Browsing by Subject "Spectral analysis"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Determination of Optimum Time for the Application of Surface Treatments to Asphalt Concrete Pavements - Phase II
    (Minnesota Department of Transportation, 2008-06) Marasteanu, Mihai; Velasquez, Raul; Herb, William; Tweet, John; Turos, Mugur; Watson, Mark; Stefan, Heinz G.
    Significant resources can be saved if reactive type of maintenance activities are replaced by proactive activities that could significantly extend the pavements service lives. Due to the complexity and the multitude of factors affecting the pavement deterioration process, the current guidelines for applying various maintenance treatments are based on empirical observations of the pavement surface condition with time. This report presents the results of a comprehensive research effort to identify the optimum timing of surface treatment applications by providing a better understanding of the fundamental mechanisms that control the deterioration process of asphalt pavements. Both traditional and nontraditional pavement material characterization methods were carried out. The nontraditional methods consisted of X-Ray Photoelectron Spectroscopy (XPS) for quantifying aging, while for microcracks detection, electron microprobe imaging test (SEM) and fluorescent dyes for inspection of cracking were investigated. A new promising area, the spectral analysis of asphalt pavements to determine aging, was also presented. Traditional methods, such as Bending Beam Rheometer (BBR), Direct Tension (DTT), Dynamic Shear Rheometer (DSR) and Fourier Transform Infrared Spectroscopy (FTIR) for asphalt binders and BBR and Semi-Circular Bending (SCB) for mixtures were used to determine the properties of the field samples studied in this effort. In addition, a substantial analysis of measured pavement temperature data from MnROAD and simulations of pavement temperature using a one-dimensional finite difference heat transfer model were performed.
  • Thumbnail Image
    Item
    Geometric methods for spectral analysis.
    (2011-11) Jiang, Xianhua
    The 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.
  • Thumbnail Image
    Item
    Matrix-valued optimal mass transportation and its applications
    (2013-11) Ning, Lipeng
    The subject of this thesis is the geometry of matrix-valued density functions. The main motivation is the need for quantitative measures to compare power spectral densities of multivariate time-series. Distance measures between statistical objects provide fundamental tools for estimation, tracking and classification. In particular, for power spectra, such notions of distance are especially relevant in modeling slowly varying time-series. To this end, power spectra estimated from short observation records are considered as data points on a statistical manifold and can be connected by a regression geodesic induced by a suitable distance measure. Moreover, metrics to compare power spectra are key in quantifying resolution in spectral analysis and in various problems in statistical estimation and smoothing.We study classical notions of distance, such as Fisher information metric, Kullback-Leibler and Itakura-Saito distance, and their multivariable generalizations. We explore the Riemannian geometric structure and derive geodesics on the corresponding statistical manifolds, we draw connections with analogous notions of distance in Quantum mechanics to compare density matrices.We introduce two formulations of matrix-valued Monge-Kantorovich optimal mass transport (OMT) problem. In the first formulation, we use a notion of non-positive transportation plan and we show that the induced Wasserstein metric is weakly continuous. The second formulation leads to a rotation-aware distance measure between the end-point power spectra that takes into account the transference of power over frequencies as well as the rotation of the principle directions. In this, we show that the optimal transportation plan is no longer supported on a monotonically increasing thin set. Applications to spectral tracking and spectral morphing highlight the relevance of the proposed distance.