Image Processing Reference

In-Depth Information

A fundamental theoretical concern was mentioned earlier that we do not

ignore (but also do not mention again): imaging from inverse scattered data is

in general an ill-posed problem. By definition (Hadamard, 1923) a problem is

considered ill posed if one (or more) of the following is true:

1. A solution does not exist.

2. A solution is nonunique.

3. The solution is unstable.

The first condition mentioned that a solution does not exist is definitely

a possibility for these types of problems. For this discussion, it is assumed

that a solution does exist, at least for the class of targets under consideration,

because if the converse were true, further work would be pointless. There

could be classes of targets where condition (i) above would be terminal. Even

if a solution does exist, it is highly possible, albeit probable, that the solution is

not unique. This means that two different sampled targets could produce the

same scattered field patterns for a finite number of receivers. This being the

case, the solution must be chosen from a solution space of possible solutions

utilizing some global minimum, which could be problematic. This unique-

ness problem could be amplified if the number of receivers is significantly

low, which in a sense would lower the available degrees of freedom. The issue

of degrees of freedom, its definition for this type of problem, and its effects

will be examined more in Chapter 6.

The final “ill-posed” condition is that of stability (or a lack thereof). The

stability of any inverse problem is a direct function of the system response as

defined in Figure 1.3, where
y
=
hx
. In general, the inverse problem is stable

if and only if
h
−1
exists and is stable. It is known that if
h
is continuous and if

h
−1
exists, then
h
−1
is also continuous. This in itself is not necessarily enough

to ensure stability in general since the noise collected in performing measure-

ments may also lead to instabilities and discontinuities being inserted in the

valid data. It has been shown that, in mathematical theory and treatment for

these types of problems,
h
can be represented by an integral equation which

leads to
y
=
hx
being a Fredholm integral equation of the first kind having a

square integral (Hilbert-Schmidt) (Boas, 2011) kernel which in its most gen-

eral form is written as

b

=
∫

yt

()

=

hx syt

()

()

(, )()

tsxs s

d

(1.3)

a

where
y
(
t
) is the system output,
u
(
t
,
s
) is the system response, and
x
(
s
) is the

object function or scatterer. This means that a minute error introduced in the

measured data could quite possibly introduce a rather large error in the recon-

structed results. For instance, if a solution
x
is perturbed by a delta function

of the form

x

h

y

Figure 1.3

Definition of the system variables.

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