Image Processing Reference
In-Depth Information
superscript
L
is a label reminding us that we are fitting a line to the spectrum. The
line-fitting process is the same as identifying iso-values of parallel planes in the
r
domain. If the energy of the Fourier transform is interpreted as the mass density,
then
e
L
(
k
) is the inertia of a mass with respect to the axis
k
. Because the integral
Eq. (12.5) is a norm, i.e.,
e
L
(
k
)=
d
L
F, d
L
F
d
L
F
2
=
(12.6)
e
L
(
k
min
) vanishes if and only if
F
is concentrated to a line.
The distance function is given by:
(
d
L
(
,
k
))
2
=
T
k
)
k
2
ω
ω
−
(
ω
=
ω
−
T
k
)
k
T
ω
−
T
k
)
k
(
ω
(
ω
T
k
is a scalar and identical
Using matrix multiplication rules and remembering that
ω
to
k
T
k
2
=
k
T
k
=1,the
quadratic form
:
ω
and
,
k
))
2
=
k
T
I
ω
T
k
(
d
L
(
T
ω
ω
−
ωω
is obtained. Thus Eq. (12.5) is expressed as
e
L
(
k
)=
k
T
Jk
(12.7)
with
⎛
⎞
J
(1
,
1)
J
(1
,
2)
J
(1
,
3)
J
(2
,
1)
J
(2
,
2)
J
(2
,
3)
J
(3
,
1)
J
(3
,
2)
J
(3
,
3)
⎝
⎠
J
=
where
J
(
i, j
)'s are given by
J
(
i, i
)=
ω
j
|
|
2
dE
3
F
(
r
)
(12.8)
E
3
j
=
i
and
|
2
dE
3
J
(
i, j
)=
−
ω
i
ω
j
|
F
(
ω
)
when
i
=
j.
(12.9)
E
3
Notice that the matrix
J
is symmetric per construction. The minimization problem
formulated in Eq. (12.5) is solved by
k
corresponding to the least eigenvalue of the
inertia matrix,
J
, of the Fourier domain [231]. All eigenvalues are real and non-
negative and the smallest eigenvalue is the minimum of
e
L
. The matrix
J
contains
sufficient information to allow computation of the optimal
k
in the TLS error sense
Eq. (12.5). As is its 2D counterpart, even this matrix is a tensor because it represents
a physical property,
inertia
.
The obtained direction will be unique if the least eigenvalue has the multipicity
2
1. When the multiplicity of the least eigenvalue is 2, there is no unique axis
k
,but
2
An
N ×N
matrix has
N
eigenvalues. If two eigenvalues are equal then that eigenvalue has
the multiplicity 2. If three eigenvalues are equal then the multiplicity is 3, and so on.
