Image Processing Reference
In-Depth Information
Accordingly this case has a nullspace with dimension 0. There is a point symmetry
in the distribution of the energy to the effect that no direction is different than other
directions.
We define the tensors
U
1
=
u
1
u
T
,
1
U
2
=
u
2
u
T
,
(12.34)
2
U
3
=
u
3
u
T
,
3
and note that these are orthogonal in the sense of Eq. (3.45) and therefore constitute
a basis. Accordingly, Eq. (12.23) yields:
S
=
λ
1
U
1
+
λ
2
U
2
+
λ
3
U
3
(12.35)
which can be interpreted as an orthogonal expansion of the structure tensor, and the
eigenvalues are the coordinates encoding the tensor
S
in terms of the basis. Inversely,
we could try to synthesize an
S
by varying the coordinates, but we would then need
to obey
(12.36)
This means that the coordinates cannot be chosen independent of each other in a
synthesis process. To simplify the synthesis, we could define a new set of coordinates
that are independent of each other as follows:
0
≤
λ
3
≤
λ
2
≤
λ
1
λ
1
=
λ
1
−
0
≤
λ
2
≤
λ
=
C
λ
2
=
λ
2
−
0
≤
λ
3
⇔
0
λ
(12.37)
0
≤
λ
3
=
λ
3
with
⎛
⎞
⎛
⎞
⎛
⎞
λ
λ
λ
3
1
10
01
−
1
001
−
λ
λ
λ
3
⎝
⎠
,
⎝
⎠
,
⎝
⎠
λ
=
C
=
λ
=
(12.38)
λ
simply encode the increments between the subsequent eigen-
values and can therefore be chosen freely, as long as they are positive or zero. The
structure tensors must thus be chosen in a cone bounded by a certain basis tensor
corresponding to the new coordinates. To find the new basis we only need to follow
the standard rule of linear algebra. Given the coordinate transformation matrix
C
,
the basis tranformation matrix equals
C
−
1
:
The new coordinates
[
U
1
U
2
U
3
]=[
U
1
U
2
U
3
]
C
−
1
(12.39)
so that we have
⎛
⎝
⎞
⎠
111
011
001
U
1
=
U
1
U
2
=
U
1
+
U
2
U
3
=
U
1
+
U
2
+
U
3
C
−
1
=
⇒
(12.40)
Consequently, the structure tensor decomposition, Eq. (12.23), can always be rear-
ranged in terms of independent coordinates as
