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Ultrasonic technology is commonly used in the eld of Non-Destructive Testing (NDT) of metal structures such as steel, aluminium, etc. Compared to ultrasonic bulk waves that travel in infinite media with no boundary influence, Ultrasonic Guided Waves (UGWs) require a structural boundary for propagation such that they can be used to inspect and monitor long elements of a structure from a single position. The greatest challenges for any UGW system are the plethora of wave modes arising from the geometry of the structural element which propagate with a range of frequency dependent velocities and the interpretation of these combined signals reflected by discontinuities in the structural element. In this thesis, a technique is developed which facilitates the measurement of Time of Arrival (ToA) and group velocity dispersion curves of wave modes for one dimensional structures as far as wave propagation is concerned. A second technique is also presented which employs the dispersion curves to deliver enhanced range measurements in complex multimodal UGW responses. Ultimately, the aforementioned techniques are used as a part of the analysis of previously unreported signals arising from interactions of UGWs with piezoelectric transducers. The first signal processing technique is presented which used a combination of frequency-sweep measurement, sampling rate conversion and the Fourier transform. The technique is applied to synthesized and experimental data in order to identify different wave modes in complex UGW signals. It is demonstrated that the technique has the capability to derive the ToA and group velocity dispersion curve of the wave modes of interest. The second signal processing technique uses broad band excitation, dispersion compensation and cross-correlation. The technique is applied to synthesized and experimental data in order to identify different wave modes in complex UGW signals. It is demonstrated that the technique noticeably improves the Signal to Noise Ratio (SNR) of the UGW response using a priori knowledge of the dispersion curve. It is also able to derive accurate quantitative information about the ToA and the propagation distance. During the development of the aforementioned signal processing techniques, some unwanted wave-packets are identified in the UGW responses which are found to be induced by the coupling of a shear mode piezoelectric transducer at the free edge of the waveguide. Accordingly, the effect of the force on the piezoelectric transducers and the corresponding reflections and mode conversions are studied experimentally. The aforementioned signal processing techniques are also employed as a part of the study. A Finite Element Analysis (FEA) procedure is also presented which can potentially improve the theoretical predictions and converge to results found in experimental routines. The approach enhances the con dence in the FEA models compared to traditional approaches. The outcome of the research conducted in this thesis paves the way to enhance the reliability of UGW inspections by utilizing the signal processing techniques and studying the multimodal responses.

A multiple-sensor optical shape measurement system (SMS) based on the principle of white-light fringe projection has been developed and commercialised by Loughborough University and Phase Vision Ltd for over 10 years. The use of the temporal phase unwrapping technique allows precise and dense shape measurements of complex surfaces; and the photogrammetry-based calibration technique offers the ability to calibrate multiple sensors simultaneously in order to achieve 360° measurement coverage. Nevertheless, to enhance the applicability of the SMS in industrial environments, further developments are needed (i) to improve the calibration speed for quicker deployment, (ii) to broaden the application range from shape measurement to deformation field measurement, and (iii) to tackle practically-challenging surfaces of which specular components may disrupt the acquired data and result in spurious measurements. The calibration process typically requires manual positioning of an artefact (i.e., reference object) at many locations within the view of the sensors. This is not only timeconsuming but also complicated for an operator with average knowledge of metrology. This thesis introduces an automated artefact positioning system which enables automatic and optimised distribution of the artefacts, automatic prediction of their whereabouts to increase the artefact detection speed and robustness, and thereby greater overall calibration performance. This thesis also describes a novel technique that integrates the digital image correlation (DIC) technique into the present fringe projection SMS for the purpose of simultaneous shape and deformation field measurement. This combined technique offers three key advantages: (a) the ability to deal with geometrical discontinuities which are commonly present on mechanical surfaces and currently challenging to most deformation measurement methods, (b) the ability to measure 3D displacement fields with a basic single-camera single-projector SMS with no additional hardware components, and (c) the simple implementation on a multiple-sensor hardware platform to achieve complete coverage of large-scale and complex samples, with the resulting displacement fields automatically lying in a single global coordinate system. A displacement measurement accuracy of ≃ 1/12,000 of the measurement volume, which is comparable to that of an industry-standard DIC system, has been achieved. The applications of this novel technique to several structural tests of aircraft wing panels on-site at the research centre of Airbus UK in Filton are also presented. Mechanical components with shiny surface finish and complex geometry may introduce another challenge to present fringe projection techniques. In certain circumstances, multiple reflections of the projected fringes on an object surface may cause ambiguity in the phase estimation process and result in incorrect coordinate measurements. This thesis presents a new technique which adopts a Fourier domain ranging (FDR) method to correctly identifying multiple phase signals and enables unambiguous triangulation for a measured coordinate. Experiments of the new FDR technique on various types of surfaces have shown promising results as compared to the traditional phase unwrapping techniques.

Organosilanes are widely used to bond organic materials such as polymers to inorganic materials in polymer composites. However, the mechanism of adhesion is poorly understood. One postulated mechanism is the interdiffusion of the silane and polymer, along with condensation of the silane to form an interpenetrating polymer network (IPN). The techniques of attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR) and Raman confocal microscopy have been used to study the diffusion, hydrolysis and condensation of three organosilanes in silicon / PVC / silane laminates. These processes are thought to contribute to the formation of the IPN, and hence to be vital for adhesion. The organosilanes studied were [3-(amino)propyl]trimethoxysilane, also known as A1110, [3-(phenylamino)propyl]trimethoxysilane, known as Y9669, and [3-(mercapto)propyl]triethoxysilane, known as A1891.ATR-FTIR was shown to be an excellent technique for studying the kinetics of silane diffusion through PVC films. It was shown that at room temperature, no diffusion through unplasticised PVC films occurred. At 70 °C, however, diffusion occurred readily for Y9669 and A1891. In plasticised PVC films, diffusion was observed for all three silanes at room temperature. It was shown that the diffusion occurred more quickly with higher plasticiser concentrations, and hence lower glass transition temperatures. The kinetics of diffusion was found to fit a dual mode sorption model. Hydrolysis of the silanes was also followed by infrared spectroscopy, and the kinetics of hydrolysis and condensation were shown to be highly dependent upon silane type, the concentration of water, and the presence of an acid catalyst. The hydrolysis of the silanes was found to slow their diffusion through both plasticised and unplasticised PVC films. It was shown that the presence of water in the films caused the hydrolysis of the silanes in situ. Raman depth profiles were measured of the films before, during and after diffusion. The spatial resolution was shown to be adversely affected by refraction at the air / PVC interface. It was shown that it is possible to deconvolve the confocal response of the microscope from the depth profiles, resulting in greater spatial resolution. Hydrolysis of the silanes was followed in solution by Raman spectroscopy, and it was found that each of the three silanes showed different rates of hydrolysis and condensation. It was shown that it was also possible to follow the kinetics of diffusion by Raman microscopy, and the results agreed well with those shown by ATR-FTIR spectroscopy.

This thesis is concerned with the development of a practical digital image processing system for recording and subsequent reconstruction of the magnitude and phase of an optical wave-front arriving from a coherently illuminated object disturbance. Since the wave-fronts of concern are coherent, the magnitude and phase of such waves are generally independent functions in the sense that the knowledge of one is not sufficient to uniquely deduce the other. To uniquely reconstruct and characterize optical disturbances both the magnitude and phase are required. In general, all recording media respond only to light intensity and no difficulty is encountered in recording the intensity and therefore the magnitude, because it is the square root of the intensity.

We explore lattice point counting and the method of stationary phase through the lens of questions about the number of lattice points on and near surfaces with vanishing curvature. Our focus is on spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic Heisenberg norms for α ≥ 2. We accomplish this through a transformative process that takes a number theory question about counting lattice points and translates it into that of an analytical estimation of measure. This process relies on truncating and scaling the n-dimensional integer lattice to produce a fractal-like set. By introducing a measure on this resulting set and using elementary Fourier analysis, the counting problem is transformed into one of bounding an energy integral.This process uses principles of fractal geometry and oscillatory integrals. Primary challenges that arise are the presence of vanishing curvature and uneven dilations. Following a discussion and formal estimate of the curvature of the Heisenberg spheres, we utilize the method of stationary phase to compute a bound on the Fourier transform of their surface measures.Our work is inspired by that of Iosevich and Taylor (2011) and Garg, Nevo, and Taylor (2015). We present an extension of the main result in the former to surfaces with vanishing curvature. Furthermore, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.Additionally, we present a mini-course on the basics of stationary phase—a quick-start guide to stationary phase in practice. This includes a discussion of the formulation of oscillatory integrals and their solutions with a focus on the impact of geometric properties (e.g. curvature) on the estimates for the decay of the Fourier transform. It further serves as a supplement to [Shakarchi and Stein, Functional Analysis: Chapter 8, Sections 2 and 3], providing detailed expositions of the proofs of fundamental results.We conclude with a discussion of potential avenues for improvement and interesting problems that could be considered utilizing our methods.

