Image Processing Reference

In-Depth Information

assume |
A
|≪1, allows us to write
G
= 1 +
A
~ exp(
A
), thereby satisfying this

minimum phase condition.

Therefore, if to our band-limited function
F
(
z
) we add another function
G
(
z
)

which we refer to as a reference function and
G
(
z
) has no zeros in upper-half

plane, then the function
F
(
z
) +
G
(
z
) will have no zeros in upper-half plane, thus

satisfying Rouche's theorem and the minimum phase condition. The addition

of a reference function to an analytic function only moves the zeros from

upper-half plane to lower-half plane without destroying them (Fiddy, 1987). If

a reference function is chosen as a constant, then we can always find a contour

in the upper-half plane along which the magnitude of the added function is

greater than the magnitude of
F
(
z
). Increasing the constant moves the contour

across the real axis and thus pushing zeros to the lower-half plane. It therefore

follows that one can preprocess by adding a finite constant or reference point

to make it minimum phase before taking its logarithm.

8.4 pRepRoCeSSIng dAtA

The preprocessing step requires the data available in
k
-space to be made

causal,
V
〈Ψ〉
c
. This can be done by moving available scattered field data into

one quadrant of
k
-space, that is, data are nonzero only in one quadrant.

The next step is to add a reference point at the origin of the causal data

V
〈Ψ〉
c
in
k
-space. This corresponds to adding a linear phase factor to
V
Ψ in

the object domain. In order to satisfy the minimum phase condition the ref-

erence point
R
does not need to have an arbitrarily large amplitude (a suffi-

cient condition), but simply be just large enough to ensure that phase of
V
Ψ

is continuous and lies within the bounds of −π and +π. A very large reference

point with amplitude
R
such that VΨ
R
1 leads to the Fourier transform of

log(
R
+
V
〈Ψ〉) → log(1 +
V
〈Ψ〉/
R
) being approximately equal to
V
〈Ψ〉/
R
indicating

that we have satisfied the minimum phase condition but we have not provided a

function of
V
〈Ψ〉 that will result in the successful filtering in the cepstral domain.

In this case, the cepstrum of
V
〈Ψ〉 contains the same information that we

originally had in
k
-space. This corresponds to
R
= 0 and the unwanted har-

monics in the cepstrum makes filtering impossible. The optimal choice of
R

is an amplitude which is just large enough to ensure that the phase of
V
〈Ψ〉 is

unwrapped and lies between −π and +π. It has been shown (Fiddy and Shahid,

2003; McGahan and Kleinman, 1997) that in order to enforce a minimum

phase condition, the reference should satisfy

(8.13)

R
≥Ψ
max

that is, the reference point needs to be larger than the maximum value of the

scattered field amplitude to satisfy Rouche's theorem. It is evident that the

inequality given in Equation (8.13) is a sufficient condition to enforce mini-

mum phase condition. In the forward scattering problem, this is intuitively

satisfying since it is equivalent to requiring that the scattering is relatively

weak compared to a larger amplitude background or incident plane wave, as

demanded by the first Born approximation. The analogy with holography is

also immediate since interference with a reference wave, especially when off-

axis with respect to the scattered field, ensures that the phase of the scattered

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