Image Processing Reference
In-Depth Information
u
(
i
)
∗
u
(
i
) is a real number which is strictly positive unless
u
(
i
)=0, in which case
u
(
i
)
∗
u
(
i
)=0. Accordingly,
u
2
=
u
,
u
=
i
|
u
(
i
)
|
2
(3.28)
is strictly positive except when
u
=
0
. Conversely, when
=0, the vector
u
equals the null vector so that even condition 4 on scalar products is fulfilled.
Evidently, this scalar product definition can be used also for
E
N
because
E
N
is a subset of
C
N
. Using this scalar product yields a natural generalization of the
customary length in
E
2
and
E
3
. Because of this, the norm associated with the scalar
product is also called the Euclidean norm even if
u
is complex and has a dimension
higher than 3. Another common name for this norm is the
u
L
2
norm.
2
In applied mathematics Hilbert spaces are used for approximation purposes. A
frequently used technique is to leave out some basis elements the space, that a priori
are known to have little impact on the problem solution. To do similar approxima-
tions in image analysis, we need to introduce the concept of orthogonality.
3.5 Orthogonal Expansion
In our
E
3
space, orthogonality is easy to imagine: Two vectors have the right an-
π
2
gle (
) between them. Orthogonal vectors occur most frequently in human-made
objects or environments. In a ceiling we normally have four corners, each being an
intersection of three orthogonal lines. The topics we read normally rectangular, and
at each corner there are orthogonal vectors. Fish-sticks do not swim around in the
ocean (perhaps because they have orthogonal vectors), but instead they are encoun-
tered in the human-made reality, etc.
In fact, orthogonality is even encountered in nonvisible spaces constructed by
man, the Hilbert spaces. In these spaces, which include nonfinite dimensional spaces
such as function spaces, one can express the concepts of distance and length. Ad-
ditionally, one can express the concepts of orthogonality and angle, as we discuss
below. First, let us introduce the definition of
orthogonality
.
Definition 3.3.
Two vectors are orthogonal if their scalar product vanishes
u
,
v
=0
(3.29)
In
E
3
, if we have three
orthogonal vectors
we can express all points in the space
easily by means of these vectors, the basis vectors. This is one of the reasons why
orthogonality appears in the human-made world, although it does not explain why
the fish are, post mortem, forced to appear as rectangular (frozen) blocks. The other
important reason, which does explain the fish-sticks, is that orthogonality reduces or
eliminates redundancy. Humans can store and transport more fish if fish have corners
with orthogonal vectors. For the same reason, pixels in images are quadratic and
2
It is pronounced as “L two norm”.
