Image Processing Reference

In-Depth Information

δ

xs

()

=

ε

sin(

2

π

cs

)

(1.4)

where
c
= 1, 2, 3, … and ε is a constant. Then, the resulting perturbation in the

output
y
(
t
) would be given as

b

∫

δ

yt

()

=

ε

ut s

(, )sin(

2d

π

css

)

(1.5)

a

where
c
= 1, 2, 3, …. Now if the Riemann-Lebesgue lemma (Renardy and

Rogers, 2004) is applied, it follows that

δ
y
→ 0

as

c
→ ∞

Now, if the integer “
c
” becomes large enough, the term ||δ
x
|| / || δ
y
|| can

similarly become large as well, which results in Equation 1.3 becoming dis-

continuous over the domain of interest bringing us back to condition (iii) of

an ill-posed problem. This situation can of course be controlled somewhat by

taking precautions to reduce noise in the measurements and by strategically

selecting the sample points. This is also complicated by the fact that there is a

nonlinear relationship between the scatterer and the scattered field which can

also make it very challenging to find a closed form solution for the 2-D inverse

scattering problem. It is true that there are various approximations that have

been developed to linearize these types of problems, but these techniques are

by definition only valid for a limited class of targets.

One additional approach to this type of problem which will be examined in

detail later is the use of nonlinear filtering techniques based upon homomor-

phic filtering to address the strong(er) scattering cases in general. This method

is used in conjunction with diffraction tomography and the Born approxima-

tion to recover meaningful images of strongly scattering targets. As will be

shown later, in applying this approach, the data are preprocessed to ensure

that they are causal and minimum phase.

ReFeRenCeS

Belkebir, K. and Saillard, M. 2001. Special section on testing inversion algo-

rithms against experimental data,
Inverse Problems
, 17, 1565-1571.

Belkebir, K. and Saillard, M. 2005. Special section on testing inversion algo-

rithms against experimental data: Inhomogeneous targets,
Inverse

Problems
, 21, S1-S3.

Boas, R. P. 2011.
Entire Functions
(2nd edition). New York: Academic Press.

Devaney, A. J. 1978. Nonuniqueness in the inverse scattering problem.
Journal

of Mathematical Physics
,
19
, 1526-1535.

Devaney, A. J. 1983. A computer simulation of diffraction tomography.
IEEE

Transactions in Biomedical Engineering
,
30
, 377-386.

Hadamard, J. 1923.
Lectures on Cauchy's Problem in Linear Partial Differential

Equations
(Dover Phoenix edition). New York: Dover Publications.

Kaveh, M. and Soumekh, M. 1985. Algorithms and error analysis for difraction

tomography using the Born and Rytov approximations. In Boerner W.-M. et al.

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