<s>
In	O
the	O
areas	O
of	O
computer	B-Application
vision	I-Application
,	O
image	B-General_Concept
analysis	I-General_Concept
and	O
signal	O
processing	O
,	O
the	O
notion	O
of	O
scale-space	B-Algorithm
representation	I-Algorithm
is	O
used	O
for	O
processing	O
measurement	O
data	O
at	O
multiple	O
scales	O
,	O
and	O
specifically	O
enhance	O
or	O
suppress	O
image	O
features	O
over	O
different	O
ranges	O
of	O
scale	O
(	O
see	O
the	O
article	O
on	O
scale	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
A	O
special	O
type	O
of	O
scale-space	B-Algorithm
representation	I-Algorithm
is	O
provided	O
by	O
the	O
Gaussian	O
scale	B-Algorithm
space	I-Algorithm
,	O
where	O
the	O
image	O
data	O
in	O
N	O
dimensions	O
is	O
subjected	O
to	O
smoothing	O
by	O
Gaussian	O
convolution	B-Language
.	O
</s>
<s>
Most	O
of	O
the	O
theory	O
for	O
Gaussian	O
scale	B-Algorithm
space	I-Algorithm
deals	O
with	O
continuous	O
images	O
,	O
whereas	O
one	O
when	O
implementing	O
this	O
theory	O
will	O
have	O
to	O
face	O
the	O
fact	O
that	O
most	O
measurement	O
data	O
are	O
discrete	O
.	O
</s>
<s>
Hence	O
,	O
the	O
theoretical	O
problem	O
arises	O
concerning	O
how	O
to	O
discretize	O
the	O
continuous	O
theory	O
while	O
either	O
preserving	O
or	O
well	O
approximating	O
the	O
desirable	O
theoretical	O
properties	O
that	O
lead	O
to	O
the	O
choice	O
of	O
the	O
Gaussian	O
kernel	O
(	O
see	O
the	O
article	O
on	O
scale-space	B-Algorithm
axioms	I-Algorithm
)	O
.	O
</s>
<s>
The	O
Gaussian	O
scale-space	B-Algorithm
representation	I-Algorithm
of	O
an	O
N-dimensional	O
continuous	O
signal	O
,	O
</s>
<s>
is	O
obtained	O
by	O
convolving	B-Language
fC	O
with	O
an	O
N-dimensional	O
Gaussian	O
kernel	O
:	O
</s>
<s>
When	O
applying	O
the	O
scale	B-Algorithm
space	I-Algorithm
concept	O
to	O
a	O
discrete	O
signal	O
fD	O
,	O
different	O
approaches	O
can	O
be	O
taken	O
.	O
</s>
<s>
For	O
larger	O
values	O
of	O
ε	O
,	O
however	O
,	O
there	O
are	O
many	O
better	O
alternatives	O
to	O
a	O
rectangular	O
window	B-Language
function	I-Language
.	O
</s>
<s>
With	O
this	O
frequency-domain	O
approach	O
,	O
the	O
scale-space	B-Algorithm
properties	O
transfer	O
exactly	O
to	O
the	O
discrete	O
domain	O
,	O
or	O
with	O
excellent	O
approximation	O
using	O
periodic	O
extension	O
and	O
a	O
suitably	O
long	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
to	O
approximate	O
the	O
discrete-time	O
Fourier	O
transform	O
of	O
the	O
signal	O
being	O
smoothed	O
.	O
</s>
<s>
Moreover	O
,	O
higher-order	O
derivative	O
approximations	O
can	O
be	O
computed	O
in	O
a	O
straightforward	O
manner	O
(	O
and	O
preserving	O
scale-space	B-Algorithm
properties	O
)	O
by	O
applying	O
small	O
support	O
central	O
difference	O
operators	O
to	O
the	O
discrete	O
scale	B-Algorithm
space	I-Algorithm
representation	I-Algorithm
.	O
</s>
<s>
As	O
with	O
the	O
sampled	O
Gaussian	O
,	O
a	O
plain	O
truncation	O
of	O
the	O
infinite	O
impulse	O
response	O
will	O
in	O
most	O
cases	O
be	O
a	O
sufficient	O
approximation	O
for	O
small	O
values	O
of	O
ε	O
,	O
while	O
for	O
larger	O
values	O
of	O
ε	O
it	O
is	O
better	O
to	O
use	O
either	O
a	O
decomposition	O
of	O
the	O
discrete	O
Gaussian	O
into	O
a	O
cascade	O
of	O
generalized	O
binomial	O
filters	O
or	O
alternatively	O
to	O
construct	O
a	O
finite	O
approximate	O
kernel	O
by	O
multiplying	O
by	O
a	O
window	B-Language
function	I-Language
.	O
</s>
<s>
Since	O
computational	O
efficiency	O
is	O
often	O
important	O
,	O
low-order	O
recursive	B-Algorithm
filters	I-Algorithm
are	O
often	O
used	O
for	O
scale-space	B-Algorithm
smoothing	O
.	O
</s>
<s>
For	O
example	O
,	O
Young	O
and	O
van	O
Vliet	O
use	O
a	O
third-order	O
recursive	B-Algorithm
filter	I-Algorithm
with	O
one	O
real	O
pole	O
and	O
a	O
pair	O
of	O
complex	O
poles	O
,	O
applied	O
forward	O
and	O
backward	O
to	O
make	O
a	O
sixth-order	O
symmetric	O
approximation	O
to	O
the	O
Gaussian	O
with	O
low	O
computational	O
complexity	O
for	O
any	O
smoothing	O
