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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