CPGD: Cadzow Plug-and-Play Gradient Descent for Generalised FRI

Finite rate of innovation (FRI) is a powerful reconstruction framework enabling the recovery of sparse Dirac streams from uniform low-pass filtered samples. An extension of this framework, called generalised FRI (genFRI), has been recently proposed for handling cases with arbitrary linear measurement models. In this context, signal reconstruction amounts to solving a joint constrained optimisation problem, yielding estimates of both the Fourier series coefficients of the Dirac stream and its so-called annihilating filter, involved in the regularisation term. This optimisation problem is however highly non convex and non linear in the data. Moreover, the proposed numerical solver is computationally intensive and without convergence guarantee. In this work, we propose an implicit formulation of the genFRI problem. To this end, we leverage a novel regularisation term which does not depend explicitly on the unknown annihilating filter yet enforces sufficient structure in the solution for stable recovery. The resulting optimisation problem is still non convex, but simpler since linear in the data and with less unknowns. We solve it by means of a provably convergent proximal gradient descent (PGD) method. Since the proximal step does not admit a simple closed-form expression, we propose an inexact PGD method, coined as Cadzow plug-and-play gradient descent (CPGD). The latter approximates the proximal steps by means of Cadzow denoising, a well-known denoising algorithm in FRI. We provide local fixed-point convergence guarantees for CPGD. Through extensive numerical simulations, we demonstrate the superiority of CPGD against the state-of-the-art in the case of non uniform time samples.Comment: 16 pages, 8 figure

HVOX: Scalable Interferometric Synthesis and Analysis of Spherical Sky Maps

Analysis and synthesis are key steps of the radio-interferometric imaging process, serving as a bridge between visibility and sky domains. They can be expressed as partial Fourier transforms involving a large number of non-uniform frequencies and spherically-constrained spatial coordinates. Due to the data non-uniformity, these partial Fourier transforms are computationally expensive and represent a serious bottleneck in the image reconstruction process. The W-gridding algorithm achieves log-linear complexity for both steps by applying a series of 2D non-uniform FFTs (NUFFT) to the data sliced along the so-called $w$ frequency coordinate. A major drawback of this method however is its restriction to direction-cosine meshes, which are fundamentally ill-suited for large field of views. This paper introduces the HVOX gridder, a novel algorithm for analysis/synthesis based on a 3D-NUFFT. Unlike W-gridding, the latter is compatible with arbitrary spherical meshes such as the popular HEALPix scheme for spherical data processing. The 3D-NUFFT allows one to optimally select the size of the inner FFTs, in particular the number of W-planes. This results in a better performing and auto-tuned algorithm, with controlled accuracy guarantees backed by strong results from approximation theory. To cope with the challenging scale of next-generation radio telescopes, we propose moreover a chunked evaluation strategy: by partitioning the visibility and sky domains, the 3D-NUFFT is decomposed into sub-problems which execute in parallel, while simultaneously cutting memory requirements. Our benchmarking results demonstrate the scalability of HVOX for both SKA and LOFAR, considering state-of-the-art challenging imaging setups. HVOX is moreover computationally competitive with W-gridder, despite the absence of domain-specific optimizations in our implementation

Towards More Accurate and Efficient Beamformed Radio Interferometry Imaging

The Square Kilometre Array (SKA) will form the largest radio telescope ever built, generating on the order of one terabyte of data per second. To reduce the data flow sent to the central processor, hierarchical designs have been proposed: the data is primarily collected in groups of antennas, and summed coherently by beamforming. Historically, Fourier analysis has played a prominent role in radio astronomy interferometry, legitimated by the celebrated van Cittert-Zernike theorem. We show that, in the case of modern hierarchical designs, beamformed data has a less intimate, and thus more complicated relationship to the Fourier domain. Unsatisfactory attempts have been proposed to compensate, which implicitly retain the Fourier framework, and are limited to directive beamforming. We show that when stepping away from Fourier, we can embed the data in a more natural domain originating from the telescope configuration and the specific beamforming technique. This leads to a new, more accurate, imaging pipeline. Standard techniques such as w-projection, and gridding are no longer needed, as the reconstruction is performed on the celestial sphere. The proposed imager operates in two steps. First, a preconditioning based on the Gram-Schmidt orthogonalization procedure is performed, in order to facilitate the computation of the pseudoinverse sky estimate. Then, from this, the LASSO estimate is approximated very efficiently. The quality of this approximation is shown to be linked directly to the effective support of the instrument point spread function. Due to the greater flexibility of this framework, information-maximising beamforming techniques such as randomised beamforming can be readily incorporated. Moreover, we use the Bonferroni method to construct global confidence intervals onto the Gram-Schmidt least squares estimate, and use them to test the statistical significance of each pixel. The complexity of the proposed technique is assessed and compared to the the state-of-the-art combined CLEAN and A-projection algorithm. In the case of LOFAR, we show that our algorithm can be from 2 to 34 times faster. The accuracy and sensitivity of the new technique is also shown, for simulated data, to be superior

