FRIDA: FRI-Based DOA Estimation for Arbitrary Array Layouts

In this paper we present FRIDA---an algorithm for estimating directions of arrival of multiple wideband sound sources. FRIDA combines multi-band information coherently and achieves state-of-the-art resolution at extremely low signal-to-noise ratios. It works for arbitrary array layouts, but unlike the various steered response power and subspace methods, it does not require a grid search. FRIDA leverages recent advances in sampling signals with a finite rate of innovation. It is based on the insight that for any array layout, the entries of the spatial covariance matrix can be linearly transformed into a uniformly sampled sum of sinusoids.Comment: Submitted to ICASSP201

Looking beyond Pixels:Theory, Algorithms and Applications of Continuous Sparse Recovery

Sparse recovery is a powerful tool that plays a central role in many applications, including source estimation in radio astronomy, direction of arrival estimation in acoustics or radar, super-resolution microscopy, and X-ray crystallography. Conventional approaches usually resort to discretization, where the sparse signals are estimated on a pre-defined grid. However, sparse signals do not line up conveniently on any grid in reality. While the discrete setup usually leads to a simple optimization problem that can be solved with standard tools, there are two noticeable drawbacks: (i) Because of the model mismatch, the effective noise level is increased; (ii) The minimum reachable resolution is limited by the grid step-size. Because of the limitations, it is essential to develop a technique that estimates sparse signals in the continuous-domain--in essence seeing beyond pixels. The aims of this thesis are (i) to further develop a continuous-domain sparse recovery framework based on finite rate of innovation (FRI) sampling on both theoretical and algorithmic aspects; (ii) adapt the proposed technique to several applications, namely radio astronomy point source estimation, direction of arrival estimation in acoustics, and single image up-sampling; (iii) show that the continuous-domain sparse recovery approach can surpass the instrument resolution limit and achieve super-resolution. We propose a continuous-domain sparse recovery technique by generalizing the FRI sampling framework to cases with non-uniform measurements. We achieve this by identifying a set of unknown uniform sinusoidal samples and the linear transformation that links the uniform samples of sinusoids to the measurements. The continuous-domain sparsity constraint can be equivalently enforced with a discrete convolution equation of these sinusoidal samples. The sparse signal is reconstructed by minimizing the fitting error between the given and the re-synthesized measurements subject to the sparsity constraint. Further, we develop a multi-dimensional sampling framework for Diracs in two or higher dimensions with linear sample complexity. This is a significant improvement over previous methods, which have a complexity that increases exponentially with dimension. An efficient algorithm has been proposed to find a valid solution to the continuous-domain sparse recovery problem such that the reconstruction (i) satisfies the sparsity constraint; and (ii) fits the measurements (up to the noise level). We validate the flexibility and robustness of the FRI-based continuous-domain sparse recovery in both simulations and experiments with real data. We show that the proposed method surpasses the diffraction limit of radio telescopes with both realistic simulation and real data from the LOFAR radio telescope. In addition, FRI-based sparse reconstruction requires fewer measurements and smaller baselines to reach a similar reconstruction quality compared with conventional methods. Next, we apply the proposed approach to direction of arrival estimation in acoustics. We show that accurate off-grid source locations can be reliably estimated from microphone measurements with arbitrary array geometries. Finally, we demonstrate the effectiveness of the continuous-domain sparsity constraint in regularizing an otherwise ill-posed inverse problem, namely single-image super-resolution. By incorporating image edge models, the up-sampled image retains sharp edges and is free from ringing artifacts

