New generations of imaging devices aim to produce high resolution and high dynamic range images. In this context, the associated high dimensional inverse problems can become extremely challenging from an algorithmic view point. Moreover, the imaging procedure can be affected by unknown calibration kernels. This leads to the need of performing joint image reconstruction and calibration, and thus of solving non-convex blind deconvolution problems. In this work, we focus on the case where the observed object is affected by smooth calibration kernels in the context of radio astronomy, and we leverage a block-coordinate forward-backward algorithm, specifically designed to minimize non-smooth non-convex and high dimensional objective functions.
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.
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.
KEYWORDS: Associative arrays, Visibility, Modulation, Signal to noise ratio, Compressed sensing, Interferometers, Fourier transforms, Spatial frequencies, Radio interferometry, Detection theory
We consider the problem of reconstruction of astrophysical signals probed by radio interferometers with baselines
bearing a non-negligible component in the pointing direction. The visibilities measured essentially identify with
a noisy and incomplete Fourier coverage of the product of the planar signals with a linear chirp modulation. We
analyze the related spread spectrum phenomenon and suggest its universality relative to the sparsity dictionary,
in terms of the achievable quality of reconstruction through the Basis Pursuit problem. The present manuscript
represents a summary of recent work.
Conference Committee Involvement (6)
Wavelets and Sparsity XVII
6 August 2017 | San Diego, California, United States
Wavelets and Sparsity XVI
10 August 2015 | San Diego, California, United States
Wavelets and Sparsity XV
26 August 2013 | San Diego, California, United States
Wavelets and Sparsity XIV
21 August 2011 | San Diego, California, United States
Wavelets XIII
2 August 2009 | San Diego, California, United States
Wavelets XII
26 August 2007 | San Diego, California, United States
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