Image Processing Reference

In-Depth Information

constraint on the linear system and allows the user to choose a solution from

many available solutions. Green's function remains unchanged during the

entire iterative process. More details of this method can be found in Wang

and Chew (1989).

7.3 dIStoRted boRn IteRAtIve Method (dbIM)

The distorted Born iterative method was also proposed by Wang and Chew

(1990) as an improvement over BIM (Wang and Chew, 1989). Similar to BIM,

it starts by solving for the first-order scattering object
V
B
1
(
r
by using the

first Born approximation, and the homogeneous Green's function with rela-

tive permittivity of unity is used initially. The next step is to use this object

function
V
B
1
(
r
to solve the forward scattering problem using the method of

moments (Richmond, 1965) and to calculate the field inside the object and

at the receiver points. Using
V
B
1
(
r
, the point source response in the object

for every observation point is computed. In BIM, Green's function was kept

constant throughout the iterative process, whereas in DBIM Green's function

G
1
(, )

rr
′ is calculated with the last reconstructed permittivity profile as the

background permittivity. The estimated Green's function and field are substi-

tuted in the integral equation

∫

·

·

Ψ

(

rr

,

)

=−

kV

2

(

r

′

)

Ψ

(

r r

′

,

)

G

( ,)

rr

′

d

r

′

(7. 3)

s

inc

inc

0

D

The calculated scattered field is then subtracted from the field at receiv-

ers and the inverse scattering problem is solved for the correction of the last

reconstructed profile Feedback from the previous profile is used to generate

a new profile The forward scattering problem is solved again using this new

profile and the computed scattered field is compared with the measured scat-

tered field If the relative residual error (RRE) is less than the criterion defined

then the process terminates; else it continues. Wang and Chew (1989) defined

RRE as

M

∑

Ψ

measured

()

r

−

Ψ

sim

()

j

()

r

s

a

s

a

RRE

=

a

=

1

(7.4)

∑

M

Ψ

measured

()

r

s

a

a

=

1

where
j
is the iteration cycle. The convergence rate of the distorted Born itera-

tive method is faster than that of the Born iterative method. However, the Born

iterative method is more tolerant to noise than the distorted Born iterative

method. Depending upon the nature of the problem, either the BIM or DBIM

can be utilized.

7.4 ConjugAte gRAdIent Method (CgM)

The conjugate gradient method (Harada et al., 1995; Lobel et al., 1996) is an

iterative technique for solving the inverse scattering problem using an opti-

mization procedure. In this method, a functional is defined as a norm of

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