1,884 research outputs found
Convex recovery from interferometric measurements
This note formulates a deterministic recovery result for vectors from
quadratic measurements of the form for some
left-invertible . Recovery is exact, or stable in the noisy case, when the
couples are chosen as edges of a well-connected graph. One possible way
of obtaining the solution is as a feasible point of a simple semidefinite
program. Furthermore, we show how the proportionality constant in the error
estimate depends on the spectral gap of a data-weighted graph Laplacian. Such
quadratic measurements have found applications in phase retrieval, angular
synchronization, and more recently interferometric waveform inversion
Compressed sensing imaging techniques for radio interferometry
Radio interferometry probes astrophysical signals through incomplete and
noisy Fourier measurements. The theory of compressed sensing demonstrates that
such measurements may actually suffice for accurate reconstruction of sparse or
compressible signals. We propose new generic imaging techniques based on convex
optimization for global minimization problems defined in this context. The
versatility of the framework notably allows introduction of specific prior
information on the signals, which offers the possibility of significant
improvements of reconstruction relative to the standard local matching pursuit
algorithm CLEAN used in radio astronomy. We illustrate the potential of the
approach by studying reconstruction performances on simulations of two
different kinds of signals observed with very generic interferometric
configurations. The first kind is an intensity field of compact astrophysical
objects. The second kind is the imprint of cosmic strings in the temperature
field of the cosmic microwave background radiation, of particular interest for
cosmology.Comment: 10 pages, 1 figure. Version 2 matches version accepted for
publication in MNRAS. Changes includes: writing corrections, clarifications
of arguments, figure update, and a new subsection 4.1 commenting on the exact
compliance of radio interferometric measurements with compressed sensin
A Deterministic Theory for Exact Non-Convex Phase Retrieval
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for
phase retrieval and identify a novel sufficient condition for universal exact
recovery through the lens of low rank matrix recovery theory. Via a perspective
in the lifted domain, we show that the convergence of the WF iterates to a true
solution is attained geometrically under a single condition on the lifted
forward model. As a result, a deterministic relationship between the accuracy
of spectral initialization and the validity of {the regularity condition} is
derived. In particular, we determine that a certain concentration property on
the spectral matrix must hold uniformly with a sufficiently tight constant.
This culminates into a sufficient condition that is equivalent to a restricted
isometry-type property over rank-1, positive semi-definite matrices, and
amounts to a less stringent requirement on the lifted forward model than those
of prominent low-rank-matrix-recovery methods in the literature. We
characterize the performance limits of our framework in terms of the tightness
of the concentration property via novel bounds on the convergence rate and on
the signal-to-noise ratio such that the theoretical guarantees are valid using
the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin
Distributed and parallel sparse convex optimization for radio interferometry with PURIFY
Next generation radio interferometric telescopes are entering an era of big
data with extremely large data sets. While these telescopes can observe the sky
in higher sensitivity and resolution than before, computational challenges in
image reconstruction need to be overcome to realize the potential of
forthcoming telescopes. New methods in sparse image reconstruction and convex
optimization techniques (cf. compressive sensing) have shown to produce higher
fidelity reconstructions of simulations and real observations than traditional
methods. This article presents distributed and parallel algorithms and
implementations to perform sparse image reconstruction, with significant
practical considerations that are important for implementing these algorithms
for Big Data. We benchmark the algorithms presented, showing that they are
considerably faster than their serial equivalents. We then pre-sample gridding
kernels to scale the distributed algorithms to larger data sizes, showing
application times for 1 Gb to 2.4 Tb data sets over 25 to 100 nodes for up to
50 billion visibilities, and find that the run-times for the distributed
algorithms range from 100 milliseconds to 3 minutes per iteration. This work
presents an important step in working towards computationally scalable and
efficient algorithms and implementations that are needed to image observations
of both extended and compact sources from next generation radio interferometers
such as the SKA. The algorithms are implemented in the latest versions of the
