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Biomedical Signal and Image Processing
that it is theoretically impossible to solve for f(x, y) from only one integral equation
such as Equation 13.2. A simple justification of such a claim is as follows: Someone
has two numbers in mind and wants you to guess these numbers. As a hint, he or
she provides you with the summation of these numbers. Is this hint sufficient to
guess every one of those two numbers? Obviously, the answer is no. Getting back
to our problem and knowing that integration is nothing but the summation over
the differential elements, one cannot find a function f(x, y) from the result of its
integral. Then, the question is “how can we solve this problem?” The beauty of CT
lies within the simple idea that one can estimate f(x, y) if she or he repeats measurements at different positions and different angles. This is loosely similar to asking
the person who has the two numbers in mind to provide you with some other algebraic fact about his or her numbers, for example, to tell you what is two times one
number plus the other one. With having another set of algebraic equation, you can
solve a system of two equations with two variables to come up with the numbers. In
tomography, many measurements at many positions and many directions (as shown
in Figure 13.5) are used to create a set of equations that can be solved for f(x, y).
In medical tomography, two types of beam system are used. In the first type,
called “parallel beam,” all beams for a given angle are in parallel (Figure 13.5),
while in the second type, often referred to as “fan beam,” the beams for a particular
angle fan out from the source (Figure 13.6). In this book, we focus on parallel beam
formulation, but the extension of the formulation to the fan-beam system is rather
more complicated and is not covered in this book.
Despite the differences in physical properties, solving inverse problems in path
integrals based on several measurements constitutes the principle of tomography.
As a result, in the following section, a detailed description of the techniques to solve
the integral equation in attenuation tomography is given.
FIGURE 13.5 Measurements at different positions and directions for tomography.
FIGURE 13.6 Fan-beam system.
Biomedical Signal and Image Processing
that it is theoretically impossible to solve for f(x, y) from only one integral equation
such as Equation 13.2. A simple justification of such a claim is as follows: Someone
has two numbers in mind and wants you to guess these numbers. As a hint, he or
she provides you with the summation of these numbers. Is this hint sufficient to
guess every one of those two numbers? Obviously, the answer is no. Getting back
to our problem and knowing that integration is nothing but the summation over
the differential elements, one cannot find a function f(x, y) from the result of its
integral. Then, the question is “how can we solve this problem?” The beauty of CT
lies within the simple idea that one can estimate f(x, y) if she or he repeats measurements at different positions and different angles. This is loosely similar to asking
the person who has the two numbers in mind to provide you with some other algebraic fact about his or her numbers, for example, to tell you what is two times one
number plus the other one. With having another set of algebraic equation, you can
solve a system of two equations with two variables to come up with the numbers. In
tomography, many measurements at many positions and many directions (as shown
in Figure 13.5) are used to create a set of equations that can be solved for f(x, y).
In medical tomography, two types of beam system are used. In the first type,
called “parallel beam,” all beams for a given angle are in parallel (Figure 13.5),
while in the second type, often referred to as “fan beam,” the beams for a particular
angle fan out from the source (Figure 13.6). In this book, we focus on parallel beam
formulation, but the extension of the formulation to the fan-beam system is rather
more complicated and is not covered in this book.
Despite the differences in physical properties, solving inverse problems in path
integrals based on several measurements constitutes the principle of tomography.
As a result, in the following section, a detailed description of the techniques to solve
the integral equation in attenuation tomography is given.
FIGURE 13.5 Measurements at different positions and directions for tomography.
FIGURE 13.6 Fan-beam system.