Overactive Bladder (OAB) syndrome affects nearly 33 million adults in the US alone and results in severe health-related quality-of-life issues. Detrusor Overactivity (DO) is thought to be an important underlying cause of OAB and bladder cystometry – a test to measure bladder pressures during filling and voiding phases, is the clinical standard for diagnosis of DO. However, cystometry has not evolved for decades, and analysis of cystometric data is still largely subjective, dependent on physician interpretation and, mostly focused on overt contractile events and voiding behavior. Patient reported outcome measures, which are subjective and non-quantitative, have instead filled this diagnostic gap, resulting in challenges for both clinicians and patients. This thesis addresses the limitations of current bladder diagnostics by 1) developing a scalable workflow to measure and quantify cystometric signals during filling and 2) applying the workflow and measurement techniques across various clinical and preclinical in vitro and in vivo models and studies to improve utility of cystometric outcomes and increase reliability and accuracy of clinically diagnosing DO.As a first step, a scalable workflow was developed from an initial set of clinical cystometry data sets to quantify bladder pressure signals during filling, with the goal of rapid near-term translation to routine clinical application. This workflow was used to analyze the acquired signals from preclinical studies and existing clinical data sets, which demonstrate its scalability and utility across different model systems. Initially, in vitro porcine tissue models were considered for measurement of contractile activity of bladder tissue strips. Numerous studies were conducted to mimic physiological and pathological conditions during bladder filling, by tissue stretching and addition of carbachol and their impacts on the contractile activities of the strips were measured and quantified. To understand bladder physiology in a clinical setting, retrospective clinical studies were conducted where clinical cystometry data sets were analyzed with a focus on filling-related bladder pressure signals. Further, the analysis was extended to DO and non-DO patients where the success of the workflow was demonstrated in not just differentiating the two groups but being able to do so with half the duration of available signals, thus showing potential to reduce cystometry duration by 50% as well as eliminating the need to initiate voiding. To bridge the gap between in vitro and clinical studies, preclinical in vivo large animal sheep studies were conducted. Leveraging the physiological and anatomical similarities of sheep and humans, multiple sets of filling cystometry data were captured from sheep in both awake behaving and anesthetized states and the data was analyzed using the workflow. Results showed similarities between in vivo signals, and those of clinical non-DO patients as well as in vitro tissue studies. Further, these studies demonstrated that sheep are viable preclinical models to mimic bladder physiology and pathologies, and test different therapeutic efficacies. This thesis demonstrates the utility of measuring and quantifying filling-related signals of the bladder in diagnosis of bladder dysfunction. Further, parallels between preclinical and clinical models of bladder dysfunction are shown. The concepts described here reflect the need for quantification of clinical data and point to a much-needed shift in clinical diagnosis of bladder dysfunction from subjective interpretation to objective quantified outcomes.

Abstract: Kernel methods are often used for nonlinear regression and classification in machine learning because they are computationally cheap and accurate. Fourier basis and wavelet basis are the bases that can efficiently approximate the kernel functions. In previous research, Bayesian approximate kernel regression with Fourier transform has been proposed. With the proposed method, we use the analytic properties of the reproducing kernel Hilbert space (RKHS) to define a linear vector space that captures nonlinear structures. We map the data into a low-dimensional randomized feature space using Fourier transform and convert kernel function into operations of a linear machine. A Bayesian approximate kernel regression model is then formulated with the application of a generalized kernel model and the Bayesian method. We replace Fourier transform with wavelet transform in randomized feature space to approximate kernel functions. We formulate a new Bayesian approximate kernel model with wavelet transform and use the Gibbs sampler to compute the parameters of the model. We then make a comparison between the performance of Fourier based and wavelet-based Bayesian approximate kernels solving both regression and classification problems.

There is a need to improve methods of monitoring patients with traumatic brain injury (TBI) in hospital settings. Current monitoring techniques and diagnosis methods are expensive, invasive, do not provide continuous measures, expose the patient to radiation, are ambiguous in the information they provide, and/or cannot be implemented at the bedside. These techniques measure imperative markers of brain function including intracranial pressure (ICP), cerebral blood flow (CBF), and oxygenation in the brain, among others. Hospitals not only require a practical method for real-time monitoring of patients at the bedside, but also meaningful metrics that characterize TBIs, since the variety of methods results in complex and ambiguous criteria for defining TBIs. Trends in the literature show a reliance on functional assessments such as the Glasgow Coma Scale (GCS) to define TBIs, however studies have shown its complexity and context-dependence in predicting outcome. Real-time cerebral assessment is currently focused on ICP monitoring, but increased attention to hemodynamic measures to improve patient outcomes warrants new technologies and metrics. Therefore, in this thesis, a novel metric of low-frequency oscillations (LFOs) from hemodynamic monitoring is proposed to provide a more objective characterization of TBI. Literature suggests that these LFOs originate from the regulation of regional changes in CBF and energetic metabolism and not from systemic regulation of the cardiovascular system, making it a representative metric of brain function. Overall, this thesis will contribute to the clinical understanding of TBI through optical imaging-derived LFOs. To achieve this overall goal, the followings have been investigated between non-TBI and TBI groups: 1) quantification of absolute concentration changes of hemodynamic parameters, 2) assessment of the LFO spectrums of the oxygenated hemoglobin signals to identify prevalent LFOs, and finally 3) quantification of the average power in predefined LFO slow bands. The results indicate that optical imaging can provide noninvasive neurovascular biomarkers for continuous assessment of TBI patients.

As machine learning models become more sophisticated, and biometric data becomes more readily available through new non-invasive technologies, it becomes increasingly possible to gain access to interesting biometric data that could revolutionize Human Computer Interaction. In this research, we propose a framework to assess and quantify human preference (like or dislike) on presenting various external visual stimuli. Our framework relies on an Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) based model and on electroencephalogram (EEG) signals analysis to predict Like or Dislike preference of human subjects when presented with various marketing images.

Abstract: In distributed computing models, where some helper nodes assist the master in a large-scale computation, a big challenge is when these helpers straggle. The straggling of even a single helper node can significantly increase the processing time. Therefore, coded distributed computing has been proposed as a solution in many recent studies. Unfortunately, in some scenarios, the decoding complexity at the master is so significant that it undermines the benefit of distributing the computation. To allow for distributed computing in these cases, we propose a multi-layer coding strategy that allows some helpers to assist with the decoding. In this thesis, we focus on two scenarios, (1) when an extra layer of helpers are introduced to help with the master's decoding, and (2) fitting a distributed coding scheme when the computation naturally requires multiple layers with shuffling (for example FFT). In both scenarios, we propose a fully-coded structure that tolerates straggling in each step. Moreover, using low-complexity codes such as Raptor codes to further reduce the master's decoding load is studied.

Velocity measurement inside of a wind tunnel is an extremely useful quantitative data for a multitude of reasons. One major reason is that velocity has a mathematical relationship with dynamic pressure which in turn influences all the aerodynamic forces on the test model. Many devices and methods exist for measuring velocity inside wind tunnels. At the same time, Doppler wind lidar (light detection and ranging) has been used for decades to make air speed measurements outdoors at long ranges. Lidar has been proven effective for many applications, and it has the potential to solve many of the problems faced by current velocimetry techniques inside wind tunnels. Despite this, minimal research has been performed with Doppler wind lidars inside wind tunnels. While multiple commercial systems exist for making air speed measurements at longer ranges, there are currently no widely available commercial devices designed to work well inside wind tunnels. In this research, initial work is described for the design and development of a continuous wave (CW), coherent wind lidar system. The system is for use as an alternative non-intrusive velocimetry method inside wind tunnels relying on the Doppler effect. A scaled down wind lidar designed to operate at much shorter ranges than current commercial wind lidars can be simpler, less expensive, and require less power. A first iteration of the design was constructed for proof of concept testing with a small-scale wind tunnel at low speeds (7.5-9 m/s). Testing showed that the lidar system could take one-dimensional speed measurements of seeded flow that closely matched Pitot static tube data. When not adding tracer particles to the flow, the lidar return signal was not strong enough for the photodetector used to measure the beat frequency. This research is focused on the process for designing the Doppler wind lidar system, constructing the experimental setup, and studying methods for data analysis. Results of testing presented in the paper and lessons learned were used to create a second iteration of the wind lidar design that can be built for future testing. Not all data analysis methods and experiments described herein were successful, but this documentation will be helpful to future researchers for improving the design and continuing to make progress on a much needed device for wind tunnel velocimetry.

