12,890 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
Signal Recovery in Perturbed Fourier Compressed Sensing
In many applications in compressed sensing, the measurement matrix is a
Fourier matrix, i.e., it measures the Fourier transform of the underlying
signal at some specified `base' frequencies , where is the
number of measurements. However due to system calibration errors, the system
may measure the Fourier transform at frequencies
that are different from the base frequencies and where
are unknown. Ignoring perturbations of this nature can lead to major errors in
signal recovery. In this paper, we present a simple but effective alternating
minimization algorithm to recover the perturbations in the frequencies \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations can be expressed
in terms of a small number of unique parameters . We demonstrate that
in such cases, the method leads to excellent quality results that are several
times better than baseline algorithms (which are based on existing off-grid
methods in the recent literature on direction of arrival (DOA) estimation,
modified to suit the computational problem in this paper). Our results are also
robust to noise in the measurement values. We also provide theoretical results
for (1) the convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included.
More challenging experiments now include
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