Yves Wiaux

Sparsity Averaging Reweighted Analysis (SARA): a novel algorithm for radio-interferometric imaging

We propose a novel algorithm for image reconstruction in radio interferometry. The ill-posed inverse problem associated with the incomplete Fourier sampling identified by the visibility measurements is regularized by the assumption of average signal sparsity over representations in multiple wavelet bases. The algorithm, defined in the versatile framework of convex optimization, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We show through simulations that the proposed approach outperforms state-of-the-art imaging methods in the field, which are based on the assumption of signal sparsity in a single basis only.

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.

On the computation of directional scale-discretized wavelet transforms on the sphere

We review scale-discretized wavelets on the sphere, which are directional and allow one to probe oriented structure in data defined on the sphere. Furthermore, scale-discretized wavelets allow in practice the exact synthesis of a signal from its wavelet coefficients. We present exact and efficient algorithms to compute the scale-discretized wavelet transform of band-limited signals on the sphere. These algorithms are implemented in the publicly available S2DW code. We release a new version of S2DW that is parallelized and contains additional code optimizations. Note that scale-discretized wavelets can be viewed as a directional generalization of needlets. Finally, we outline future improvements to the algorithms presented, which can be achieved by exploiting a new sampling theorem on the sphere developed recently by some of the authors.

Data compression on the sphere

Large data-sets defined on the sphere arise in many fields. In particular, recent and forthcoming observations of the anisotropies of the cosmic microwave background (CMB) made on the celestial sphere contain approximately three and fifty mega-pixels respectively. The compression of such data is therefore becoming increasingly important. We develop algorithms to compress data defined on the sphere. A Haar wavelet transform on the sphere is used as an energy compression stage to reduce the entropy of the data, followed by Huffman and run-length encoding stages. Lossless and lossy compression algorithms are developed. We evaluate compression performance on simulated CMB data, Earth topography data and environmental illumination maps used in computer graphics. The CMB data can be compressed to approximately 40% of its original size for essentially no loss to the cosmological information content of the data, and to approximately 20% if a small cosmological information loss is tolerated. For the topographic and illumination data compression ratios of approximately 40:1 can be achieved when a small degradation in quality is allowed. We make our SZIP program that implements these compression algorithms available publicly.

Sampling theorems and compressive sensing on the sphere

We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.

Structured sparsity through reweighting and application to diffusion MRI

We consider the problem of multiple correlated sparse signals reconstruction and propose a new implementation of structured sparsity through a reweighting scheme. We present a particular application for diffusion Magnetic Resonance Imaging data and show how this procedure can be used for fibre orientation reconstruction in the white matter of the brain. In that framework, our structured sparsity prior can be used to exploit the fundamental coherence between fibre directions in neighbour voxels. Our method approaches the ℓ0 minimisation through a reweighted ℓ1-minimisation scheme. The weights are here defined in such a way to promote correlated sparsity between neighbour signals.