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
hx syt
(, )()
tsxs s
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
Figure 1.3
Definition of the system variables.
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