Image Processing Reference
In-Depth Information
One can easily verify that the problem of solving the operator Equation . is equivalent to finding
an element x ∈E
such that the functional
)=∥
R
(
x
Ax
y
E
(.)
is minimized. Note that the minimizing element x ∈E
always exists even when the original Equa-
tion . does not have a solution. In any case, if the right-hand side of Equation . is not exact, that
is, if we replace y by y δ such that
y
y δ
E
<
δwhereδisasmallvalue,anewelement x δ
∈E
will
minimize the functional
=∥
R δ
(
x
)
Ax
y δ
E .
(.)
However, the new solution x δ is not necessarily close to the first solution x even if δ tends to zero.
In other words, lim δ
x
x δ
E
 when the operator equation Ax
=
y is ill-posed.
9.3.2 Regularization Methods for Solving Ill-Posed Linear Operator Equations
Hadamard [] thought that ill-posed problems are a pure mathematical phenomenon and that all
real-life problems are well-posed. However, in the second half of the th century, a number of very
important real-life problems were found to be ill-posed. In particular, as we just discussed, ill-posed
problemsarisewhenonetriestoreversethecause-efectrelationstoindunknowncausesfrom
known consequences. Even if the cause-effect relationship forms a one-to-one mapping, the prob-
lem of inverting it can be ill-posed. he discovery of various “regularization methods” by Tikhonov,
Ivanov, and Phillips in the early s made it possible to construct a sequence of “well-posed
solutions” that converges to the desired one.
Regularization theory was one of the first signs of existence of “intelligent inference.” It demon-
strated that whereas the “self-evident” methods of solving an operator equation might not work, the
“non-self-evident” methods of regularization theory do. he influence of the philosophy created by
the theory of regularization is very deep. Both the regularization philosophy and the regularization
techniques became widely disseminated in many areas of science and engineering [,].
9.3.2.1 Tikhonov's Method
In the early s, it was discovered by Tikhonov [,] that if instead of the functional R δ
(
x
)
one
minimizes
)=∥
R reg
(
x
Ax
y δ
E
+
ξ
(
δ
)
S
(
x
)
,
(.)
where S
is
an appropriately chosen constant (whose value depends on the “noise” level δ), then one obtains a
sequence of solutions x δ that converges to the desired one as δ tends to zero. For the above result to
be valid, it is required that
(
x
)
is a “stabilizing functional” (that belongs to a certain class of functionals) and ξ
(
δ
)
. The problem of minimizing R reg
(
x
)
be well-posed for fixed values of δ and ξ
(
δ
)
.
x
. lim
δ
x δ
E
whenξ
(
δ
)
is chosen appropriately.
To save in notation, we write
a
b
E
to denote the “distance” between the two elements a , b
∈E
whether the metric
space
is a normed space too, our notation is self-evident. Otherwise, it should be interpreted
only as a “symbol” for the distance between a and b .
E
is a normed space or not. If
E
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