<s>
The	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
of	O
a	O
signal	O
obtained	O
by	O
Gaussian	B-Error_Name
smoothing	I-Error_Name
satisfies	O
a	O
number	O
of	O
special	O
properties	O
,	O
scale-space	B-Algorithm
axioms	I-Algorithm
,	O
which	O
make	O
it	O
into	O
a	O
special	O
form	O
of	O
multi-scale	O
representation	O
.	O
</s>
<s>
There	O
are	O
,	O
however	O
,	O
also	O
other	O
types	O
of	O
"	O
multi-scale	B-Algorithm
approaches	I-Algorithm
"	O
in	O
the	O
areas	O
of	O
computer	B-Application
vision	I-Application
,	O
image	B-Algorithm
processing	I-Algorithm
and	O
signal	O
processing	O
,	O
in	O
particular	O
the	O
notion	O
of	O
wavelets	O
.	O
</s>
<s>
For	O
one-dimensional	O
signals	O
,	O
there	O
exists	O
quite	O
a	O
well-developed	O
theory	O
for	O
continuous	O
and	O
discrete	O
kernels	O
that	O
guarantee	O
that	O
new	O
local	O
extrema	O
or	O
zero-crossings	O
cannot	O
be	O
created	O
by	O
a	O
convolution	B-Language
operation	O
.	O
</s>
<s>
For	O
continuous	O
signals	O
,	O
it	O
holds	O
that	O
all	O
scale-space	B-Algorithm
kernels	O
can	O
be	O
decomposed	O
into	O
the	O
following	O
sets	O
of	O
primitive	O
smoothing	O
kernels	O
:	O
</s>
<s>
For	O
discrete	O
signals	O
,	O
we	O
can	O
,	O
up	O
to	O
trivial	O
translations	O
and	O
rescalings	O
,	O
decompose	O
any	O
discrete	O
scale-space	B-Algorithm
kernel	O
into	O
the	O
following	O
primitive	O
operations	O
:	O
</s>
<s>
From	O
this	O
classification	O
,	O
it	O
is	O
apparent	O
that	O
we	O
require	O
a	O
continuous	O
semi-group	O
structure	O
,	O
there	O
are	O
only	O
three	O
classes	O
of	O
scale-space	B-Algorithm
kernels	O
with	O
a	O
continuous	O
scale	O
parameter	O
;	O
the	O
Gaussian	O
kernel	O
which	O
forms	O
the	O
scale-space	B-Algorithm
of	O
continuous	O
signals	O
,	O
the	O
discrete	O
Gaussian	O
kernel	O
which	O
forms	O
the	O
scale-space	B-Algorithm
of	O
discrete	O
signals	O
and	O
the	O
time-causal	O
Poisson	O
kernel	O
that	O
forms	O
a	O
temporal	O
scale-space	B-Algorithm
over	O
discrete	O
time	O
.	O
</s>
<s>
For	O
temporal	O
data	O
,	O
the	O
one-sided	O
truncated	O
exponential	O
kernels	O
and	O
the	O
first-order	O
recursive	O
filters	O
provide	O
a	O
way	O
to	O
define	O
time-causal	O
scale-spaces	B-Algorithm
that	O
allow	O
for	O
efficient	O
numerical	O
implementation	O
and	O
respect	O
causality	O
over	O
time	O
without	O
access	O
to	O
the	O
future	O
.	O
</s>
<s>
The	O
first-order	O
recursive	O
filters	O
also	O
provide	O
a	O
framework	O
for	O
defining	O
recursive	O
approximations	O
to	O
the	O
Gaussian	O
kernel	O
that	O
in	O
a	O
weaker	O
sense	O
preserve	O
some	O
of	O
the	O
scale-space	B-Algorithm
properties	O
.	O
</s>
