<s>
Scale-space	B-Algorithm
theory	I-Algorithm
is	O
a	O
framework	O
for	O
multi-scale	B-Algorithm
signal	O
representation	O
developed	O
by	O
the	O
computer	B-Application
vision	I-Application
,	O
image	B-Algorithm
processing	I-Algorithm
and	O
signal	O
processing	O
communities	O
with	O
complementary	O
motivations	O
from	O
physics	O
and	O
biological	O
vision	O
.	O
</s>
<s>
It	O
is	O
a	O
formal	O
theory	O
for	O
handling	O
image	O
structures	O
at	O
different	O
scales	O
,	O
by	O
representing	O
an	O
image	O
as	O
a	O
one-parameter	O
family	O
of	O
smoothed	B-Application
images	O
,	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
,	O
parametrized	O
by	O
the	O
size	O
of	O
the	O
smoothing	B-Application
kernel	O
used	O
for	O
suppressing	O
fine-scale	O
structures	O
.	O
</s>
<s>
The	O
parameter	O
in	O
this	O
family	O
is	O
referred	O
to	O
as	O
the	O
scale	O
parameter	O
,	O
with	O
the	O
interpretation	O
that	O
image	O
structures	O
of	O
spatial	O
size	O
smaller	O
than	O
about	O
have	O
largely	O
been	O
smoothed	B-Application
away	O
in	O
the	O
scale-space	B-Algorithm
level	O
at	O
scale	O
.	O
</s>
<s>
The	O
main	O
type	O
of	O
scale	B-Algorithm
space	I-Algorithm
is	O
the	O
linear	O
(	O
Gaussian	O
)	O
scale	B-Algorithm
space	I-Algorithm
,	O
which	O
has	O
wide	O
applicability	O
as	O
well	O
as	O
the	O
attractive	O
property	O
of	O
being	O
possible	O
to	O
derive	O
from	O
a	O
small	O
set	O
of	O
scale-space	B-Algorithm
axioms	I-Algorithm
.	O
</s>
<s>
The	O
corresponding	O
scale-space	B-Algorithm
framework	O
encompasses	O
a	O
theory	O
for	O
Gaussian	O
derivative	B-Algorithm
operators	O
,	O
which	O
can	O
be	O
used	O
as	O
a	O
basis	O
for	O
expressing	O
a	O
large	O
class	O
of	O
visual	O
operations	O
for	O
computerized	O
systems	O
that	O
process	O
visual	O
information	O
.	O
</s>
<s>
The	O
notion	O
of	O
scale	B-Algorithm
space	I-Algorithm
applies	O
to	O
signals	O
of	O
arbitrary	O
numbers	O
of	O
variables	O
.	O
</s>
<s>
where	O
the	O
semicolon	O
in	O
the	O
argument	O
of	O
implies	O
that	O
the	O
convolution	B-Language
is	O
performed	O
only	O
over	O
the	O
variables	O
,	O
while	O
the	O
scale	O
parameter	O
after	O
the	O
semicolon	O
just	O
indicates	O
which	O
scale	O
level	O
is	O
being	O
defined	O
.	O
</s>
<s>
This	O
definition	O
of	O
works	O
for	O
a	O
continuum	O
of	O
scales	O
,	O
but	O
typically	O
only	O
a	O
finite	O
discrete	O
set	O
of	O
levels	O
in	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
would	O
be	O
actually	O
considered	O
.	O
</s>
<s>
The	O
scale	O
parameter	O
is	O
the	O
variance	O
of	O
the	O
Gaussian	O
filter	O
and	O
as	O
a	O
limit	O
for	O
the	O
filter	O
becomes	O
an	O
impulse	O
function	O
such	O
that	O
that	O
is	O
,	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
at	O
scale	O
level	O
is	O
the	O
image	O
itself	O
.	O
</s>
<s>
As	O
increases	O
,	O
is	O
the	O
result	O
of	O
smoothing	B-Application
with	O
a	O
larger	O
and	O
larger	O
filter	O
,	O
thereby	O
removing	O
more	O
and	O
more	O
of	O
the	O
details	O
that	O
the	O
image	O
contains	O
.	O
</s>
<s>
When	O
faced	O
with	O
the	O
task	O
of	O
generating	O
a	O
multi-scale	B-Algorithm
representation	O
