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
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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