In recent years, networked systems have become prevalent in communications, computing, sensing, and many other areas. In a network composed of spatially distributed agents, network-wide synchronization of information about the physical environment and the network configuration must be maintained using measurements collected locally by the agents. Registration is a process for connecting the coordinate frames of multiple sets of data. This poses numerous challenges, particularly due to availability of direct communication only between neighboring agents in the network. These are exacerbated by uncertainty in the measurements and also by imperfect communication links. This research explored statistically based registration in a sensor network. The approach developed exploits measurements of offsets formed as differences of state values between pairs of agents that share a link in the network graph. It takes into account that the true offsets around any closed cycle in the network graph must sum to zero.

Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods can be computationally intensive, so I consider their behavior when only a portion of the sampled data is used. I show that the results of these methods converge as the sampling resolution increases, and use this to suggest a method of downsampling to estimate λ. I then present numerical results showing that this method can be feasible, and propose future avenues of inquiry.

Foveal sensors employ a small region of high acuity (the foveal region) surrounded by a periphery of lesser acuity. Consequently, the output map that describes their sensory acuity is nonlinear, rendering the vast corpus of linear system theory inapplicable immediately to the state estimation of a target being tracked by such a sensor. This thesis treats the adaptation of the Kalman filter, an iterative optimal estimator for linear-Gaussian dynamical systems, to enable its application to the nonlinear problem of foveal sensing. Results of simulations conducted to evaluate the effectiveness of this algorithm in tracking a target are presented, culminating in successful tracking for motion in two dimensions.

Radio communication has become the dominant form of correspondence in modern society. As the demand for high speed communication grows, the problems associated with an expanding consumer base and limited spectral access become more difficult to address. One communications system in which people commonly find themselves is the multiple access cellular network. Users operate within the same geographical area and bandwidth, so providing access to every user requires advanced processing techniques and careful subdivision of spectral access. This is known as the multiple access problem. This paper addresses this challenge in the context of airborne transceivers operating at high altitudes and long ranges. These operators communicate by transmitting a signal through a target scattering field on the ground without a direct line of sight to the receiver. The objective of this investigation is to develop a model for this communications channel, identify and quantify the relevant characteristics, and evaluate the feasibility of using it to effectively communicate.

A distributed sensor network (DSN) is a set of spatially scattered intelligent sensors designed to obtain data across an environment. DSNs are becoming a standard architecture for collecting data over a large area. We need registration of nodal data across the network in order to properly exploit having multiple sensors. One major problem worth investigating is ensuring the integrity of the data received, such as time synchronization. Consider a group of match filter sensors. Each sensor is collecting the same data, and comparing the data collected to a known signal. In an ideal world, each sensor would be able to collect the data without offsets or noise in the system. Two models can be followed from this. First, each sensor could make a decision on its own, and then the decisions could be collected at a ``fusion center'' which could then decide if the signal is present or not. The fusion center can then decide if the signal is present or not based on the number true-or-false decisions that each sensor has made. Alternatively, each sensor could relay the data that it collects to the fusion center, and it could then make a decision based on all of the data that it then receives. Since the fusion center would have more information to base its decision on in the latter case--as opposed to the former case where it only receives a true or false from each sensor--one would expect the latter model to perform better. In fact, this would be the gold standard for detection across a DSN. However, there is random noise in the world that causes corruption of data collection, especially among sensors in a DSN. Each sensor does not collect the data in the exact same way or with the same precision. We classify these imperfections in data collections as offsets, specifically the offset present in the data collected by one sensor with respect to the rest of the sensors in the network. Therefore, reconsider the two models for a DSN described above. We can naively implement either of these models for data collection. Alternatively, we can attempt to estimate the offsets between the sensors and compensate. One could see how it would be expected that estimating the offsets within the DSN would provide better overall results than not finding estimators. This thesis will be structured as follows. First, there will be an extensive investigation into detection theory and the impact that different types of offsets have on sensor networks. Following the theory, an algorithm for estimating the data offsets will be proposed correct for the offsets. Next, we will look at Monte Carlo simulation results to see the impact on sensor performance of data offsets in comparison to a sensor network without offsets present. The algorithm is then implemented, and further experiments will demonstrate sensor performance with offset detection.

Electric field imaging allows for a low cost, compact, non-invasive, non-ionizing alternative to other methods of imaging. It has many promising industrial applications including security, safely imaging power lines at construction sites, finding sources of electromagnetic interference, geo-prospecting, and medical imaging. The work presented in this dissertation concerns low frequency electric field imaging: the physics, hardware, and various methods of achieving it.

