Image Processing Reference
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A problem for which one or more of the above conditions are violated is called ill-posed. Note
that the conditions mentioned above do not make a precise definition for well-posedness. To make a
precise definition in a concrete situation, one has to specify the notion of a solution, which data are
considered admissible, and which topology is used for measuring continuity.
If a problem is well-posed, then it stands a good chance of solution on a computer using a stable
algorithm. If it is not well-posed, it needs to be reformulated for numerical treatment. Typically, this
involves including additional assumptions, such as smoothness of solution. his process is known as
“regularization.” he theory of regularization is well developed for linear inverse problems and will
be introduced in Section ...
9.3.1 Ill-Posed Linear Operator Equations
Consider the linear operator equation
Ax
=
y
(.)
defined by the continuous operator
A
that maps the elements
x
of a metric space
E
into elements
y
of the metric space
. In the early s, noted French mathematician Jacques Hadamard observed
that under some (very general) circumstances the problem of solving the operator Equation . is
ill-posed. his is because, even if there exists a unique solution
x
E
that satisfies the equality .,
a small deviation on the right-hand side can cause large deviations in the solution. The following
example illustrates this issue.
∈E
Example .
Let A denote a Fredholm integral operator of the first kind. hus, we define
b
△
=
∫
(
Ax
)(
s
)
K
(
s
,
t
)
x
(
t
)
d
t
.
(.)
a
he kernel K
(
s
,
t
)
is continuous on
[
ab
]×[
ab
]
and maps a function x
(
t
)
continuous on
[
ab
]
to a
function y
(
s
)
also continuous on
[
ab
]
. We observe that the continuous function
b
)=
∫
g
ω
(
s
K
(
s
,
t
)
sin
(
ω
t
)
d
t
,
(.)
a
whichisformedbymeansofthekernelK
(
s
,
t
)
possesses the property
lim
ω
g
ω
(
s
)=
,
for everys
∈[
a
,
b
]
.
(.)
→∞
The above property is a consequence of the fact that the Fourier series coefficients of a continuous func-
tion tend to zero at high frequencies. See, for example,
([]
,Chapter,SectionI
)
.Now,considerthe
integral equation
Ax
=
y
+
g
ω
,
(.)
where y is given and g
ω
is defined in Equation .. As the above equation is linear, it follows using
Equation . that its solution x
(
t
)
has the form
x
∗
(
x
(
t
)=
t
)+
sin
(
ω
t
)
,
(.)
where x
∗
(
y. For sufficiently large
ω
, the right-hand
side of Equation . differs from the right-hand side of Equation . only by the
small
amount g
ω
t
)
is a solution to the original integral equation Ax
=
(
s
)
,
while its solution differs from that of Equation . by the amount
sin
.hus,theproblemofsolving
Equation . where A is a Fredholm integral operator of the first kind is ill-posed.
(
ω
t
)
♢
