Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe

Matrix Coherence and the Nystrom Method

The Nystrom method is an efficient technique to speed up large-scale learning applications by generating low-rank approximations. Crucial to the performance of this technique is the assumption that a matrix can be well approximated by working exclusively with a subset of its columns. In this work we relate this assumption to the concept of matrix coherence and connect matrix coherence to the performance of the Nystrom method. Making use of related work in the compressed sensing and the matrix completion literature, we derive novel coherence-based bounds for the Nystrom method in the low-rank setting. We then present empirical results that corroborate these theoretical bounds. Finally, we present more general empirical results for the full-rank setting that convincingly demonstrate the ability of matrix coherence to measure the degree to which information can be extracted from a subset of columns

Compressed Sensing in Multi-Hop Large-Scale Wireless Sensor Networks Based on Routing Topology Tomography

Data acquisition from multi-hop large-scale outdoor wireless sensor network (WSN) deployments for environmental monitoring is full of challenges. This is because of the severe resource constraints on tiny battery-operated motes (e.g., bandwidth, memory, power, and computing capacity), the data acquisition volume from large-scale WSNs, and the highly dynamic wireless link conditions in outdoor harsh communication environments. We present a novel compressed sensing approach, which can recover the sensing data at the sink with high fidelity when a very few data packets need to be collected, leading to a significant reduction of the network transmissions and thus an extension of the WSN lifetime. Interplaying with the dynamic WSN routing topology, the proposed approach is both efficient and simple to implement on the resource-constrained motes without motes' storing of any part of the random projection matrix, as opposed to other existing compressed sensing-based schemes. We further propose a systematic method via machine learning to find a suitable representation basis, for any given WSN deployment and data field, which is both sparse and incoherent with the random projection matrix in compressed sensing for data collection. We validate our approach and evaluate its performance using a real-world outdoor multihop WSN testbed deployment in situ. The results demonstrate that our approach significantly outperforms existing compressed sensing approaches by reducing data recovery errors by an order of magnitude for the entire WSN observation field while drastically reducing wireless communication costs at the same time

Fast Compressed Automatic Target Recognition for a Compressive Infrared Imager

Many military systems utilize infrared sensors which allow an operator to see targets at night. Several of these are either mid-wave or long-wave high resolution infrared sensors, which are expensive to manufacture. But compressive sensing, which has primarily been demonstrated in medical applications, can be used to minimize the number of measurements needed to represent a high-resolution image. Using these techniques, a relatively low cost mid-wave infrared sensor can be realized which has a high effective resolution. In traditional military infrared sensing applications, like targeting systems, automatic targeting recognition algorithms are employed to locate and identify targets of interest to reduce the burden on the operator. The resolution of the sensor can increase the accuracy and operational range of a targeting system. When using a compressive sensing infrared sensor, traditional decompression techniques can be applied to form a spatial-domain infrared image, but most are iterative and not ideal for real-time environments. A more efficient method is to adapt the target recognition algorithms to operate directly on the compressed samples. In this work, we will present a target recognition algorithm which utilizes a compressed target detection method to identify potential target areas and then a specialized target recognition technique that operates directly on the same compressed samples. We will demonstrate our method on the U.S. Army Night Vision and Electronic Sensors Directorate ATR Algorithm Development Image Database which has been made available by the Sensing Information Analysis Center