Image Processing Reference
In-Depth Information
·
·
�
Ψ
Ψ
(,
rr
rr
)
-
‚
=
Ψ
(,
rr
)
log( )
V
r
inc
log()
V
r
+
log
inc
(,
·
)
Ψ
(,
rr
·
)
inc
inc
inc
inc
(8.2)
·
�
�
Ψ
Ψ
(,
rr
rr
)
-
‚
[
]
+
inc
+
i
arg
V
()
r
arg
·
(,
)
inc
inc
If the Fourier transform is now taken of Equation 8.2, the results will be the
complex cepstrum representation of the target or
V
(
r
)Ψ/Ψ
inc
(Childers et al.,
1977; Raghuramireddy and Unbehauen, 1985). With the data now in this form,
the phase information of the complex data is retained which is essential in
doing any type of useful filtering processes. It should be noted that there are
numerous potential problems that can occur when calculating the complex
log(
V
Ψ) and taking its Fourier transform. There can be unwanted harmon-
ics introduced by taking the complex logarithm of sampled data where their
magnitude is less than 1 and approaches zero, which could lead to aliasing in
the cepstral domain (Childers et al., 1977). There is also the common concern
of branch cuts associated with singularities caused by zeros in the data repre-
sentation of
V
(
r
)Ψ since the logarithm function of zero is strictly undefined
Thus, the complex logarithmic function can be multivalued if the imaginary
part of the data representing
V
(
r
)Ψ exceeds 2π. When this happens, this results
in what is called phase wrapping, which can be extremely difficult to deal
with, especially in two or more dimensions. These phase wrappings pres-
ent discontinuities which generate many frequencies in the Fourier domain,
being associated with zeros in the field data corresponding to dislocated wave
fronts (Fiddy and Shahid, 2003).
8.2 CepStRAl FIlteRIng wIth MInIMuM phASe
In previous research (Shahid, 2009), it has been demonstrated that if the ceps-
tral data are made to be causal and minimum phase, and a spatial filter in
the cepstral domain is chosen properly, the ability to eliminate, or at least
greatly reduce or attenuate, the presence of the 〈
·
·
inc inc inc
term
can be achieved leading to a much better reconstructed image of
V
(
r
) than that
obtained using the Born approximation alone. The first two characteristics
mentioned above, that is, forcing the data to be causal and to be minimum
phase, are crucial in the success of this approach. These conditions are imper-
ative as demonstrated in Shahid (2009) and they form the basis for the success
of this approach. This approach will be thoroughly examined and extended
in this topic; so this being the case, it seems prudent to review the aspects of
this approach here.
The issue of processing the data to be causal is a simple one in that this can
be achieved by shifting all of the data into the first quadrant of an expanded
space whose origin is defined to be at its center. This is basically reduced to
a function of “re-indexing” the indices of the data in the computer code. The
second aspect mentioned above is that of making the data to be minimum
phase. In 1-D this concept is well understood as discussed in Dudgeon and
Mersereau (1984). In this case, a 1-D signal,
f
(
x
)
,
is said to be minimum phase
if and only if its Fourier transform,
F
(
z
), has a zero-free upper-half plane. In
other words,
F
(
z
) has no zeros for
v
> 0 where
z
=
u
+
iv
and
Ψ
(,
rr
)/
Ψ
(,
rr
)
〉
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