one	O
may	O
ask	O
:	O
could	O
any	O
filter	O
g	O
of	O
low-pass	O
type	O
and	O
with	O
a	O
parameter	O
t	O
which	O
determines	O
its	O
width	O
be	O
used	O
to	O
generate	O
a	O
scale	B-Algorithm
space	I-Algorithm
?	O
</s>
<s>
The	O
answer	O
is	O
no	O
,	O
as	O
it	O
is	O
of	O
crucial	O
importance	O
that	O
the	O
smoothing	B-Application
filter	O
does	O
not	O
introduce	O
new	O
spurious	O
structures	O
at	O
coarse	O
scales	O
that	O
do	O
not	O
correspond	O
to	O
simplifications	O
of	O
corresponding	O
structures	O
at	O
finer	O
scales	O
.	O
</s>
<s>
In	O
the	O
scale-space	B-Algorithm
literature	O
,	O
a	O
number	O
of	O
different	O
ways	O
have	O
been	O
expressed	O
to	O
formulate	O
this	O
criterion	O
in	O
precise	O
mathematical	O
terms	O
.	O
</s>
<s>
The	O
conclusion	O
from	O
several	O
different	O
axiomatic	O
derivations	B-Algorithm
that	O
have	O
been	O
presented	O
is	O
that	O
the	O
Gaussian	O
scale	B-Algorithm
space	I-Algorithm
constitutes	O
the	O
canonical	O
way	O
to	O
generate	O
a	O
linear	O
scale	B-Algorithm
space	I-Algorithm
,	O
based	O
on	O
the	O
essential	O
requirement	O
that	O
new	O
structures	O
must	O
not	O
be	O
created	O
when	O
going	O
from	O
a	O
fine	O
scale	O
to	O
any	O
coarser	O
scale	O
.	O
</s>
<s>
Conditions	O
,	O
referred	O
to	O
as	O
scale-space	B-Algorithm
axioms	I-Algorithm
,	O
that	O
have	O
been	O
used	O
for	O
deriving	O
the	O
uniqueness	O
of	O
the	O
Gaussian	O
kernel	O
include	O
linearity	O
,	O
shift	O
invariance	O
,	O
semi-group	O
structure	O
,	O
non-enhancement	O
of	O
local	O
extrema	O
,	O
scale	O
invariance	O
and	O
rotational	O
invariance	O
.	O
</s>
<s>
In	O
the	O
works	O
,	O
the	O
uniqueness	O
claimed	O
in	O
the	O
arguments	O
based	O
on	O
scale	O
invariance	O
has	O
been	O
criticized	O
,	O
and	O
alternative	O
self-similar	O
scale-space	B-Algorithm
kernels	O
have	O
been	O
proposed	O
.	O
</s>
<s>
The	O
Gaussian	O
kernel	O
is	O
,	O
however	O
,	O
a	O
unique	O
choice	O
according	O
to	O
the	O
scale-space	B-Algorithm
axiomatics	O
based	O
on	O
causality	O
or	O
non-enhancement	O
of	O
local	O
extrema	O
.	O
</s>
<s>
Equivalently	O
,	O
the	O
scale-space	B-Algorithm
family	O
can	O
be	O
defined	O
as	O
the	O
solution	O
of	O
the	O
diffusion	O
equation	O
(	O
for	O
example	O
in	O
terms	O
of	O
the	O
heat	O
equation	O
)	O
,	O
</s>
<s>
This	O
formulation	O
of	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
L	O
means	O
that	O
it	O
is	O
possible	O
to	O
interpret	O
the	O
intensity	O
values	O
of	O
the	O
image	O
f	O
as	O
a	O
"	O
temperature	O
distribution	O
"	O
in	O
the	O
image	O
plane	O
and	O
that	O
the	O
process	O
that	O
generates	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
as	O
a	O
function	O
of	O
t	O
corresponds	O
to	O
heat	O
diffusion	O
in	O
the	O
image	O
plane	O
over	O
time	O
t	O
(	O
assuming	O
the	O
thermal	O
conductivity	O
of	O
the	O
material	O
equal	O
to	O
the	O
arbitrarily	O
chosen	O
constant	O
½	O
)	O
.	O
</s>
<s>
Although	O
this	O
connection	O
may	O
appear	O
superficial	O
for	O
a	O
reader	O
not	O
familiar	O
with	O
differential	O
equations	O
,	O
it	O
is	O
indeed	O
the	O
case	O
that	O
the	O
main	O
scale-space	B-Algorithm
