Image Processing Reference
In-Depth Information
In Fig. 3.6, we show the nine basis vectors obtained via Eq. (3.47), and the re-
construction coefficients of the example matrix:
⎛
⎞
124
232
002
⎝
⎠
F
=
(3.51)
The matrix is expanded in this basis and is reconstructed by a weighted summation
of the same basis with the weighting coefficients
c
i
.
We generalize the result of the example as a theorem.
Theorem 3.1.
Let
u
i
∈
=
u
i
u
j
=0
E
N
be orthogonal to each other, i.e.,
u
i
,
u
j
=
j
. Then the (tensor) products of these,
U
k
=
u
i
u
j
, are also orthogonal to
each other, i.e.,
for
i
=
mn
U
k
(
m, n
)
U
l
(
m, n
)=0
when
k
=
l
.
We can construct images where image values are not gray values but tensors.
A gray image can thus be viewed as a
field
with
zero-order tensors
. A color image
can be viewed as a
field
having
first-order tensors
as values. In Sect. 10.3 we will
discuss the direction as a
second-order tensor
. An image depicting the local direction
will thus be a
field of second-order tensors
. Evidently, even fields of tensors are
Hilbert spaces as they are multi-index arrays, which are Hilbert spaces. We restate
this property for the sake of completeness.
Lemma 3.4.
With the following scalar products:
U
k
,
U
l
=
ij
A
(
i, j, k
)
∗
B
(
i, j, k
)
A
,
B
(3.52)
k
=
ij
A
(
i, j, k, l
)
∗
B
(
i, j, k, l
)
A
,
B
(3.53)
kl
(where
k, l
run over the components of the tensor indices, and
i, j
run over the dis-
crete points of the image) tensor fields up to the second degree having the same
dimension constitute Hilbert spaces.
3.7 Schwartz Inequality, Angles and Similarity of Images
We explore here the concept of “angle” which we already met, at least verbally in
connection with the “orthogonality” which has visual conotations with the right an-
gles. For this too, we need the scalar product. First, we present an important inequal-
ity, known as the Schwartz inequality.
Theorem 3.2 (Schwartz inequality I).
The
Schwartz inequality
,
|
u
,
v
| ≤
u
v
(3.54)
holds for Hilbert spaces.
