Image Processing Reference
InDepth Information
field is encoded in the magnitude via a logarithmic Hilbert transform. An
important point, however, is that if one can compute a good estimate for
V
〈Ψ〉
from measured data, then one could numerically add in this reference wave
prior to homomorphic filtering It becomes a postdata processing step and not
an experimental requirement.
In order to choose an appropriate reference point, the starting point is to
make the amplitude of the reference point the same as the maximum ampli
tude of
V
〈Ψ〉 and then systematically increase its amplitude until the phase of
Born reconstructed
V
〈Ψ〉 lies between −π and +π. Figure 8.2 shows the result
of this procedure applied to a cylinder of radius 1λ.
This value for
R
permits us to write
2
�

‚
�

‚
V
Ψ
V
Ψ
1
2
V
Ψ
(8.14)
log(
RV
+ →+
Ψ
)
log
1
:
−
+
R
R
R
or one could also write
�
V
Ψ

‚
=
�
V
R
R
V

‚
=
V
R
�
R
V

‚
�

‚
+
log
1 +
log
+
log
+
Ψ
(8.15)
Ψ
log
R
(a)
(b)
6
30
4
25
2
ε
r
= 2.3
20
0
15
2
10
4
5
6
2
λ
6
4
2
0246
mm
Minimum phase
V
Ψ
(c)
3
6
2
4
2
1
0
0
2
1
4
2
6
3
mm
246
6
4
2
Figure 8.2
Scattering cylinder. (a) Cylinder with 2
λ
diameter and relative permittivity of 2.3, (b) Born recon
struction, and (c) minimum phase after adding a suitable reference point.
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