Nested Sampling and its Applications in Stable Compressive Covariance Estimation and Phase Retrieval with Near-Minimal Measurements

Compressed covariance sensing using quadratic samplers is gaining increasing interest in recent literature. Covariance matrix often plays the role of a sufficient statistic in many signal and information processing tasks. However, owing to the large dimension of the data, it may become necessary to obtain a compressed sketch of the high dimensional covariance matrix to reduce the associated storage and communication costs. Nested sampling has been proposed in the past as an efficient sub-Nyquist sampling strategy that enables perfect reconstruction of the autocorrelation sequence of Wide-Sense Stationary (WSS) signals, as though it was sampled at the Nyquist rate. The key idea behind nested sampling is to exploit properties of the difference set that naturally arises in quadratic measurement model associated with covariance compression. In this thesis, we will focus on developing novel versions of nested sampling for low rank Toeplitz covariance estimation, and phase retrieval, where the latter problem finds many applications in high resolution optical imaging, X-ray crystallography and molecular imaging. The problem of low rank compressive Toeplitz covariance estimation is first shown to be fundamentally related to that of line spectrum recovery. In absence if noise, this connection can be exploited to develop a particular kind of sampler called the Generalized Nested Sampler (GNS), that can achieve optimal compression rates. In presence of bounded noise, we develop a regularization-free algorithm that provably leads to stable recovery of the high dimensional Toeplitz matrix from its order-wise minimal sketch acquired using a GNS. Contrary to existing TV-norm and nuclear norm based reconstruction algorithms, our technique does not use any tuning parameters, which can be of great practical value. The idea of nested sampling idea also finds a surprising use in the problem of phase retrieval, which has been of great interest in recent times for its convex formulation via PhaseLift, By using another modified version of nested sampling, namely the Partial Nested Fourier Sampler (PNFS), we show that with probability one, it is possible to achieve a certain conjectured lower bound on the necessary measurement size. Moreover, for sparse data, an l1 minimization based algorithm is proposed that can lead to stable phase retrieval using order-wise minimal number of measurements.

Optical Wireless Communication for Mobile Platforms

The past few decades have witnessed the widespread adaptation of wireless devices such as cellular phones and Wifi-connected laptops, and demand for wireless communication is expected to continue to increase. Though radio frequency (RF) communication has traditionally dominated in this application space, recent decades have seen an increasing interest in the use of optical wireless (OW) communication to supplement RF communications. In contrast to RF communication technology, OW systems offer the use of largely unregulated electromagnetic spectrum and large bandwidths for communication. They also offer the potential to be highly secure against jamming and eavesdropping. Interest in OW has become especially keen in light of the maturation of light-emitting diode (LED) technology. This maturation, and the consequent emerging ubiquity of LED technology in lighting systems, has motivated the exploration of LEDs for wireless communication purposes in a wide variety of applications. Recent interest in this field has largely focused on the potential for indoor local area networks (LANs) to be realized with increasingly common LED-based lighting systems. We envision the use of LED-based OW to serve as a supplement to RF technology in communication between mobile platforms, which may include automobiles, robots, or unmanned aerial vehicles (UAVs). OW technology may be especially useful in what are known as RF-denied environments, in which RF communication may be prohibited or undesirable. The use of OW in these settings presents major challenges. In contrast to many RF systems, OWsystems that operate at ranges beyond a few meters typically require relatively precise alignment. For example, some laser-based optical wireless communication systems require alignment precision to within small fractions of a degree. This level of alignment precision can be difficult to maintain between mobile platforms. Additionally, the use of OW systems in outdoor settings presents the challenge of interference from ambient light, which can be much brighter than any LED transmitter. This thesis addresses these challenges to the use of LED-based communication between mobile platforms. We propose and analyze a dual-link LED-based system that uses one link with a wide transmission beam and relaxed alignment constraints to support a more narrow, precisely aligned, higher-data-rate link. The use of an optical link with relaxed alignment constraints to support the alignment of a more precisely aligned link motivates our exploration of a panoramic imaging receiver for estimating the range and bearing of neighboring nodes. The precision of such a system is analyzed and an experimental system is realized. Finally, we present an experimental prototype of a self-aligning LED-based link.