Modélisation numérique des formes d’équilibre d’un globule rouge

Nous nous proposons dans cette étude de valider le modèle introduit par Canham et Helfrich afin de décrire les formes d’équilibre statique d’un globule rouge. Selon ce modèle, la forme du globule rouge est solution d’un problème d’optimisation sous contrainte: minimisation de l’énergie de Canham Helfrich pour un volume et une aire fixés. Après avoir formaliser le problème matématiquement, nous dérivons la conditionn d’optimalité menant à une équation différentielle ordinaire non linéaire vérifiée par la forme du globule rouge. Nous traitons les cas bidimensionnel et tridimensionnel axismétrique. Nous présentons ensuite la méthodologie employée pour résoudre numériquement le problème et exploitons les résultats des simulations

Statistics on Manifolds applied to Shape Theory

In this report, we use a variety of tools from differential geometry to propose a nonlinear extension of the principal components analysis (PCA) into manifolds setting. This extension, that we shall call principal geodesics analysis (PGA), attempts to find analogs of the principal components by introducing the principal geodesic components. We then construct the shape space of triangles Σ^3_2 and find a convenient parametrization of it. Finally, we apply the PGA procedure previously designed to analyze the variability of a sample of shapes, randomly chosen onto the shape space of triangles

Shape Optimization of an Hydrofoil by Isogeometric Analysis

We use Isogeometric Analysis as a framework for NURBS-based shape optimization of hydrofoils. We present geometrical representations by NURBS and some of their properties to design an hydrofoil. Then, we consider an irrotational flow around an hydrofoil and solve the Laplace equation in the stream function formulation. Finally, we perform the shape optimization of the hydrofoil by considering the stream function formulation as the state problem and different objective functionals

Semi-Automatic Transcription Tool for Ancient Manuscripts

In this work, we investigate various techniques from the fields of shape analysis and image processing in order to construct a semi-automatic transcription tool for ancient manuscripts. First, we design a shape matching procedure using shape contexts, introduced in [1], and exploit this procedure to compute different distances between two arbitrary shapes/words. Then, we use Fischer discrimination to combine these distances in a single similarity measure and use it to naturally represent the words on a similarity graph. Finally, we investigate an unsupervised clustering analysis on this graph to create groups of semantically similar words and propose an uncertainty measure associated with the attribution of one word to a group. The clusters together with the uncertainty measure form the core of the semi-automatic transcription tool, that we test on a dataset of 42 words. The average classification accuracy achieved with this technique on this dataset is of 86%, which is quiet satisfying. This tool allows to reduce the actual number of words we need to type to transcript a document of 70%

Statistical Inference in Positron Emission Tomography

In this report, we investigate mathematical algorithms for image reconstruction in the context of positron emission tomography (a medical diagnosis technique). We first take inspiration from the physics of PET to design a mathematical model tailored to the problem. We think of positron emissions as an output of an indirectly observed Poisson process and formulate the link between the emissions and the scanner records through the Radon transform. This model allows us to express the image reconstruction in terms of a standard problem in statistical estimation from incomplete data. Then, we investigate different algorithms as well as stopping criterion, and compare their relative efficiency

Flexarray: Random phased array layouts for analytical spatial filtering

We propose a method for designing phased-arrays according to a given, analytically-specified, target beamshape. Building on the flexibeam framework, antenna locations are sampled from a probabilistic density function. Naturally scalable with the number of antennas, it is also computationally efficient and numerically stable, as it relies on analytical derivation

Détermination d’une orbite autour de L2 pour la mission CHEOPS

Afin de déterminer l’orbite qu’un satellite d’observation d’exoplanètes devait adopter, il a été nécessaire de considérer les points de Lagrange et plus particulièrement le deuxième de ces points. Dans les pages suivantes, nous localisons les trois premiers de ces points (L1, L2 et L3), expliquons en quoi ils sont importants et démontrons la quasi-stabilité de L1 et L2. Ensuite, nous analysons les différents types d’orbites quasi-périodiques qui existent autour de L2 et construisons une orbite de Lyapunov grâce au logiciel STK/Astrogator, ce qui permet de modéliser la mission et de souligner ses avantages et ses inconvénients