Image Processing Reference

In-Depth Information

Seven

Alternate Inverse Methods

7.1 IteRAtIve MethodS

Iterative techniques to solve the inverse scattering problem have gained tre-

mendous attention in the past 25 years (Belkebir and Tijhuis, 2001; Chun et al.,

2005; Crocco et al., 2005; Dubois et al., 2005; Estataico et al., 2005; Harada

et al., 1995; 1990; Isernia et al., 2004; Rieger et al., 1999; Wang and Chew,

1989). Several iterative methods have been proposed but only a few of them

have achieved some level of success. We will briefly examine the Born itera-

tive method, distorted Born iterative method, conjugate gradient method, and

the prior discrete Fourier transform method which serve as the basis for many

other iterative methods and approaches.

7.2 boRn IteRAtIve Method (bIM)

Wang and Chew (1989) first proposed the Born iterative method. The first Born

approximation is traditionally considered limited for strong scattering objects

because of strong diffraction effects; therefore, the inherent nonlinearity of

the integral equation in Equation 4.28 has to be taken into account. The start-

ing point of BIM is to first acquire an initial estimate,
V
B
1
()
r
of the object

by using the first Born approximation. The estimated
V
B
1
(
r
is then used to

compute the field inside the scattering volume and at the receiver points. The

BIM uses a point-matching method with the pulse basis function to solve the

forward scattering problem (Wang and Chew, 1989). The estimated field com-

puted in the above step is substituted into Equation 4.7 as follows:

kV

·

·

Ψ

(

rr

,

)

=

Ψ

()

r

−

2

(

r rr rr r

′

)

Ψ

(

′,

)

G

(

,

′

)

d

′

(7.1)

inc

inc

inc

0

D

to calculate the next order scattering function
V
B
2
()
r
The second-order scat-

tering object
V
B
2
(
r
is used to solve the scattering problem for the field inside

the object and at the observation points. The simulated field Ψ
sim
is then com-

pared with the measured scattered field data Ψ
s
measured
shown here

·

·

D

=

Ψ

sim

(

rr

,

)

−

Ψ

measured

(

rr

,

)

(7. 2)

s

inc

s

inc

and if the difference between them is less than 5% then the iteration can be

terminated, otherwise one continues with the iterations. The BIM also uses

a regularization method to address the nonuniqueness and instability of the

inverse scattering problem. The regularization method imposes an additional

97

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