formulation	O
in	O
terms	O
of	O
non-enhancement	O
of	O
local	O
extrema	O
is	O
expressed	O
in	O
terms	O
of	O
a	O
sign	O
condition	O
on	O
partial	O
derivatives	B-Algorithm
in	O
the	O
2+	O
1-D	O
volume	O
generated	O
by	O
the	O
scale	B-Algorithm
space	I-Algorithm
,	O
thus	O
within	O
the	O
framework	O
of	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
Furthermore	O
,	O
a	O
detailed	O
analysis	O
of	O
the	O
discrete	O
case	O
shows	O
that	O
the	O
diffusion	O
equation	O
provides	O
a	O
unifying	O
link	O
between	O
continuous	O
and	O
discrete	O
scale	B-Algorithm
spaces	I-Algorithm
,	O
which	O
also	O
generalizes	O
to	O
nonlinear	O
scale	B-Algorithm
spaces	I-Algorithm
,	O
for	O
example	O
,	O
using	O
anisotropic	B-Algorithm
diffusion	I-Algorithm
.	O
</s>
<s>
Hence	O
,	O
one	O
may	O
say	O
that	O
the	O
primary	O
way	O
to	O
generate	O
a	O
scale	B-Algorithm
space	I-Algorithm
is	O
by	O
the	O
diffusion	O
equation	O
,	O
and	O
that	O
the	O
Gaussian	O
kernel	O
arises	O
as	O
the	O
Green	O
's	O
function	O
of	O
this	O
specific	O
partial	O
differential	O
equation	O
.	O
</s>
<s>
The	O
motivation	O
for	O
generating	O
a	O
scale-space	B-Algorithm
representation	I-Algorithm
of	O
a	O
given	O
data	O
set	O
originates	O
from	O
the	O
basic	O
observation	O
that	O
real-world	O
objects	O
are	O
composed	O
of	O
different	O
structures	O
at	O
different	O
scales	O
.	O
</s>
<s>
For	O
a	O
computer	B-Application
vision	I-Application
system	O
analysing	O
an	O
unknown	O
scene	O
,	O
there	O
is	O
no	O
way	O
to	O
know	O
a	O
priori	O
what	O
scales	O
are	O
appropriate	O
for	O
describing	O
the	O
interesting	O
structures	O
in	O
the	O
image	O
data	O
.	O
</s>
<s>
Taken	O
to	O
the	O
limit	O
,	O
a	O
scale-space	B-Algorithm
representation	I-Algorithm
considers	O
representations	O
at	O
all	O
scales	O
.	O
</s>
<s>
Another	O
motivation	O
to	O
the	O
scale-space	B-Algorithm
concept	O
originates	O
from	O
the	O
process	O
of	O
performing	O
a	O
physical	O
measurement	O
on	O
real-world	O
data	O
.	O
</s>
<s>
The	O
scale-space	B-Algorithm
theory	I-Algorithm
on	O
the	O
other	O
hand	O
explicitly	O
incorporates	O
the	O
need	O
for	O
a	O
non-infinitesimal	O
size	O
of	O
the	O
image	O
operators	O
as	O
an	O
integral	O
part	O
of	O
any	O
measurement	O
as	O
well	O
as	O
any	O
other	O
operation	O
that	O
depends	O
on	O
a	O
real-world	O
measurement	O
.	O
</s>
<s>
There	O
is	O
a	O
close	O
link	O
between	O
scale-space	B-Algorithm
theory	I-Algorithm
and	O
biological	O
vision	O
.	O
</s>
<s>
Many	O
scale-space	B-Algorithm
operations	O
show	O
a	O
high	O
degree	O
of	O
similarity	O
with	O
receptive	O
field	O
profiles	O
recorded	O
from	O
the	O
mammalian	O
retina	O
and	O
the	O
first	O
stages	O
in	O
the	O
visual	O
cortex	O
.	O
</s>
<s>
In	O
these	O
respects	O
,	O
the	O
scale-space	B-Algorithm
framework	O
can	O
be	O
seen	O
as	O
a	O
theoretically	O
well-founded	O
paradigm	O
for	O
early	O
vision	O
,	O
which	O
in	O
addition	O
has	O
been	O
thoroughly	O
tested	O
by	O
algorithms	O
and	O
experiments	O
.	O
</s>
<s>
At	O
any	O
scale	O
in	O
scale	B-Algorithm
space	I-Algorithm
,	O
we	O
can	O
apply	O
local	O
derivative	B-Algorithm
operators	O
