<s>
In	O
mathematics	O
,	O
deconvolution	B-Algorithm
is	O
the	O
operation	O
inverse	O
to	O
convolution	B-Language
.	O
</s>
<s>
Both	O
operations	O
are	O
used	O
in	O
signal	O
processing	O
and	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
it	O
may	O
be	O
possible	O
to	O
recover	O
the	O
original	O
signal	O
after	O
a	O
filter	O
(	O
convolution	B-Language
)	O
by	O
using	O
a	O
deconvolution	B-Algorithm
method	O
with	O
a	O
certain	O
degree	O
of	O
accuracy	O
.	O
</s>
<s>
Deconvolution	B-Algorithm
offers	O
a	O
solution	O
to	O
this	O
problem	O
.	O
</s>
<s>
The	O
foundations	O
for	O
deconvolution	B-Algorithm
and	O
time-series	O
analysis	O
were	O
largely	O
laid	O
by	O
Norbert	O
Wiener	O
of	O
the	O
Massachusetts	O
Institute	O
of	O
Technology	O
in	O
his	O
book	O
Extrapolation	O
,	O
Interpolation	O
,	O
and	O
Smoothing	O
of	O
Stationary	O
Time	O
Series	O
(	O
1949	O
)	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
objective	O
of	O
deconvolution	B-Algorithm
is	O
to	O
find	O
the	O
solution	O
f	O
of	O
a	O
convolution	B-Language
equation	O
of	O
the	O
form	O
:	O
</s>
<s>
If	O
we	O
know	O
g	O
,	O
or	O
at	O
least	O
know	O
the	O
form	O
of	O
g	O
,	O
then	O
we	O
can	O
perform	O
deterministic	O
deconvolution	B-Algorithm
.	O
</s>
<s>
There	O
are	O
several	O
deconvolution	B-Algorithm
techniques	O
,	O
depending	O
on	O
the	O
choice	O
of	O
the	O
measurement	O
error	O
and	O
deconvolution	B-Algorithm
parameters	O
.	O
</s>
<s>
The	O
lower	O
the	O
signal-to-noise	O
ratio	O
,	O
the	O
worse	O
the	O
estimate	O
of	O
the	O
deconvolved	B-Algorithm
signal	O
will	O
be	O
.	O
</s>
<s>
However	O
,	O
if	O
at	O
least	O
some	O
knowledge	O
exists	O
of	O
the	O
type	O
of	O
noise	O
in	O
the	O
data	O
(	O
for	O
example	O
,	O
white	O
noise	O
)	O
,	O
the	O
estimate	O
of	O
ƒ	O
can	O
be	O
improved	O
through	O
techniques	O
such	O
as	O
Wiener	B-Algorithm
deconvolution	I-Algorithm
.	O
</s>
<s>
Deconvolution	B-Algorithm
(	O
raw	O
)	O
collapses	O
into	O
a	O
filter	O
reversing	O
when	O
the	O
measurement	O
error	O
is	O
very	O
low	O
(	O
ideal	O
case	O
)	O
.	O
</s>
<s>
Raw	O
deconvolution	B-Algorithm
can	O
be	O
performed	O
in	O
the	O
Laplace	O
domain	O
.	O
</s>
<s>
By	O
computing	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
recorded	O
signal	O
h	O
and	O
the	O
system	O
response	O
function	O
g	O
you	O
get	O
H	O
and	O
G	O
,	O
with	O
G	O
as	O
the	O
transfer	O
function	O
.	O
</s>
<s>
Finally	O
,	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
function	O
F	O
is	O
taken	O
to	O
find	O
the	O
estimated	O
deconvolved	B-Algorithm
signal	O
f	O
.	O
Note	O
that	O
G	O
is	O
at	O
the	O
denominator	O
and	O
could	O
amplify	O
elements	O
of	O
the	O
error	O
model	O
if	O
present	O
.	O
</s>
<s>
The	O
concept	O
of	O
deconvolution	B-Algorithm
had	O
an	O
early	O
application	O
in	O
reflection	O
seismology	O
.	O
</s>
<s>
This	O
model	O
assumes	O
that	O
the	O
recorded	O
seismogram	O
s(t )	O
is	O
the	O
convolution	B-Language
of	O
an	O
Earth-reflectivity	O
function	O
e(t )	O
and	O
a	O
seismic	O
wavelet	O
w(t )	O
from	O
a	O
point	O
source	O
,	O
where	O
t	O
represents	O
recording	O
time	O
.	O
</s>
<s>
By	O
assuming	O
that	O
the	O
reflectivity	O
is	O
white	O
,	O
we	O
can	O
assume	O
that	O
the	O
power	B-Algorithm
spectrum	I-Algorithm
of	O
the	O
reflectivity	O
is	O
constant	O
,	O
and	O
that	O
the	O
power	B-Algorithm
spectrum	I-Algorithm
of	O
the	O
seismogram	O
is	O
the	O
spectrum	O
of	O
the	O
wavelet	O
multiplied	O
by	O
that	O
constant	O
.	O
</s>
<s>
If	O
we	O
assume	O
that	O
the	O
wavelet	O
is	O
minimum	O
phase	O
,	O
we	O
can	O
recover	O
it	O
by	O
calculating	O
the	O
minimum	O
phase	O
equivalent	O
of	O
the	O
power	B-Algorithm
spectrum	I-Algorithm
we	O
just	O
found	O
.	O
</s>
<s>
In	O
practice	O
,	O
since	O
we	O
are	O
dealing	O
with	O
noisy	O
,	O
finite	O
bandwidth	O
,	O
finite	O
length	O
,	O
discretely	B-Algorithm
sampled	I-Algorithm
datasets	O
,	O
the	O
above	O
procedure	O
only	O
