Alternate Inverse Methods - Introduction to Imaging from Scattered Fields - page 97

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

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