<s>
In	O
mathematics	O
,	O
symmetric	B-Algorithm
convolution	I-Algorithm
is	O
a	O
special	O
subset	O
of	O
convolution	B-Language
operations	O
in	O
which	O
the	O
convolution	B-Language
kernel	O
is	O
symmetric	B-Algorithm
across	O
its	O
zero	O
point	O
.	O
</s>
<s>
Many	O
common	O
convolution-based	O
processes	O
such	O
as	O
Gaussian	B-Error_Name
blur	I-Error_Name
and	O
taking	O
the	O
derivative	B-Algorithm
of	O
a	O
signal	O
in	O
frequency-space	O
are	O
symmetric	B-Algorithm
and	O
this	O
property	O
can	O
be	O
exploited	O
to	O
make	O
these	O
convolutions	B-Language
easier	O
to	O
evaluate	O
.	O
</s>
<s>
The	O
convolution	B-Language
theorem	O
states	O
that	O
a	O
convolution	B-Language
in	O
the	O
real	O
domain	O
can	O
be	O
represented	O
as	O
a	O
pointwise	O
multiplication	O
across	O
the	O
frequency	O
domain	O
of	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Since	O
sine	O
and	O
cosine	O
transforms	O
are	O
related	O
transforms	O
a	O
modified	O
version	O
of	O
the	O
convolution	B-Language
theorem	O
can	O
be	O
applied	O
,	O
in	O
which	O
the	O
concept	O
of	O
circular	B-Algorithm
convolution	I-Algorithm
is	O
replaced	O
with	O
symmetric	B-Algorithm
convolution	I-Algorithm
.	O
</s>
<s>
Using	O
these	O
transforms	O
to	O
compute	O
discrete	O
symmetric	B-Algorithm
convolutions	I-Algorithm
is	O
non-trivial	O
since	O
discrete	O
sine	O
transforms	O
(	O
DSTs	O
)	O
and	O
discrete	B-General_Concept
cosine	I-General_Concept
transforms	I-General_Concept
(	O
DCTs	B-General_Concept
)	O
can	O
be	O
counter-intuitively	O
incompatible	O
for	O
computing	O
symmetric	B-Algorithm
convolution	I-Algorithm
,	O
i.e.	O
</s>
<s>
symmetric	B-Algorithm
convolution	I-Algorithm
can	O
only	O
be	O
computed	O
between	O
a	O
fixed	O
set	O
of	O
compatible	O
transforms	O
.	O
</s>
<s>
In	O
order	O
to	O
compute	O
symmetric	B-Algorithm
convolution	I-Algorithm
effectively	O
,	O
one	O
must	O
know	O
which	O
particular	O
frequency	O
domains	O
(	O
which	O
are	O
reachable	O
by	O
transforming	O
real	O
data	O
through	O
DSTs	O
or	O
DCTs	B-General_Concept
)	O
the	O
inputs	O
and	O
outputs	O
to	O
the	O
convolution	B-Language
can	O
be	O
and	O
then	O
tailor	O
the	O
symmetries	O
of	O
the	O
transforms	O
to	O
the	O
required	O
symmetries	O
of	O
the	O
convolution	B-Language
.	O
</s>
<s>
The	O
following	O
table	O
documents	O
which	O
combinations	O
of	O
the	O
domains	O
from	O
the	O
main	O
eight	O
commonly	O
used	O
DST	O
I-IV	O
and	O
DCT	B-General_Concept
I-IV	O
satisfy	O
where	O
represents	O
the	O
symmetric	B-Algorithm
convolution	I-Algorithm
operator	O
.	O
</s>
<s>
Convolution	B-Language
is	O
a	O
commutative	O
operator	O
,	O
and	O
so	O
and	O
are	O
interchangeable	O
.	O
</s>
<s>
Forward	O
transforms	O
of	O
,	O
and	O
,	O
through	O
the	O
transforms	O
specified	O
should	O
allow	O
the	O
symmetric	B-Algorithm
convolution	I-Algorithm
to	O
be	O
computed	O
as	O
a	O
pointwise	O
multiplication	O
,	O
with	O
any	O
excess	O
undefined	O
frequency	O
amplitudes	O
set	O
to	O
zero	O
.	O
</s>
<s>
Possibilities	O
for	O
symmetric	B-Algorithm
convolutions	I-Algorithm
involving	O
DSTs	O
and	O
DCTs	B-General_Concept
V-VIII	O
derived	O
from	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transforms	I-Algorithm
(	O
DFTs	O
)	O
of	O
odd	O
logical	O
order	O
can	O
be	O
determined	O
by	O
adding	O
four	O
to	O
each	O
type	O
in	O
the	O
above	O
tables	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
advantages	O
to	O
computing	O
symmetric	B-Algorithm
convolutions	I-Algorithm
in	O
DSTs	O
and	O
DCTs	B-General_Concept
in	O
comparison	O
with	O
the	O
more	O
common	O
circular	B-Algorithm
convolution	I-Algorithm
with	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
using	O
a	O
DCT-II	O
,	O
a	O
symmetric	B-Algorithm
signal	O
need	O
only	O
have	O
the	O
positive	O
half	O
DCT-II	O
transformed	O
,	O
since	O
the	O
frequency	O
domain	O
will	O
implicitly	O
construct	O
the	O
mirrored	O
data	O
comprising	O
the	O
other	O
half	O
.	O
</s>
<s>
This	O
enables	O
larger	O
convolution	B-Language
kernels	O
to	O
be	O
used	O
with	O
the	O
same	O
cost	O
as	O
smaller	O
kernels	O
circularly	O
convolved	O
on	O
the	O
DFT	O
.	O
</s>
<s>
Also	O
the	O
boundary	O
conditions	O
implicit	O
in	O
DSTs	O
and	O
DCTs	B-General_Concept
create	O
edge	O
effects	O
that	O
are	O
often	O
more	O
in	O
keeping	O
with	O
neighbouring	O
data	O
than	O
the	O
periodic	O
effects	O
introduced	O
by	O
using	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
