Image Processing Reference
In-Depth Information
∞
∏
∞
Fx
(
+= −
=−
iy
)
e
cz
(
1/
(
z z
))
j
j
where the
z
j
's are the locations of the
j
th complex zero of the function
F
(
z
).
In this way, a band-limited function of a single complex variable is uniquely
defined by its complex zeros. The zeros occur at an infinite number of isolated
points in the complex plane, and, in general, these points will not be expected
to coincide with the real axis.
To extend the zero description to functions of more than one complex vari-
able, the Plancherel-Polya theorem states that, for an object of compact sup-
port, the
p
-
dimensional Fourier transform
F
(
z
1
,
z
2
, ….,
z
p
) is an entire function
of exponential type.
F
(
z
) can be uniquely represented by a product of the form
N
z
z
�
-
‚
∏
1
Fz z
(, )
≈
e
(
cz
+
cz
)
1
−
11
22
12
g
()
2
j
=
1
known as the Osgood product shown here for two complex variables and
where it is possible that
N
= 1.
Thus, in two or more dimensions,
F
(
z
) is represented by the product of
either a finite or infinite set of factors. However,
if F is
irreducible, then
N
= 1,
and arg(
F
) is uniquely defined by the magnitude of
F.
This does not necessar-
ily make that unique phase easy to determine!
A logical consequence of Fourier-based models for fields and waves is the
inherent analyticity of the functions being considered. We will address in the
next section the practical consequences of having measured noisy discrete
data samples, but for now let us focus on these analytic properties. It is well
known that the Fourier transform
F
(
x
) of a function
f
(
t
) will be entire (i.e.,
analytic throughout the complex
z
=
x
+
iy
complex plane) if
f
(
t
) has compact
support.
F
(
x
) is also known as a band-limited function. In the time domain
this simply means that the function is turned on and off, and if
t
is a spatial
variable, then edges or boundaries of some object or aperture will serve the
same purpose. Indeed, the presence of just one “edge” or, in the time domain,
a causality condition requiring
f
(
t
) = 0, for
t
< 0 ensures that
F
(
z
) is regular in
the upper half of the complex
z
plane. Known as Titchmarsh's theorem, it was
pointed out in the 1920s that a logical equivalence to a causality condition is
that Cauchy's integral formula can be written for a contour,
C
, comprising the
real
x
axis and an infinite semicircle in the upper half plane the contribution
from which can be neglected. This results in the real and imaginary parts of
F
(
x
) being related by Hilbert transformation.
∞
Re[( )]
fx
xx
x
1
′
∫
Im[( )]
fx
=
d
′
π
′ −
−∞
∞
∫
1
Im[( )]
fx
xx
′
Re[( )]
fx
=
P
d
x
′
(B.1)
π
′ −
−∞
These integral transforms allow the real part of
F
to be computed from the
imaginary part and vice versa, and these transforms can be generalized to
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