SOPT (https://github.com/astro-informatics/sopt) and PURIFY
(https://github.com/astro-informatics/purify) software packages {(Versions
3.1.0)}, which have been released alongside of this article.Comment: 25 pages, 5 figure
PURIFY: a new algorithmic framework for next-generation radio-interferometric imaging
In recent works, compressed sensing (CS) and convex opti- mization techniques have been applied to radio-interferometric imaging showing the potential to outperform state-of-the-art imaging algorithms in the field. We review our latest contributions [1, 2, 3], which leverage the versatility of convex optimization to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to significant acceleration of the reconstruction and high-dimensional data scalability. The new algorithmic structure promoted in a new software PURIFY (beta version) relies on the simultaneous-direction method of multipliers (SDMM). The performance of various sparsity priors is evaluated through simulations in the continuous visibility setting, confirming the superiority of our recent average sparsity approach SARA
Compressed sensing for wide-field radio interferometric imaging
For the next generation of radio interferometric telescopes it is of
paramount importance to incorporate wide field-of-view (WFOV) considerations in
interferometric imaging, otherwise the fidelity of reconstructed images will
suffer greatly. We extend compressed sensing techniques for interferometric
imaging to a WFOV and recover images in the spherical coordinate space in which
they naturally live, eliminating any distorting projection. The effectiveness
of the spread spectrum phenomenon, highlighted recently by one of the authors,
is enhanced when going to a WFOV, while sparsity is promoted by recovering
images directly on the sphere. Both of these properties act to improve the
quality of reconstructed interferometric images. We quantify the performance of
compressed sensing reconstruction techniques through simulations, highlighting
the superior reconstruction quality achieved by recovering interferometric
images directly on the sphere rather than the plane.Comment: 15 pages, 8 figures, replaced to match version accepted by MNRA
PURIFY: a new approach to radio-interferometric imaging
In a recent article series, the authors have promoted convex optimization algorithms for radio-interferometric imaging in the framework of compressed sensing, which leverages sparsity regularization priors for the associated inverse problem and defines a minimization problem for image reconstruction. This approach was shown, in theory and through simulations in a simple discrete visibility setting, to have the potential to outperform significantly CLEAN and its evolutions. In this work, we leverage the versatility of convex optimization in solving minimization problems to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to significant acceleration of the reconstruction and high-dimensional data scalability. The new algorithmic structure promoted relies on the simultaneous-direction method of multipliers (SDMM), and contrasts with the current major-minor cycle structure of CLEAN and its evolutions, which in particular cannot handle the state-of-the-art minimization problems under consideration where neither the regularization term nor the data term are differentiable functions. We release a beta version of an SDMM-based imaging software written in C and dubbed PURIFY (http://basp-group.github.io/purify/) that handles various sparsity priors, including our recent average sparsity approach SARA. We evaluate the performance of different priors through simulations in the continuous visibility setting, confirming the superiority of SARA
A randomised primal-dual algorithm for distributed radio-interferometric imaging
Next generation radio telescopes, like the Square Kilometre Array, will
acquire an unprecedented amount of data for radio astronomy. The development of
fast, parallelisable or distributed algorithms for handling such large-scale
data sets is of prime importance. Motivated by this, we investigate herein a
convex optimisation algorithmic structure, based on primal-dual
forward-backward iterations, for solving the radio interferometric imaging
problem. It can encompass any convex prior of interest. It allows for the
distributed processing of the measured data and introduces further flexibility
by employing a probabilistic approach for the selection of the data blocks used
at a given iteration. We study the reconstruction performance with respect to
the data distribution and we propose the use of nonuniform probabilities for
the randomised updates. Our simulations show the feasibility of the
randomisation given a limited computing infrastructure as well as important
computational advantages when compared to state-of-the-art algorithmic
structures.Comment: 5 pages, 3 figures, Proceedings of the European Signal Processing
Conference (EUSIPCO) 2016, Related journal publication available at
https://arxiv.org/abs/1601.0402
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