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
)
, t
<
δ. δ.The notation
B(
X
)
B(
X
)
.
δ
In this case, ξ
should converge to zero “strictly slower” than δ . In more precise terms, lim δ
(
δ
)
) =
musthold.
ξ
(
δ
 
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