Image Processing Reference

In-Depth Information

,
·

·

Ψ(

rr

)

=

e
ik

r

−

r

(4.32)

inc

inc

In theory, for the first Born approximation to be valid (i.e., for an object to be

classified as a weak scatterer) a necessary condition is that the product of the

target's permittivity, its characteristic dimension, and wave number should be

much less than unity (Li and Wang, 2010). This is typically expressed math-

ematically as

(4.33)

kV

(
r
1

a

where
k
is the wave number; the object and permittivity are related by

V
(
r
) = ε
r
(
r
) − 1 where, in this example, ε
r
(
r
) is the relative permittivity of the

target and '
a
' specifies some measure of the physical size of the target. It

should be noted that the absolute value of this product should be less than 1 to

account for new meta-materials that may have a permittivity that is negative. It

is widely expected and believed by many that as the dimensions of the object

increase, or the magnitude of absolute value of the permittivity fluctuations

increases, the first Born approximation becomes increasingly poor. While this

does seem to be the case in general, this assumption will be examined in more

detail, and the limits of this will be better deined later in this topic.

In order to examine the Born approximation in more detail, Equation 4.32

can be substituted into Equation 4.31 which yields the linearized version of

the inversion problem as follows:

∫

·

·

··

f

BA

(,

kk

rr

)

=

k

2

Ve

(

r

′

)

−

i

k

(

rr

⋅

)

⋅

r

′

d

r

′

(4.34)

inc

inc

D

which illustrates where the Ewald circles (introduced later) originate and

where the scattered field within the first Born approximation is expressed as

1

8

∫

·

Ψ

(

rr

,

)

=

(

r

′

)

··

d

′

BA

e

ikr

(

+

π

/

42

)

k

Ve

−

i

k

(

rr r

⋅

)

⋅

′

r

(4.35)

inc

s

inc

π

kr

D

Equation 4.35 gives the Fourier relationship between the target or scat-

tering object,
V
(
r
), and the measured scattering amplitude at the receivers

f

·

(,

kk

rr

·

).

If the integral in Equation 4.35 is isolated as follows

BA

inc

(

)

8

π

kr e

k

−+

ikr

(

π

/

4

)

∫

·

(4.36)

Ve

()

r

′

k
··

d

r

′

=

Ψ

(

r r

,

c
)

−

i

(

rr r

⋅

)

⋅

′

BA

inc

s

in

2

D

the right-hand side of Equation 4.36 is what is implemented in MATLAB
®

code later in this text. This gives a representation not of
V
(
r
), but it gives an

estimate for the product
V
(
r
)Ψ/Ψ
inc
. The challenge is now to determine how to

identify Ψ and effectively remove or minimize it to recover a valid representa-

tion of the target,
V
(
r
).

One challenge with the Born approximation is the inconsistency mentioned

by Ramm (1990). Ramm demonstrated that the target or scattering object
V
(
r
)

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