Electric fields have historically been notoriously difficult to work with due to how intrinsically noisy the data is in electric field sensors. As a first contribution, an in-depth study demonstrates just how prevalent electric field noise is. In field tests, various cables were placed underneath power lines. Despite being shielded, the 60 Hz power line signal readily penetrated several types of cables.

The challenges of high noise levels were largely addressed by connecting the output of an electric field sensor to a lock-in amplifier. Using the more accurate means of collecting electric field data, D-dot sensors were arrayed in a compact grid to resolve electric field images as a second contribution. This imager has successfully captured electric field images of live concealed wires and electromagnetic interference.

An active method was developed as a third contribution. In this method, distortions created by objects when placed in a known electric field are read. This expands the domain of what can be imaged because the object does not need to be a time-varying electric field source. Images of dielectrics (e.g. bodies of water) and DC wires were captured using this new method.

The final contribution uses a collection of one-dimensional electric field images, i.e. projections, to reconstruct a two-dimensional image. This was achieved using algorithms based in computed tomography such as filtered backprojection. An algebraic approach was also used to enforce sparsity regularization with the L1 norm, further improving the quality of some images.

Modern systems that measure dynamical phenomena often have limitations as to how many sensors can operate at any given time step. This thesis considers a sensor scheduling problem in which the source of a diffusive phenomenon is to be localized using single point measurements of its concentration. With a linear diffusion model, and in the absence of noise, classical observability theory describes whether or not the system's initial state can be deduced from a given set of linear measurements. However, it does not describe to what degree the system is observable. Different metrics of observability have been proposed in literature to address this issue. Many of these methods are based on choosing optimal or sub-optimal sensor schedules from a predetermined collection of possibilities. This thesis proposes two greedy algorithms for a one-dimensional and two-dimensional discrete diffusion processes. The first algorithm considers a deterministic linear dynamical system and deterministic linear measurements. The second algorithm considers noise on the measurements and is compared to a Kalman filter scheduling method described in published work.

The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to \emph{represent, compare, and manipulate} such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the \emph{true} space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression.

Resource allocation in communication networks aims to assign various resources such as power, bandwidth and load in a fair and economic fashion so that the networks can be better utilized and shared by the communicating entities. The design of efficient resource-allocation algorithms is, however, becoming more and more challenging due to the precipitously increasing scale of the networks. This thesis strives to understand how to design such low-complexity algorithms with performance guarantees.

In the first part, the link scheduling problem in wireless ad hoc networks is considered. The scheduler is charge of finding a set of wireless data links to activate at each time slot with the considerations of wireless interference, traffic dynamics, network topology and quality-of-service (QoS) requirements. Two different yet essential scenarios are investigated: the first one is when each packet has a specific deadline after which it will be discarded; the second is when each packet traverses the network in multiple hops instead of leaving the network after a one-hop transmission. In both scenarios the links need to be carefully scheduled to avoid starvation of users and congestion on links. One greedy algorithm is analyzed in each of the two scenarios and performance guarantees in terms of throughput of the networks are derived.

In the second part, the load-balancing problem in parallel computing is studied. Tasks arrive in batches and the duty of the load balancer is to place the tasks on the machines such that minimum queueing delay is incurred. Due to the huge size of modern data centers, sampling the status of all machines may result in significant overhead. Consequently, an algorithm based on limited queue information at the machines is examined and its asymptotic delay performance is characterized and it is shown that the proposed algorithm achieves the same delay with remarkably less sampling overhead compared to the well-known power-of-two-choices algorithm.

Two messages of the thesis are the following: greedy algorithms can work well in a stochastic setting; the fluid model can be useful in "derandomizing" the system and reveal the nature of the algorithm.

Modern measurement schemes for linear dynamical systems are typically designed so that different sensors can be scheduled to be used at each time step. To determine which sensors to use, various metrics have been suggested. One possible such metric is the observability of the system. Observability is a binary condition determining whether a finite number of measurements suffice to recover the initial state. However to employ observability for sensor scheduling, the binary definition needs to be expanded so that one can measure how observable a system is with a particular measurement scheme, i.e. one needs a metric of observability. Most methods utilizing an observability metric are about sensor selection and not for sensor scheduling. In this dissertation we present a new approach to utilize the observability for sensor scheduling by employing the condition number of the observability matrix as the metric and using column subset selection to create an algorithm to choose which sensors to use at each time step. To this end we use a rank revealing QR factorization algorithm to select sensors. Several numerical experiments are used to demonstrate the performance of the proposed scheme.