to	O
the	O
scale-space	B-Algorithm
representation	I-Algorithm
:	O
</s>
<s>
Due	O
to	O
the	O
commutative	O
property	O
between	O
the	O
derivative	B-Algorithm
operator	O
and	O
the	O
Gaussian	O
smoothing	B-Application
operator	O
,	O
such	O
scale-space	B-Algorithm
derivatives	B-Algorithm
can	O
equivalently	O
be	O
computed	O
by	O
convolving	O
the	O
original	O
image	O
with	O
Gaussian	O
derivative	B-Algorithm
operators	O
.	O
</s>
<s>
For	O
this	O
reason	O
they	O
are	O
often	O
also	O
referred	O
to	O
as	O
Gaussian	O
derivatives	B-Algorithm
:	O
</s>
<s>
The	O
uniqueness	O
of	O
the	O
Gaussian	O
derivative	B-Algorithm
operators	O
as	O
local	O
operations	O
derived	O
from	O
a	O
scale-space	B-Algorithm
representation	I-Algorithm
can	O
be	O
obtained	O
by	O
similar	O
axiomatic	O
derivations	B-Algorithm
as	O
are	O
used	O
for	O
deriving	O
the	O
uniqueness	O
of	O
the	O
Gaussian	O
kernel	O
for	O
scale-space	B-Algorithm
smoothing	B-Application
.	O
</s>
<s>
These	O
Gaussian	O
derivative	B-Algorithm
operators	O
can	O
in	O
turn	O
be	O
combined	O
by	O
linear	O
or	O
non-linear	O
operators	O
into	O
a	O
larger	O
variety	O
of	O
different	O
types	O
of	O
feature	O
detectors	O
,	O
which	O
in	O
many	O
cases	O
can	O
be	O
well	O
modelled	O
by	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
Specifically	O
,	O
invariance	O
(	O
or	O
more	O
appropriately	O
covariance	O
)	O
to	O
local	O
geometric	O
transformations	O
,	O
such	O
as	O
rotations	O
or	O
local	O
affine	O
transformations	O
,	O
can	O
be	O
obtained	O
by	O
considering	O
differential	O
invariants	O
under	O
the	O
appropriate	O
class	O
of	O
transformations	O
or	O
alternatively	O
by	O
normalizing	O
the	O
Gaussian	O
derivative	B-Algorithm
operators	O
to	O
a	O
locally	O
determined	O
coordinate	O
frame	O
determined	O
from	O
e.g.	O
</s>
<s>
a	O
preferred	O
orientation	O
in	O
the	O
image	O
domain	O
,	O
or	O
by	O
applying	O
a	O
preferred	O
local	O
affine	O
transformation	O
to	O
a	O
local	O
image	O
patch	O
(	O
see	O
the	O
article	O
on	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
for	O
further	O
details	O
)	O
.	O
</s>
<s>
When	O
Gaussian	O
derivative	B-Algorithm
operators	O
and	O
differential	O
invariants	O
are	O
used	O
in	O
this	O
way	O
as	O
basic	O
feature	O
detectors	O
at	O
multiple	O
scales	O
,	O
the	O
uncommitted	O
first	O
stages	O
of	O
visual	O
processing	O
are	O
often	O
referred	O
to	O
as	O
a	O
visual	O
front-end	O
.	O
</s>
<s>
This	O
overall	O
framework	O
has	O
been	O
applied	O
to	O
a	O
large	O
variety	O
of	O
problems	O
in	O
computer	B-Application
vision	I-Application
,	O
including	O
feature	O
detection	O
,	O
feature	B-General_Concept
classification	I-General_Concept
,	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
image	B-Algorithm
matching	I-Algorithm
,	O
motion	O
estimation	O
,	O
computation	O
of	O
shape	O
cues	O
and	O
object	O
recognition	O
.	O
</s>
<s>
The	O
set	O
of	O
Gaussian	O
derivative	B-Algorithm
operators	O
up	O
to	O
a	O
certain	O
order	O
is	O
often	O
referred	O
to	O
as	O
the	O
N-jet	B-Algorithm
and	O
constitutes	O
a	O
basic	O
type	O
of	O
feature	O
within	O
the	O
scale-space	B-Algorithm
