Image Processing Reference

In-Depth Information

integral equation of the first kind. In order to solve this equation for
V
(
r
), we

must know or be able to adequately approximate what Ψ(
r
) is inside the target

domain
D
.

4.3 FIRSt boRn AppRoxIMAtIon

One of the more well-known and common approaches in imaging from scat-

tered fields using the diffraction tomography method is the Born approxima-

tion. In general, this method assumes that the target is a weakly scattering

object, and it is generally used in conjunction with the data inversion method

described in the previous section. Many current inverse scattering algorithms

utilize this approach even when it is not strictly valid to do so (Avish and

Slaney, 1988; Lin and Fiddy, 1990). As already indicated, when using this

approach, the problem is linearized and establishes a Fourier relationship

between the measured scattered field and some function of the target's scatter-

ing properties from which we hope to compute an image. A brief introduction

to the first Born approximation is given here.

Recalling from Equation 4.6 that the total measured scattered field at the

receivers can be generally expressed as

ΨΨ Ψ

()

r

=

()

r

+

()

r

(4.27)

inc

s

this can be expanded in terms of an inhomogeneous Fredholm integral equa-

tion of the first kind (Morse and Feshbach, 1953) as follows:

1

8

∫

·

·
inc
)
r

Ψ

(

rr

,

)

=

e

ik

rr

·

⋅

+

e

ikr

(

+

π

/

42

)

k

Ve

(

r

′

)

i

k

rr

·

Ψ

(

r r

′

,

′

⋅

′

inc

(4.28)

inc

π

kr

D

The first term in the above equation is the contribution from the incident

(or illuminating) wave. This term can be premeasured, that is, data obtained

when no target is present, and subtracted out, which leaves only the second

term in above equation which is the scattered field expressed as follows:

1

8

∫

·

·

Ψ

(

rr

,

)

=

e

ikr

(

+

π

/

42

)

k

Ve

(

r

′

)

i

k

rr

·

⋅

Ψ

(

r r

′

,

)d

r

′

′

(4.29)

s

inc

inc

π

kr

D

1

8

·

··

Ψ

(

rr

,

)

=

e

ikr

(

+

π

/

4

)

f kk

(

rr

,

)

(4.30)

s

inc

inc

π

kr

·

where
fk k

(,

rr

·
inc
is the scattering amplitude, which is defined as

)

∫

··

·

fk k

(,

rr

)

=

k

2

Ve

(

r

′

)

i

k

rr

·

⋅

Ψ

(

r r

′

,

)

d

r

′

′

(4.31)

inc

inc

D

The basis for the Born approximation is that the total field, Ψ, inside the tar-

get can be approximated by the incident field in the above integral in Equation

4.31 as follows:

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