33,116 research outputs found
Phase Retrieval for Sparse Signals: Uniqueness Conditions
In a variety of fields, in particular those involving imaging and optics, we
often measure signals whose phase is missing or has been irremediably
distorted. Phase retrieval attempts the recovery of the phase information of a
signal from the magnitude of its Fourier transform to enable the reconstruction
of the original signal. A fundamental question then is: "Under which conditions
can we uniquely recover the signal of interest from its measured magnitudes?"
In this paper, we assume the measured signal to be sparse. This is a natural
assumption in many applications, such as X-ray crystallography, speckle imaging
and blind channel estimation. In this work, we derive a sufficient condition
for the uniqueness of the solution of the phase retrieval (PR) problem for both
discrete and continuous domains, and for one and multi-dimensional domains.
More precisely, we show that there is a strong connection between PR and the
turnpike problem, a classic combinatorial problem. We also prove that the
existence of collisions in the autocorrelation function of the signal may
preclude the uniqueness of the solution of PR. Then, assuming the absence of
collisions, we prove that the solution is almost surely unique on 1-dimensional
domains. Finally, we extend this result to multi-dimensional signals by solving
a set of 1-dimensional problems. We show that the solution of the
multi-dimensional problem is unique when the autocorrelation function has no
collisions, significantly improving upon a previously known result.Comment: submitted to IEEE TI
An elementary approach for the phase retrieval problem
If the phase retrieval problem can be solved by a method similar to that of
solving a system of linear equations under the context of FFT, the time
complexity of computer based phase retrieval algorithm would be reduced. Here I
present such a method which is recursive but highly non-linear in nature, based
on a close look at the Fourier spectrum of the square of the function norm. In
a one dimensional problem it takes steps of calculation to recover the
phases of an N component complex vector. This method could work in 1, 2 or even
higher dimensional finite Fourier analysis without changes in the behavior of
time complexity. For one dimensional problem the performance of an algorithm
based on this method is shown, where the limitations are discussed too,
especially when subject to random noises which contains significant high
frequency components.Comment: 4 pages, 4 figure
Common pulse retrieval algorithm: a fast and universal method to retrieve ultrashort pulses
We present a common pulse retrieval algorithm (COPRA) that can be used for a
broad category of ultrashort laser pulse measurement schemes including
frequency-resolved optical gating (FROG), interferometric FROG, dispersion
scan, time domain ptychography, and pulse shaper assisted techniques such as
multiphoton intrapulse interference phase scan (MIIPS). We demonstrate its
properties in comprehensive numerical tests and show that it is fast, reliable
and accurate in the presence of Gaussian noise. For FROG it outperforms
retrieval algorithms based on generalized projections and ptychography.
Furthermore, we discuss the pulse retrieval problem as a nonlinear
least-squares problem and demonstrate the importance of obtaining a
least-squares solution for noisy data. These results improve and extend the
possibilities of numerical pulse retrieval. COPRA is faster and provides more
accurate results in comparison to existing retrieval algorithms. Furthermore,
it enables full pulse retrieval from measurements for which no retrieval
algorithm was known before, e.g., MIIPS measurements
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
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