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Yves Wiaux

Publications and source records attributed to Yves Wiaux.

At least 55 records · Page 3Linked to original sources
arXiv2017

Robust dimensionality reduction for interferometric imaging of Cygnus A

Extremely high data rates expected in next-generation radio interferometers necessitate a fast and robust way to process measurements in a big data context. Dimensionality reduction can alleviate computational load needed to process these data, in terms of both computing speed and memory usage. In this article, we present image reconstruction results from highly reduced radio-interferometric data, following our previously proposed data dimensionality reduction method, $\mathrm{R}_{\mathrm{sing}}$, based on studying the distribution of the singular values of the measurement operator. This method comprises a simple weighted, subsampled discrete Fourier transform of the dirty image. Additionally, we show that an alternative gridding-based reduction method works well for target data sizes of the same order as the image size. We reconstruct images from well-calibrated VLA data to showcase the robustness of our proposed method down to very low data sizes in a 'real data' setting. We show through comparisons with the conventional reduction method of time- and frequency-averaging, that our proposed method produces more accurate reconstructions while reducing data size much further, and is particularly robust when data sizes are aggressively reduced to low fractions of the image size. $\mathrm{R}_{\mathrm{sing}}$ can function in a block-wise fashion, and could be used in the future to process incoming data by blocks in real-time, thus opening up the possibility of performing 'on-line' imaging as the data are being acquired. MATLAB code for the proposed dimensionality reduction method is available on GitHub.

astro-ph.IM↗
arXiv2017

Sparse image reconstruction on the sphere: analysis and synthesis

We develop techniques to solve ill-posed inverse problems on the sphere by sparse regularisation, exploiting sparsity in both axisymmetric and directional scale-discretised wavelet space. Denoising, inpainting, and deconvolution problems, and combinations thereof, are considered as examples. Inverse problems are solved in both the analysis and synthesis settings, with a number of different sampling schemes. The most effective approach is that with the most restricted solution-space, which depends on the interplay between the adopted sampling scheme, the selection of the analysis/synthesis problem, and any weighting of the l1 norm appearing in the regularisation problem. More efficient sampling schemes on the sphere improve reconstruction fidelity by restricting the solution-space and also by improving sparsity in wavelet space. We apply the technique to denoise Planck 353 GHz observations, improving the ability to extract the structure of Galactic dust emission, which is important for studying Galactic magnetism.

cs.IT↗
arXiv2017

Non-convex optimization for self-calibration of direction-dependent effects in radio interferometric imaging

Radio interferometric imaging aims to estimate an unknown sky intensity image from degraded observations, acquired through an antenna array. In the theoretical case of a perfectly calibrated array, it has been shown that solving the corresponding imaging problem by iterative algorithms based on convex optimization and compressive sensing theory can be competitive with classical algorithms such as CLEAN. However, in practice, antenna-based gains are unknown and have to be calibrated. Future radio telescopes, such as the SKA, aim at improving imaging resolution and sensitivity by orders of magnitude. At this precision level, the direction-dependency of the gains must be accounted for, and radio interferometric imaging can be understood as a blind deconvolution problem. In this context, the underlying minimization problem is non-convex, and adapted techniques have to be designed. In this work, leveraging recent developments in non-convex optimization, we propose the first joint calibration and imaging method in radio interferometry, with proven convergence guarantees. Our approach, based on a block-coordinate forward-backward algorithm, jointly accounts for visibilities and suitable priors on both the image and the direction-dependent effects (DDEs). As demonstrated in recent works, sparsity remains the prior of choice for the image, while DDEs are modelled as smooth functions of the sky, i.e. spatially band-limited. Finally, we show through simulations the efficiency of our method, for the reconstruction of both images of point sources and complex extended sources. MATLAB code is available on GitHub.

astro-ph.IM↗
arXiv2017

The w-effect in interferometric imaging: from a fast sparse measurement operator to super-resolution

