Image Processing Reference
In-Depth Information
Exercise 3.3.
In two dimensions, how many points are there such that their coordi-
nate vectors have the norm (i) 5, (ii) 0?
HINT: Can we count them in both cases?
Once it is defined on a
vector space
, an important usage of the norm is in the
computation of the distance between two points of that space. The difference
u
−
v
(3.14)
is a vector if the vectors
u
and
v
belong to the same vector space, because in such
a space addition and scaling are well defined and the space is closed. The distance
between two points is defined as
u
−
v
(3.15)
This is natural because when
=0, then we know that the two vectors are one
and the same. This follows from the properties of the norm, namely that if the norm
is the null then the vector is null vector,
u
u
−
v
v
=
0
, which is the same as
u
=
v
.
There are two other properties that a norm must have before it can properly be
called a norm:
−
•
When a vector is scaled by a scalar, its norm must scale with the magnitude of
the scalar.
•
The vector space must yield the
triangle inequality
under the norm, i.e., the short-
est distance between two points is the norm of the vector joining them.
We summarize the
norm properties
that every norm, regardless of the vector
space on which it is defined, must have as follows
0
≤
u
,
Nonnegativity
(3.16)
u
=0
⇔
u
=0
,
Nullness
(3.17)
α
u
=
|
α
|
u
,
Scaling
(3.18)
u
+
v
≤
u
+
v
.
Triangle inequality
(3.19)
We already know that discrete color images constitute a vector space. By using
the rules above, we can add a norm to a vector space that can be used to represent
discrete images having vector-valued pixels.
Exercise 3.4.
Let
A
be an image with vector-valued pixels. Show that the expression
=[
ijk
A
∗
(
i, j, k
)
A
(
i, j, k
)]
1
/
2
A
(3.20)
where
∗
is the complex conjugate, is a norm.
HINT: This rule obeys the triangle inequality under the addition and scaling rules of
Eq. (3.11)-(3.12).
