Image Processing Reference
In-Depth Information
Ordinarily, one uses the same grid as the
f
j
to compute its discrete symmetry deriva-
tives, so that the discrete symmetry derivative of
f
is obtained by convolving the
discrete
f
k
with the following filter, theorem 11.2.
r
j
2
2
σ
p
1
2
πσ
p
exp(
n
2
,σ
p
}
(
r
j
)=(
D
x
+
iD
y
)
n
Γ
{
)
2
2
exp(
r
j
2
2
σ
p
=(
−
1
σ
p
1
2
πσ
p
)
n
(
x
+
iy
)
n
)
(14.16)
2
In analogy with Chapter 10, where we computed the elements of the structure
tensor (the group direction tensor for 2-folded symmetry), we conclude that even the
group direction tensor [(
D
x
+
iD
y
)
n
2
f
]
2
is band-limited because (
D
x
+
iD
y
)
n
2
f
is
band-limited. Accordingly, there is an interpolation function
μ
2
that can reconstruct
the former from its discrete elements as
[(
D
x
+
iD
y
)
n
2
f
]
2
(
r
)=
j
[(
D
x
+
iD
y
)
n
2
f
]
2
(
r
j
)
μ
2
(
r
−
r
j
)
(14.17)
Then, assuming a Gaussian as an interpolator, an element of the group direction
tensor for
n
-folded symmetry:
I
n,
0
=
E
2
[(
D
x
+
iD
y
)
n
2
f
]
2
(
r
j
)
μ
2
(
r
−
r
j
)
dxdy
j
2
f
]
2
(
r
j
)
=
j
[(
D
x
+
iD
y
)
n
μ
2
(
r
−
r
j
)
dxdy
E
2
=
j
[(
D
x
+
iD
y
)
n
2
f
]
2
(
r
j
)
μ
2
(
r
j
)
dxdy
(14.18)
is obtained by an ordinary Gaussian smoothing of the (pixelwise) square of the com-
plex image delivered by the computation scheme given in Eq. (14.14).
Following a similar reasoning, we can obtain the remaining tensor element,
I
n
2
,
,
n
2
which is always real and nonnegative. We summarize our findings as a theorem.
Theorem 14.2 (Group direction tensor II).
The complex group direction number,
k
n
min
=exp(
iθ
min
)
, associated with an
n
-folded symmetric line set fitted to the
power spectrum in the TLS error sense and the extremal TLS errors of the fit are
given by
[(
D
x
+
iD
y
)
n
1
4
π
2
I
n,
0
=(
e
(
k
n
e
(
k
n
min
)) exp
iθ
min
=
2
f
(
r
)]
2
d
r
)
−
max
n
2
,σ
p
}
[
Γ
{
f
]
2
=
∗
(14.19)
1
4
π
2
(
D
x
+
iD
y
)
n
=(
e
(
k
n
max
)+
e
(
k
n
min
2
d
r
I
n
2
)) =
|
2
f
(
r
)
|
,
n
2
n
2
,σ
p
}
=
|Γ
{
∗ f |
2
(14.20)
n
2
,σ
p
}
is defined according to (11.97).
where
Γ
{
