Recent Developments in Signal Approximation and Reconstruction

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Background

The past decade has seen the emergence of novel techniques for signal reconstruction from few measurements. This resurgence due in part to the development of compressed sensing is a fast-evolving field which has had impact in many areas from communication engineering to high-dimensional geometry and geometric functional analysis.

The goal of this Research Topic is to give an up-to-date overview of the various theoretical and numerical recovery techniques. Recent literature lacks a general survey of tools and results to handle the latest developments in this area in the past decade. Many sampling and reconstruction frameworks have emerged since the first development of compressed sensing.

* Phase reconstruction tries to reconstruct a given signal for which only its magnitude is being sampled. This has application for instance in X-Ray tomography.

* Low-rank and tensor approximation naturally extends the ideas of compressed sensing to matrices and high-dimensional tensors. A particular example of such approximation schemes appears in the case of CUR decomposition that has been widely used in recent years for clustering and learning.

* Signal processing on graphs. Compressed sensing on graphs, in particular, is interested in recovering sparse vectors representing the properties of the edges from a graph. Precisely, the measurements are constrained to follow connected paths over the underlying graph.

* Recovery under unknown noise level has been an active area of research recently. Motivated by the fact that usual l1 minimization and / or iterative methods require some knowledge of the noise level in the measurements, it is only meaningful to try to understand what happens when the noise is falsely estimated.

* Dynamical sampling, recently introduced, investigates the sampling of dynamical systems and leverages the number of samples at a given time by later samples. One may be interested in system identification, in the number of sampling points required and their localization as well as the times for sampling.

* Recovery in general Hilbert spaces and generalized sampling comes as a generation of the compressed sensing problem for the continuous signal.

* Distributed sensing has recently evolved by merging the fields of frames and fusion frames with compressed sensing ideas. It can be shown that the pieces of information to be recovered can be obtained in a distributed fashion, while reducing the complexity of each sensor.

* Quantized, saturated and one-bit sensing are problems of importance when dealing with actual physical implementations of compressed sensing systems. The measurements are rarely perfect and the noise introduced by analog to digital converters has to be analyzed.

* Structured sparse recovery, also referred to as model-based compressed sensing, where the underlying redundancy has an extra structure that can be exploited to improve recovery both in terms of speed and accuracy of algorithms.

* Non-linear compressed sensing. Unlike standard compressed sensing and sparse recovery, the measurements operators are not linear. This too has a lot of real applications.

This Research Topic is also interested in successful applications of sparsity, low-rank, etc... promoting techniques in the natural sciences which may include results from (but not limited to) bio-mathematics, uncertainty quantification, and sparse polynomial chaos approximations.

Keywords: Compressed sensing, sparse approximation, low rank approximation, phase reconstruction, 1-Bit compressed sensing

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