scale	O
.	O
</s>
<s>
The	O
transfer	O
function	O
,	O
H1	O
,	O
of	O
a	O
symmetric	O
pole-pair	O
recursive	B-Algorithm
filter	I-Algorithm
is	O
closely	O
related	O
to	O
the	O
discrete-time	O
Fourier	O
transform	O
of	O
the	O
discrete	O
Gaussian	O
kernel	O
via	O
first-order	O
approximation	O
of	O
the	O
exponential	O
:	O
</s>
<s>
Scale-space	B-Algorithm
axioms	I-Algorithm
that	O
are	O
still	O
satisfied	O
by	O
these	O
filters	O
are	O
:	O
</s>
<s>
This	O
recursive	B-Algorithm
filter	I-Algorithm
method	O
and	O
variations	O
to	O
compute	O
both	O
the	O
Gaussian	O
smoothing	O
as	O
well	O
as	O
Gaussian	O
derivatives	O
has	O
been	O
described	O
by	O
several	O
authors	O
.	O
</s>
<s>
When	O
computing	O
several	O
derivatives	O
in	O
the	O
N-jet	B-Algorithm
simultaneously	O
,	O
discrete	O
scale-space	B-Algorithm
smoothing	O
with	O
the	O
discrete	O
analogue	O
of	O
the	O
Gaussian	O
kernel	O
,	O
or	O
with	O
a	O
recursive	B-Algorithm
filter	I-Algorithm
approximation	O
,	O
followed	O
by	O
small	O
support	O
difference	O
operators	O
,	O
may	O
be	O
both	O
faster	O
and	O
more	O
accurate	O
than	O
computing	O
recursive	O
approximations	O
of	O
each	O
derivative	O
operator	O
.	O
</s>
<s>
For	O
small	O
scales	O
,	O
a	O
low-order	O
FIR	O
filter	O
may	O
be	O
a	O
better	O
smoothing	O
filter	O
than	O
a	O
recursive	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
The	O
FIR	O
filter	O
's	O
zeros	O
can	O
be	O
combined	O
with	O
the	O
recursive	B-Algorithm
filter	I-Algorithm
's	O
poles	O
to	O
make	O
a	O
general	O
high-quality	O
smoothing	O
filter	O
.	O
</s>
<s>
The	O
FIR	O
filter	O
transfer	O
function	O
is	O
closely	O
related	O
to	O
the	O
discrete	O
Gaussian	O
's	O
DTFT	O
,	O
just	O
as	O
was	O
the	O
recursive	B-Algorithm
filter	I-Algorithm
's	O
.	O
</s>
<s>
These	O
FIR	O
and	O
pole-zero	O
filters	O
are	O
valid	O
scale-space	B-Algorithm
kernels	O
,	O
satisfying	O
the	O
same	O
axioms	O
as	O
the	O
all-pole	O
recursive	B-Algorithm
filters	I-Algorithm
.	O
</s>
<s>
Regarding	O
the	O
topic	O
of	O
automatic	O
scale	O
selection	O
based	O
on	O
normalized	O
derivatives	O
,	O
pyramid	B-Algorithm
approximations	I-Algorithm
are	O
frequently	O
used	O
to	O
obtain	O
real-time	O
performance	O
.	O
</s>
<s>
The	O
appropriateness	O
of	O
approximating	O
scale-space	B-Algorithm
operations	O
within	O
a	O
pyramid	B-Algorithm
originates	O
from	O
the	O
fact	O
that	O
repeated	O
cascade	O
smoothing	O
with	O
generalized	O
binomial	O
kernels	O
leads	O
to	O
equivalent	O
smoothing	O
kernels	O
that	O
under	O
reasonable	O
conditions	O
approach	O
the	O
Gaussian	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
binomial	O
kernels	O
(	O
or	O
more	O
generally	O
the	O
class	O
of	O
generalized	O
binomial	O
kernels	O
)	O
can	O
be	O
shown	O
to	O
constitute	O
the	O
unique	O
class	O
of	O
finite-support	O
kernels	O
that	O
guarantee	O
non-creation	O
of	O
local	O
extrema	O
or	O
zero-crossings	O
with	O
increasing	O
scale	O
(	O
see	O
the	O
article	O
on	O
multi-scale	B-Algorithm
approaches	I-Algorithm
for	O
details	O
)	O
.	O
</s>
<s>
For	O
one-dimensional	O
kernels	O
,	O
there	O
is	O
a	O
well-developed	O
theory	O
of	O
multi-scale	B-Algorithm
approaches	I-Algorithm
,	O
concerning	O
filters	O
that	O
do	O
not	O
create	O
new	O
local	O
extrema	O
or	O
new	O
zero-crossings	O
with	O
increasing	O
scales	O
.	O
</s>
<s>
There	O
are	O
many	O
other	O
multi-scale	O
signal	O
processing	O
,	O
image	O
processing	O
and	O
data	O
compression	O
techniques	O
,	O
using	O
wavelets	O
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	B-Algorithm
requirements	I-Algorithm
as	O
scale	B-Algorithm
space	I-Algorithm
descriptions	O
do	O
;	O
that	O
is	O
,	O
they	O
do	O
not	O
depend	O
on	O
a	O
coarser	O
scale	O
not	O
generating	O
a	O
new	O
extremum	O
that	O
was	O
not	O
present	O
at	O
a	O
finer	O
scale	O
(	O
in	O
1D	O
)	O
or	O
non-enhancement	O
of	O
local	O
extrema	O
between	O
adjacent	O
scale	O
levels	O
(	O
in	O
any	O
number	O
of	O
dimensions	O
)	O
.	O
</s>
