Browsing by Subject "Geodesics"
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Item Geometric methods for spectral analysis.(2011-11) Jiang, XianhuaThe purpose of this thesis is to study the geometry of power spectra and develop geometric methods for spectral analysis, modeling, and filtering of time series. We first study notions of distance suitable to quantify mismatch between probability distributions, power spectra, and covariance matrices. To this end, we review and relate relevant concepts from statistical inference, information theory, signal processing, and quantum mechanics. We introduce several alternative metrics for scalar as well as multivariate power spectra, and explore concepts from Riemannian geometry, in particular geodesics and means, to model slowly varying time series, and to interpolate and fuse statistics from different sources. We first study mismatch between power spectra in the context of linear prediction. We introduce two alternative metrics. The first quantifies the degradation of predictive error variance when one power spectrum is used to design an optimal filter which is then applied to a process corresponding to another power spectrum. The second metric reflects the flatness of the innovations process. For this second metric, we provide closed-form expressions for the induced geodesics and geodesic distances. The theory we develop applies to multivariate power spectra. The metrics based on the prediction paradigm lack the important property of weak continuity. To this end we explore an alternative framework based on the transportation problem. In this, energy content has a physical significance and metrics quantify transportation cost between frequencies. Throughout we explore geometric methods on the Riemannian manifold of power spectra, in a way analogous to the familiar Euclidean geometry. In particular, we use the notion of a geodesic to model the evolution of power spectral densities for slowly varying time series. More specifically, we develop the idea of geodesic path-fitting as an analogue of the least squares line-fitting in the Euclidean space. Further, we investigate means or averages of distributions and of positive semi-definite matrices arising in signal processing, computer vision and pattern recognition applications. The transportation mean, as well as the median, for normalized scalar power spectra are provided analytically. For multivariate Gaussian distributions, we show that the transportation mean requires solving a linear matrix inequality problem, which is computationally tractable. Furthermore, linear structural constraints on the means, based on prior knowledge, can be easily incorporated and solved efficiently using the linear matrix inequality formulation. We highlight the relevance of the geometric framework with several applications in signal processing, such as spectral tracking, speech morphing, filtering, and spectral averaging.Item Regge Finite Elements with Applications in Solid Mechanics and Relativity(2018-05) Li, LizaoThis thesis proposes a new family of finite elements, called generalized Regge finite elements, for discretizing symmetric matrix-valued functions and symmetric 2-tensor fields. We demonstrate its effectiveness for applications in computational geometry, mathematical physics, and solid mechanics. Generalized Regge finite elements are inspired by Tullio Regge’s pioneering work on discretizing Einstein’s theory of general relativity. We analyze why current discretization schemes based on Regge’s original ideas fail and point out new directions which combine Regge’s geometric insight with the successful framework of finite element analysis. In particular, we derive well-posed linear model problems from general relativity and propose discretizations based on generalized Regge finite elements. While the first part of the thesis generalizes Regge’s initial proposal and enlarges its scope to many other applications outside relativity, the second part of this thesis represents the initial steps towards a stable structure-preserving discretization of the Einstein’s field equation.