This thesis explores load monitoring on dc micro-grids specifically applied to Naval shipboard power systems. More electronic loads are being placed on Naval ships and, increasingly, more of these loads are pulsed-power in nature; drawing pulsating current from the grid. This presents a challenge for conventional fault monitoring devices as many fault currents are also pulsating in nature and can be difficult to differentiate from a desirable pulse event. Short-time Fourier transform is the technique preferred in this work for spectral analysis of the current signals. In addition to event based monitoring, another unsupervised control is being employed to continuously check the frequency content of the current to look for arcing faults caused by loose electrical connections. The objective of the dissertation is to develop a load monitoring algorithm that records the current drawn by these loads and is able to detect events and differentiate between desirable events and faults using their frequency content in run-time. The algorithm is realized on an micro-controller unit and validated on a low-voltage dc test-bed.

We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove a quantized version of the Fourier Slice Theorem and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the families of compatible Riemannian and Lorentzian metrics $g_\epsilon$ and $g_{i\epsilon}$, respectively, taming the sub-Riemannian metric and prove that their associated X-ray transforms are injective for all $\epsilon>0$. In the latter case, we show in particular that a function on $g_{i\epsilon}$-spacetime is determined by integrals over spacelike geodesics. These results provide explicit examples of injective X-ray transforms in a geometry with an abundance of conjugate points and with metrics of mixed curvature.

We study global existence and asymptotic behavior of the solutions to models for chemotaxis systems with chemo attractants and repellents in three dimensions. Chemo attractants and repellents may be called chemo agents. For Part I, we use the logistic model for the mass. The interactions between chemo agents and the mass are taken into account. For Part II, we consider the case when mass is conserved and we use the Lotka-Volterra type model for chemo agents. To accomplish this, we use the Fourier transform and energy method. We show the existence of global solutions by the energy method. Also, we establish $L^q$ time-decay for the linear homogeneous system by using the Fourier transform and finding Green's matrix. Then, we find $L^q$ time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.

The focus of this dissertation is on the dependent structure and limit theorems of high dimensional probability theory. In this dissertation, we investigate two related topics: the Central Limit Theorem (CLT) for stationary random fields (multi-indexed random variables) and for Fourier transform of stationary random fields. We show that the CLT for stationary random processes under a sharp projective condition introduced by Maxwell and Woodroofe in 2000 \cite{MW00} can be extended to random fields. To prove this result, new theorems are established herein for triangular arrays of martingale differences which have interest in themselves. Later on, to exploit the richness of martingale techniques, we establish the necessary and sufficient conditions for martingale approximation of random fields, which extend to random fields many corresponding results for random sequences (e.g. \cite{DMV07}). Besides, a stronger form of convergence, the quenched convergence, is investigated and a quenched CLT is obtained under some projective criteria. The discrete Fourier transform of random fields, $(X_{\bm{k}})_{\bm{k}\in\mathbb{Z}^d}$ $(d\geq 2)$, where $\mathbb{Z}$ is the set of integers, is defined as the rotated sum of the random fields. Being one of the important tools to prove the CLT for Fourier transform of random fields, the law of large numbers (LLN) is obtained for discrete Fourier transform of random sequences under a very mild regularity condition. Then the central limit theorem is studied for Fourier transform of random fields, where the dependence structure is general and no restriction is assumed on the rate of convergence to zero of the covariances.

The use of fingerprints for personal identification has been around for thousands of years (first established in ancient China and India). Fingerprint identification is based on two basic premises that the fingerprint is unique to an individual and the basic characteristics such as ridge pattern do not change over time. Despite extensive research, there are still mathematical challenges in characterization of fingerprints, matching and compression. We develop a new mathematical model in the spatial domain for globally modeling loop pattern fingerprints. Although it is based on the well-known AM-FM (amplitude modulation and frequency modulation) image representation, the model is constructed by a global mathematical function for the continuous phase and it provides a flexible parametric model for loop pattern fingerprints. In sharp contrast to the existing methods, we estimate spatial parameters from the spectral domain by combining the exact values of frequencies with their orientations perpendicular to the fingerprint ridge flow. In addition, to compress fingerprint images and test background Gaussian white noise, we propose a new method based on periodogram spacings. We obtain the joint pdf of these m-dependent random variables at Fourier frequencies and derive the asymptotic distribution of the test statistic.

The big data era poses great challenges as well as opportunities for researchers to develop efficient statistical approaches to analyze massive data. Sufficient dimension reduction is such an important tool in modern data analysis and has received extensive attention in both academia and industry. In this dissertation, we introduce inverse regression estimators using Fourier transforms, which is superior to the existing SDR methods in two folds, (1) it avoids the slicing of the response variable, (2) it can be readily extended to solve the high dimensional data problem. For the ultra-high dimensional problem, we investigate both eigenvalue decomposition and minimum discrepancy approaches to achieve optimal solutions and also develop a novel and efficient optimization algorithm to obtain the sparse estimates. We derive asymptotic properties of the proposed estimators and demonstrate its efficiency gains compared to the traditional estimators. The oracle properties of the sparse estimates are derived. Simulation studies and real data examples are used to illustrate the effectiveness of the proposed methods. Wavelet transform is another tool that effectively detects information from time-localization of high frequency. Parallel to our proposed Fourier transform methods, we also develop a wavelet transform version approach and derive the asymptotic properties of the resulting estimators.

Among functions f majorized by indicator functions of sets E of measure 1, which functions have maximal Fourier transform in L^p? We investigate the existence of maximizers, using a concentration compactness approach and ingredients from additive combinatorics to establish properties of maximizing sequences. For exponents q sufficiently close to even integers, we exploit variational techniques and combinatorial results to identify all maximizers. This follows from establishing a sharper version of an associated inequality: if the input f, where |f| is less than or equal to the indicator function of a measure 1 set E, has a certain structure, then the Fourier transform of f in L^q is at least a certain quantitative distance from being optimal.

Mass spectrometry (MS) is an important analytical tool for obtaining information about the identity, structure and function of molecules over a wide range of sizes, from small molecules up to large protein complexes. Conventional MS techniques work by measuring the mass to charge ratio (m/z) of an ensemble of ions, then using the spacing between different m/z peaks to find the ion charge states so that the masses can be determined. This approach can be difficult to use for samples with a molecular mass beyond ~1 MDa or made up of a heterogeneous mixture of molecules with similar masses, which are typically detected as a single broad, unresolved m/z peak for which the charge cannot be obtained. These samples can instead be weighed with single ion MS, in which the m/z and charge of each ion is measured individually, so other ions do not interfere with the mass measurement. The work described in this dissertation is focused on developing a new instrument for single ion MS that uses charge detection mass spectrometry (CDMS) to better analyze samples that are difficult to measure with conventional MS. New instrumentation and data analysis techniques to improve the precision of CDMS mass measurements and use CDMS for new types of measurements are demonstrated. In CDMS, ions are detected by the charge they induce as they fly through a conducting tube. A new type of CDMS detector with four detector tubes inside an ion trap is used to increase the number of measurements of each ion and reduce the uncertainty in the m/z and charge by signal averaging. A method to measure the ion energy during the trapping time from the signal pattern the ion produces was also developed. This enables the energy and mass of the ion to be determined after collisions with the background gas and after fragmentation events, as in tandem MS. The energy lost to collisions can be used to obtain information about the collision rate and size of the ions, as in ion mobility spectrometry. Monitoring the ion energy also makes it possible to normalize charge measurements for the effect of energy, improving the precision of mass measurements in CDMS.

Abstract: This thesis focuses on formulating and analyzing non linear models for microbial dynamics vis-a-vis human and environmental health. Firstly, we develop and investigate a stoichiometric organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. This mechanistic biodegradation model lead to reliable and suggestive ecological insights in the preservation and restoration of our fragile ecosystems. Questions we attempt to answer include: (i) What mechanisms allow microbes and resources to persist uniformly or go extinct? (ii) How do grazing and dead microbial residues affect decomposition? (iii) How can the rate of decomposition be maximized or minimized? Secondly, we designed a greenhouse gas biogenesis model, which may be used to (i) predict the volume of greenhouse gasses emitted at any given time in an oil sands tailing pond and an end pit lake, (ii) calculate the time required to produce a given volume of cumulative greenhouse gases from them and (iii) estimate how long it will take for an oil sands tailing pond and an end pit lake to stop emitting greenhouse gases. Lastly, we formulate and analyze directly and indirectly transmitted infectious disease models. The questions aim to answer include: (i) Why are there irregularities in seasonal patterns of outbreaks amongst different countries? (ii) How can we estimate the transmission function of an infectious disease from a given incidence or prevalence data set? (iii) What is the estimated value of the basic reproduction number in affected regions? (iv) How can we control the period and intensity of pathogenic disease outbreaks?

