Digital Signal Processing Reference
In-Depth Information
domain. For this, we define the directions
according to pairs of symmetric
points from the boundary of the 2D discrete Fourier spectra.
We now briefly discuss the strategy for inverting our discrete Radon Transform.
Our analytical reconstruction procedure works as follows:
1. Compute the 1D FFT transform for each set
R
ω
s(
[
p,q
]
,
·
)
to obtain
P
ω
[
[
p,q
]
.
p,q
]
ˆ
2. Substitute the sampled value of
s
on the lattice where the points fall on lines
L
ω
[
ˆ
with the sampled value of
s
on the square lattice:
ˆ
s
[
p,q
]
f
1
,f
2
=
P
ω
p,q
]
(k)
[
p,q
]
ω
2
qf
1
−
pf
2
|≤
1 the length of
L
ω
[
p,q
]
such that
|
for 0
<k<K
+
1 with
K
+
and for all the directions
[
p,q
]
.
Due to the redundancy, some Fourier coefficients belong to more than one dis-
crete line. In this case, the Fourier value is defined by the mean average:
R
1
ˆ
s
[
p
r
,q
r
]
(f
1
,f
2
)
ˆ
s(f
1
,f
2
)
=
(15.32)
q
r
f
1
−
p
r
f
2
|≤
ω
such that
2
.
R
is the number of times the pixel
(f
1
,f
2
)
belongs to
a discrete line. It depends on the frequency (it is more important at low frequencies)
and the type of discrete lines.
|
3. Apply the 2D IFFT transform.
The previous procedure allows us to obtain an exact reconstruction if the set of
directions of lines provide a complete cover of the square lattice: analytical Radon
transform followed by backprojection analytical Radon transform is a one-to-one
transform (in this case all the coefficients
s
[
p
r
,q
r
]
(f
1
,f
2
)
in Eq. (
15.32
) are equal to
ˆ
the original value of
s(f
1
,f
2
)
.
ˆ
15.5.3 Discrete Radon Based Riesz Transform
We propose now to use this discrete Radon representation to perform monogenic
analysis. Computing of a discrete Radon based monogenic analysis can now be
done as follows:
•
Apply 2D bandpass filtering to select some scale (an isotropic bandpass 2D filter-
ing). As we have seen, bandpass filtering is natural in monogenic analysis that is
usually presented either in a scale-space formalism or in a wavelet transform;
•
Process the discrete analytical Radon transform of the filtering signal
s
;
•
Process the Hilbert transform of every projection
s
θ
;
Multiply every projection by
e
j
θ
(by using the computation of
θ
explained
above);
•
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