<s>
In	O
image	B-Algorithm
processing	I-Algorithm
and	O
computer	B-Application
vision	I-Application
,	O
a	O
scale	B-Algorithm
space	I-Algorithm
framework	O
can	O
be	O
used	O
to	O
represent	O
an	O
image	O
as	O
a	O
family	O
of	O
gradually	O
smoothed	O
images	O
.	O
</s>
<s>
This	O
framework	O
is	O
very	O
general	O
and	O
a	O
variety	O
of	O
scale	B-Algorithm
space	I-Algorithm
representations	I-Algorithm
exist	O
.	O
</s>
<s>
A	O
typical	O
approach	O
for	O
choosing	O
a	O
particular	O
type	O
of	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
is	O
to	O
establish	O
a	O
set	O
of	O
scale-space	B-Algorithm
axioms	I-Algorithm
,	O
describing	O
basic	O
properties	O
of	O
the	O
desired	O
scale-space	B-Algorithm
representation	I-Algorithm
and	O
often	O
chosen	O
so	O
as	O
to	O
make	O
the	O
representation	O
useful	O
in	O
practical	O
applications	O
.	O
</s>
<s>
Once	O
established	O
,	O
the	O
axioms	O
narrow	O
the	O
possible	O
scale-space	B-Algorithm
representations	I-Algorithm
to	O
a	O
smaller	O
class	O
,	O
typically	O
with	O
only	O
a	O
few	O
free	O
parameters	O
.	O
</s>
<s>
A	O
set	O
of	O
standard	O
scale	B-Algorithm
space	I-Algorithm
axioms	I-Algorithm
,	O
discussed	O
below	O
,	O
leads	O
to	O
the	O
linear	O
Gaussian	O
scale-space	B-Algorithm
,	O
which	O
is	O
the	O
most	O
common	O
type	O
of	O
scale	B-Algorithm
space	I-Algorithm
used	O
in	O
image	B-Algorithm
processing	I-Algorithm
and	O
computer	B-Application
vision	I-Application
.	O
</s>
<s>
The	O
linear	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
of	O
signal	O
obtained	O
by	O
smoothing	O
with	O
the	O
Gaussian	O
kernel	B-Algorithm
satisfies	O
a	O
number	O
of	O
properties	O
'	O
scale-space	B-Algorithm
axioms	I-Algorithm
 '	O
that	O
make	O
it	O
a	O
special	O
form	O
of	O
multi-scale	O
representation	O
:	O
</s>
<s>
In	O
fact	O
,	O
it	O
can	O
be	O
shown	O
that	O
the	O
Gaussian	O
kernel	B-Algorithm
is	O
a	O
unique	O
choice	O
given	O
several	O
different	O
combinations	O
of	O
subsets	O
of	O
these	O
scale-space	O
axioms:xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx1	O
[	O
Lindeberg	O
,	O
T	O
.	O
Generalized	O
axiomatic	O
scale-space	B-Algorithm
theory	I-Algorithm
,	O
Advances	O
in	O
Imaging	O
and	O
Electron	O
Physics	O
,	O
Elsevier	O
,	O
volume	O
178	O
,	O
pages	O
1-96	O
,	O
2013	O
.	O
</s>
<s>
most	O
of	O
the	O
axioms	O
(	O
linearity	O
,	O
shift-invariance	O
,	O
semigroup	O
)	O
correspond	O
to	O
scaling	O
being	O
a	O
semigroup	O
of	O
shift-invariant	O
linear	O
operator	O
,	O
which	O
is	O
satisfied	O
by	O
a	O
number	O
of	O
families	O
integral	B-Algorithm
transforms	I-Algorithm
,	O
while	O
"	O
non-creation	O
of	O
local	O
extrema	O
"	O
for	O
one-dimensional	O
signals	O
or	O
"	O
non-enhancement	O
of	O
local	O
extrema	O
"	O
for	O
higher-dimensional	O
signals	O
are	O
the	O
crucial	O
axioms	O
which	O
relate	O
scale-spaces	B-Algorithm
to	O
