An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition]

This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Our intent in this article is to overview the basic CS theory that emerged in the works [1]–[3], present the key mathematical ideas underlying this theory, and survey a couple of important results in the field. Our goal is to explain CS as plainly as possible, and so our article is mainly of a tutorial nature. One of the charms of this theory is that it draws from various subdisciplines within the applied mathematical sciences, most notably probability theory. In this review, we have decided to highlight this aspect and especially the fact that randomness can — perhaps surprisingly — lead to very effective sensing mechanisms. We will also discuss significant implications, explain why CS is a concrete protocol for sensing and compressing data simultaneously (thus the name), and conclude our tour by reviewing important applications

On Grid Compressive Sampling for Spherical Field Measurements in Acoustics

We derive a compressive sampling method for acoustic field reconstruction using field measurements on a predefined spherical grid that has theoretically guaranteed relations between signal sparsity, measurement number, and reconstruction accuracy. This method can be used to reconstruct band-limited spherical harmonic or Wigner $D$-function series (spherical harmonic series are a special case) with sparse coefficients. Contrasting typical compressive sampling methods for Wigner $D$-function series that use arbitrary random measurements, the new method samples randomly on an equiangular grid, a practical and commonly used sampling pattern. Using its periodic extension, we transform the reconstruction of a Wigner $D$-function series into a multi-dimensional Fourier domain reconstruction problem. We establish that this transformation has a bounded effect on sparsity level and provide numerical studies of this effect. We also compare the reconstruction performance of the new approach to classical Nyquist sampling and existing compressive sampling methods. In our tests, the new compressive sampling approach performs comparably to other guaranteed compressive sampling approaches and needs a fraction of the measurements dictated by the Nyquist sampling theorem. Moreover, using one-third of the measurements or less, the new compressive sampling method can provide over 20 dB better denoising capability than oversampling with classical Fourier theory.Comment: 19 pages 14 figure

Compressive Sensing with Wigner DD-functions on Subsets of the Sphere

In this paper, we prove a compressive sensing guarantee for restricted measurement domains on the rotation group, $\mathrm{SO}(3)$. We do so by first defining Slepian functions on a measurement sub-domain $R$ of the rotation group $\mathrm{SO}(3)$. Then, we transform the inverse problem from the measurement basis, the bounded orthonormal system of band-limited Wigner $D$-functions on $\mathrm{SO}(3)$, to the Slepian functions in a way that limits increases to signal sparsity. Contrasting methods using Wigner $D$-functions that require measurements on all of $\mathrm{SO}(3)$, we show that the orthogonality structure of the Slepian functions only requires measurements on the sub-domain $R$, which is select-able. Due to the particulars of this approach and the inherent presence of Slepian functions with low concentrations on $R$, our approach gives the highest accuracy when the signal under study is well concentrated on $R$. We provide numerical examples of our method in comparison with other classical and compressive sensing approaches. In terms of reconstruction quality, we find that our method outperforms the other compressive sensing approaches we test and is at least as good as classical approaches but with a significant reduction in the number of measurements

Enhancing Sparsity by Reweighted ℓ(1) Minimization

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing

Sparsity and Incoherence in Compressive Sampling

We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization recovers $x^0$ exactly when the number of measurements exceeds $$m\geq \mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n,$$ where $S$ is the number of nonzero components in $x^0$, and $\mu$ is the largest entry in $U$ properly normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The smaller $\mu$, the fewer samples needed. The result holds for ``most'' sparse signals $x^0$ supported on a fixed (but arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero entry on $T$ and the observed values of $Ux^0$ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples

A First Analysis of the Stability of Takens' Embedding

Takens' Embedding Theorem asserts that when the states of a hidden dynamical system are confined to a low-dimensional attractor, complete information about the states can be preserved in the observed time-series output through the delay coordinate map. However, the conditions for the theorem to hold ignore the effects of noise and time-series analysis in practice requires a careful empirical determination of the sampling time and number of delays resulting in a number of delay coordinates larger than the minimum prescribed by Takens' theorem. In this paper, we use tools and ideas in Compressed Sensing to provide a first theoretical justification for the choice of the number of delays in noisy conditions. In particular, we show that under certain conditions on the dynamical system, measurement function, number of delays and sampling time, the delay-coordinate map can be a stable embedding of the dynamical system's attractor

The restricted isometry property for random block diagonal matrices

In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. It is by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIP under certain conditions on the number of measurements. Their use can be limited in practice, however, due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. These issues have recently motivated considerable effort towards studying the RIP for structured random matrices. In this paper, we study the RIP for block diagonal measurement matrices where each block on the main diagonal is itself a sub-Gaussian random matrix. Our main result states that such matrices can indeed satisfy the RIP but that the requisite number of measurements depends on certain properties of the basis in which the signals are sparse. In the best case, these matrices perform nearly as well as dense Gaussian random matrices, despite having many fewer nonzero entries

Joint Elastic Side-Scattering Lidar and Raman Lidar Measurements of Aerosol Optical Properties in South East Colorado

We describe an experiment, located in south-east Colorado, USA, that measured aerosol optical depth profiles using two Lidar techniques. Two independent detectors measured scattered light from a vertical UV laser beam. One detector, located at the laser site, measured light via the inelastic Raman backscattering process. This is a common method used in atmospheric science for measuring aerosol optical depth profiles. The other detector, located approximately 40km distant, viewed the laser beam from the side. This detector featured a 3.5m2 mirror and measured elastically scattered light in a bistatic Lidar configuration following the method used at the Pierre Auger cosmic ray observatory. The goal of this experiment was to assess and improve methods to measure atmospheric clarity, specifically aerosol optical depth profiles, for cosmic ray UV fluorescence detectors that use the atmosphere as a giant calorimeter. The experiment collected data from September 2010 to July 2011 under varying conditions of aerosol loading. We describe the instruments and techniques and compare the aerosol optical depth profiles measured by the Raman and bistatic Lidar detectors.Comment: 34 pages, 16 figure