<s>
The	O
Landweber	B-Algorithm
iteration	I-Algorithm
or	O
Landweber	O
algorithm	O
is	O
an	O
algorithm	O
to	O
solve	O
ill-posed	B-Algorithm
linear	O
inverse	O
problems	O
,	O
and	O
it	O
has	O
been	O
extended	O
to	O
solve	O
non-linear	O
problems	O
that	O
involve	O
constraints	O
.	O
</s>
<s>
The	O
linear	O
version	O
assumes	O
that	O
for	O
a	O
linear	B-Architecture
operator	I-Architecture
A	O
.	O
</s>
<s>
However	O
,	O
if	O
A	O
is	O
ill-conditioned	B-Algorithm
,	O
the	O
explicit	O
solution	O
is	O
a	O
poor	O
choice	O
since	O
it	O
is	O
sensitive	O
to	O
any	O
noise	O
in	O
the	O
data	O
y	O
.	O
</s>
<s>
and	O
hence	O
the	O
algorithm	O
is	O
a	O
special	O
case	O
of	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
For	O
ill-posed	B-Algorithm
problems	I-Algorithm
,	O
the	O
iterative	O
method	O
needs	O
to	O
be	O
stopped	O
at	O
a	O
suitable	O
iteration	O
index	O
,	O
because	O
it	O
semi-converges	O
.	O
</s>
<s>
Using	O
the	O
Landweber	B-Algorithm
iteration	I-Algorithm
as	O
a	O
regularization	O
algorithm	O
has	O
been	O
discussed	O
in	O
the	O
literature	O
.	O
</s>
<s>
Since	O
this	O
is	O
special	O
type	O
of	O
gradient	B-Algorithm
descent	I-Algorithm
,	O
there	O
currently	O
is	O
not	O
much	O
benefit	O
to	O
analyzing	O
it	O
on	O
its	O
own	O
as	O
the	O
nonlinear	O
Landweber	O
,	O
but	O
such	O
analysis	O
was	O
performed	O
historically	O
by	O
many	O
communities	O
not	O
aware	O
of	O
unifying	O
frameworks	O
.	O
</s>
<s>
can	O
be	O
solved	O
by	O
the	O
constrained	O
,	O
nonlinear	O
Landweber	B-Algorithm
iteration	I-Algorithm
,	O
given	O
by	O
:	O
</s>
<s>
where	O
is	O
the	O
projection	O
onto	O
the	O
set	O
C	O
.	O
Convergence	B-Algorithm
is	O
guaranteed	O
when	O
.	O
</s>
<s>
This	O
is	O
again	O
a	O
special	O
case	O
of	O
projected	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
(	O
which	O
is	O
a	O
special	O
case	O
of	O
the	O
forward	O
–	O
backward	O
algorithm	O
)	O
as	O
discussed	O
in	O
.	O
</s>
<s>
Since	O
the	O
method	O
has	O
been	O
around	O
since	O
the	O
1950s	O
,	O
it	O
has	O
been	O
adopted	O
and	O
rediscovered	O
by	O
many	O
scientific	O
communities	O
,	O
especially	O
those	O
studying	O
ill-posed	B-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
It	O
has	O
also	O
been	O
used	O
in	O
the	O
computer	B-Application
vision	I-Application
community	O
and	O
the	O
signal	O
restoration	O
community	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
in	O
image	B-Algorithm
processing	I-Algorithm
,	O
since	O
many	O
image	O
problems	O
,	O
such	O
as	O
deconvolution	B-Algorithm
,	O
are	O
ill-posed	B-Algorithm
.	O
</s>
