Image Processing Reference
In-Depth Information
Fig. 12.1. The
green axes
show the 3D Euclidean space in which a linearly symmetric image
is defined, the
dashed triangular plane
.The
red axes
show the corresponding 3D Fourier
transform domain axes in which a concentration of the energy to the axis represented by the
vector
k
occurs
Lemma 12.1.
A linearly symmetric image,
f
(
r
)=
g
(
k
T
0
r
)
, has a Fourier transform
concentrated to a line through the origin:
T
k
0
)
δ
(
T
u
1
)
δ
(
T
u
2
)
T
u
N−
1
)
F
(
ω
)=
G
(
ω
ω
ω
···
δ
(
ω
where
k
0
,
u
1
···
u
N−
1
are orthonormal vectors in
E
N
, and
δ
is the Dirac distribu-
tion.
G
is the one-dimensional Fourier transform of
g
.
The lemma states that the function
g
(
k
T
0
r
), which is in general a “spread” func-
tion, is compressed to a line, even for functions defined on spaces with higher di-
mension than two. To detect linearly symmetric functions is consequently the same
as to check whether or not the energy is concentrated to a line in the Fourier domain.
To simplify the discussion, we assume that
f
is defined on
E
3
and we attempt to fit
an axis to
F
, through the origin of the corresponding 3D Fourier transform domain.
e
L
(
k
)=
E
3
(
d
L
(
,
k
))
2
|
|
2
dE
3
min
k
=1
ω
F
(
ω
)
(12.5)
ω
=(
ω
1
,ω
2
,ω
3
)
T
represents the frequency coordinates corresponding to
the spatial coordinates
r
=(
x
1
,x
2
,x
3
)
T
. The
dE
3
where
is equal to
dω
1
dω
2
dω
3
, and the