Neonatal abstinence syndrome (NAS) is a condition caused by in-utero exposure to opioids, and its occurrence is increasing nationwide. NAS patients are newborns who can experience withdrawal symptoms including tremors, poor feeding, and respiratory distress. Presently, the Finnegan Scoring System, a subjective rating scale, is commonly used to judge the patient's condition and determine appropriate treatment methods. This project sought to develop a sensor system that is capable of objectively assessing symptoms of withdrawal, including tremors and high pitched cry. The system developed is composed of five wireless accelerometers, for attachment to a subject's limbs and chest, and an external microphone. The sensor system is targeted toward quantifying limb movements of the subject and recording audio information that includes samples of the subject's cry. The sensor system was used as part of a research study, and data was collected from recruited participants. A total of 29 out of 30 desired participants were enrolled and studied as part of the data collection process. Gathered data was analyzed using MATLAB, with motion data being searched for tremor activity in NAS participants, and cry samples searched for unique characteristics. Results generated indicate that detection of tremors was successful, and that the average fundamental frequency of cry differs between the NAS and non-NAS participants. Future considerations for this project include expanding to measure more symptoms, and system refinement to minimize the number of sensors.

This dissertation consists of three essays on testing structural changes and constant conditional dependence via the Fourier transform. The first essay, "A Model-free Consistent Test for Structural Change in Regression Possibly with Endogeneity", proposes a consistent test for structural change in a nonparametric times series regression model based on the Fourier transform. It is well known that structural instability leads to misleading inference and imprecise prediction of stationary time series models. I propose a model-free consistent test for structural change in regression by testing the instability of the Fourier transform of data. This novel approach avoids smoothed nonparametric estimation of the unknown regression function and so is free of the "curse of dimensionality" problem, especially when the dimension of regressors is high. As a result, the proposed test is asymptotically more powerful against a class of local alternatives than Vogt's (2015) nonparametric test for structural changes, which is the only consistent test for structural changes in a nonparametric regression model in the existing literature. The nonparametric tests of Hidalgo (1995) and Su and Xiao (2008) are asymptotically more powerful than the proposed test against certain smooth local alternatives, but they are not consistent tests and are asymptotically less powerful against a class of non-smooth local alternatives. Unlike the existing literature, I allow for endogenous and discrete regressors. By using a proper choice of weighting functions for the transform parameters in the Fourier transform, I avoid numerical integration so that the test statistic is easy to compute. The test statistic has a convenient asymptotic N(0,1) distribution under the null hypothesis of no structural change and is consistent against a large class of smooth structural changes as well as abrupt structural breaks with unknown break dates. A Monte Carlo study and an empirical application show that the test performs reasonably well in finite samples. In the second essay titled "Consistent Testing for Structural Change in Time Series Regression Models via the Fourier Transform", I focus on testing structural changes in a linear time series regression model via the Discrete Fourier Transform (DFT). The intuition is straightforward: if the true model parameters are time-varying, then the conventional estimation methods like OLS or 2SLS will fail to estimate the unknown parameters consistently. The estimated residuals will contain the time-varying local feature of model parameters. The Discrete Fourier Transform of estimated residuals will contain this information and reveal it in the frequency domain. One can then infer the existence of structural changes regardless of whether they are smooth or abrupt. Compared to the existing consistent tests for structural change, my test avoids smoothed nonparametric estimation of the unknown time-varying parameters. The rate of the local alternatives that our test can detect is Root-T. Furthermore, my test is robust to unknown structural change in explanatory variables and instrumental variables, which makes the test widely applicable, especially in macroeconomic models. Simulation studies demonstrate its good finite sample performance. I apply my test to examine the stability of the hybrid New Keynesian Phillips Curve and find evidence of structural changes in 1980 to 2001 which is treated as a stable period by Zhang et al. (2008) and Hall et al. (2012). The third essay, "Testing Constancy of Conditional Joint Dependence", proposes an omnibus test for the constancy of conditional joint dependence on some state variables. The test statistic is constructed by comparing the generalized conditional covariance function and the generalized unconditional covariance function. It detects if the dependence strength between any two random variables varies with a certain factor that we are interested in. I show that by using a special weighting function proposed by Szekely et al. (2007), the test statistic can be easily computed without using numerical simulation. Also by nonparametric regression, I show that the test statistic is both computationally and asymptotically invariant to possible high dimensional data. Furthermore, the test statistic is asymptotically pivotal and follows a convenient asymptotic N(0,1) distribution under the null hypothesis. I also show that the test statistic can apply to many other testing frameworks with proper transformation. A simulation study shows that the test works well in finite samples.

In 1958, L. Carleson showed that a set E is removable for Holder continuous solutions of the Laplacian operator if and only if its (n-2+α)-dimensional Hausdorff measure equals zero, where α is the Holder exponent of the solution. In this thesis, we extend Carleson's result to a certain of non-local pseudo-differential operator called the fractional Laplacian.

In this thesis we discuss linear dependence of translations which is intimately related to the zero divisor conjecture. We also discuss the square integrable representations of the generalized Wyle-Heisenberg group in ² dimensions and its relations with Gabor's question from Gabor Analysis in the light of the time-frequency equation. We study the zero divisor conjecture in relation to the reduced *-algebras and operator norm *-algebras. For certain classes of groups we address the zero divisor conjecture by providing an isomorphism between the the reduced *-algebra and the operator norm *-algebra. We also provide an isomorphism between the algebra of weak closure and the von Neumann algebra under mild conditions. Finally, we prove some theorems about the injectivity of some spaces as ℂ modules for some groups .

With the recent proliferation of 3D sensors such as Light Detection and Ranging (LIDAR), it is essential to develop feature representation methods that can best characterize the point clouds produced by these devices. When these devices are employed in targeting and surveillance of human actions from both ground and aerial platforms, the corresponding point clouds of body shape often comprise low-resolution, disjoint, and irregular patches of points resulted from self-occlusions and viewing angle variations. The prevailing method of depth image analysis has the limitation of relying on 2D features that are not native representation of 3D spatial relationships. On the other hand, many existing 3D shape descriptors cannot work effectively with these degenerated point clouds because of their dependency on 360-degree dense and smooth point clouds. In this research, a new degeneracy-tolerable, multi-scale 3D shape descriptor based on the discrete orthogonal Tchebichef moment, named Tchebichef moment shape descriptor (TMSD), is proposed as an alternative for single-view partial point cloud representation and characterization. It has the advantage of decomposing a complex 3D surface or volumetric distribution into orthogonal moments in a much compact subspace that is independent of learning datasets, thereby supports accurate, robust, and consistent shape search and pattern recognition in the embedded subspace. Complimentary to the proposed descriptor, a new voxelization and normalization scheme is proposed to achieve translation, scale, and resolution invariance, which may be less of a concern in the traditional full-body 3D shape analysis but are crucial requirements for discerning partial point clouds. To evaluate the effectiveness of TMSD and voxelization algorithms for static pose shape search and dynamic action recognition, we built a first-of-its-kind multi-subject pose shape baseline consisting of simulated LIDAR captures of actions at different viewing angles. Compared to the other existing public datasets, our baseline has more subjects and viewing angle variations to support solid algorithm development and evaluation. Using the pose shape baseline, we developed single-view nearest neighbor (NN) search for pose shape retrieval using TMSD. We proved the lower bounding distance condition under the orthonormality of Tchebichef moment, which prevents false dismissal by any subspace queries. Our experimental results show that 3D TMSD performs significantly better than 3D Fourier transform (3D DFT) and slightly better than 3D wavelet transform (3D DWT). It is also more flexible than 3D DWT for multi-scale representation because it does not have the restriction of dyadic sampling. The action recognition was built on the Naive Bayes classifiers using temporal statistics of a 'bag of pose shapes'. Our experiments demonstrate that the 3D TMSD-based classification of action and viewing angle outperforms the similar classification based on the depth image analysis using the popular 2D features of the histograms of oriented gradients. In other experiments, we demonstrated our approach's scale invariance by showing consistent query and classification performance across a wide range of spatial scales, down to the extremely small scale of 6% of the original point clouds, at which level the 2D depth image analysis tends to degrade significantly. We also validated performance against varying viewing angles on both azimuth and elevation directions, which has an important implication for aerial sensor platforms. In summary, many of the performance advantages shown by TMSD are fundamentally due to its sound mathematical properties. Through the direct 3D encoding of point cloud distribution, our research offers a promising new approach for analyzing low-quality, single-view 3D sensor data, other than the usual approach of 2D-based depth image analysis.

