<s>
Image	B-Algorithm
derivatives	I-Algorithm
can	O
be	O
computed	O
by	O
using	O
small	O
convolution	B-Language
filters	O
of	O
size	O
2×2	O
or	O
3×3	O
,	O
such	O
as	O
the	O
Laplacian	B-Algorithm
,	O
Sobel	B-Algorithm
,	O
Roberts	B-Algorithm
and	O
Prewitt	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
derivative	O
kernel	O
and	O
is	O
the	O
pixel	O
values	O
in	O
a	O
region	O
of	O
the	O
image	O
and	O
is	O
the	O
operator	O
that	O
performs	O
the	O
convolution	B-Language
.	O
</s>
<s>
The	O
derivative	O
kernels	O
,	O
known	O
as	O
the	O
Sobel	B-Algorithm
operator	I-Algorithm
are	O
defined	O
as	O
follows	O
,	O
for	O
the	O
and	O
directions	O
respectively	O
:	O
</s>
<s>
where	O
here	O
denotes	O
the	O
2-dimensional	O
convolution	B-Language
operation	O
.	O
</s>
<s>
Farid	O
and	O
Simoncelli	O
propose	O
to	O
use	O
a	O
pair	O
of	O
kernels	O
,	O
one	O
for	O
interpolation	O
and	O
another	O
for	O
differentiation	O
(	O
compare	O
to	O
Sobel	B-Algorithm
above	O
)	O
.	O
</s>
<s>
Derivative	O
filters	O
based	O
on	O
arbitrary	O
cubic	B-Algorithm
splines	I-Algorithm
was	O
presented	O
by	O
Hast	O
.	O
</s>
<s>
He	O
showed	O
how	O
both	O
first	O
and	O
second	O
order	O
derivatives	O
can	O
be	O
computed	O
more	O
correctly	O
using	O
cubic	O
or	O
trigonometric	O
splines	B-Algorithm
.	O
</s>
<s>
Also	O
keep	O
in	O
mind	O
that	O
convolution	B-Language
is	O
commutative	O
,	O
so	O
that	O
the	O
order	O
of	O
the	O
two	O
kernels	O
does	O
not	O
matter	O
and	O
it	O
is	O
also	O
possible	O
to	O
insert	O
a	O
second	O
order	O
derivative	O
as	O
well	O
as	O
a	O
first	O
order	O
derivative	O
kernel	O
.	O
</s>
<s>
Any	O
cubic	O
filter	O
can	O
be	O
applied	O
and	O
used	O
for	O
computing	O
the	O
image	O
derivates	O
using	O
the	O
above	O
equations	O
,	O
such	O
as	O
Bézier	O
,	O
Hermite	B-Algorithm
or	O
B-splines	B-Algorithm
.	O
</s>
<s>
Steerable	B-Algorithm
filters	I-Algorithm
can	O
be	O
used	O
for	O
computing	O
derivatives	O
Moreover	O
,	O
Savitzky	O
and	O
Golay	O
propose	O
a	O
least-squares	O
polynomial	O
smoothing	O
approach	O
,	O
which	O
could	O
be	O
used	O
for	O
computing	O
derivatives	O
and	O
Luo	O
et	O
al	O
discuss	O
this	O
approach	O
in	O
further	O
detail	O
.	O
</s>
