3.4 The scale parameter
In order to avoid the summing of squares, one often uses the following parametrization: 2
s
2
t, so the
Gaussian kernel get a particular short form. In N dimensions:G
ND
H
x
¸
, t
L
=
1
þþþþþþþþþþþþþþþþ
H
pt
L
N
2
e
-
x
2
þþþþþþþþ
t
. It is this t that emerges
in the diffusion equation
L
þþþþþþþ
t
=
2
L
þþþþþþþþþ
x
2
+
2
L
þþþþþþþþþ
y
2
+
2
L
þþþþþþþþþ
z
2
. It is often referred to as ’scale’ (like in: differentiation to
scale,
L
þþþþþþþ
), but a better name is variance.
To make the self-similarity of the Gaussian kernel explicit, we can introduce a new dimensionless spatial
parameter, x
ê
=
x
þþþþþþþþþþþþþ
s
!!!!
2
. We say that we have reparametrized the x-axis. Now the Gaussian kernel becomes:
g
n
H
x
ê
;
s
L
=
1
þþþþþþþþþþþþþþþþþ
s
!!!!!! !!
2
p
e
-
x
ê
2
, or g
n
H
x
ê
; t
L
=
1
þþþþþþþþþþþþþþþþ
H
pt
L
N
2
e
-
x
ê
2
. In other words: if we walk along the spatial axis in
footsteps expressed in scale-units (
s
’s), all kernels are of equal size or ’width’ (but due to the normalization
constraint not necessarily of the same amplitude). We now have a ’natural’ size of footstep to walk over the
spatial coordinate: a unit step in x is now
s
!!!!!
2 , so in more blurred images we make bigger steps. We call
this basic Gaussian kernel the natural Gaussian kernel g
n
H
x
ê
;
s
L
. The new coordinate x
ê
=
x
þþþþþþþþþþþþþ
s
!!!!
2
is called the
natural coordinate. It eliminates the scale factor
s
from the spatial coordinates, i.e. it makes the Gaussian
kernels similar, despite their different inner scales. We will encounter natural coordinates many times hereafter.
The spatial extent of the Gaussian kernel ranges from -
to +
, but in practice it has negligeable values for x
larger then a few (say 5)
s
. The numerical value at x=5
s
, and the area under the curve from x=5
s
to infinity
(recall that the total area is 1):
gauss
@
5, 1
D
N
Integrate
@
gauss
@
x, 1
D
,
8
x, 5, Infinity
<D
N
1.48672
10
-
6
2.86652
10
-
7
The larger we make the standard deviation
s
, the more the image gets blurred. In the limit to infinity, the
image becomes homogenous in intensity. The final intensity is the average intensity of the image. This is true
for an image with infinite extent, which in practice will never occur, of course. The boundary has to be taken
into account. Actually, one can take many choices what to do at the boundary, it is a matter of concensus.
Boundaries are discussed in detail in chapter 5, where practical issues of computer implementation are
discussed.
3.5 Relation to generalized functions
The Gaussian kernel is the physical equivalent of the mathematical point. It is not strictly local, like the
mathematical point, but semi-local. It has a Gaussian weighted extent, indicated by its inner scale
s
. Because
scale-space theory is revolving around the Gaussian function and its derivatives as a physical differential
operator (in more detail explained in the next chapter), we will focus here on some mathematical notions that
are directly related, i.e. the mathematical notions underlying sampling of values from functions and their
derivatives at selected points (i.e. that is why it is referred to as sampling). The mathematical functions
involved are the generalized functions, i.e. the Delta-Dirac function, the Heavyside function and the error
function. We study in the next section these functions in more detail.
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