Local feature detection and local feature description have been frequently utilised as an integral system to accomplish diverse computer vision applications due to its invariance capabilities against image distortions caused by geometric and photometric transformations. The applications include image retrieval, image registration and object recognition. In computer vision, however, local feature detection and description should be treated as independent processes since the former identifies repeatable keypoints in images while the latter analyses local image patches around detected keypoints to represent those keypoints. Following the principle of separable processes, this research concentrates on local feature description and suggests further improvements. A variety of local feature descriptors has been devised based on statistical analysis in spatial domain such as creating a histogram of image gradients. To address the overdevelopment of these approaches, a descriptor that relies on spectral analysis in frequency domain is proposed. The proposed descriptor, which is coined as Radial Fourier Analysis (RFA) descriptor, transforms the image gradients of local image patches to frequency domain, then its frequency response is decomposed and analysed to provide a robust and distinctive representation for keypoints. Concurrent with the use of spectral analysis in the frequency domain, a keypoint orientation estimator is designed in this domain to enhance the rotational invariance of RFA descriptor. The keypoint orientation estimator employs the starting point normalisation of Fourier coefficients to deduce rotating angles that ensure the local image patches of keypoint correspondences are aligned at the same orientation. As such, the performance of RFA descriptor is further enhanced with this accurate estimation of keypoint orientation. Through comprehensive experiments and comparisons, RFA descriptor had demonstrated its outstanding performances against various image distortions. In particular, the descriptor gives reliable performances in dealing with images that are degraded by blur, rotation, illumination and JPG compression changes. It also indicates that the proposed RFA descriptor and keypoint orientation estimator combine perfectly to yield a powerful local feature descriptor. All these show that spectral analysis serves well as an alternative approach for statistical analysis of local feature description and keypoint orientation estimation.

The theory of stretching and compressing of short light pulses by the chirped volume Bragg gratings (CBG) is reviewed based on spectral decomposition of short pulses and on the wavelength-dependent coupled wave equations. The analytic theory of diffraction efficiency of a CBG with constant chirp and approximate theory of time delay dispersion are presented. Based on those, we performed comparison of the approximate analytic results with the exact numeric coupled-wave modeling. We also study theoretically various definitions of laser beam width in a given cross-section. Quality of the beam is characterized by the dimensionless beam propagation products (?x???_x)?? , which are different for each of the 21 definitions. We study six particular beams and introduce an axially-symmetric self-MFT (mathematical Fourier transform) function, which may be useful for the description of diffraction-quality beams. Furthermore, we discuss various saturation curves and their influence on the amplitudes of recorded gratings. Special attention is given to multiplexed volume Bragg gratings (VBG) aimed at recording of several gratings in the same volume. The best shape of a saturation curve for production of the strongest gratings is found to be the threshold-type curve. Both one-photon and two-photon absorption mechanism of recording are investigated. Finally, by means of the simulation software we investigate forced airflow cooling of a VBG heated by a laser beam. Two combinations of a setup are considered, and a number of temperature distributions and thermal deformations are obtained for different rates of airflows. Simulation results are compared to the experimental data, and show good mutual agreement.

Discrete (mm- to m-scale) mylonitic shear zones in the northeastern Harquahala metamorphic core complex, Arizona, show evidence of fluid-mineral interactions catalyzing deformation and metamorphism. Many contain a deformed central epidote vein with adjacent bleached haloes and flanking paired shear zones that indicate significant fluid-rock interaction during deformation. An integration of structural and geochemical methods was employed to understand timing, metamorphic conditions, and physiochemical processes responsible for producing the discrete shear zones. Field and microstructural evidence suggest the zones initiated on antecedent fractures. Electron backscatter diffraction (EBSD) analyses show a significant coaxial contribution to the shear, and quartz deformation predominately by prism slip, along with some rhomb slip, suggesting amphibolite-facies conditions during shearing. Fourier Transform Infrared spectroscopy analyses of quartz reveal higher water contents within shear zones than within country rocks, indicating fluid infiltration synchronous with shearing. Stable isotope analyses of quartz and feldspar from mylonites are consistent with an igneous or metamorphic fluid origin. Microstructural observations suggest that the zone morphology with epidote veins, bleached haloes, and flanking discrete paired shear zones was developed predominantly from reaction softening mechanisms. The increase in deformation from bleached rock to flanking shear zones is marked by progressive modal increases in biotite and myrmekite, and modal decreases in K-feldspar, and locally epidote and titanite. Myrmekitic textures recrystallized readily and resulted in progressively greater grain size reduction of feldspar, which aided in the progressive alignment and linkage of the biotite grains, which together concentrated the deformation in bands. Volume reduction resulting from some of the metamorphic reactions may have led to a positive feedback cycle among fluid infiltration, metamorphism and deformation. U-Pb isotope analyses of syn-metamorphic titanite yield an age of ~70 Ma, suggesting the shear zones formed during cooling of the Late Cretaceous (75.5±1.3 Ma) Brown’s Canyon pluton, consistent with their top-to-the-southwest sense of shear, rather than during top-to-the-northeast directed Miocene metamorphic core complex exhumation. Petrography, EBSD analyses, and U-Pb dating of titanite from other (non-discrete) mylonites in the area imply most formed synchronously with the discrete shear zone mylonites. Only rare, scattered mylonites show features consistent with metamorphic core complex exhumation.

In this master's thesis, several digital image processing techniques are explored for potential in evaluation of Brightness mode (B-mode) ultrasound images. Currently, many processing techniques are utilized during ultrasound visualization in cardiovascular applications, mammography, and three-dimensional ultrasound systems. However, approaches that serve to aid the clinician in diagnostic assessment of tendinous and ligamentous injuries are more limited. Consequently, the methods employed below are aimed at reducing the dependence on clinician judgment alone to assess the healing stage and mechanical properties of tendinous injuries. Initial focus concentrated on the use of entropy in texture analysis to relate a tendon's appearance in an ultrasound image to its mechanical integrity. Confounding effects such as motion artifacts and region of interest selection by the user limited the applicability of small regions selected for analysis, but general trends were observed when the entire visualized tendon or superficial background region was selected. Entropy calculations suggested a significant change in texture pattern for tendinous regions compared to the selected background regions. In order to reduce the impact of motion artifacts and dependence of the texture analysis on manual identification of regions of interest, a Matlab® script was developed intended to isolate the tendinous regions of interest for further analysis. Methods for segmentation employed relied on a moving window Fourier Transform to compare local parameters in the image to a predefined window of tendinous tissue. Further assessment of each local region benefited from parameterization of the local window's properties that focused on capturing indicators of mean pixel intensity, local variation in pixel intensity, and local directionality consistency derived from the spatial frequency patterns observed in the Fourier Transforms via comparison by the circular Earth Mover's Distance. Results of the segmentation algorithm developed indicated the presence of directional consistency within the tendinous regions, and changes in the speckle pattern were observed for the image derived from mean intensity and local pixel intensity variation. However, non-tendinous regions were also identified for their directional consistency, limiting the applicability of the current process in tendinous region isolation. The results obtained for calculations of the circular Earth Mover's Distance improved slightly with the inclusion of temporal averaging and image registration, but still require improvement before implementation in clinical applications can be realized.

Transverse fiber modes are the distribution of electromagnetic fields in the cross-section of an optical fiber through which light propagates in the fiber. Research is being conducted to explore the usage of fiber modes with the purpose of increasing the capacity and reach of optical fibers. Mode Division Multiplexing, an area in which reseach is being conducted with the same purpose, is a multiplexing scheme used in optical networks that maps data channels onto different modes, and multiplexes these modes into one fiber for transmission. This thesis focuses on the study of modes in multi-mode fibers, and the simulation of mode creation in gradient-index multi-mode fibers. Starting from published related works, equations for electric fields in step-index and gradient-index multi-mode fibers are derived, and examples of intensity profiles in several modes are shown. For the first time, we seek to study modal cross-talk between modes. Given the creation process, not all power is coupled into the desired mode, resulting in some power being coupled into undesired modes. We also seek to increase the power coupling between the created mode and the desired mode. We report a 23% power coupling increase up from previous works. Finally, we seek to study the effect of noise introduced by the components used in the process of creating modes. We report an approximate 30dB signal-to-noise ratio is sufficient to create a mode with maximum power coupling.

Distributions of $C^k$ functions over $\ZP$ have been characterized using formal power series by Amice. These characterizations allow us to find a sharp bound on the growth of $C^k$ distributions. In the case of $C^1$ distributions it is shown that two distributions that agree on the functions of the form $\chi_{a+p^n\ZP}$ have Fourier Transforms that differ by the product of the logarithm and the Fourier Transform of a measure.The development of a Radon-Nikodym derivative is established for $C^1$ distributions under certain restrictions. This gives a result that is similar to a Lebesgue decomposition. This generalizes a theorem of Barbacioru.

We develop some aspects of the theory of D-modules on schemes and indschemes of pro-finite type. These notions are used to define D-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. We also extend the Fourier-Deligne transform to Tate vector spaces. Let N be the maximal unipotent subgroup of a reductive group G. For a non-degenerate character c of N((t)), and a category C acted upon by N((t)), there are two possible notions of the category of (N((t)),c)-objects: the invariant category and the coinvariant category. These are the Whittaker categories of C, which are in general not equiva- lent. However, there is always a natural functor T from the coinvariant category to the invariant category. We conjecture that T is an equivalence, provided that the N((t))-action on C is the restriction of a G((t))-action. We prove this conjecture for G=GLn and show that the Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme) of G((t)).