framework	O
.	O
</s>
<s>
In	O
an	O
analogous	O
fashion	O
,	O
corner	B-Algorithm
detectors	I-Algorithm
and	O
ridge	O
and	O
valley	O
detectors	O
can	O
be	O
expressed	O
as	O
local	O
maxima	O
,	O
minima	O
or	O
zero-crossings	B-Algorithm
of	O
multi-scale	B-Algorithm
differential	O
invariants	O
defined	O
from	O
Gaussian	O
derivatives	B-Algorithm
.	O
</s>
<s>
The	O
algebraic	O
expressions	O
for	O
the	O
corner	O
and	O
ridge	O
detection	O
operators	O
are	O
,	O
however	O
,	O
somewhat	O
more	O
complex	O
and	O
the	O
reader	O
is	O
referred	O
to	O
the	O
articles	O
on	O
corner	B-Algorithm
detection	I-Algorithm
and	O
ridge	O
detection	O
for	O
further	O
details	O
.	O
</s>
<s>
Scale-space	B-Algorithm
operations	O
have	O
also	O
been	O
frequently	O
used	O
for	O
expressing	O
coarse-to-fine	O
methods	O
,	O
in	O
particular	O
for	O
tasks	O
such	O
as	O
image	B-Algorithm
matching	I-Algorithm
and	O
for	O
multi-scale	B-Algorithm
image	I-Algorithm
segmentation	I-Algorithm
.	O
</s>
<s>
Following	O
this	O
approach	O
of	O
gamma-normalized	O
derivatives	B-Algorithm
,	O
it	O
can	O
be	O
shown	O
that	O
different	O
types	O
of	O
scale	O
adaptive	O
and	O
scale	O
invariant	O
feature	O
detectors	O
can	O
be	O
expressed	O
for	O
tasks	O
such	O
as	O
blob	B-Algorithm
detection	I-Algorithm
,	O
corner	B-Algorithm
detection	I-Algorithm
,	O
ridge	O
detection	O
,	O
edge	B-Algorithm
detection	I-Algorithm
and	O
spatio-temporal	O
interest	O
point	O
detection	O
(	O
see	O
the	O
specific	O
articles	O
on	O
these	O
topics	O
for	O
in-depth	O
descriptions	O
of	O
how	O
these	O
scale-invariant	O
feature	O
detectors	O
are	O
formulated	O
)	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
scale	O
levels	O
obtained	O
from	O
automatic	O
scale	O
selection	O
can	O
be	O
used	O
for	O
determining	O
regions	O
of	O
interest	O
for	O
subsequent	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
to	O
obtain	O
affine	O
invariant	O
interest	O
points	O
or	O
for	O
determining	O
scale	O
levels	O
for	O
computing	O
associated	O
image	B-General_Concept
descriptors	I-General_Concept
,	O
such	O
as	O
locally	O
scale	O
adapted	O
N-jets	B-Algorithm
.	O
</s>
<s>
by	O
computing	O
local	O
image	B-General_Concept
descriptors	I-General_Concept
(	O
N-jets	B-Algorithm
or	O
local	O
histograms	O
of	O
gradient	O
directions	O
)	O
at	O
scale-adapted	O
interest	O
points	O
obtained	O
from	O
scale-space	B-Algorithm
extrema	O
of	O
the	O
normalized	O
Laplacian	O
operator	O
(	O
see	O
also	O
scale-invariant	B-Algorithm
feature	I-Algorithm
transform	I-Algorithm
)	O
or	O
the	O
determinant	O
of	O
the	O
Hessian	O
(	O
see	O
also	O
SURF	B-Algorithm
)	O
;	O
see	O
also	O
the	O
Scholarpedia	O
article	O
on	O
the	O
for	O
a	O
more	O
general	O
outlook	O
of	O
object	O
recognition	O
approaches	O
based	O
on	O
receptive	O
field	O
responses	O
in	O
terms	O
Gaussian	O
derivative	B-Algorithm
operators	O
or	O
approximations	O
thereof	O
.	O
</s>
<s>
An	O
image	B-Algorithm
pyramid	I-Algorithm
is	O
a	O
discrete	O
representation	O
in	O
which	O
a	O
scale	B-Algorithm
space	I-Algorithm
is	O
sampled	O
in	O
both	O
space	O
and	O
scale	O
.	O
</s>
<s>
When	O
properly	O
