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acase@tulane.edu / 1 / Kui Zhang

In computer vision, object recognition involves segmentation of the image into separate components. One way to do this is to detect the edges of the components. Several algorithms for edge detection exist and one of the most sophisticated is the Canny edge detector.Canny [2] designed an optimal edge detector for images which are corrupted with noise. He suggested that a Gaussian filter be applied to the image and edges be sought in the smoothed image. The directional derivative of the Gaussian is obtained, then convolved with the image. The direction, n, involved is normal to the edge direction. Edges are assumed to exist where the result is a local extreme, i.e., where∂2 (g * f) = 0.(0.1)_____∂n2In the above, g(x, y) is the Gaussian, f (x, y) is the image function and The direction of n is an estimate of the direction of the gradient of the true edge. In this thesis, we discuss the computational algorithm of the Canny edge detector and its implementation. Our experimental results show that the Canny edge detection scheme is robust enough to perform well over a wide range of signal-to-noise ratios. In most cases the Canny edge detector performs much better than the other edge detectors. / Department of Mathematical Sciences

This thesis deals with two topics in asymptotic statistics. A concept of asymptotic optimality for sequential tests of statistical hypotheses is introduced. Sequential Probability Ratio Tests are shown to have asymptotic optimality properties corresponding to their usual optimality properties. Secondly, the asymptotic power of Pearson's chi-square test for goodness of fit is derived in a new way. The main tool for evaluating asymptotic performance of tests is the likelihood ratio of two hypotheses. In situations examined here the likelihood ratio based on a sample of size ⁿ has a limiting distribution as ⁿ → ∞ and the limit is also a likelihood ratio. To calculate limiting values of various performance criteria of statistical tests the calculations can be made using the limiting likelihood ratio. / Science, Faculty of / Statistics, Department of / Graduate

The asymptotic behaviour of nonoscillatory solutions of second order nonlinear ordinary differential equations is studied. Necessary and sufficient conditions are given for the existence of positive solutions with specified asymptotic behaviour at infinity. Existence of nonoscillatory solutions is established using the Schauder-Tychonoff fixed point theorem. Techniques such as factorization of linear disconjugate operators are employed to reveal the similar nature of asymptotic solutions of nonlinear differential equations to that of linear equations. Some examples illustrating the asymptotic theory of ordinary differential equations are given. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Mixed linear models are a time honored method of analyzing correlated data. However, there is still no method of calculating exact confidence intervals or p-values for an arbitrary parameter in any mixed linear model. Instead, researchers must use either specialized approximate and exact tests that have been developed for particular models or rely on likelihood based approximate tests and confidence intervals which may be unreliable in problems with small sample sizes. This thesis develops procedures to improve small sample likelihood based inference in these important models. The first manuscript develops I.M. Skovgaard's modified directed likelihood for mixed linear models and shows how it is a general, accurate, and easy to apply method of improving inference in mixed linear models. In the second manuscript, O.E. Barndorff-Nielsen's approximate modified profile likelihood is applied to mixed linear models. This modified profile likelihood is a sensible generalization of the commonly used residual likelihood and can be applied if either a fixed or a covariance parameter is of interest. The final manuscript discusses how the design of a mixed linear model effects the accuracy of Skovgaard's modified likelihood and suggests a useful decomposition of that statistic. / Graduation date: 1998

In this master thesis, it is proposed to solve in the large thedifferential equationZ2(d2y/dz2) + z (dy/dz) (b0+b1zm) + (c0+c1zm)y = 0Here, m is an arbitrary positive integer, the variable z is complex as are the constants bi, ci (i = 0, 1). It is also assumed that the roots of the indicial equation about the regular singular point z=0 are such that their difference is incongruent to zero modulo m.

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Under consideration are two-point boundary value problems for a system of second order differential equations which contains a small parameter multiplying the highest dereivatives. We prove the existence of solutions exhibiting left and right boundary layers by constructing upper and lower solutions of the system. The behavior of the solutions as the parameter tends to zero is also established. Of special interest is the existence of a compound boundary layer (i.e., one involving two scales) at the left endpoint of the interval.