by Yuen Wai Ching. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 54-55). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Group Analysis --- p.9 / Chapter 2.1 --- Groups and Differential Equations --- p.9 / Chapter 2.2 --- Prolongation --- p.11 / Chapter 2.3 --- The Prolongation Formula --- p.14 / Chapter 3 --- Symmetry Group For the Eikonal Equation --- p.17 / Chapter 4 --- An Optimal System For the Eikonal Equation --- p.25 / Chapter 5 --- Group Invariant Solutions --- p.33 / Chapter 5.1 --- Straight Lines --- p.33 / Chapter 5.2 --- Stationary Solutions --- p.33 / Chapter 5.3 --- Traveling Waves --- p.34 / Chapter 5.4 --- Circles --- p.37 / Chapter 5.5 --- Spirals --- p.38 / Chapter 6 --- Appendix --- p.50 / A Group Analysis for some Geometric Evolution Equations --- p.4 / Bibliography

Typescript (Photocopy). / Digitized by Kansas Correctional Industries

Thesis (Ph. D.)--John Hopkins University, 1912. / "Reprinted from American journal of mahtematics, vol. XXXVI, no. 1." Biographical.

This thesis presents a theory of multi-scale, curvature and torsion based shape representation for planar and space curves. The theory presented has been developed to satisfy various criteria considered useful for evaluating shape representation methods in computer vision. The criteria are: invariance, uniqueness, stability, efficiency, ease of implementation and computation of shape properties. The regular representation for planar curves is referred to as the curvature scale space image and the regular representation for space curves is referred to as the torsion scale space image. Two variants of the regular representations, referred to as the renormalized and resampled curvature and torsion scale space images, have also been proposed. A number of experiments have been carried out on the representations which show that they are very stable under severe noise conditions and very useful for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation. Planar or space curves are described at varying levels of detail by convolving their parametric representations with Gaussian functions of varying standard deviations. The curvature or torsion of each such curve is then computed using mathematical equations which express curvature and torsion in terms of the convolutions of derivatives of Gaussian functions and parametric representations of the input curves. Curvature or torsion zero-crossing points of those curves are then located and combined to form one of the representations mentioned above. The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. This thesis contains a number of theorems about evolution and arc length evolution of planar and space curves along with their proofs. Some of these theorems demonstrate that evolution and arc length evolution do not change the physical interpretation of curves as object boundaries and others are in fact statements on the global properties of planar and space curves during evolution and arc length evolution and their representations. Other theoretical results shed light on the local behavior of planar and space curves just before and just after the formation of a cusp point during evolution and arc length evolution. Together these results provide a sound theoretical foundation for the representation methods proposed in this thesis. / Science, Faculty of / Computer Science, Department of / Graduate

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Includes bibliographical references. / Ph.D.

Thesis (M. Sc.)--University of Canterbury, 2009. / Typescript (photocopy). "September 2009." Includes bibliographical references (p. 115-116) and index. Also available via the World Wide Web.

Halphen's Theorem states that, "A necessary and sufficient condition for every dynamical trajectory in a positional field of force in E3 to be planar is that the field of force is either parallel or central." This result has been known for some time, however only the sufficiency part of the theorem is widely documented. A new analytic proof of the necessity part of Halphen's Theorem was developed. The details of this proof motivated the new concepts of a flat point in a field of force and a flat point on a dynamical trajectory in a positional field of force. / Ph. D.

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.

A spherical wave AVO response is investigated by measuring ultrasonic reflection amplitudes from a water/Plexiglas interface. The experimental results show substantial deviation from the plane-wave reflection coefficients at large angles. However there is an excellent agreement between experimental data and full-wave numerical simulations performed with the reflectivity algorithm. By comparing the spherical-wave AVO response, modeled with different frequencies, to the plane-wave response, I show that the differences between the two are of such magnitude that three-term AVO inversion based on AVA curvature can be erroneous. I then propose an alternative approach to use critical angle information extracted from AVA curves, and show that this leads to a significant improvement of the estimation of elastic parameters. Azimuthal variation of the AVO response of a vertically fractured model also shows good agreement with anisotropic reflectivity simulations, especially in terms of extracted critical angles which indicated that (1) reflection measurements are consistent with the transmission measurements; (2) the anisotropic numerical simulation algorithm is capable of simulating subtle azimuthal variations with excellent accuracy; (3) the methodology of picking critical angles on seismograms using the inflection point is robust, even in the presence of random and/or systematic noise.