constructed	O
,	O
the	O
ratio	O
of	O
the	O
sample	O
rates	O
in	O
space	O
and	O
scale	O
are	O
held	O
constant	O
so	O
that	O
the	O
impulse	O
response	O
is	O
identical	O
in	O
all	O
levels	O
of	O
the	O
pyramid	B-Algorithm
.	O
</s>
<s>
Fast	O
,	O
O(N )	O
,	O
algorithms	O
exist	O
for	O
computing	O
a	O
scale	O
invariant	O
image	B-Algorithm
pyramid	I-Algorithm
,	O
in	O
which	O
the	O
image	O
or	O
signal	O
is	O
repeatedly	O
smoothed	B-Application
then	O
subsampled	O
.	O
</s>
<s>
Values	O
for	O
scale	B-Algorithm
space	I-Algorithm
between	O
pyramid	B-Algorithm
samples	O
can	O
easily	O
be	O
estimated	O
using	O
interpolation	O
within	O
and	O
between	O
scales	O
and	O
allowing	O
for	O
scale	O
and	O
position	O
estimates	O
with	O
sub	O
resolution	O
accuracy	O
.	O
</s>
<s>
In	O
a	O
scale-space	B-Algorithm
representation	I-Algorithm
,	O
the	O
existence	O
of	O
a	O
continuous	O
scale	O
parameter	O
makes	O
it	O
possible	O
to	O
track	O
zero	B-Algorithm
crossings	I-Algorithm
over	O
scales	O
leading	O
to	O
so-called	O
deep	O
structure	O
.	O
</s>
<s>
For	O
features	O
defined	O
as	O
zero-crossings	B-Algorithm
of	O
differential	O
invariants	O
,	O
the	O
implicit	O
function	O
theorem	O
directly	O
defines	O
trajectories	B-Application
across	O
scales	O
,	O
and	O
at	O
those	O
scales	O
where	O
bifurcations	O
occur	O
,	O
the	O
local	O
behaviour	O
can	O
be	O
modelled	O
by	O
singularity	O
theory	O
.	O
</s>
<s>
Extensions	O
of	O
linear	O
scale-space	B-Algorithm
theory	I-Algorithm
concern	O
the	O
formulation	O
of	O
non-linear	O
scale-space	B-Algorithm
concepts	O
more	O
committed	O
to	O
specific	O
purposes	O
.	O
</s>
<s>
These	O
non-linear	O
scale-spaces	B-Algorithm
often	O
start	O
from	O
the	O
equivalent	O
diffusion	O
formulation	O
of	O
the	O
scale-space	B-Algorithm
concept	O
,	O
which	O
is	O
subsequently	O
extended	O
in	O
a	O
non-linear	O
fashion	O
.	O
</s>
<s>
It	O
should	O
be	O
noted	O
,	O
however	O
,	O
that	O
not	O
all	O
of	O
these	O
non-linear	O
scale-spaces	B-Algorithm
satisfy	O
similar	O
"	O
nice	O
"	O
theoretical	O
requirements	O
as	O
the	O
linear	O
Gaussian	O
scale-space	B-Algorithm
concept	O
.	O
</s>
<s>
Hence	O
,	O
unexpected	O
artifacts	O
may	O
sometimes	O
occur	O
and	O
one	O
should	O
be	O
very	O
careful	O
of	O
not	O
using	O
the	O
term	O
"	O
scale-space	B-Algorithm
"	O
for	O
just	O
any	O
type	O
of	O
one-parameter	O
family	O
of	O
images	O
.	O
</s>
<s>
A	O
first-order	O
extension	O
of	O
the	O
isotropic	O
Gaussian	O
scale	B-Algorithm
space	I-Algorithm
is	O
provided	O
by	O
the	O
affine	O
(	O
Gaussian	O
)	O
scale	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
One	O
motivation	O
for	O
this	O
extension	O
originates	O
from	O
the	O
common	O
need	O
for	O
computing	O
image	B-General_Concept
descriptors	I-General_Concept
subject	O
for	O
real-world	O
objects	O
that	O
are	O
viewed	O
under	O
a	O
perspective	O
camera	O
model	O
.	O
</s>
<s>
To	O
handle	O
such	O
non-linear	O
deformations	O
locally	O
,	O
partial	O
invariance	O
(	O
or	O
more	O
correctly	O
covariance	O
)	O
to	O
local	O
affine	O
deformations	O
can	O
be	O
achieved	O
by	O
considering	O
affine	O
Gaussian	O
