12,441 research outputs found
Phase Retrieval with Application to Optical Imaging
This review article provides a contemporary overview of phase retrieval in
optical imaging, linking the relevant optical physics to the information
processing methods and algorithms. Its purpose is to describe the current state
of the art in this area, identify challenges, and suggest vision and areas
where signal processing methods can have a large impact on optical imaging and
on the world of imaging at large, with applications in a variety of fields
ranging from biology and chemistry to physics and engineering
Non-Convex Phase Retrieval from STFT Measurements
The problem of recovering a one-dimensional signal from its Fourier transform
magnitude, called Fourier phase retrieval, is ill-posed in most cases. We
consider the closely-related problem of recovering a signal from its phaseless
short-time Fourier transform (STFT) measurements. This problem arises naturally
in several applications, such as ultra-short laser pulse characterization and
ptychography. The redundancy offered by the STFT enables unique recovery under
mild conditions. We show that in some cases the unique solution can be obtained
by the principal eigenvector of a matrix, constructed as the solution of a
simple least-squares problem. When these conditions are not met, we suggest
using the principal eigenvector of this matrix to initialize non-convex local
optimization algorithms and propose two such methods. The first is based on
minimizing the empirical risk loss function, while the second maximizes a
quadratic function on the manifold of phases. We prove that under appropriate
conditions, the proposed initialization is close to the underlying signal. We
then analyze the geometry of the empirical risk loss function and show
numerically that both gradient algorithms converge to the underlying signal
even with small redundancy in the measurements. In addition, the algorithms are
robust to noise
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