1,255 research outputs found
PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
Suppose we wish to recover a signal x in C^n from m intensity measurements of
the form ||^2, i = 1, 2,..., m; that is, from data in which phase
information is missing. We prove that if the vectors z_i are sampled
independently and uniformly at random on the unit sphere, then the signal x can
be recovered exactly (up to a global phase factor) by solving a convenient
semidefinite program---a trace-norm minimization problem; this holds with large
probability provided that m is on the order of n log n, and without any
assumption about the signal whatsoever. This novel result demonstrates that in
some instances, the combinatorial phase retrieval problem can be solved by
convex programming techniques. Finally, we also prove that our methodology is
robust vis a vis additive noise
Phase Retrieval via Matrix Completion
This paper develops a novel framework for phase retrieval, a problem which
arises in X-ray crystallography, diffraction imaging, astronomical imaging and
many other applications. Our approach combines multiple structured
illuminations together with ideas from convex programming to recover the phase
from intensity measurements, typically from the modulus of the diffracted wave.
We demonstrate empirically that any complex-valued object can be recovered from
the knowledge of the magnitude of just a few diffracted patterns by solving a
simple convex optimization problem inspired by the recent literature on matrix
completion. More importantly, we also demonstrate that our noise-aware
algorithms are stable in the sense that the reconstruction degrades gracefully
as the signal-to-noise ratio decreases. Finally, we introduce some theory
showing that one can design very simple structured illumination patterns such
that three diffracted figures uniquely determine the phase of the object we
wish to recover
Multiple Illumination Phaseless Super-Resolution (MIPS) with Applications To Phaseless DOA Estimation and Diffraction Imaging
Phaseless super-resolution is the problem of recovering an unknown signal
from measurements of the magnitudes of the low frequency Fourier transform of
the signal. This problem arises in applications where measuring the phase, and
making high-frequency measurements, are either too costly or altogether
infeasible. The problem is especially challenging because it combines the
difficult problems of phase retrieval and classical super-resolutionComment: To appear in ICASSP 201
Signal reconstruction from the magnitude of subspace components
We consider signal reconstruction from the norms of subspace components
generalizing standard phase retrieval problems. In the deterministic setting, a
closed reconstruction formula is derived when the subspaces satisfy certain
cubature conditions, that require at least a quadratic number of subspaces.
Moreover, we address reconstruction under the erasure of a subset of the norms;
using the concepts of -fusion frames and list decoding, we propose an
algorithm that outputs a finite list of candidate signals, one of which is the
correct one. In the random setting, we show that a set of subspaces chosen at
random and of cardinality scaling linearly in the ambient dimension allows for
exact reconstruction with high probability by solving the feasibility problem
of a semidefinite program
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