kernels	O
with	O
their	O
shapes	O
determined	O
by	O
the	O
local	O
image	O
structure	O
,	O
see	O
the	O
article	O
on	O
affine	B-Algorithm
shape	I-Algorithm
adaptation	I-Algorithm
for	O
theory	O
and	O
algorithms	O
.	O
</s>
<s>
Indeed	O
,	O
this	O
affine	O
scale	B-Algorithm
space	I-Algorithm
can	O
also	O
be	O
expressed	O
from	O
a	O
non-isotropic	O
extension	O
of	O
the	O
linear	O
(	O
isotropic	O
)	O
diffusion	O
equation	O
,	O
while	O
still	O
being	O
within	O
the	O
class	O
of	O
linear	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
There	O
exists	O
a	O
more	O
general	O
extension	O
of	O
the	O
Gaussian	O
scale-space	B-Algorithm
model	O
to	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	B-Algorithm
theory	I-Algorithm
also	O
comprises	O
other	O
types	O
of	O
variabilities	O
caused	O
by	O
geometric	O
transformations	O
in	O
the	O
image	O
formation	O
process	O
,	O
including	O
variations	O
in	O
viewing	O
direction	O
approximated	O
by	O
local	O
affine	O
transformations	O
,	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>
This	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>
There	O
are	O
strong	O
relations	O
between	O
scale-space	B-Algorithm
theory	I-Algorithm
and	O
wavelet	O
theory	O
,	O
although	O
these	O
two	O
notions	O
of	O
multi-scale	B-Algorithm
representation	O
have	O
been	O
developed	O
from	O
somewhat	O
different	O
premises	O
.	O
</s>
<s>
There	O
has	O
also	O
been	O
work	O
on	O
other	O
multi-scale	B-Algorithm
approaches	I-Algorithm
,	O
such	O
as	O
pyramids	B-Algorithm
and	O
a	O
variety	O
of	O
other	O
kernels	O
,	O
that	O
do	O
not	O
exploit	O
or	O
require	O
the	O
same	O
requirements	O
as	O
true	O
scale-space	B-Algorithm
descriptions	O
do	O
.	O
</s>
<s>
There	O
are	O
interesting	O
relations	O
between	O
scale-space	B-Algorithm
representation	I-Algorithm
and	O
biological	O
vision	O
and	O
hearing	O
.	O
</s>
<s>
that	O
can	O
be	O
well	O
modelled	O
by	O
linear	O
Gaussian	O
derivative	B-Algorithm
operators	O
,	O
in	O
some	O
cases	O
also	O
complemented	O
by	O
a	O
non-isotropic	O
affine	O
scale-space	B-Algorithm
model	O
,	O
a	O
spatio-temporal	O
scale-space	B-Algorithm
model	O
and/or	O
non-linear	O
combinations	O
of	O
such	O
linear	O
operators	O
.	O
</s>
<s>
Regarding	O
biological	O
hearing	O
there	O
are	O
receptive	O
field	O
profiles	O
in	O
the	O
inferior	O
colliculus	O
and	O
the	O
primary	O
auditory	O
cortex	O
that	O
can	O
be	O
well	O
modelled	O
by	O
spectra-temporal	O
receptive	O
fields	O
that	O
can	O
be	O
well	O
modelled	O
by	O
Gaussian	O
derivates	O
over	O
logarithmic	O
frequencies	O
and	O
windowed	O
Fourier	O
transforms	O
over	O
time	O
with	O
the	O
window	O
functions	O
being	O
temporal	O
scale-space	B-Algorithm
kernels	O
.	O
</s>
<s>
In	O
the	O
area	O
of	O
classical	O
computer	B-Application
vision	I-Application
,	O
scale-space	B-Algorithm
theory	I-Algorithm
has	O
established	O
itself	O
as	O
a	O
theoretical	O
framework	O
for	O
early	O
vision	O
,	O
with	O
Gaussian	O
derivatives	B-Algorithm
constituting	O
a	O
canonical	O
model	O
for	O
the	O
first	O
layer	O
of	O
receptive	O
fields	O
.	O
</s>
<s>
With	O
the	O
introduction	O
