Image Processing Reference
In-Depth Information
go one step further, these fields are entire functions of the exponential type
(Burge et al., 1976), which by definition means that they satisfy the Cauchy-
Riemann equation for all finite z = x + iy as expressed here:
Re
F
x
Im
F
=
(8.6)
y
Re
F
−∂
Im
F
=
(8.7)
y
x
There are of course very strict constraints on these types of functions as
to how their amplitude varies or increases and how their zero crossings are
distributed in general. In 1-D this is well understood and these concepts are,
for the most part, indirectly applicable in the 2-D realm.
8.3 geneRAtIng the MInIMuM phASe FunCtIon
Dudgeon and Mersereau (1984) state that a 2-D minimum phase function is
one that is absolutely summable and the inverse and complex cepstrum of
which are also absolutely summable and have the same region of support.
This region of support also has to be a convex region of some kind. It has
not been possible to find general properties for classes of functions for which
these conditions can be satisfied, and this condition appears to be sufficient
but not necessary. Some specific examples exist of well-conditioned cepstra,
as a result of the incorporation of a background or reference wave on the func-
tion whose logarithm is to be Fourier transformed. Taking the logarithm of a
band-limited function produces a band-limited function and hence a ceps-
trum of finite support only for minimum phase functions.
At this point, a brief digression on dispersion relations may be helpful. It
is well known that the real and imaginary parts of the Fourier transform of
a causal function f ( x ) are related to each other by a Hilbert transform. This
follows directly from Titchmarsh's theorem (Fiddy and Shahid, 2003) and is
a consequence of the finite support or causal nature of the Fourier transform.
The Hilbert transform, HT, is an integral transform that is a principal value
integral, which is implicitly solved as a contour integral. The contour is the
real axis and a semicircle in either the upper or lower half of the complex
plane, which, if Jordan's lemma is satisfied, can be ignored. The HT relation-
ship then takes the form
+∞
1
π
Re
Fu
uu u
[( )]
[
] =
Im
Fu
()
P
d
(8.8)
−∞
+∞
1
π
Im
Fu
uu u
[( )]
[
] =−
Re
Fu
()
P
d
(8.9)
′ −
−∞
where P is the Cauchy principal value. Closure of the contour without contri-
butions to the value of the integral from the residues is necessary and so the
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