Modern radio telescopes, such as the Square Kilometre Array (SKA), will probe the radio sky over large fields-of-view, which results in large w-modulations of the sky image. This effect complicates the relationship between the measured visibilities and the image under scrutiny. In algorithmic terms, it gives rise to massive memory and computational time requirements. Yet, it can be a blessing in terms of reconstruction quality of the sky image. In recent years, several works have shown that large w-modulations promote the spread spectrum effect. Within the compressive sensing framework, this effect increases the incoherence between the sensing basis and the sparsity basis of the signal to be recovered, leading to better estimation of the sky image. In this article, we revisit the w-projection approach using convex optimisation in realistic settings, where the measurement operator couples the w-terms in Fourier and the de-gridding kernels. We provide sparse, thus fast, models of the Fourier part of the measurement operator through adaptive sparsification procedures. Consequently, memory requirements and computational cost are significantly alleviated, at the expense of introducing errors on the radio-interferometric data model. We present a first investigation of the impact of the sparse variants of the measurement operator on the image reconstruction quality. We finally analyse the interesting super-resolution potential associated with the spread spectrum effect of the w-modulation, and showcase it through simulations. Our C++ code is available online on GitHub.

astro-ph.IM↗
arXiv2017

A regularized tri-linear approach for optical interferometric imaging

In the context of optical interferometry, only undersampled power spectrum and bispectrum data are accessible. It poses an ill-posed inverse problem for image recovery. Recently, a tri-linear model was proposed for monochromatic imaging, leading to an alternated minimization problem. In that work, only a positivity constraint was considered, and the problem was solved by an approximated Gauss-Seidel method. In this paper, we propose to improve the approach on three fundamental aspects. Firstly, we define the estimated image as a solution of a regularized minimization problem, promoting sparsity in a fixed dictionary using either an $\ell_1$ or a weighted-$\ell_1$ regularization term. Secondly, we solve the resultant non-convex minimization problem using a block-coordinate forward-backward algorithm. This algorithm is able to deal both with smooth and non-smooth functions, and benefits from convergence guarantees even in a non-convex context. Finally, we generalize our model and algorithm to the hyperspectral case, promoting a joint sparsity prior through an $\ell_{2,1}$ regularization term. We present simulation results, both for monochromatic and hyperspectral cases, to validate the proposed approach.

astro-ph.IM↗
arXiv2017

Directional spin wavelets on the sphere

We construct a directional spin wavelet framework on the sphere by generalising the scalar scale-discretised wavelet transform to signals of arbitrary spin. The resulting framework is the only wavelet framework defined natively on the sphere that is able to probe the directional intensity of spin signals. Furthermore, directional spin scale-discretised wavelets support the exact synthesis of a signal on the sphere from its wavelet coefficients and satisfy excellent localisation and uncorrelation properties. Consequently, directional spin scale-discretised wavelets are likely to be of use in a wide range of applications and in particular for the analysis of the polarisation of the cosmic microwave background (CMB). We develop new algorithms to compute (scalar and spin) forward and inverse wavelet transforms exactly and efficiently for very large data-sets containing tens of millions of samples on the sphere. By leveraging a novel sampling theorem on the rotation group developed in a companion article, only half as many wavelet coefficients as alternative approaches need be computed, while still capturing the full information content of the signal under analysis. Our implementation of these algorithms is made publicly available.

cs.IT↗
arXiv2017

An accelerated splitting algorithm for radio-interferometric imaging: when natural and uniform weighting meet

