The Phase Retrieval Problem - Introduction to Imaging from Scattered Fields - page 199

transformation, while retaining the current estimate for the phase. The itera-
tions would cease after some noise-determined number of steps or when the
changes at each step were less than some acceptable threshold.
The original Gerchberg-Saxton algorithm (Fiddy and Shahid, 2013) was
modified by Fienup to use the Fourier modulus and object support and non-
negativity constraints on f ( t ); it became known as “error-reduction” iteration.
The nonconvexity of the Fourier magnitude constraint leads to stagnation of
this algorithm due to local minima or fixed points that satisfy one constraint
but not the other. Stagnation could manifest itself by the presence of the twin
images shown earlier, superimposed stripes on the image due to phase wraps
near real zeros of F or an inappropriate choice of p ( t ). These stagnation mecha-
nisms can occur in all phase retrieval methods and various generalizations of
these iterative schemes were proposed to avoid it. The most successful modi-
fications have been algorithms where the constraints in the object domain are
not rigidly imposed, but are relaxed. For example, taking linear combinations
of estimates that are generated can help to avoid stagnation at local minima in
the cost function. Fienup developed a very robust approach and developed a
family of “input - output” procedures motivated by ideas from control theory.
The only limitation of the hybrid input - output map is that the support con-
straint cannot always be applied, as in crystallography, for example.
ReFeRenCeS
Carter, W. H. 1983. Inverse Optics: Proc. , Devaney, A. J. (ed.), SPIE 413, pp. 65-73.
Fiddy, M. A. and Shahid, U. 2013. Legacies of the Gerchberg-Saxton algorithm.
Ultramicroscopy , 134 , 48-54.