Image Processing Reference

In-Depth Information

more than one
x
dimension. The Kramers-Kronig relations in the temporal

frequency domain are logically tied to the causal nature of a medium's polar-

izability. Interestingly, if one could ensure that log
F
(
x
) = lo g|
F
(
x
)| +
i
φ(
x
) were

also regular in the upper half of the complex plane, the phase problem would

be solved since we could write:

∞

∫

1

π

Re[( )]

Fx

xx
x

′

φ()

x

=

Im[log()]

F x

=

P

d

′

(B.2)

′ −

−∞

Even more interesting is that in 1-D problems, most functions
F
(
z
) we

encounter have isolated points in the complex plane at which |
F
| = 0. Indeed,

much work has been done to study how functions
f
(
t
) influence the distribu-

tion of these zeros, of which there is an infinite number for a band-limited

function. Zeros in the upper half plane lead to singularities inside the con-

tour
C
, and residues need to be found to correct the phase calculated using

Equation B.2. If
F
(
z
) is a band-limited function, it can be represented either by

its complex amplitude values measured at the Shannon or Nyquist sampling

rate, or by the coordinate locations at which these zeros occur.

From |
F
| measurements, one has only the knowledge of the zero locations

to within a sign ambiguity for their imaginary coordinates,
y
j
. With
N
complex

zeros one can create ~2
N
−1
different functions
F
(
x
) since
F
and its complex

conjugate have complex zeros that are located symmetrically about the real

x
-axis. Hence, there are 2
N
−1
functions
f
(
t
) that are consistent with the mea-

sured |
F
|.

However, if one knew
a priori
that
F
(
z
) has a zero free half plane, then

Equation B.2 will produce the correct phase. If one applies Equation B.2 regard-

less, that is, without the prior knowledge of a zero free half plane, then due

to the ambiguity in the sign of |
y
j
|, the solution for
f
(
t
) that one obtains using

the phase calculated from Equation B.2 is the so-called minimum phase func-

tion. In 1974, this fact suggested that one simply needs to preprocess a wave

prior to detection by adding a known reference wave that satisfied Rouche's

theorem and hence ensure that this new superposition of fields represented

a minimum phase function. Rouche's theorem states that if function
g
has
N

zeros in some region of the complex plane, and function
h
has
M
<
N
zeros in

the same region, then if around that boundary, |
h
| > |
g
| then
g
+
h
will have

M
zeros in the region. For
h
= constant, then
M
= 0 providing a mechanism

to enforce a zero free region. There is a clear connection here with making a

hologram, as mentioned above.

Many iterative phase retrieval methods have emerged over the years and

many comparisons between them made. Until a few years ago, the consensus

was that several of these methods could be made to succeed but that there

remained a degree of uncertainly or an occasional lack of confidence is the

results obtained as mentioned above.

The Gerchberg-Saxton algorithm was one of the first such iterative algo-

rithms and it can be described as follows. Given an object function
f
(
t
) and its

Fourier transform,
F
(
x
), the objective of the algorithm is to iteratively recover

complex
f
and
F
by imposing consistency with measured |
f
| and |
F
|. The iter-

ations would begin by initially choosing a random phase function followed

by reintroduction of the magnitudes after each forward or inverse Fourier

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