Digital Signal Processing Reference
In-Depth Information
Note that
∞
−∞
exp
j
2
π
lU
G
(
U
,θ)
|
U
|
dU
is the inverse fourier transformation of
the function
G
(
U
,θ)
|
U
|
for constant
θ
. This can be achieved as the convolution of
inverse fourier transform (IFT) of
G
(
U
,θ)
and IFT of
|
U
|
. Note that IFT of
G
(
U
,θ)
is
R
(
l
,θ)
for constant
θ
(refer (
1.2
)). Final reconstruction formula fromparallel beam
radon transformation
R
(
l
,θ)
is represented as follows.
π
f
(
x
,
y
)
=
R
(
l
,θ)
∗
IFT
(
|
U
|
)
d
θ
(1.15)
0
1.2.1 Discrete Realization of (
1.15
)
Let
Z
(θ,
x
,
y
)
be the value obtained by the convolution
R
(
l
,θ)
∗
IFT
(
|
U
|
)
at
l
=
x
cos
θ
+
y
sin
θ
for the particular
θ,
x
and
y
. Integrating
Z
(θ,
x
,
y
)
over the complete
range of
. Let the discrete
version of the continuous image
f
be represented as
f
d
. Note that
f
d
is the image
matrix that are having finite range for
x
and
y
with origin in the middle of the image.
It is noted that the set of co-ordinates
θ
(0 to
π
) with constant
x
and
y
gives the value of
f
at
(
x
,
y
)
(
x
,
y
)
whose
l
is constant for the particular
θ
is the straight line
x
cos
(θ)
+
y
sin
(θ)
=
l
. Thus to obtain
f
d
, the following steps
are followed.
1. Create the zero matrix
r
whose size is exactly same as that of
f
d
.
2. Let z(l,
θ
(
,θ)
∗
(
|
|
)
) be the value obtained by the convolution
R
l
IFT
U
at
l
for
.
3. Fill the matrix
r
with
z
the particular
θ
(
l
,θ)
in all the co-ordinates
(
x
,
y
)
that satisfies
x
cos
(θ)
+
y
sin
(θ)
=
l
, which is the straight line tilted with an angle
θ
in the anticlockwise
direction.
4. Repeat step 2 for all
l
ranging from
−
l
max
to
l
max
with fixed
θ
. Let the obtained
matrix be represented as
r
. Note that the lines corresponding to different
l
for
θ
the particular
θ
are parallel to each other.
5. Compute
r
for all values of
θ
with some resolution for
θ
.
θ
6. Thus
fd
(discrete version of
f
) is obtained as
θ
=
π
θ
=
0
r
.
θ
Instead of filling the matrix with the particular value in the co-ordinates corre-
sponding to the particular line tilted with an angle
anticlockwise, fill the matrix
in the particular column and tilt the matrix by an angle
θ
θ
anticlockwise. Thus the
matrix
f
d
can also be obtained as follows.
1. Create the zero matrix
r
.
2. Fill the first row of the matrix
r
with the vector
R
.
3. All other rows of the matrix
r
are also filled up with the same vectors.
4. Rotate the matrix
r
by an angle
(
l
,θ)
∗
IFT
(
|
U
|
)
θ
anticlockwise to obtain
r
θ
.
5. Compute
r
θ
for all values of
θ
with some resolution for
θ
.
is obtained as
θ
=
π
θ
=
6. Thus
f
(
x
,
y
)
0
r
θ
.
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