Next generation radio-interferometers, like the Square Kilometre Array, will acquire tremendous amounts of data with the goal of improving the size and sensitivity of the reconstructed images by orders of magnitude. The efficient processing of large-scale data sets is of great importance. We propose an acceleration strategy for a recently proposed primal-dual distributed algorithm. A preconditioning approach can incorporate into the algorithmic structure both the sampling density of the measured visibilities and the noise statistics. Using the sampling density information greatly accelerates the convergence speed, especially for highly non-uniform sampling patterns, while relying on the correct noise statistics optimises the sensitivity of the reconstruction. In connection to CLEAN, our approach can be seen as including in the same algorithmic structure both natural and uniform weighting, thereby simultaneously optimising both the resolution and the sensitivity. The method relies on a new non-Euclidean proximity operator for the data fidelity term, that generalises the projection onto the $\ell_2$ ball where the noise lives for naturally weighted data, to the projection onto a generalised ellipsoid incorporating sampling density information through uniform weighting. Importantly, this non-Euclidean modification is only an acceleration strategy to solve the convex imaging problem with data fidelity dictated only by noise statistics. We showcase through simulations with realistic sampling patterns the acceleration obtained using the preconditioning. We also investigate the algorithm performance for the reconstruction of the 3C129 radio galaxy from real visibilities and compare with multi-scale CLEAN, showing better sensitivity and resolution. Our MATLAB code is available online on GitHub.

astro-ph.IM↗
arXiv2017

A Fourier dimensionality reduction model for big data interferometric imaging

Data dimensionality reduction in radio interferometry can provide savings of computational resources for image reconstruction through reduced memory footprints and lighter computations per iteration, which is important for the scalability of imaging methods to the big data setting of the next-generation telescopes. This article sheds new light on dimensionality reduction from the perspective of compressed sensing theory and studies its interplay with imaging algorithms designed in the context of convex optimization. We propose a post-gridding linear data embedding to the space spanned by the left singular vectors of the measurement operator, providing a dimensionality reduction below image size. This embedding preserves the null space of the measurement operator and hence also its sampling properties as per compressed sensing theory. We show that this can be approximated by first computing the dirty image and then applying a weighted subsampled discrete Fourier transform to obtain the final reduced data vector. This Fourier dimensionality reduction model ensures a fast implementation of the full measurement operator, essential for any iterative image reconstruction method. The proposed reduction also preserves the i.i.d. Gaussian properties of the original measurement noise. For convex optimization-based imaging algorithms, this is key to justify the use of the standard L2-norm as the data fidelity term. Our simulations confirm that this dimensionality reduction approach can be leveraged by convex optimization algorithms with no loss in imaging quality relative to reconstructing the image from the complete visibility data set. Reconstruction results in simulation settings with no direction dependent effects or calibration errors show promising performance of the proposed dimensionality reduction. MATLAB code implementing the proposed reduction method is available on GitHub.

astro-ph.IM↗
arXiv2017

PURIFYing real radio interferometric observations

Next-generation radio interferometers, such as the Square Kilometre Array (SKA), will revolutionise our understanding of the universe through their unprecedented sensitivity and resolution. However, standard methods in radio interferometry produce reconstructed interferometric images that are limited in quality and they are not scalable for big data. In this work we apply and evaluate alternative interferometric reconstruction methods that make use of state-of-the-art sparse image reconstruction algorithms motivated by compressive sensing, which have been implemented in the PURIFY software package. In particular, we implement and apply the proximal alternating direction method of multipliers (P-ADMM) algorithm presented in a recent article. We apply PURIFY to real interferometric observations. For all observations PURIFY outperforms the standard CLEAN, where in some cases PURIFY provides an improvement in dynamic range by over an order of magnitude. The latest version of PURIFY, which includes the developments presented in this work, is made publicly available.

astro-ph.IM↗
arXiv2017

Gauss-Legendre Sampling on the Rotation Group

We propose a Gauss-Legendre quadrature based sampling on the rotation group for the representation of a band-limited signal such that the Fourier transform (FT) of a signal can be exactly computed from its samples. Our figure of merit is the sampling efficiency, which is defined as a ratio of the degrees of freedom required to represent a band-limited signal in harmonic domain to the number of samples required to accurately compute the FT. The proposed sampling scheme is asymptotically as efficient as the most efficient scheme developed very recently. For the computation of FT and inverse FT, we also develop fast algorithms of complexity similar to the complexity attained by the fast algorithms for the existing sampling schemes. The developed algorithms are stable, accurate and do not have any pre-computation requirements. We also analyse the computation time and numerical accuracy of the proposed algorithms and show, through numerical experiments, that the proposed Fourier transforms are accurate with errors on the order of numerical precision.

