Image Processing Reference
In-Depth Information
S
(
i, j
)=
|
2
dE
3
ω
i
ω
j
|
F
(
ω
)
(12.13)
E
3
Comparing Eq. (12.13) with Eqs. (12.8) and (12.9) yields an invertible algebraic
relationship between the matrices
S
and
J
, arising from line-fitting and plane-fitting
problems, respectively.
S
(12.14)
Here “Trace(
S
)”isthe
trace of a matrix
that is the sum of all eigenvalues of
S
.It
can be computed conveniently by summing up
S
's diagonal elements. Like
J
,the
matrix
S
is also positive semidefinite, and the solution to the plane-fitting problem is
given by the least eigenvalue of
S
and its corresponding eigenvector(s). The matrix
S
defined by Eq. (12.13) is the
structure tensor
in
n
dimensions. Both of the matri-
ces
S
and
J
are tensors because they encode physical qualities, i.e., the scatter and
the inertia along with the extremal axes of a mass distribution. The scatter and the
inertia remain the same, regardless the coordinate frames they are measured in. The
axes do not change relative the mass distribution. Relative to coordinate frames not
attached to the mass distribution, their representation varies up to a rotation, which
is a viewpoint transformation.
J
= Trace(
S
)
I
−
Lemma 12.3.
The tensors
J
and
S
have common eigenvectors, that is,
Ju
=
λ
u
⇔
Su
=
λ
u
(12.15)
with
λ
= Trace(
S
)
−
λ
(12.16)
The lemma is a consequence of the fact that
J
and
S
commute but can also be proven
immediately by utilizing the relationship Eq. (12.14) and operating with
J
on
u
,
which is assumed to be an eigenvector of
S
.
According to the lemma, fitting a line to
F
or fitting a plane to
F
can be achieved
by a quadratic form using the same tensor, i.e., either of
S
or
J
. Then, the following
lemma holds, too.
Lemma 12.4.
Let the spectrum be
3
D and that the eigenvalues of the structure tensor
S
are enumerated in ascending order,
0
≤
λ
3
≤
λ
2
≤
λ
1
. If and only if
0=
λ
3
<λ
2
(12.17)
the FT values are zero outside of a
plane
through the origin. The normal of this
plane is given by
u
3
, the least significant eigenvector of
S
. On the plane, the FT
values equal the
2
D FT of the values of
f
collected from any plane with normal
u
3
.
Similarly, if and only if
(12.18)
the FT is concentrated to a
line
through origin. The direction of this line is given by
u
1
, the most significant eigenvector of
S
. The FT values on the plane are given by
the
1
DFTof
f
lying on any of the lines with the direction
u
1
.
0=
λ
3
=
λ
2
<λ
1
