Structured random measurements in signal processing

Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.Comment: 22 pages, 2 figure

Compressed sensing in astronomy and remote sensing: a data fusion perspective

Recent advances in signal processing have focused on the use of sparse representations in various applications. A new field of interest based on sparsity has recently emerged: compressed sensing. This theory is a new sampling framework that provides an alternative to the well-known Shannon sampling theory. In this paper we investigate how compressed sensing (CS) can provide new insights into astronomical data compression. In a previous study1 we gave new insights into the use of Compressed Sensing (CS) in the scope of astronomical data analysis. More specifically, we showed how CS is flexible enough to account for particular observational strategies such as raster scans. This kind of CS data fusion concept led to an elegant and effective way to solve the problem ESA is faced with, for the transmission to the earth of the data collected by PACS, one of the instruments onboard the Herschel spacecraft which will launched in late 2008/early 2009. In this paper, we extend this work by showing how CS can be effectively used to jointly decode multiple observations at the level of map making. This allows us to directly estimate large areas of the sky from one or several raster scans. Beyond the particular but important Herschel example, we strongly believe that CS can be applied to a wider range of applications such as in earth science and remote sensing where dealing with multiple redundant observations is common place. Simple but illustrative examples are given that show the effectiveness of CS when decoding is made from multiple redundant observations

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

Time for dithering: fast and quantized random embeddings via the restricted isometry property

Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect the restricted isometry property (RIP) can induce a quantized random embedding with controllable multiplicative and additive distortions with respect to the pairwise distances of the data points beings considered. In other words, linear matrices having fast matrix-vector multiplication algorithms (e.g., based on partial Fourier ensembles or on the adjacency matrix of unbalanced expanders) can be readily used in the definition of fast quantized embeddings with small distortions. This implication is made possible by applying right after the linear map an additive and random "dither" that stabilizes the impact of the uniform scalar quantization operator applied afterwards. For different categories of RIP matrices, i.e., for different linear embeddings of a metric space $(\mathcal K \subset \mathbb R^n, \ell_q)$ in $(\mathbb R^m, \ell_p)$ with $p,q \geq 1$, we derive upper bounds on the additive distortion induced by quantization, showing that it decays either when the embedding dimension $m$ increases or when the distance of a pair of embedded vectors in $\mathcal K$ decreases. Finally, we develop a novel "bi-dithered" quantization scheme, which allows for a reduced distortion that decreases when the embedding dimension grows and independently of the considered pair of vectors.Comment: Keywords: random projections, non-linear embeddings, quantization, dither, restricted isometry property, dimensionality reduction, compressive sensing, low-complexity signal models, fast and structured sensing matrices, quantized rank-one projections (31 pages

How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT

We introduce phase-diagram analysis, a standard tool in compressed sensing, to the X-ray CT community as a systematic method for determining how few projections suffice for accurate sparsity-regularized reconstruction. In compressed sensing a phase diagram is a convenient way to study and express certain theoretical relations between sparsity and sufficient sampling. We adapt phase-diagram analysis for empirical use in X-ray CT for which the same theoretical results do not hold. We demonstrate in three case studies the potential of phase-diagram analysis for providing quantitative answers to questions of undersampling: First we demonstrate that there are cases where X-ray CT empirically performs comparable with an optimal compressed sensing strategy, namely taking measurements with Gaussian sensing matrices. Second, we show that, in contrast to what might have been anticipated, taking randomized CT measurements does not lead to improved performance compared to standard structured sampling patterns. Finally, we show preliminary results of how well phase-diagram analysis can predict the sufficient number of projections for accurately reconstructing a large-scale image of a given sparsity by means of total-variation regularization.Comment: 24 pages, 13 figure