Digital Signal Processing Reference
In-Depth Information
Fig. 15.7
(
a
) Cover of the Fourier domain with the Euclidean line
L
[
p,q
]
.(
b
) Redundancy on the
cover of the Fourier lattice by closed naive lines, (
c
) by supercover lines
that has a central symmetry and that forms a “good” approximation of the corre-
sponding Euclidean straight line (i.e., direction). We chose discrete analytical 2D
lines. It defines not a unique line but a family of lines with a thickness parameter,
called arithmetical thickness. The arithmetical thickness provides a control over the
transform redundancy factor and properties such as the connectivity of the straight
line.
The discrete analytical lines we use for our transform are defined as follows [
1
]:
(x
1
,x
2
)
2
|
ω
2
L
ω
[
p,q
]
=
∈ Z
qx
1
−
px
2
|≤
arctan
(
p
)
) and
ω
, a function of
(p,q)
, the arithmetical thickness. It is easy to see that these discrete
analytical lines
L
ω
[
2
with
[
p,q
]∈Z
the direction of the Radon projection (we have
θ
=
have a central symmetry regardless of the value of
ω
.The
arithmetical thickness
ω
is an important parameter that controls, among other things,
the connectivity of the discrete line
L
ω
[
p,q
]
[
2
]: naive lines with
ω
=
max
(
|
p
|
,
|
q
|
)
p,q
]
where
L
ω
[
, where
L
ω
[
=|
|+|
|
is 8-connected and the supercover lines
ω
p
q
is
p,q
]
p,q
]
4-connected, for example.
We use the Fourier domain for the computation of our discrete Radon transform:
Fourier coefficients of
s
are extracted along the discrete analytical line
L
ω
[
(the
extracted points of the line are ordered in a natural way) and we take the 1D inverse
discrete Fourier transform on each value of the direction
ˆ
p,q
]
[
p,q
]
to obtain the Radon
projection
R
ω
s(
)
.
Figure
15.7
illustrates the cover of the Fourier lattice for two different types of
discrete lines. The gray value of the pixel represents the redundancy in the projection
(number of times a pixel belongs to a discrete line). One isolated line is drawn to
illustrate the shape of the discrete lines depending on its arithmetical thickness.
At last, the set of discrete directions
[
p,q
]
,
·
for a complete representation has to be
determined. The set of line segments must cover all the square lattice in the Fourier
[
p,q
]
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