The Phase Retrieval Problem - Introduction to Imaging from Scattered Fields - page 198

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