Image Processing Reference
In-Depth Information
“resizing” the basis, to “add” up to the vector
x
. Assuming that we know how to
resize, and add vectors, this idea is expressed as follows.
x
=
x
(1)
e
1
+
x
(2)
e
2
+
x
(3)
e
3
(3.2)
In related literature as well as later in this topic, the basis set attached to an origin
is also called a
coordinate frame,
or just
frame
when there is no risk of interference
with other concepts. Since a basis set is the same for all points in the space,
e
i
do
not have to be written each time one needs to represent a vector. This is why they
are omitted in (3.1). Because any point
P
in
E
3
can be represented by a triplet of
real numbers relative to an origin and a basis set, as in Eq. (3.2), all points of
E
3
are
“vector” quantities. In that, the point
P
's being a vector depends on another point,
which is often not mentioned, the origin,
O
. Without
O
the point
P
would not have
been a vector. Indeed vectors are quantities with directions so that any two points
A
and
B
in
E
3
define a vector, also written as
−−
AB
or
AB
(or even as
AB
when there
is no risk of confusion) in an alternative notation to
x
.
The above concept does not need to be limited to our 3D space. The vectors
e
i
,
and thereby
x
, do not need to be basis vectors in our ordinary 3D space. They can be
any abstract quantities or “points” as long as the set they are part of is a set that has
the vector space properties defined below:
Definition 3.1.
A set of quantities
{
x
}
is called a
vector space
if the members obey
two rules
1.
Scaling with a scalar
is defined and the space is
closed
under this operation,
also called
vector scaling
;
2.
Addition
is defined and the space is closed under this operation, also called
vector addition
.
The term
closed
represents the fact that the result of an operation (vector scaling
or vector addition) never leaves the original space (the set
{
x
}
), no matter which
quantities are involved in the operation.
Scaling and addition in our 3D Euclidean space are visually meaningful. Scaling
represents making the vectors “longer” or “shorter” while adding two vectors means
that we concatenate them at their ends by translation (moving one of the vectors
without rotation). In a given basis of the
E
3
, the mechanism of scaling is to multiply
all three vector components with the scalar
α
, while the mechanism performing addi-
tion is to add the three vector components. In other words, given a representation of
E
3
, there is a method of performing scaling and addition that obeys the scaling and
addition properties without leaving the original space. Accordingly,
E
3
is a vector
space.
Using the same idea, we can construct similar scaling and addition mechanisms
for
N
-tuple arrays with
N>
3. The
N
-tuple of real numbers,
⎛
⎞
⎠
x
(1)
x
(2)
.
x
(
N
)
⎝
(3.3)
