Biomedical Engineering Reference
In-Depth Information
where
r
=
x
cos
θ
+
y
sin
θ
(2.7)
and
l
r
,θ
represents a straight line that has a perpendicular distance
r
from the
origin and is at an angle
θ
with respect to the
x
-axis. It can be shown that an
object can be uniquely reconstructed if its projections at various angles are
known [4, 30]. Here,
p
(
r
,θ
) is also referred to as
line integral
. It can also be
shown that the Fourier transform of a one-dimensional projection at a given
angle describes a line in the two-dimensional Fourier transform of
f
(
x
,
y
)at
the same angle. This is known as the central slice theorem, which relates the
Fourier transform of the object and the Fourier transform of the object's Radon
transform or projection. The original object can be reconstructed by taking
the inverse Fourier transform of the two-dimensional signal which contains
superimposed one-dimensional Fourier transform of the projections at different
angles, and this is the so-called Fourier reconstruction method. A great deal of
interpolation is required to fill the Fourier space evenly in order to avoid artifacts
in the reconstructed images. Yet in practice, an equivalent but computationally
less demanding approach to the Fourier reconstruction method is used which
determines
f
(
x
,
y
) in terms of
p
(
r
,θ
) as:
π
∞
f
(
x
,
y
)
=
p
(
r
,θ
)
ψ
(
r
−
s
)
ds d
θ
(2.8)
0
−∞
where
ψ
(
r
) is a filter function that is convolved with the projection function
in the spatial domain. Ramachandran and Lakshminarayanan [31] showed that
exact reconstruction of
f
(
x
,
y
) can be achieved if the filter function
ψ
(
r
)in
equation (2.8) is chosen as
if
ω
≤
ω
◦
|
ω
|
(2.9)
{
ψ
}=
0
otherwise
where
{
ψ
}
represents the Fourier transform of
ψ
(
r
) and
ω
◦
is the highest
frequency component in
f
(
x
,
y
). The filter function
ψ
(
r
) in the spatial domain
can be expressed as:
sin 2
πω
◦
r
2
πω
◦
r
sin
πω
◦
r
πω
◦
r
2
2
◦
2
◦
ψ
(
r
)
=
2
ω
(2.10)
−
ω
This method of reconstruction is referred to as the
filtered-backprojection
,or
