, physicians need to “see” the inside of a body as a two-dimensional (2-D)
or three-dimensional (3-D) image noninvasively, i.e., without having to cut the skin,
organ, or tissue. The popularity and widespread use of all the existing imaging
systems in the recent decades witness to the importance of CT.
From the simple definition of CT given earlier, one might think that this is not
a feasible task and no math can handle it. In order to see if this task might be possible even without any math, let us make a quick journey to our younger age. Piggy
banks are not only popular play objects among children but also the best examples
to witness to the feasibility of CT. Children are often curious to know how full or
empty their piggy banks are, i.e., they try to estimate the contents of their black
box (piggy bank) at least roughly. In order to do so, they shake (i.e., “stimulate”)
their piggy bank and get two types of feedback: tactile and audio information.
Based on the way the shacked piggy bank “touch” and “sound,” children decide
not only whether it is full or not but also whether the contents are mainly notes or
coins. This CT process happens in the brain apparently without any math as most
of these children do not even know how to add numbers!
The reader might think that the examples of CT are limited to early medical imaging,
but CT problems are encountered in many areas of science such as nondestructive tests
249