Image Processing Reference
In-Depth Information
As in the case of
E
3
, we should get used to the idea that a function, e.g.,
f
(
t
),
is a point in the function space. This is probably the hardest part of all, since many
students do not have experience other than to imagine functions as curves placed
over the
x
-axis or surfaces placed over an (
x, y
)-plane and so on. So, we need to
imagine that the entire graph is a point among its likes in a function space. This idea
is transmitted by writing a
function
as
f
, without its argument(s), which also serves
to keep from becoming distracted by the (
t
) or whatever
function arguments
that
usually follow
f
. If one has a cosine or arctan function, for example, these will be
points in an appropriate function space. Observe that we are not changing the fact
that a function is an abstract rule that tells how to produce a
function value
(e.g., the
number that a function delivers) when coordinate arguments are given to it. When
one has a unique rule (a function), then one has a unique point (in a vector space
of functions to be discussed below) that corresponds to that function. If every point
is a function then the origin of the function space must also be a function, which is
perfectly true. The origin of the function space, the null function, is a function that
represents a unique rule: “deliver the function value zero for all arguments”. What is
more, it is contained in all vector spaces of functions!
4.2 Addition and Scaling in Vector Spaces of Functions
The next step is to have a way to obtain a new “point” (function) by a rule of scaling,
function scaling
. This will yield a different function (rule) than the one we started
with. Since we are looking for a rule, we have to define it as such and assign a new
function symbol to it. Let us call it
g
:
g
=
αf
(4.1)
We know
g
if we know a rule how to obtain its value when an argument is presented
to it. Since we know the rule for
f
, (this is
f
(
t
)), the rule to obtain the value of
g
can
be simply defined as to multiply the function value of
f
at
t
with the scalar
α
:
g
(
t
)=
αf
(
t
)
(4.2)
This is, of course, the way we are used to multiplying functions with scalars from
calculus. But now we have additionally the interpretation that this is an operation
making the corresponding vector “longer”.
We also need a way to add two vectors in the function spaces.
Function addition
is also defined in the old fashion: The new rule
h
,
h
=
f
+
g
(4.3)
is obtained by using the already known rules regarding
f
and
g
:
h
(
t
)=
f
(
t
)+
g
(
t
)
(4.4)
