<p> Chronic pain affects more than 100 million Americans and more than 1.5 billion people worldwide. Pain is a multidimensional construct, expressed through a variety of means. Facial expressions are one such type of pain expression. Automatic facial expression recognition, and in particular pain expression recognition, are fields that have been studied extensively. However, nothing has explored the possibility of an automatic pain quantification algorithm, able to output pain levels based upon a facial image. </p><p> Developed for a remote monitoring context, a computational pain quantification algorithm has been developed and validated by two distinct sets of data. The second set of data also included associated data for the fields of age, gender, culture and cause of pain. These four fields were investigated for their effect on automatic pain quantification, determining that age and gender have a definite impact and should be involved in the algorithm, while culture and cause require further investigation.</p>

<p> The past decade has seen an abundance of applications that utilize sensors to collect data. One such example is a gigapixel image, which combines a multitude of high-quality images into a panorama capable of viewing hundreds of acres. The resulting datasets can be quite large, making analysis time consuming and resource intensive. Moreover, coverage of such broad environments can mean numerous sensor feeds to which one must attend. A suitable approach for analysis and sense-making of such data is to focus on &ldquo;interesting&rdquo; samples of data, namely regions of interest, or ROI. ROIs are especially useful in wide-area sensing situations that return datasets that are largely similar from one instance to the next, but also possess small differences. Identifying subtle changes is relevant to certain scenarios in surveillance, such as the evidence of human activity. Several ROI detection techniques exist in the research literature. My work focuses on ROI detection tuned to subtle differences for images at varying zoom levels. My thesis consists of developing a method that identifies regions of interest for subtle changes in images. In this pursuit, my contributions will address key questions including the characterization of image information dynamics through introduction of dynamic zoom, the definition and measurement of subtlety, and an approach for scoring and selecting ROIs. This work will provide an automated attention mechanism for zoomed images, but is also applicable to domains include satellite imagery and cyber security. </p><p>

<p>Network analysis provides tools for addressing fundamental applications in graphs such as webpage ranking, protein-function prediction, and product categorization and recommendation. As real-world networks grow to have millions of nodes and billions of edges, the scalability of network analysis algorithms becomes increasingly important. Whereas many standard graph algorithms rely on matrix-vector operations that require exploring the entire graph, this thesis is concerned with graph algorithms that are local (that explore only the graph region near the nodes of interest) as well as the localized behavior of global algorithms. We prove that two well-studied matrix functions for graph analysis, PageRank and the matrix exponential, stay localized on networks that have a skewed degree sequence related to the power-law degree distribution common to many real-world networks. Our results give the first theoretical explanation of a localization phenomenon that has long been observed in real-world networks. We prove our novel method for the matrix exponential converges in sublinear work on graphs with the specified degree sequence, and we adapt our method to produce the first deterministic algorithm for computing the related heat kernel diffusion in constant-time. Finally, we generalize this framework to compute any graph diffusion in constant time. </p>

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<p> From the early days of AI, computers have been programmed to play games against human players. Most of the AI work has sought to build world-champion programs to play turn-based games such as Chess and Checkers, however computer games increasingly provide for entertaining real-time play. In this dissertation, we present an extension of recursive game theory, which can be used to analyze games involving simultaneous movement. We include an algorithm which can be used to practically solve recursive games, and present a proof of its correctness. We also define a game theory of lowered expectations to deal with situations where game theory currently fails to give players a definitive strategy, and demonstrate its applicability using several example games.</p>

<p> Imaging the interior structure of comets and asteroids in 3D holds the key for understand- ing early Solar System and planetary processes, aids mitigation of collisional hazards, and enables future space investigation. 3D wavefield extrapolation of time-domain finite differ- ences, which is referred to as reverse-time migration (RTM), is a tool to provide high-quality images of the complex 3D-internal structure of the target. Instead of a type of acquisition that separately deploys one orbiting and one landing satellite, I discuss dual orbiter systems, where transmitter and receiver satellites orbit around the asteroid target at different speeds. The dual orbiter acquisition can provide multi-offset data that improve the image quality by illuminating the target from different directions and by attenuating coherent noise caused by wavefield multi-pathing. Shot-record imaging requires dense and evenly distributed receiver coordinates to fully image the interior structure at every source-location. </p><p> I illustrate a 3D imaging method on a complex asteroid model based on the asteroid 433 Eros using realistic data generated from different acquisition designs for the dual orbiter system. In realistic 3D acquisition, the distribution and number of receivers are limited by the acquisition time, revolving speed and direction of both the transmitter and receiver satellites, and the rotation of the asteroid. The migrated image quality depends on different acquisition parameters (i.e., source frequency bandwidth, acquisition time, the spinning rate of the asteroid) and the intrinsic asteroid medium parameters (i.e., the asteroid attenuation factor and an accurate velocity model). </p><p> A critical element in reconstructing the interior of an asteroid is to have different ac- quisition designs, where the transmitter and receivers revolve quasi-continuously in different inclinational and latitudinal directions and offer evenly distributed receiver coordinates in the shot-record domain. Among different acquisition designs, the simplest orbit (where the transmitter satellite is fixed in the longitudinal plane and the receiver plane gradually shifts in the latitudinal direction around the asteroid target) offers the best data coverage and requires the least energy to shift the satellite. To obtain reasonable coverage for successfully imaging the asteroid interior, the selected acquisition takes up to eight months. However, this mission is attainable because the propulsion requirements are small due to the slow (&lt; 10 cm/s) orbital velocities around a kilometer-sized asteroid.</p>

<p> Observation, estimation and prediction are universal challenges that become especially difficult when the system under consideration is dynamical and chaotic. Chaos injects dynamical noise into the estimation process that must be suppressed to satisfy the necessary conditions for success: namely, synchronization of the estimate and the observed data. The ability to control the growth of errors is constrained by the spatiotemporal resolution of the observations, and often exhibits critical thresholds below which the probability of success becomes effectively zero. This thesis examines the connections between these limits and basic issues of complexity, conditioning, and instability in the observation and forecast models. The results suggest several new ideas to improve the collaborative design of combined observation, analysis, and forecast systems. Among these, the most notable is perhaps the fundamental role that symplectic structure plays in the remarkable observational efficiency of Kalman-based estimation methods.</p><p>

<p> Many problems in physics and engineering require the solution of the heat equation in moving geometry. Integral representations are particularly appropriate in this setting since they satisfy the governing equation automatically and, in the homogeneous case, require the discretization of the space-time boundary alone. Unlike methods based on direct discretization of the partial differential equation, they are unconditonally stable. Moreover, while a naive implementation of this approach is impractical, several efforts have been made over the past few years to reduce the overall computational cost. Of particular note are Fourier-based methods which achieve optimal complexity so long as the time step <i>&Delta;t</i> is of the same order as <i> &Delta;x,</i> the mesh size in the spatial variables. As the time step goes to zero, however, the cost of the Fourier-based fast algorithms grows without bound. A second difficulty with existing schemes has been the lack of efficient, high-order local-in-time quadratures for layer heat potentials. </p><p> In this dissertation, we present a new method for evaluating heat potentials that makes use of a spatially adaptive mesh instead of a Fourier series, a new version of the fast Gauss transform, and a new hybrid asymptotic/numerical method for local-in-time quadrature. The method is robust and efficient for any <i>&Delta;t,</i> with essentially optimal computational complexity. We demonstrate its performance with numerical examples and discuss its implications for subsequent work in diffusion, heat flow, solidification and fluid dynamics. </p><p>

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