cs.IT↗
arXiv2017

Fast nonnegative least squares through flexible Krylov subspaces

Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace method. The new method can be easily applied to image reconstruction problems affected by both Gaussian and Poisson noise, where the components of the solution represent nonnegative intensities. {Theoretical insight is given, and} numerical experiments and comparisons are displayed in order to validate the new method, which delivers results of equal or better quality than many state-of-the-art methods for nonnegative least squares solvers, with a significant speedup.

math.NA↗
arXiv2016

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.

astro-ph.IM↗
arXiv2016

Scalable splitting algorithms for big-data interferometric imaging in the SKA era

In the context of next generation radio telescopes, like the Square Kilometre Array, the efficient processing of large-scale datasets is extremely important. Convex optimisation tasks under the compressive sensing framework have recently emerged and provide both enhanced image reconstruction quality and scalability to increasingly larger data sets. We focus herein mainly on scalability and propose two new convex optimisation algorithmic structures able to solve the convex optimisation tasks arising in radio-interferometric imaging. They rely on proximal splitting and forward-backward iterations and can be seen, by analogy with the CLEAN major-minor cycle, as running sophisticated CLEAN-like iterations in parallel in multiple data, prior, and image spaces. Both methods support any convex regularisation function, in particular the well studied l1 priors promoting image sparsity in an adequate domain. Tailored for big-data, they employ parallel and distributed computations to achieve scalability, in terms of memory and computational requirements. One of them also exploits randomisation, over data blocks at each iteration, offering further flexibility. We present simulation results showing the feasibility of the proposed methods as well as their advantages compared to state-of-the-art algorithmic solvers. Our Matlab code is available online on GitHub.

astro-ph.IM↗
arXiv2016

Localisation of directional scale-discretised wavelets on the sphere

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.

cs.IT↗
arXiv2015

Why CLEAN when you can PURIFY? A new approach for next-generation radio-interferometric imaging

In recent works, sparse models and convex optimization techniques have been applied to radio-interferometric (RI) imaging showing the potential to outperform state-of-the-art imaging algorithms in the field. In this talk, I will review our latest contributions in RI imaging, which leverage the versatility of convex optimization to both handle realistic continuous visibilities and offer a highly parallelizable structure paving the way to high-dimensional data scalability. Firstly, I will review our recently proposed average sparsity approach, SARA, which relies on the observation that natural images exhibit strong average sparsity over multiple coherent bases. Secondly, I will discuss efficient implementations of SARA, and sparse regularization problems in general, for large-scale imaging problems in a new toolbox dubbed

astro-ph.IM↗
arXiv2014

On spin scale-discretised wavelets on the sphere for the analysis of CMB polarisation

A new spin wavelet transform on the sphere is proposed to analyse the polarisation of the cosmic microwave background (CMB), a spin $\pm 2$ signal observed on the celestial sphere. The scalar directional scale-discretised wavelet transform on the sphere is extended to analyse signals of arbitrary spin. The resulting spin scale-discretised wavelet transform probes the directional intensity of spin signals. A procedure is presented using this new spin wavelet transform to recover E- and B-mode signals from partial-sky observations of CMB polarisation.

astro-ph.IM↗
arXiv2014

A Compressed Sensing Framework for Magnetic Resonance Fingerprinting

Inspired by the recently proposed Magnetic Resonance Fingerprinting (MRF) technique, we develop a principled compressed sensing framework for quantitative MRI. The three key components are: a random pulse excitation sequence following the MRF technique; a random EPI subsampling strategy and an iterative projection algorithm that imposes consistency with the Bloch equations. We show that theoretically, as long as the excitation sequence possesses an appropriate form of persistent excitation, we are able to accurately recover the proton density, T1, T2 and off-resonance maps simultaneously from a limited number of samples. These results are further supported through extensive simulations using a brain phantom.

cs.IT↗