yields	O
an	O
approximation	O
of	O
the	O
filter	O
required	O
to	O
deconvolve	O
the	O
data	O
.	O
</s>
<s>
However	O
,	O
by	O
formulating	O
the	O
problem	O
as	O
the	O
solution	O
of	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
and	O
using	O
Levinson	B-Algorithm
recursion	I-Algorithm
,	O
we	O
can	O
relatively	O
quickly	O
estimate	O
a	O
filter	O
with	O
the	O
smallest	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
possible	O
.	O
</s>
<s>
We	O
can	O
also	O
do	O
deconvolution	B-Algorithm
directly	O
in	O
the	O
frequency	O
domain	O
and	O
get	O
similar	O
results	O
.	O
</s>
<s>
In	O
optics	O
and	O
imaging	O
,	O
the	O
term	O
"	O
deconvolution	B-Algorithm
"	O
is	O
specifically	O
used	O
to	O
refer	O
to	O
the	O
process	O
of	O
reversing	O
the	O
optical	O
distortion	O
that	O
takes	O
place	O
in	O
an	O
optical	O
microscope	O
,	O
electron	O
microscope	O
,	O
telescope	O
,	O
or	O
other	O
imaging	O
instrument	O
,	O
thus	O
creating	O
clearer	O
images	O
.	O
</s>
<s>
It	O
is	O
usually	O
done	O
in	O
the	O
digital	O
domain	O
by	O
a	O
software	O
algorithm	O
,	O
as	O
part	O
of	O
a	O
suite	O
of	O
microscope	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
techniques	O
.	O
</s>
<s>
Deconvolution	B-Algorithm
is	O
also	O
practical	O
to	O
sharpen	B-Algorithm
images	O
that	O
suffer	O
from	O
fast	O
motion	O
or	O
jiggles	O
during	O
capturing	O
.	O
</s>
<s>
Early	O
Hubble	O
Space	O
Telescope	O
images	O
were	O
distorted	O
by	O
a	O
flawed	O
mirror	O
and	O
were	O
sharpened	O
by	O
deconvolution	B-Algorithm
.	O
</s>
<s>
Since	O
the	O
original	O
convolution	B-Language
discards	O
data	O
,	O
some	O
algorithms	O
use	O
additional	O
data	O
acquired	O
at	O
nearby	O
focal	O
points	O
to	O
make	O
up	O
some	O
of	O
the	O
lost	O
information	O
.	O
</s>
<s>
Regularization	O
in	O
iterative	B-Algorithm
algorithms	O
(	O
as	O
in	O
expectation-maximization	B-Algorithm
algorithms	I-Algorithm
)	O
can	O
be	O
applied	O
to	O
avoid	O
unrealistic	O
solutions	O
.	O
</s>
<s>
This	O
procedure	O
is	O
called	O
blind	B-Algorithm
deconvolution	I-Algorithm
.	O
</s>
<s>
Blind	B-Algorithm
deconvolution	I-Algorithm
is	O
a	O
well-established	O
image	B-Algorithm
restoration	I-Algorithm
technique	O
in	O
astronomy	O
,	O
where	O
the	O
point	O
nature	O
of	O
the	O
objects	O
photographed	O
exposes	O
the	O
PSF	O
thus	O
making	O
it	O
more	O
feasible	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
in	O
fluorescence	O
microscopy	O
for	O
image	B-Algorithm
restoration	I-Algorithm
,	O
and	O
in	O
fluorescence	O
spectral	O
imaging	O
for	O
spectral	O
separation	O
of	O
multiple	O
unknown	O
fluorophores	O
.	O
</s>
<s>
The	O
most	O
common	O
iterative	B-Algorithm
algorithm	O
for	O
the	O
purpose	O
is	O
the	O
Richardson	B-Algorithm
–	I-Algorithm
Lucy	I-Algorithm
deconvolution	I-Algorithm
algorithm	O
;	O
the	O
Wiener	B-Algorithm
deconvolution	I-Algorithm
(	O
and	O
approximations	O
)	O
are	O
the	O
most	O
common	O
non-iterative	O
algorithms	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
as	O
shown	O
in	O
the	O
figure	O
,	O
deconvolution	B-Algorithm
of	O
the	O
modeled	O
PSF	O
and	O
the	O
terahertz	O
image	O
can	O
give	O
a	O
higher	O
resolution	O
representation	O
of	O
the	O
terahertz	O
image	O
.	O
</s>
<s>
Typical	O
use	O
of	O
deconvolution	B-Algorithm
is	O
in	O
tracer	O
kinetics	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
measuring	O
a	O
hormone	O
concentration	O
in	O
the	O
blood	O
,	O
its	O
secretion	O
rate	O
can	O
be	O
estimated	O
by	O
deconvolution	B-Algorithm
.	O
</s>
<s>
Deconvolution	B-Algorithm
has	O
been	O
applied	O
extensively	O
to	O
absorption	O
spectra	O
.	O
</s>
<s>
Deconvolution	B-Algorithm
maps	O
to	O
division	O
in	O
the	O
Fourier	B-Algorithm
co-domain	I-Algorithm
.	O
</s>
<s>
This	O
allows	O
deconvolution	B-Algorithm
to	O
be	O
easily	O
applied	O
with	O
experimental	O
data	O
that	O
are	O
subject	O
to	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
