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Person# Thierry Blu

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Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy

Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.

Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advan

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Thierry Blu, Paul Hurley, Hanjie Pan, Matthieu Martin Jean-André Simeoni, Martin Vetterli

Context. Two main classes of imaging algorithms have emerged in radio interferometry: the CLEAN algorithm and its multiple variants, and compressed-sensing inspired methods. They are both discrete in nature, and estimate source locations and intensities on a regular grid. For the traditional CLEAN-based imaging pipeline, the resolution power of the tool is limited by the width of the synthesized beam, which is inversely proportional to the largest baseline. The finite rate of innovation (FRI) framework is a robust method to find the locations of point-sources in a continuum without grid imposition. The continuous formulation makes the FRI recovery performance only dependent on the number of measurements and the number of sources in the sky. FRI can theoretically find sources below the perceived tool resolution. To date, FRI had never been tested in the extreme conditions inherent to radio astronomy: weak signal / high noise, huge data sets, large numbers of sources. Aims. The aims were (i) to adapt FRI to radio astronomy, (ii) verify it can recover sources in radio astronomy conditions with more accurate positioning than CLEAN, and possibly resolve some sources that would otherwise be missed, (iii) show that sources can be found using less data than would otherwise be required to find them, and (v) show that FRI does not lead to an augmented rate of false positives. Methods. We implemented a continuous domain sparse reconstruction algorithm in Python. The angular resolution performance of the new algorithm was assessed under simulation, and with visibility measurements from the LOFAR telescope. Existing catalogs were used to confirm the existence of sources. Results. We adapted the FRI framework to radio interferometry, and showed that it is possible to determine accurate off-grid point source locations and their corresponding intensities. In addition, FRI-based sparse reconstruction required less integration time and smaller baselines to reach a comparable reconstruction quality compared to a conventional method. The achieved angular resolution is higher than the perceived instrument resolution, and very close sources can be reliably distinguished. The proposed approach has cubic complexity in the total number (typically around a few thousand) of uniform Fourier data of the sky image estimated from the reconstruction. It is also demonstrated that the method is robust to the presence of extended-sources, and that false-positives can be addressed by choosing an adequate model order to match the noise level.

2017Thierry Blu, Hanjie Pan, Martin Vetterli

It is a classic problem to estimate continuous-time sparse signals, like point sources in a direction-of-arrival problem, or pulses in a time-of-flight measurement. The earliest occurrence is the estimation of sinusoids in time series using Prony's method. This is at the root of a substantial line of work on high resolution spectral estimation. The estimation of continuous-time sparse signals from discrete-time samples is the goal of the sampling theory for finite rate of innovation (FRI) signals. Both spectral estimation and FRI sampling usually assume uniform sampling. But not all measurements are obtained uniformly, as exemplified by a concrete radioastronomy problem we set out to solve. Thus, we develop the theory and algorithm to reconstruct sparse signals, typically sum of sinusoids, from non-uniform samples. We achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements. These uniform samples are known to satisfy the annihilation equations. A valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint. Thanks to this new approach, we unify a variety of FRI-based methods. We demonstrate the versatility and robustness of the proposed approach with five FRI reconstruction problems, namely Dirac reconstructions with irregular time or Fourier domain samples, FRI curve reconstructions, Dirac reconstructions on the sphere and point source reconstructions in radioastronomy. The proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled Fourier measurements under severe noise conditions.

Thierry Blu, Hanjie Pan, Martin Vetterli

Estimating Diracs in continuous two or higher dimensions is a fundamental problem in imaging. Previous approaches extended one dimensional methods, like the ones based on finite rate of innovation (FRI) sampling, in a separable manner, e.g., along the horizontal and vertical dimensions separately in 2D. The separate estimation leads to a sample complexity of O(K^D) for K Diracs in D dimensions, despite that the total degrees of freedom only increase linearly with respect to D. We propose a new method that enforces the continuous-domain sparsity constraints simultaneously along all dimensions, leading to a reconstruction algorithm with linear sample complexity O(K), or a gain of O(K^{D-1}) over previous FRI-based methods. The multi-dimensional Dirac locations are subsequently determined by the intersections of hypersurfaces (e.g., curves in 2D), which can be computed algebraically from the common roots of polynomials. We first demonstrate the performance of the new multi-dimensional algorithm on simulated data: multi-dimensional Dirac location retrieval under noisy measurements. Then we show results on real data: radio astronomy point source reconstruction (from LOFAR telescope measurements) and the direction of arrival estimation of acoustic signals (using Pyramic microphone arrays).

2018