11 research outputs found

    TR-2008003: Unified Nearly Optimal Algorithms for Structured Integer Matrices and Polynomials

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    We seek the solution of banded, Toeplitz, Hankel, Vandermonde, Cauchy and other structured linear systems of equations with integer coefficients. By combining Hensel’s symbolic lifting with either divide-and-conquer algorithms or numerical iterative refinement, we unify the solution for all these structures. We yield the solution in nearly optimal randomized Boolean time, which covers both solution and its correctness verification. Our algorithms and nearly optimal time bounds are extended to the computation of the determinant of a structured integer matrix, its rank and a basis for its null space as well as to some fundamental computations with univariate polynomials that have integer coefficients. Furthermore, we allow to perform lifting modulo a properly bounded power of two t

    Maximum Likelihood Estimation of Exponentials in Unknown Colored Noise for Target Identification in Synthetic Aperture Radar Images

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    This dissertation develops techniques for estimating exponential signals in unknown colored noise. The Maximum Likelihood (ML) estimators of the exponential parameters are developed. Techniques are developed for one and two dimensional exponentials, for both the deterministic and stochastic ML model. The techniques are applied to Synthetic Aperture Radar (SAR) data whose point scatterers are modeled as damped exponentials. These estimated scatterer locations (exponentials frequencies) are potential features for model-based target recognition. The estimators developed in this dissertation may be applied with any parametrically modeled noise having a zero mean and a consistent estimator of the noise covariance matrix. ML techniques are developed for a single instance of data in colored noise which is modeled in one dimension as (1) stationary noise, (2) autoregressive (AR) noise and (3) autoregressive moving-average (ARMA) noise and in two dimensions as (1) stationary noise, and (2) white noise driving an exponential filter. The classical ML approach is used to solve for parameters which can be decoupled from the estimation problem. The remaining nonlinear optimization to find the exponential frequencies is then solved by extending white noise ML techniques to colored noise. In the case of deterministic ML, the computationally efficient, one and two-dimensional Iterative Quadratic Maximum Likelihood (IQML) methods are extended to colored noise. In the case of stochastic ML, the one and two-dimensional Method of Direction Estimation (MODE) techniques are extended to colored noise. Simulations show that the techniques perform close to the Cramer-Rao bound when the model matches the observed noise

    Maximum Likelihood Estimation of Exponentials in Unknown Colored Noise for Target in Identification Synthetic Aperture Radar Images

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    This dissertation develops techniques for estimating exponential signals in unknown colored noise. The Maximum Likelihood ML estimators of the exponential parameters are developed. Techniques are developed for one and two dimensional exponentials, for both the deterministic and stochastic ML model. The techniques are applied to Synthetic Aperture Radar SAR data whose point scatterers are modeled as damped exponentials. These estimated scatterer locations exponentials frequencies are potential features for model-based target recognition. The estimators developed in this dissertation may be applied with any parametrically modeled noise having a zero mean and a consistent estimator of the noise covariance matrix. ML techniques are developed for a single instance of data in colored noise which is modeled in one dimension as 1 stationary noise, 2 autoregressive AR noise and 3 autoregressive moving-average ARMA noise and in two dimensions as 1 stationary noise, and 2 white noise driving an exponential filter. The classical ML approach is used to solve for parameters which can be decoupled from the estimation problem. The remaining nonlinear optimization to find the exponential frequencies is then solved by extending white noise ML techniques to colored noise. In the case of deterministic ML, the computationally efficient, one and two-dimensional Iterative Quadratic Maximum Likelihood IQML methods are extended to colored noise. In the case of stochastic ML, the one and two-dimensional Method of Direction Estimation MODE techniques are extended to colored noise. Simulations show that the techniques perform close to the Cramer-Rao bound when the model matches the observed noise

    Author index for volumes 101–200

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    Superfast Inference for Stationary Gaussian Processes in Particle Tracking Microrheology

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    Particle tracking of passive microscopic species has become the experimental measurement of choice in diverse applications, where either the material volumes are limited, or the materials themselves are so soft that they deform uncontrollably under the stresses and strains of traditional instruments. As such, the results of countless biological and rheological analyses hinge pivotally on extracting reliable dynamical information from large datasets of particle trajectory recordings. However, to do this in a statistically and computationally efficient manner presents a number of important challenges. Addressing some of these challenges is the focus of the present work. In Chapter 2, we present a superfast set of tools for parametric inference in single-particle tracking. Parametric likelihoods for particle trajectory measurements typically consist of stationary Gaussian time series, for which traditional fast inference algorithms scale as N-square in the number of observations. We present a superfast algorithm for parametric inference for stationary Gaussian processes and propose novel superfast algorithms for score and Hessian calculations. This effectively enables superfast inference for stationary Gaussian process via a wide array of frequentist and Bayesian methods. In Chapters 3 and 4, we use the superfast toolkit to address two outstanding problems prevalent in many particle tracking analyses. The first is that particle position measurements are generally contaminated by various forms of high-frequency errors. Failure to account for these errors leads to considerable bias in estimation results. In Chapter 3 we propose a novel strategy to filter high-frequency noise from measurements of particle positions. Our filters are shown theoretically to cover a vast range of high-frequency noise regimes and lead to an efficient computational estimator of model coefficients. Analyses of numerous experimental and simulated datasets suggest that our filtering approach performs remarkably well. The second problem we address is the considerable heterogeneity of typical biological fluids in which particle tracking experiments are conducted. In Chapter 4, we propose a simple metric by which to quantify the degree of heterogeneity of a fluid, along with a computationally efficient estimator and statistical test against the hypothesis that the fluid is homogeneous. The thesis is concluded by outlining several directions for future research

    Parameter estimation of models with many damped complex exponentials

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    Parameter estimation techniques for data modelled as a sum of damped complex exponentials are proving to be a successful alternative to Fourier transform methods for spectral estimation

    Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

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    This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization

    Model-based Analysis and Processing of Speech and Audio Signals

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