smoothing	O
(	O
formally	O
,	O
parabolic	O
partial	O
differential	O
equations	O
)	O
,	O
and	O
hence	O
select	O
for	O
the	O
Gaussian	O
.	O
</s>
<s>
The	O
Gaussian	O
kernel	B-Algorithm
is	O
also	O
separable	O
in	O
Cartesian	O
coordinates	O
,	O
i.e.	O
</s>
<s>
Separability	O
is	O
,	O
however	O
,	O
not	O
counted	O
as	O
a	O
scale-space	B-Algorithm
axiom	O
,	O
since	O
it	O
is	O
a	O
coordinate	O
dependent	O
property	O
related	O
to	O
issues	O
of	O
implementation	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
requirement	O
of	O
separability	O
in	O
combination	O
with	O
rotational	O
symmetry	O
per	O
se	O
fixates	O
the	O
smoothing	O
kernel	B-Algorithm
to	O
be	O
a	O
Gaussian	O
.	O
</s>
<s>
There	O
exists	O
a	O
generalization	O
of	O
the	O
Gaussian	O
scale-space	B-Algorithm
theory	I-Algorithm
to	O
more	O
general	O
affine	O
and	O
spatio-temporal	O
scale-spaces	B-Algorithm
.	O
</s>
<s>
In	O
addition	O
to	O
variabilities	O
over	O
scale	O
,	O
which	O
original	O
scale-space	B-Algorithm
theory	I-Algorithm
was	O
designed	O
to	O
handle	O
,	O
this	O
generalized	O
scale-space	O
theory''	O
also	O
comprises	O
other	O
types	O
of	O
variabilities	O
,	O
including	O
image	O
deformations	O
caused	O
by	O
viewing	O
variations	O
,	O
approximated	O
by	O
local	O
affine	B-Algorithm
transformations	I-Algorithm
,	O
and	O
relative	O
motions	O
between	O
objects	O
in	O
the	O
world	O
and	O
the	O
observer	O
,	O
approximated	O
by	O
local	O
Galilean	O
transformations	O
.	O
</s>
<s>
In	O
this	O
theory	O
,	O
rotational	O
symmetry	O
is	O
not	O
imposed	O
as	O
a	O
necessary	O
scale-space	B-Algorithm
axiom	O
and	O
is	O
instead	O
replaced	O
by	O
requirements	O
of	O
affine	O
and/or	O
Galilean	O
covariance	O
.	O
</s>
<s>
The	O
generalized	O
scale-space	B-Algorithm
theory	I-Algorithm
leads	O
to	O
predictions	O
about	O
receptive	O
field	O
profiles	O
in	O
good	O
qualitative	O
agreement	O
with	O
receptive	O
field	O
profiles	O
measured	O
by	O
cell	O
recordings	O
in	O
biological	O
vision	O
.	O
</s>
<s>
In	O
the	O
computer	B-Application
vision	I-Application
,	O
image	B-Algorithm
processing	I-Algorithm
and	O
signal	O
processing	O
literature	O
there	O
are	O
many	O
other	O
multi-scale	B-Algorithm
approaches	I-Algorithm
,	O
using	O
wavelets	O
and	O
a	O
variety	O
of	O
other	O
kernels	B-Algorithm
,	O
that	O
do	O
not	O
exploit	O
or	O
require	O
the	O
same	O
requirements	O
as	O
scale	B-Algorithm
space	I-Algorithm
descriptions	O
do	O
;	O
please	O
see	O
the	O
article	O
on	O
related	O
multi-scale	B-Algorithm
approaches	I-Algorithm
.	O
</s>
<s>
There	O
has	O
also	O
been	O
work	O
on	O
discrete	O
scale-space	B-Algorithm
concepts	O
that	O
carry	O
the	O
scale-space	B-Algorithm
properties	O
over	O
to	O
the	O
discrete	O
domain	O
;	O
see	O
the	O
article	O
on	O
scale	B-Algorithm
space	I-Algorithm
implementation	I-Algorithm
for	O
examples	O
and	O
references	O
.	O
</s>
