Image Processing Reference

In-Depth Information

+∞

∫

()
=
()

−∞

Fu

fxe

iux

2π

d

x

(8.3)

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

unwrapped (nondiscontinuous).

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

[

]

ˆ

(8.4)

gifft

=

log

()

F

here

F

=

fft f

()

or, another common way of expressing this is as follows

+∞

∫

1

2

[

]

ˆ

gx

()=log

Fu

()

e

d

u

i

2

π

ux

(8.5)

π

−∞

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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