of	O
deep	B-Algorithm
learning	I-Algorithm
,	O
there	O
has	O
also	O
been	O
work	O
on	O
also	O
using	O
Gaussian	O
derivatives	B-Algorithm
or	O
Gaussian	O
kernels	O
as	O
a	O
general	O
basis	O
for	O
receptive	O
fields	O
in	O
deep	O
networks	O
.	O
</s>
<s>
Using	O
the	O
transformation	O
properties	O
of	O
the	O
Gaussian	O
derivatives	B-Algorithm
and	O
Gaussian	O
kernels	O
under	O
scaling	O
transformations	O
,	O
it	O
is	O
in	O
this	O
way	O
possible	O
to	O
obtain	O
scale	O
covariance/equivariance	O
and	O
scale	O
invariance	O
of	O
the	O
deep	O
network	O
to	O
handle	O
image	O
structures	O
at	O
different	O
scales	O
in	O
a	O
theoretically	O
well-founded	O
manner	O
.	O
</s>
<s>
For	O
processing	O
pre-recorded	O
temporal	O
signals	O
or	O
video	O
,	O
the	O
Gaussian	O
kernel	O
can	O
also	O
be	O
used	O
for	O
smoothing	B-Application
and	O
suppressing	O
fine-scale	O
structures	O
over	O
the	O
temporal	O
domain	O
,	O
since	O
the	O
data	O
are	O
pre-recorded	O
and	O
available	O
in	O
all	O
directions	O
.	O
</s>
<s>
When	O
processing	O
temporal	O
signals	O
or	O
video	O
in	O
real-time	O
situations	O
,	O
the	O
Gaussian	O
kernel	O
cannot	O
,	O
however	O
,	O
be	O
used	O
for	O
temporal	O
smoothing	B-Application
,	O
since	O
it	O
would	O
access	O
data	O
from	O
the	O
future	O
,	O
which	O
obviously	O
cannot	O
be	O
available	O
.	O
</s>
<s>
For	O
temporal	O
smoothing	B-Application
in	O
real-time	O
situations	O
,	O
one	O
can	O
instead	O
use	O
the	O
temporal	O
kernel	O
referred	O
to	O
as	O
the	O
time-causal	O
limit	O
kernel	O
,	O
which	O
possesses	O
similar	O
properties	O
in	O
a	O
time-causal	O
situation	O
(	O
non-creation	O
of	O
new	O
structures	O
towards	O
increasing	O
scale	O
and	O
temporal	O
scale	O
covariance	O
)	O
as	O
the	O
Gaussian	O
kernel	O
obeys	O
in	O
the	O
non-causal	O
case	O
.	O
</s>
<s>
The	O
time-causal	O
limit	O
kernel	O
corresponds	O
to	O
convolution	B-Language
with	O
an	O
infinite	O
number	O
of	O
truncated	O
exponential	O
kernels	O
coupled	O
in	O
cascade	O
,	O
with	O
specifically	O
chosen	O
time	O
constants	O
to	O
obtain	O
temporal	O
scale	O
covariance	O
.	O
</s>
<s>
For	O
an	O
earlier	O
approach	O
to	O
handling	O
temporal	O
scales	O
in	O
a	O
time-causal	O
way	O
,	O
by	O
performing	O
Gaussian	O
smoothing	B-Application
over	O
a	O
logarithmically	O
transformed	O
temporal	O
axis	O
,	O
however	O
,	O
not	O
having	O
any	O
known	O
memory-efficient	O
time-recursive	O
implementation	O
as	O
the	O
time-causal	O
limit	O
kernel	O
has	O
,	O
see	O
,	O
</s>
<s>
When	O
implementing	O
scale-space	B-Algorithm
smoothing	B-Application
in	O
practice	O
there	O
are	O
a	O
number	O
of	O
different	O
approaches	O
that	O
can	O
be	O
taken	O
in	O
terms	O
of	O
continuous	O
or	O
discrete	O
Gaussian	O
smoothing	B-Application
,	O
implementation	O
in	O
the	O
Fourier	O
domain	O
,	O
in	O
terms	O
of	O
pyramids	B-Algorithm
based	O
on	O
binomial	O
filters	O
that	O
approximate	O
the	O
Gaussian	O
or	O
using	O
recursive	O
filters	O
.	O
</s>
<s>
More	O
details	O
about	O
this	O
are	O
given	O
in	O
a	O
separate	O
article	O
on	O
scale	B-Algorithm
space	I-Algorithm
implementation	I-Algorithm
.	O
</s>
