<s>
Deep	B-Algorithm
image	I-Algorithm
prior	I-Algorithm
is	O
a	O
type	O
of	O
convolutional	B-Architecture
neural	I-Architecture
network	I-Architecture
used	O
to	O
enhance	O
a	O
given	O
image	O
with	O
no	O
prior	O
training	O
data	O
other	O
than	O
the	O
image	O
itself	O
.	O
</s>
<s>
A	O
neural	O
network	O
is	O
randomly	O
initialized	O
and	O
used	O
as	O
prior	O
to	O
solve	O
inverse	O
problems	O
such	O
as	O
noise	O
reduction	O
,	O
super-resolution	B-Algorithm
,	O
and	O
inpainting	B-Algorithm
.	O
</s>
<s>
Inverse	O
problems	O
such	O
as	O
noise	O
reduction	O
,	O
super-resolution	B-Algorithm
,	O
and	O
inpainting	B-Algorithm
can	O
be	O
formulated	O
as	O
the	O
optimization	O
task	O
,	O
where	O
is	O
an	O
image	O
,	O
a	O
corrupted	O
representation	O
of	O
that	O
image	O
,	O
is	O
a	O
task-dependent	O
data	O
term	O
,	O
and	O
R(x )	O
is	O
the	O
regularizer	O
.	O
</s>
<s>
Deep	O
neural	O
networks	O
learn	O
a	O
generator/decoder	O
which	O
maps	O
a	O
random	O
code	B-Device
vector	I-Device
to	O
an	O
image	O
.	O
</s>
<s>
The	O
minimizer	O
(	O
typically	O
a	O
gradient	B-Algorithm
descent	I-Algorithm
)	O
starts	O
from	O
a	O
randomly	O
initialized	O
parameters	O
and	O
descends	O
into	O
a	O
local	O
best	O
result	O
to	O
yield	O
the	O
restoration	O
function	O
.	O
</s>
<s>
This	O
results	O
in	O
the	O
θ	O
parameter	O
approaching	O
a	O
good-looking	O
local	O
optimum	O
so	O
long	O
as	O
the	O
number	O
of	O
iterations	O
in	O
the	O
optimization	O
process	O
remains	O
low	O
enough	O
not	O
to	O
overfit	B-Error_Name
data	O
.	O
</s>
<s>
Super-resolution	B-Algorithm
is	O
used	O
to	O
generate	O
a	O
higher	O
resolution	O
version	O
of	O
image	O
x	O
.	O
</s>
<s>
The	O
data	O
term	O
is	O
set	O
to	O
where	O
d( ·	O
)	O
is	O
a	O
downsampling	B-Algorithm
operator	I-Algorithm
such	O
as	O
Lanczos	B-Algorithm
that	O
decimates	O
the	O
image	O
by	O
a	O
factor	O
t	O
.	O
</s>
<s>
Inpainting	B-Algorithm
is	O
used	O
to	O
reconstruct	O
a	O
missing	O
area	O
in	O
an	O
image	O
.	O
</s>
<s>
A	O
reference	O
implementation	O
rewritten	O
in	O
Python	B-Language
3.6	I-Language
with	O
the	O
PyTorch	O
0.4.0	O
library	O
was	O
released	O
by	O
the	O
author	O
under	O
the	O
Apache	O
2.0	O
license	O
:	O
</s>
<s>
A	O
TensorFlow-based	O
implementation	O
written	O
in	O
Python	B-Language
2	I-Language
and	O
released	O
under	O
the	O
CC-SA	O
3.0	O
license	O
:	O
</s>
<s>
A	O
Keras-based	O
implementation	O
written	O
in	O
Python	B-Language
2	I-Language
and	O
released	O
under	O
the	O
GPLv3	O
:	O
</s>