Christopher P. HoseitSystolic Array Implementation of the Fourier TransformAbstractAdaptive optics is a relatively new technology originally designed to eliminate negative effects of the atmosphere on telescopic images. This technology allows ground based telescopes to achieve resolutions comparable to the Hubble space telescope. Adaptive optics is the "cheaper" solution to capturing better images than sending telescopes into space. Ground based telescopes can be built with larger mirrors providing more light collecting area and ultimately better pictures. Telescopes on the ground can also be more easily maintained and upgraded than telescopes in space. Using a combination of innovative technology such as Laser Guide stars and deformable mirrors, (MEMs) devices, along with significant software and hardware engineering to implement algorithms allow adaptive optics to improve imaging with telescopes.This thesis work implements a subset of the hardware system required for the systolic array tomography algorithm on a Field Programmable Gate Array (FPGA). Previous technology analysis has concluded that a FPGA platform is suitable for implementing the tomography algorithm. The algorithm will be verified through simulation and comparing results with values generated in a Microsoft Excel script. The tomography engine is a subset of the real time control system described in [1]. Future work will involve implementing the full scale of the algorithm on all of the required hardware.viiiThis thesis presents: 1) the scalable design of the Fourier transform, 2) implementation results and simulation verification.

Graph-structured data appears in many modern applications like social networks, sensor networks, transportation networks and computer graphics. These applications are defined by an underlying graph (e.g. a social graph) with associated nodal attributes (e.g. number of ad-clicks by an individual). A simple model for such data is that of a graph signal--a function mapping every node to a scalar real value. Our aim is to develop signal processing tools for analysis of such signals de- fined over irregular graph-structured domains, analogous to classical Fourier and Wavelet analysis defined for regular structures like discrete-time sequences and two-dimensional grids.In this work, we start by reviewing the notion of a Graph Fourier Transform (GFT), which has been defined in the literature for graph signals. We examine the spatial and spectral features of circulant graphs, which accommodate linear shift-invariant operations. We describe fundamental operations such as shifting, sampling, graph-reconnection and linear filtering for signals on circulant graphs and derive associated sampling and graph-reconnection theorems. We also develop wavelet filter bank structures for multi resolution analysis of large-scale graphs.We present a method to decompose an arbitrary graph into a linear combination of circulant graphs. This helps extend fundamental operations such as sampling, filtering and multi resolution filter banks to general graphs. We present an application in the area of graph semi-supervised learning where some of the existing algorithms can be viewed as suitably designed filters defined in the GFT domain. We propose a wavelet regularized learning algorithm and evaluate the performance on some real-world datasets.

An emerging challenge in recent years for engineers, researchers, and data scientists across the globe is to acquire, store, and analyze ever-increasing amounts of data. In the past decade, a new paradigm in data acquisition called ``compressed-sensing" has emerged to cope with this data explosion. Compressed sensing exploits the fact that in many applications, although the signal of interest has a large ambient dimension, the relevant information resides in a significantly lower dimensional space. For example, the Magnetic-Resonance-Imaging (MRI) data is sparse in the wavelet-domain. In this thesis, we consider the problem of computing a sparse Discrete-Fourier-Transform of a high-dimensional signal from its time-domain samples, as a representative example of compressed-sensing problems. We use this problem to investigate the tradeoff between the number of measurements, noise robustness, and the computational complexity of the recovery algorithm in compressed sensing problems. We propose a new family of deterministic sparse sensing matrices, obtained by blending together diverse ideas from sparse graph codes, digital signal processing, and number-theoretic concepts like the Chinese-remainder-theorem (CRT). The specific sparse structure of the proposed family of measurement matrices further enables a Peeling-based Ultra-Low complexity algorithms for Sparse signal Estimation, that are accordingly dubbed PULSE algorithms. Further, using the CRT, we establish an intimate connection between the problem of computing a sparse DFT of a signal and decoding over an appropriately designed sparse graph code. This connection is then exploited 1) to design a sample efficient measurement matrix and a low-complexity peeling-style iterative recovery algorithm, and 2) to perform a rigorous analysis of the recovery algorithm by wielding powerful and well-established analytical tools like density-evolution, martingales, and expander graphs from the coding theory literature. In particular, we show that under some mild conditions a k-sparse n-length DFT of a signal can be computed using (nearly optimal) 4k measurements and O(k log k) computations. As a concrete example, when k=300, and n = 3.8 x 10^6, our algorithm achieves computational savings by a factor of more than 6000, and savings in the number of input samples by a factor of more than 3900 over the standard Fast-Fourier-Transform (FFT) algorithms. This can be a significant advantage in many existing applications and can enable new classes of applications that were not thought to be practical so far. Next, we extend these results to the case of noise-corrupted samples, computing sparse 2D-DFTs as well as to interpolation of multi-variate sparse polynomials over the complex field and finite fields. We also demonstrate an application of the proposed PULSE algorithm to acquire the Magnetic-Resonance-Imaging of the Brain. This provides some empirical evidence that PULSE algorithms are applicable for acquiring morerealistic signals. The proposed sensing framework and the recovery algorithm are sample efficient and robust against observation noise. We believe that this framework provides a possible direction towards designing efficient and low-power engineering solutions for sparse signal acquisition.

Digital signal processing and classification algorithms play a crucial role in modern day biomedical monitoring systems. Fortunately, emerging sensors and stimulators as well as specialized networking technologies have enabled biomedical devices to advance to new frontiers. Deep-brain stimulators, for instance, offer unprecedented modalities for delivering therapy to patients affected by neurological conditions, ranging from Parkinson’s disease to epilepsy; out-patient monitoring networks raise the possibility of comprehensive yet cost-scalable healthcare delivery over large populations with increasingly diverse disease states. The central need, as these systems advance towards intelligent, closed-loop operation, is the ability to detect specific physiological states of interest from signals that are available through sensors. A key challenge in closed-loop biomedical systems is the ability to detect complex physiological states from the patient data within a constrained power budget. Signal processing and data-driven machine learning techniques are major enablers for modeling and detection of such states. However, the computational power scales with the complexity of models required. This thesis considers the VLSI implementation of basic signal processing techniques such as fast Fourier transform (FFT), power spectral density (PSD) computation. Reconfigurable architectures for classification algorithms including support vector machines (SVM) and Adaboost are also presented. The proposed architectures improve performance and reduce area/power consumption. First, we present a novel methodology to design parallel pipelined FFT architectures using folding transformation and register minimization techniques. Novel parallelpipelined architectures for the computation of complex valued fast Fourier transform are derived. The proposed architectures overcome prior bottlenecks and achieve full hardware utilization. The operating frequency of the proposed architecture can be decreased which in turn reduces the power consumption. This significantly reduces power at same speed or increases speed at same power consumption level. The power consumption can be reduced up to 37% in 2-parallel architectures. Further, we propose a novel approach to develop pipelined fast Fourier transform (FFT) architectures for realvalued signals. Novel 2-parallel and 4-parallel architectures are presented for radix-23 and radix-24 algorithms. The proposed radix-23 and radix-24 architectures lead to low hardware complexity compared to a prior RFFT architecture. We propose an efficient architecture for memory-based in-place FFT/IFFT computation. A conflict-free memory addressing scheme is proposed to ensure the continuous operation of the FFT processor. The proposed architecture requires fewer computation cycles along with the low hardware cost compared to prior work. We then present a low-complexity algorithm and architecture to compute power spectral density (PSD) using the Welch method. The complexity reduction comes at the cost of slight performance loss in accuracy due to the approximation used for the implementation of the fractional delay filter. The performance loss is 6-8% using fractional delay filter with 2-3 multipliers. A novel architecture is presented based on the proposed algorithm which consumes 33% less energy compared to the original method. We propose a low-energy reconfigurable architecture for support vector machines (SVMs) based on approximate computing by exploiting the inherent error resilience in the computation. We present two design optimizations, fixed-width multiply-add and non-uniform look-up table (LUT) for exponent function to minimize power consumption and hardware complexity while retaining the classification performance. The proposed design consumes 31% less energy on average compared to a conventional design. Finally, we present a novel low-complexity patient-specific algorithm for seizure prediction using spectral power features. The proposed algorithm achieves a sensitivity of 94.375% for a total of 71 seizure events with a low false alarm rate of 0.13 per hour and 6.5% of time spent in false alarms using an average of 5 features for the Freiburg database. The low computational complexity of the proposed algorithm makes it suitable for an implantable device.

The purpose of this study was to show how mathematics can be used to analyze effects on sound. Our hope is that this may inspire student interest in mathematics.We analyzed five common industry standard effects. Research data was gathered using Mathematica and GarageBand software. Three versions of each effect were used to alter pure tone sound waves of ten different frequencies using GarageBand. Then using Mathematica's Fourier command, the frequency spectrum of each altered sound wave was generated. Through observation of each set of 30 frequency spectra, the most prominent and common pure tone components were determined. For each effect, Mathematica's Fit command was used to determine a best fit model of the magnitude of each component as a function of frequency. Our models provide descriptions of the effects that are consistent with the traditional descriptions of the industry standard effects in our study. If similar research is to be conducted, our recommendation is that more versions of each effect, a wider range of input frequencies, and a higher sampling rate would produce function models that are even more consistent with traditionally accepted effect descriptions. Furthermore, an understanding of the hardware and software design used to build effects on sound is highly recommended.