An Iterative Linear Expansion of Thresholds for l1-based Image Restoration

This paper proposes a novel algorithmic framework to solve image restoration problems under sparsity assumptions. As usual, the reconstructed image is the minimum of an objective functional that consists of a data fidelity term and an l1 regularization. However, instead of estimating the reconstructed image that minimizes the objective functional directly, we focus on the restoration process that maps the degraded measurements to the reconstruction. Our idea amounts to parameterizing the process as a linear combination of few elementary thresholding functions (LET) and solve for the linear weighting coefficients by minimizing the objective functional. It is then possible to update the thresholding functions and to iterate this process (i-LET). The key advantage of such a linear parametrization is that the problem size reduces dramatically—each time we only need to solve an optimization problem over the dimension of the linear coefficients (typically less than 10) instead of the whole image dimension. With the elementary thresholding functions satisfying certain constraints, global convergence of the iterated LET algorithm is guaranteed. Experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperform state-of-the-art algorithms in terms of both CPU time and number of iterations

Sparse Image Restoration Using Iterated Linear Expansion of Thresholds

We focus on image restoration that consists in regularizing a quadratic data-fidelity term with the standard l1 sparse-enforcing norm. We propose a novel algorithmic approach to solve this optimization problem. Our idea amounts to approximating the result of the restoration as a linear sum of basic thresholds (e.g. soft-thresholds) weighted by unknown coefficients. The few coefficients of this expansion are obtained by minimizing the equivalent low-dimensional l1-norm regularized objective function, which can be solved efficiently with standard convex optimization techniques, e.g. iterative reweighted least square (IRLS). By iterating this process, we claim that we reach the global minimum of the objective function. Experimentally we discover that very few iterations are required before we reach the convergence

Towards Generalized FRI Sampling with an Application to Source Resolution in Radioastronomy

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

Annihilation-driven Localised Image Edge Models

We propose a novel edge detection algorithm with sub-pixel accuracy based on annihilation of signals with finite rate of innovation. We show that the Fourier domain annihilation equations can be interpreted as spatial domain multiplications. From this new perspective, we obtain an accurate estimation of the edge model by assuming a simple parametric form within each localised block. Further, we build a locally adaptive global mask function (i.e, our edge model) for the whole image. The mask function is then used as an edge- preserving constraint in further processing. Numerical experiments on both edge localisations and image up-sampling show the effectiveness of the proposed approach, which out- performs state-of-the-art method

Efficient Multi-dimensional Diracs Estimation with Linear Sample Complexity

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)

LEAP: Looking beyond pixels with continuous-space EstimAtion of Point sources

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

Sparse Recovery of Strong Reflectors With an Application to Non-Destructive Evaluation

In this paper we show that it is sufficient to recover the locations of K strong reflectors within an insonified medium from three receive elements and 2K+1 samples per element. The proposed approach leverages advances in sampling signals with a finite rate of innovation along each element and rank properties from the Euclidean distance matrix construction across elements. With the proposed approach, it is not necessary to construct an image in order to identify strong reflective sources, which is why much fewer receive elements are needed. However, the assumed transmit scheme still uses a standard linear array in order to excite the entire medium with sufficient energy. The approach is validated with simulated data and a measurement that emulates a scenario in non-destructive evaluation

Hardware And Software For Reproducible Research In Audio Array Signal Processing

In our demo, we present two hardware platforms for prototyping audio array signal processing. Pyramic is a 48-channel microphone array fitted on an FPGA and Compact Six is a portable microphone array with six microphones, closer to the technical constraints of consumer electronics. A browser based interface was developed that allows the user to interact with the audio stream from the arrays in real time. The software component of this demo is a Python module with implementations of basic audio signal processing blocks and popular techniques like STFT, beamforming, and DoA. Both the hardware design files and the software are open source and freely shared. As part of a collaboration with IBM Research, their beamforming and imaging technologies will also be portrayed. The hardware will be demonstrated through an installation processing the microphone signals into light patterns on a circular LED array. The demo will be interactive and let visitors play with different algorithms for DoA (SRP, FRIDA [1], Bluebild) and beamforming (MVDR, Flexibeam [2]). The availability of an open platform with reference implementations encourages reproducible research and minimizes setup-time when testing and benchmarking new audio array signal processing algorithms. It can also serve as a useful educational tool, providing a means to work with real-life signals