260
Biomedical Signal and Image Processing
A simple solution for this problem is to make more scans, i.e., make more lines
that are closer to each other and therefore create a better representation of the high
frequencies. Even though simple in idea, this approach is often limited by the
restrictions imposed by the instrumentation and technical issues. Another solution (which is heavily used by the industry) is to design mathematical techniques
to interpolate the known points in high frequencies to find more points in high
frequency and calculate IFFT of F(u, v) on many more points to create a better
representation of f(x, y). There are very many techniques to perform this interpolation,
but, in principle, they all share the idea of using the known points to estimate the
values of the points in between them. We do not discuss these techniques here, and
interested readers can use the list of books and papers provided in this book for
further study of these methods.
13.4 SUMMARY
In this chapter, the principle ideas of CT were presented. Knowing that the basic
mathematical methods behind all tomographic modalities such as CT, MRI, PET,
and some ultrasound imaging systems are the same, covering these techniques in
this chapter covers almost all mathematical foundations of medical imaging. In the
chapters dedicated to each particular modality, we will describe how the physical
principles of each imaging modality provide tomographic equations to be solved
by the methods described in this chapter. As described in this chapter, the major
mathematical methods for solving the tomographic equations are based on a theorem
called Fourier slice theorem that translated the tomographic measurement into a
computationally simpler problem in the frequency domain.
PROBLEMS
13.1 During an attenuation tomographic measurement, assume that the two projections
at θ = 0 and θ = π/2 have resulted in the following projection functions:
t < a
⎧0 ⎪
(13.12)
P q =0 = ⎨
1 t ≥ a
⎪ ⎩
and
⎧0
t < b
⎪
(13.13)
P q p/2 =
=
⎨
1 t ≥ b
⎪ ⎩
Without using any mathematical calculations and only using heuristics, try to
visualize the object being imaged.
13.2 Study the formulation of the 2-D FT in the polar coordinates and explain how
this formulation helps with the implementation and usage of the Fourier
slice theorem for CT.
Biomedical Signal and Image Processing
A simple solution for this problem is to make more scans, i.e., make more lines
that are closer to each other and therefore create a better representation of the high
frequencies. Even though simple in idea, this approach is often limited by the
restrictions imposed by the instrumentation and technical issues. Another solution (which is heavily used by the industry) is to design mathematical techniques
to interpolate the known points in high frequencies to find more points in high
frequency and calculate IFFT of F(u, v) on many more points to create a better
representation of f(x, y). There are very many techniques to perform this interpolation,
but, in principle, they all share the idea of using the known points to estimate the
values of the points in between them. We do not discuss these techniques here, and
interested readers can use the list of books and papers provided in this book for
further study of these methods.
13.4 SUMMARY
In this chapter, the principle ideas of CT were presented. Knowing that the basic
mathematical methods behind all tomographic modalities such as CT, MRI, PET,
and some ultrasound imaging systems are the same, covering these techniques in
this chapter covers almost all mathematical foundations of medical imaging. In the
chapters dedicated to each particular modality, we will describe how the physical
principles of each imaging modality provide tomographic equations to be solved
by the methods described in this chapter. As described in this chapter, the major
mathematical methods for solving the tomographic equations are based on a theorem
called Fourier slice theorem that translated the tomographic measurement into a
computationally simpler problem in the frequency domain.
PROBLEMS
13.1 During an attenuation tomographic measurement, assume that the two projections
at θ = 0 and θ = π/2 have resulted in the following projection functions:
t < a
⎧0 ⎪
(13.12)
P q =0 = ⎨
1 t ≥ a
⎪ ⎩
and
⎧0
t < b
⎪
(13.13)
P q p/2 =
=
⎨
1 t ≥ b
⎪ ⎩
Without using any mathematical calculations and only using heuristics, try to
visualize the object being imaged.
13.2 Study the formulation of the 2-D FT in the polar coordinates and explain how
this formulation helps with the implementation and usage of the Fourier
slice theorem for CT.