In this thesis, we demonstrate, through simulation and experiment, new digital signal processing techniques that increase the throughput, and thus the sensitivity, of Cavity Ringdown Spectroscopy (CRDS). Almost all of the experiments in the literature are designed to generate single decay transients, and achieve sensitivity improvements by increasing the reflectance of the cavity mirrors. We have designed and built a laser-locked cavity ringdown spectrometer using modest reflectance mirrors where the sensitivity is achieved by increasing the data throughput and processing the ringdown data more efficiently. We report on the development of a Fourier transform based signal processing method for laser-locked Continuous Wave CRDS. Rather than analysing single ringdowns, as is the norm in traditional methods, we analyse the entire waveform output of the locked optical cavity. We have compared our method to Levenburg-Marquardt non linear least squares fitting, and have found that our method has a comparable accuracy and comparable or higher precision; moreover, the analysis time is approximately 500 times faster. We demonstrate the first laser-locked Cavity Attenuated Phase Shift measurement: we have adapted this technique, replacing the lock in amplifier and ratiometer, which were previously essential, with digital signal processing. Rather than using a traditional unstabilised cavity, we have used our laser-locked instrument to generate data with a high signal-to-noise ratio (SNR): this improvement in the SNR, along with our rapid signal processing, results in a sensitivity two orders of magnitude higher than any other CAPS experiment in the literature. We show that the sampling rate may be reduced to a point where the data processing runs faster than data is generated: this offers the prospect of real time analysis. We show that our Fourier transform based signal processing method may also be applied to the analysis of single exponential decays: we have implemented the data processing on a field programmable gate array that extracts, in real time, the ringdown time from a Pulsed CRD Spectrometer.

Whenever we have a bounded linear operator T:X → Y between two Banach spaces X, Y we can ask what nonzero elements x* ∈ X satisfy ||Tx*||=||T|| ||x*||. Such elements of X are called extremizers for the inequality ||Tx|| ≤ ||T|| ||x||. A sequence {xn}n ∈ ℕ in X satisfying ||xn||≤1 and ||Txn|| → ||T||, as n → ∞, is called an extremizing sequence. For extremizing sequences we can ask whether they are precompact after the application of symmetries of the operator T. We can also ask for the value of the operator norm of T, ||T||.The adjoint Fourier restriction operator associated to a hypersurface S with measure σ in ℝd, f ↦ (fσ)^, is known to be bounded from L2 to Lp in the case of the cone, the hyperboloid and the paraboloid in ℝd, for a certain range of exponents p ∈ [1,∞]. Existence and nonexistence of extremizers, precompactness of extremizing sequences, Euler-Lagrange equations for extremizers and best constants is what we study in the first three parts of this dissertation.In the first part we study the adjoint restriction inequality on the cone, Γ2⊆ ℝ3. We prove that an extremizing sequence for the inequality from L2(Γ2) to L6(ℝ3) is precompact up to the natural symmetries of the cone, dilations and Lorentz transformations.In the second part we study extremizers on the hyperboloid in dimensions 3 and 4. We prove that in both cases extremizers do not exist and compute the best constant in the adjoint Fourier restriction inequality.In the third part, in a joint work with Michael Christ, we consider the case of the paraboloid, or equivalently, Strichartz inequalities for the Shrödinger equation. It is shown there that a natural class of functions, the Gaussians, known to extremize the L2 → Lp adjoint Fourier restriction inequalities in dimensions 2 and 3 are no longer critical points, and thus are not extremizers, of the nonlinear functional associated to the Lq→ Lp inequalities for q ≠ 2. The case of mixed norms is also studied.In the last chapter we look at an incidence geometry problem, the problem of counting noncoplanar intersections of lines in ℝd. The problem can be seen as a discrete version of the Kakeya problem, an open problem in real analysis. There we prove a sharp upper bound for the number of transverse intersections of a collection of lines.

Exciton dynamics in semiconductor nanostructures are dominated by the effects of many-body physics. The application of coherent spectroscopic tools, such as two-dimensional Fourier transform spectroscopy (2dFTS), to the study of these systems can reveal signatures of these effects, and in combination with sophisticated theoretical modeling, can lead to more complete understanding of the behaviour of these systems. 2dFTS has previously been applied to the study of GaAs quantum well samples. In this thesis, we outline a precis of the technique before describing our own experiments using 2dFTS in a partially collinear geometry. This geometry has previously been used to study chemical systems, but we believe these experiments to be the first such performed on semiconductor samples. We extend this technique to a reflection mode 2dFTS experiment, which we believe to be the first such measurement. In order to extend the techniques of coherent spectroscopy to structured systems, we construct an experimental apparatus that permits us to control the beam geometry used to perform four-wave mixing reflection measurements. To isolate extremely weak signals from intense background fields, we extend a conventional lock-in detection scheme to one that treats the optical fields exciting the sample on an unequal footing. To the best of our knowledge, these measurements represent a novel spectroscopic tool that has not previously been described.

The efforts of my research have led to the successful construction of several instruments that have helped expand the field of microwave spectroscopy. The classic Balle-Flygare spectrometer has been modified to include two different sets of antenna to operate in the frequency ranges 6-18 GHz and 18-26 GHz, allowing it to function for a large range without having to break vacuum. This modified FTMW instrument houses two low noise amplifiers in the vacuum chamber to allow for the LNAs to be as close to the antenna as physically possible, improving sensitivity. A new innovative Balle-Flygare type spectrometer, the efficient low frequency FTMW, was conceived and built to operate at frequencies as low as 500 MHz through the use of highly curved mirrors. This is new for FTMW techniques that normally operate at 4 GHz or higher with only a few exceptions around 2 GHz. The chirped pulse FTMW spectrometer uses horn antennas to observe spectra that span 2 GHz versus the standard 1 MHz of a cavity technique. This instrument decreases the amount of time to obtain a large spectral region of relative correct intensity molecular transitions. A Nd:YAG laser ablation apparatus was attached to the classic Balle-Flygare and chirped pulse FTMW spectrometers. This allowed the study of heavy metal containing compounds. The instruments I constructed and the techniques I used have allowed the discovery of further insights into molecular chemistry. I have seen the effects of fluorinating an alkyl halide by determining the geometry of the carbon backbone of trans-1-iodoperfluoropropane and observing a ΔJ = 3 forbidden transition caused by a strong quadrupole coupling constant on this heavy molecule. The quadrupole coupling tensors of butyronitrile, a molecule observed in space, have been improved. The nuclear quadrupole coupling tensor of difluoroiodomethane was added to a list of variably fluorinated methyl halides upholding a trend for the magnitude of χzz. The study of SrS led to the determination of the Born-Oppenheimer breakdown terms and improving the precision of the SrS internuclear distance. I have also conducted the first pure rotational spectroscopic investigation on an actinide containing molecule, ThO.

Nearly all biologic tissues exhibit viscoelastic behavior. This behavior is characterized by hysteresis in the response of the material to load or strain. This information can be utilized in extrapolation of life expectancy of vascular implant materials including native tissues and synthetic materials. This behavior is exhibited in many engineering materials as well such as the polymers PTFE, polyamide, polyethylene, etc. While procedures have been developed for evaluating the engineering polymers the techniques for biologic tissues are not as mature. There are multiple reasons for this. A major one is a cultural divide between the medical and engineering communities. Biomedical engineers are beginning to fill that void. A digitally controlled drivetrain designed to evaluate both elastic and viscoelastic characteristics of biologic tissues has been developed. The initial impetus for the development of this device was to evaluate the potential for human umbilical tissue to serve as a vascular graft material. The consequence is that the load frame is configured for membrane type specimens with rectangular dimensions of no more than 25mm per side. The designed load capacity of the drivetrain is to impose an axial load of 40N on the specimen. This drivetrain is capable of assessing the viscoelastic response of the specimens by four different test modes: stress relaxation, creep, harmonic induced oscillations, and controlled strain rate tests. The fluorocarbon PTFE has mechanical properties commensurate with vascular tissue. In fact, it has been used for vascular grafts in patients who have been victims of various traumas. Hardware and software validation of the device was accomplished by testing PTFE and comparing the results to properties that have been published by both researchers and manufacturers.

