Edge effects and efficient parameter estimation for stationary random fields

We consider the estimation of the parameters of a stationary random field on d-dimensional lattice by minimizing the classical Whittle approximation to the Gaussian log likelihood. If the usual biased sample covariances are used, the estimate is efficient only in one dimension. To remove this edge effect, we introduce data tapers and show that the resulting modified estimate is efficient also in two and three dimensions. This avoids the use of the unbiased sample covariances which are in general not positive-definit

Relative entropy and the multi-variable multi-dimensional moment problem

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy $trace (\rho\log \rho)$ and a generalization of the Kullback-Leibler distance $trace (\rho \log \rho - \rho \log \sigma)$, refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore these as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.Comment: 24 pages, 3 figure

Digital Signal Processing

Contains an introduction and reports on seventeen research projects.U.S. Navy - Office of Naval Research (Contract N00014-77-C-0266)Amoco Foundation FellowshipU.S. Navy - Office of Naval Research (Contract N00014-81-K-0742)National Science Foundation (Grant ECS80-07102)U.S. Army Research Office (Contract DAAG29-81-K-0073)Hughes Aircraft Company FellowshipAmerican Edwards Labs. GrantWhitaker Health Sciences FundPfeiffer Foundation GrantSchlumberger-Doll Research Center FellowshipGovernment of Pakistan ScholarshipU.S. Navy - Office of Naval Research (Contract N00014-77-C-0196)National Science Foundation (Grant ECS79-15226)Hertz Foundation Fellowshi

Multidimensional spectral estimation using iterative methods

This thesis treats the topic of multi-dimensional autoregressive (AR) spectral estimation. An iterative algorithm for the solution of toeplitz block- toeplitz matrix equations is presented. This leads to a fast solution of the two dimensional normal equation compared with direct inversion of the autocorrelation matrix. The covariance method is used to estimate the autocorrelation function. Because the resulting matrix is not toeplitz block- toeplitz, a modified iterative algorithm is presented. Quarter-plane and nonsymmetric half-plane support are used, as well as combined quadrant averaging. Results of computer simulation show that in some cases a single iteration is sufficient to produce an acceptable spectral estimate. Because the AR parameters are estimated from previous values, this suggests the possibility to estimate spectral densities of slowly varying random processes.http://archive.org/details/multidimensional1094530666Lieutenant, United States NavyApproved for public release; distribution is unlimited

Constrained least-squares digital image restoration

The design of a digital image restoration filter must address four concerns: the completeness of the underlying imaging system model, the validity of the restoration metric used to derive the filter, the computational efficiency of the algorithm for computing the filter values and the ability to apply the filter in the spatial domain. Consistent with these four concerns, this dissertation presents a constrained least-squares (CLS) restoration filter for digital image restoration. The CLS restoration filter is based on a comprehensive, continuous-input/discrete- processing/continuous-output (c/d/c) imaging system model that accounts for acquisition blur, spatial sampling, additive noise and imperfect image reconstruction. The c/d/c model-based CLS restoration filter can be applied rigorously and is easier to compute than the corresponding c/d/c model-based Wiener restoration filter. The CLS restoration filter can be efficiently implemented in the spatial domain as a small convolution kernel. Simulated restorations are used to illustrate the CLS filter\u27s performance for a range of imaging conditions. Restoration studies based, in part, on an actual Forward Looking Infrared (FLIR) imaging system, show that the CLS restoration filter can be used for effective range reduction. The CLS restoration filter is also successfully tested on blurred and noisy radiometric images of the earth\u27s outgoing radiation field from a satellite-borne scanning radiometer used by the National Aeronautics and Space Administration (NASA) for atmospheric research

One and two dimensional maximum entropy spectral estimation

Originally published as thesis (Dept. of Electrical Engineering and Computer Science, Sc.D., 1981).Bibliography: p. 115-117.Naveed Akhtar Malik

Linear Predictive Spectral Analysis via the Lp Norm

This study involves linear predictive spectral analysis under the general LP norm; both one dimensional and two dimensional spectral estimation algorithms are developed. The objective in this study is determination of frequency resolution capability for various LP normed solutions to linear predictive spectral estimation equations. A modified residual steepest descent algorithm is utilized to generate the required solution. The research presented in this thesis could not have been accomplished without the support of the Oklahoma State University Research Consortium For Well Log Data Enhancement Via Signal Processing. The member companies of this consortium include Amococ Production Company, Area Oil and Gas Company, Cities Service Oil and Gas Corporation, Conoco, Exxon, IBM, Mobil Research and Development, Phillips Petroleum Corporation, Sohio Petroleum Company, and Texaco.Electrical Engineerin

Multiscale Methods in Image Modelling and Image Processing

The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications. 'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools. This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated