This thesis applies the methods of previous work in aerial magnetic field mapping and use in state estimation to the Virginia Tech Swing Space motion capture indoor facility. State estimation with magnetic field data acquired from a quadrotor is comparatively performed with Gaussian process regression, a multiplicative extended Kalman filter, and a particle filter to estimate the position and attitude of an uncrewed aircraft system (UAS) at any point in the motion capture testing environment. Motion capture truth data is used in the analysis. The first experimental method utilized in this thesis is Gaussian process regression. This machine learning tool allows us to create three-dimensional magnetic field maps of the indoor test space by collecting magnetic field vector data with a small UAS. Here, the maps illustrate the 3D magnetic field strengths and directions in the Virginia Tech Swing Space motion capture lab. Also, the magnetic field spatial variation of the test space is analyzed, yielding higher magnetic field gradient at lower heights above the ground. Next, the multiplicative extended Kalman filter is used with our Gaussian process regression magnetic field maps to estimate the attitude of the quadrotor. The results indicate an increase in attitude estimation accuracy when magnetic field mapping is utilized compared to when it is not. Here, results show that the addition of aerial magnetic field mapping leads to enhanced attitude estimation. Finally, the particle filter is utilized with support from our magnetic field maps to estimate the position of a small quadrotor UAS. The magnetic field maps allow us to obtain UAS position vectors by tracking UAS movement through magnetic field data. The particle filter gives three-dimensional position estimates to within 0.2 meters for five out of our eight test flights. The root mean square error is within 0.1 meters for each test flight. The effects of magnetic field spatial variation are also analyzed. The accuracy of position estimation is higher for two out the four flights in the maximum magnetic gradient area, while the accuracy is similar in both minimum and maximum gradient regions for the remaining two flights. There is evidence to support an increase in accuracy for high magnetic variation areas, but further work is needed to confirm utility for practical applications. / Master of Science / This thesis investigates airborne magnetic field mapping for the Virginia Tech Swing Space motion capture indoor facility. Position and attitude estimation with magnetic field data acquired from a small uncrewed aircraft system (UAS) is comparatively performed with multiple estimation methods. Motion capture truth data is used in analyses. The first data processing method is called Gaussian process regression. This machine learning tool allows us to create magnetic field maps of the indoor test space by averaging or regressing field estimates over collected UAS data. The maps illustrate the magnetic field strengths and directions over a three dimensional volume in the Virginia Tech Swing Space motion capture lab. Next, a multiplicative extended Kalman filter is used with our Gaussian process regression magnetic field maps to estimate UAS attitude. Results show improvement in attitude estimation accuracy when magnetic field mapping is utilized compared to when it is not. Finally, a particle filter method is utilized with our magnetic field maps to estimate UAS position. The particle filter estimates three-dimensional UAS position estimates to within 0.2 meters for five out of our eight test flights. The effects of magnetic field spatial variation are also analyzed, indicating the need for future work before magnetic field based position estimation can be practically applied.

Since its development, the minimax framework has been one of the corner stones of theoretical statistics, and has contributed to the popularity of many well-known estimators, such as the regularized M-estimators for high-dimensional problems. In this thesis, we will first show through the example of sparse Gaussian sequence model, that the theoretical results under the classical minimax framework are insufficient for explaining empirical observations. In particular, both hard and soft thresholding estimators are (asymptotically) minimax, however, in practice they often exhibit sub-optimal performances at various signal-to-noise ratio (SNR) levels. To alleviate the descrepancy, we first demonstrate that this issue can be resolved if the signal-to-noise ratio is taken into account in the construction of the parameter space. We call the resulting minimax framework the signal-to-noise ratio aware minimaxity. Then, we showcase how one can use higher-order asymptotics to obtain accurate approximations of the SNR-aware minimax risk and discover minimax estimators. Theoretical findings obtained from this refined minimax framework provide new insights and practical guidance for the estimation of sparse signals. In a broader context, we investigated the same problem for sparse linear regression. We assume the random design and allow the feature matrix to be high dimensional as 𝑿 ∈ R^{𝑛 x 𝑝} and 𝑝 ⪢ 𝑛 . This adds an extra layer of challenge to the estimation of coefficients. Previous studies have largely relied on results expressed in rate-minimaxity, where estimators are compared based on minimax risk with order-wise accuracy, without specifying the precise constant in the approximation. This lack of precision contributes to the notable gap between theoretical conclusions of the asymptotic minimax estimators and empirical findings of the sub-optimality. This thesis addresses this gap by initially refining the classical minimax result, providing a characterization of the constant in the first-order approximation. Subsequently, by following the framework of SNR-aware minimaxity we introduced before, we derived improved approximations of minimax risks under different SNR levels. Notably, these refined results demonstrated better alignment with empirical findings compared to classical minimax outcomes. As showcased in the thesis, our enhanced SNR-aware minimax framework not only offers a more accurate depiction of sparse estimation but also unveils the crucial role of SNR in the problem. This insight emerges as a pivotal factor in assessing the optimality of estimators.

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A new mathematical model for the fundamental mode of a propagating Gaussian beam is presented. The model is two-fold, consisting of a mathematical expression and a corresponding geometrical representation which interprets the expression in the light of geometrical optics. The mathematical description arises from the (0,0) order of a new family of exact, closed-form solutions to the scalar Helmholtz equation. The family consists of nonseparable functions in the oblate spheroidal coordinate system and can easily be transformed to a different set of solutions in the prolate spheroidal coordinate system, where the (0,0) order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on axis. Further, each higher-order member of the family of solutions possesses an amplitude consisting of a finite number of higher-order terms with a zero-order term that is Gaussian. The geometrical interpretation employs the skew-line generator of a hyperboloid of one sheet as a ray-like element on a contour of constant amplitude in the Gaussian beam. The geometrical characteristics of the skew line and the consequences of treating it as a ray are explored in depth. The skew line is ultimately used to build a nonorthogonal coordinate system which allows straight-line propagation of a Gaussian beam in three-dimensional space. Highlights of the research into other methods used to model a propagating Gaussian beam--such as complex rays, complex point sources and complex argument functions--are reviewed and compared with this work.

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