Abstract: The analysis of wide-angle cellular light scattering patterns is a challenging problem. Small changes to the organization, orientation, shape, and optical properties of scatterers and scattering populations can significantly alter their complex two-dimensional scattering signatures. Because of this, it is difficult to find methods that can identify medically relevant cellular properties while remaining robust to experimental noise and sample-to-sample differences. It is an important problem. Recent work has shown that changes to the internal structure of cells---specifically, the distribution and aggregation of organelles---can indicate the progression of a number of common disorders, ranging from cancer to neurodegenerative disease, and can also predict a patient's response to treatments like chemotherapy. However, there is no direct analytical solution to the inverse wide-angle cellular light scattering problem, and available simulation and interpretation methods either rely on restrictive cell models, or are too computationally demanding for routine use. This dissertation addresses these challenges from a computational vantage point. First, it explores the theoretical limits and optical basis for wide-angle scattering pattern analysis. The result is a rapid new simulation method to generate realistic organelle scattering patterns without the need for computationally challenging or restrictive routines. Pattern analysis, image segmentation, machine learning, and iterative pattern classification methods are then used to identify novel relationships between wide-angle scattering patterns and the distribution of organelles (in this case mitochondria) within a cell. Importantly, this work shows that by parameterizing a scattering image it is possible to extract vital information about cell structure while remaining robust to changes in organelle concentration, effective size, and random placement. The result is a powerful collection of methods to simulate and interpret experimental light scattering signatures. This gives new insight into the theoretical basis for wide-angle cellular light scattering, and facilitates advances in real-time patient care, cell structure prediction, and cell morphology research.

In this thesis we study the optimal stochastic control problem of the drift of a Lévy process. We show that, for a broad class of Lévy processes, the partial integro-differential Hamilton-Jacobi-Bellman equation for the value function admits classical solutions and that control policies exist in feedback form. We then explore the class of Lévy processes that satisfy the requirements of the theorem, and find connections between the uniform integrability requirement and the notions of the score function and Fisher information from information theory. Finally we present three different numerical implementations of the control problem: a traditional dynamic programming approach, and two iterative approaches, one based on a finite difference scheme and the other on the Fourier transform.

Inverse problems are common in many areas of science and engineering. Estimating dynamics of cellular transport from concentration monitored at some distance from the cells is an example of an inverse problem in a biomedical context. Abnormalities in such transport mechanisms can lead to life threatening conditions such as Cystic Fibrosis. A quantitative understanding of the dynamics of cellular transport is thus essential to gain insight into various biological processes and their anomalies. Flux density, the release rate versus time at the cell(s) surface is the physiologically relevant information. In order to obtain biologically more relevant information from microsensor recordings, it is crucial that flux density be correctly estimated from the measured concentration data. This paradigm leads to an inverse problem whose solution requires deconvolution. Existing techniques to solve deconvolution: discrete Fourier transform (DFT) and square error optimization fail to produce realistic flux dynamics. It is in this context that a novel approach, mathematical shape optimization is developed in this research to solve diffusion related deconvolution. This method is used to estimate a) the dynamics of electrolyte secretion from a monolayer of epithelial cells and b) calcium release from mucin granules of an epithelial cell monolayer. Performance of shape error optimization is quantitatively compared with DFT and square error optimization. This work also discusses the effect of sampling rate on reconstructing flux from measured concentration data. The shape optimization approach can also be modified for spectral data analysis. One specific example is monitoring glucose concentration mediated color changes observed from an in vivo optical glucose sensor. This work describes the characteristics of an optical glucose sensor that we refer to as the sliver sensor and in vitro and in vivo functionality studies of the sensor in serum and mouse models respectively.

Long-term omni-directional ambient noise was collected at several sites in the Gulf of Mexico during 2004 and 2005. The Naval Oceanographic Office deployed bottom moored Environmental Acoustic Recording System (EARS) buoys approximately 159 nautical miles south of Panama City, Florida, in water depths of 3200 meters. The hydrophone of each buoy was 265 meters above the bottom. The data duration ranged from 10-14 months. The buoys were located near a major shipping lane, with an estimated 1.5 to 4.5 ships per day passing nearby. The data were sampled at 2500 Hz and have a bandwidth of 10-1000 Hz. Data are processed in eight 1/3-octave frequency bands, centered from 25 to 950 Hz, and monthly values of the following statistical quantities are computed from the resulting eight time series of noise spectral level: mean, median, standard deviation, skewness, kurtosis and coherence time. Four hurricanes were recorded during the summer of 2004 and they have a major impact on all of the noise statistics. Noise levels at higher frequencies (400-950 Hz) peak during extremely windy months (summer hurricanes and winter storms). Standard deviation is least in the region 100-200 Hz but increases at higher frequencies, especially during periods of high wind variability (summer hurricanes). Skewness is positive from 25-400 Hz and negative from 630-950 Hz. Skewness and kurtosis are greatest near 100 Hz. Coherence time is low in shipping bands and high in weather bands, and it peaks during hurricanes. The noise coherence is also analyzed. The 14-month time series in each 1/3- octave band is highly correlated with other 1/3-octave band time series ranging from 2 octaves below to 2 octaves above the band's center frequency. Spatial coherence between hydrophones is also analyzed for hydrophone separations of 2.29, 2.56 and 4.84 km over a 10-month period. The noise field is highly coherent out to the maximum distance studied, 4.84 km. Additionally, fluctuations of each time series are analyzed to determine time scales of greatest variability. The 14-month data show clearly that variability occurs primarily over three time scales: 7-22 hours (shipping-related), 56-282 hours (2-12 days, weather-related) and over an 8-12 month period.

Image segmentation is one of the most difficult tasks in image processing. Segmentation algorithms are generally based on searching a region where pixels share similar gray level intensity and satisfy a set of defined criteria. However, the segmented region cannot be used directly for partial image retrieval. In this dissertation, a Contour Based Image Structure (CBIS) model is introduced. In this model, images are divided into several objects defined by their bounding contours. The bounding contour structure allows individual object extraction, and partial object matching and retrieval from a standard CBIS image structure. The CBIS model allows the representation of 3D objects by their bounding contours which is suitable for parallel implementation particularly when extracting contour features and matching them for 3D images require heavy computations. This computational burden becomes worse for images with high resolution and large contour density. In this essence we designed two parallel algorithms; Contour Parallelization Algorithm (CPA) and Partial Retrieval Parallelization Algorithm (PRPA). Both algorithms have considerably improved the performance of CBIS for both contour shape matching as well as partial image retrieval. To improve the effectiveness of CBIS in segmenting images with inhomogeneous backgrounds we used the phase congruency invariant features of Fourier transform components to highlight boundaries of objects prior to extracting their contours. The contour matching process has also been improved by constructing a fuzzy contour matching system that allows unbiased matching decisions. Further improvements have been achieved through the use of a contour tailored Fourier descriptor to make translation and rotation invariance. It is proved to be suitable for general contour shape matching where translation, rotation, and scaling invariance are required. For those images which are hard to be classified by object contours such as bacterial images, we define a multi-level cosine transform to extract their texture features for image classification. The low frequency Discrete Cosine Transform coefficients and Zenike moments derived from images are trained by Support Vector Machine (SVM) to generate multiple classifiers.

With the increasing threat of biological warfare and the fear of an epidemic outbreak of influenza, smallpox, and other deadly diseases, the field of epidemic modeling is becoming increasingly important in the scientific fields. The focus of this thesis will be to create a model to study the effects of the rates of reaction and the rates of diffusion within a network based on the different parameters used in the modeling of any disease. For this model, the exact parameters of a specific disease are not as crucial as the qualitative behaviors that occur from the changing parameters. The model is linearly stable when diffusion does not exist. As diffusion is incorporated, Turing instabilities occur.

The objective of this research is to develop and implement some numerical models to analyze pavement responses under vehicle loading. Firstly, to study the pavement delamination problem, the pavement structure is modeled as an elastic finite layer system subjected to vertical and horizontal loadings over circular areas. By using the finite layer method, the maximum interface shear stress are determined; the maximum interface shear stress can be used to compare with the interface shear strength obtained through simple shear testing to determine reasonable pavement design parameters to prevent delamination failure. Secondly, the responses of a linear viscoelastic pavement system, with asphalt concrete layer of viscoelastic properties, subjected to vertical circular loadings, are analyzed by finite element method using three algorithms: (i) direct time integration; (ii) Fourier transform; (iii) Laplace transform. The inverse Fast Fourier Transform algorithm and the numerical inversion of Laplace transform method of Honig and Hirdes are used. The numerical results of the quasi-static responses by the three algorithms are presented and compared with respect to their accuracy and computational efficiency. To use the viscoelastic model in the pavement analysis, the parameters of the generalized Maxwell model based on the frequency sweep test results are determined by using the software IRIS, which is then assigned as the property of the asphalt concrete layer in a typical pavement structure subjected to a standard dual tire axle loading. Results for the distributions of stress and strain at various times are presented. In the last part of the research, a preliminary study is presented for permanent deformation of asphalt concrete. A simplified one-dimensional elasto-visco-plastic model is implemented and used to analyze the visco-plastic deformation of a cylindrical asphalt concrete sample under one-dimensional loading.