Image Processing Reference
() = ()
Functions of this type have some very useful properties (Dudgeon and
Mersereau, 1984), such as
1. f ( x ) is causal.
2. The phase of F ( u ) lies between -π and +π; that is, its phase is always
3. Most of the energy in f ( x ) lies close to origin.
4. F ( u ) is absolutely summable.
Another very interesting and important aspect of minimum phase func-
tions is that these functions have a “minimum energy delay” property. This is
important because it results in a function that possesses the highest “partial”
energy among all functions that have the same Fourier magnitude (Oppenheim
et al., 1999). As stated before, a function, f ( x ), is defined as minimum phase if
its Fourier transform, F ( u ), is analytic and has a zero-free half plane.
With these definitions in hand, the concept of complex cepstrum (Bogert et al.,
1963) will now be defined and discussed. The complex cepstrum for a function
f is defined as taking the complex natural logarithm of the Fourier transform of
the function f , then taking the inverse Fourier transform. This can be written
mathematically for signal processing applications utilizing the FFT as
or, another common way of expressing this is as follows
where g is defined as the cepstrum of the input signal f . As alluded to earlier,
in a 1-D case, the minimum phase condition is required to ensure that the
Fourier transform of a causal function has a zero-free upper-half plane. This
is important because this now permits Cauchy's integral formula to be used to
relate the Fourier magnitude and the phase on the real u -axis. This relation-
ship is known as the logarithmic Hilbert transform.
As discussed in Shahid (2009), the concept of zero-free half plane in 2-D
can be problematic, to say the least. Even for a function that is separable such
as F ( u 1 ,u 2 ) = G ( u 1 ) H ( u 2 ), a zero-free upper-half plane in G ( z 1 ) automatically
leads to F ( z 1 , z 2 ) having zeros in the upper-half plane of z 2 , since we know
that H ( u 2 ) must have zeros. In practice, functions having a zero-free half plane
are rare, and there are few general conditions that are known for which these
characteristics can be imposed.
In general, in terms of Fourier-based theory and analysis of scattering and
inverse scattering, scattered and propagating fields are analytic functions due
to the fact that the scattering objects themselves are of finite spatial extent. To
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