Image Processing Reference
In-Depth Information
Consider a real-valued lower semi-continuous
∗
functional
S
(
x
)
. We shall call
S
(
x
)
a “stabilizing
functional” if it possesses the following properties:
. Solution of the operator equation
Ax
=
y
belongs to the domain of definition
D(
S
)
of
the functional
S
.
.
S
(
x
)≥
,
∀
x
∈D(
S
)
.
const
., are all compact.
It turns out that the above conditions are sufficient for the problem of minimizing
R
reg
. Level sets
{
x
∶
S
(
x
)≤
c
}
,
c
=
to be
well-posed [, p. ]. Now, the important remaining problem is to determine the functional rela-
tionship between δ and ξ
(
x
)
such that the sequence of solutions obtained by minimizing Equation
. converges to the solution of Equation . as δ tends to zero. he following theorem establishes
sufficient conditions on such a relationship:
(
δ
)
THEOREM .
[, p. ]
Let
E
and
E
be two metric spaces and let A
∶E
→E
be a continuous
andone-to-oneoperator.Supposethatfory
∈E
there exists a solution x
∈D(
S
)⊂E
to the operator
equation Ax
=
y. Let y
δ
be an element in
E
such that
∥
y
−
y
δ
∥
E
≤
δ
.Iftheparameter
ξ
(
δ
)
is chosen
such that
(i) ξ
(
δ
)→
when
δ
→
δ
)
<∞
Then the elements x
δ
(ii) lim
δ
ξ
(
δ
→
∈D(
S
)
minimizing the functional
R
reg
(
x
)=∥
Ax
−
y
δ
∥
E
+
ξ
(
δ
)
S
(
x
)
converge to the exact solution x as
δ
→
.
,which,
If
E
is a Hilbert space, the stabilizing functional
S
(
x
)
may simply be chosen as
∥
x
∥
indeed, is the original choice made by Tikhonov. In this case, the level sets of
S
will only be weakly
compact. However, the convergence of the regularized solutions will be a strong one in view of the
properties of Hilbert spaces. he conditions imposed on the parameter ξ
(
x
)
(
δ
)
are, nevertheless, more
stringent than those stated in the above theorem.
†
9.3.2.2 Residual Method
The results presented above are fundamentals in Tikhonov's theory of regularization. Tikhonov's
theory, however, is only one of several proposed schemes for solving ill-posed problems. An impor-
tant variation known as Residual Method was introduced by Phillips []. In Phillips' method, one
minimize the functional
△
=
R
P
(
x
)
S
(
x
)
subject to the constraint
∥
Ax
−
y
δ
∥
E
≤
µ,
where µ is a fixed constant. he stabilizing functional
S
(
x
)
isdeinedasintheprevioussubsection.
∗
N
N
if for any
t
Afunction
f
∶
R
→[−∞
,
∞]
is called lower semi-continuous at
X
∈
R
<
f
(
X
)
there exists δ
>
such that for all
y
represents a ball with center at
X
and radius δ. δ.This definition
generalizes to functional spaces by using the appropriate metric in defining
∈B(
X
,δ
)
,
t
<
δ. δ.The notation
B(
X
,δ
)
B(
X
,δ
)
.
δ
†
In this case, ξ
should converge to zero “strictly slower” than δ
. In more precise terms, lim
δ
(
δ
)
)
=
musthold.
→
ξ
(
δ