Thesis (Ph. D.)--University of Chicago, 1918. / Vita. "Private edition, distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from the American journal of mathematics, vol. XLIII, no. 1, January, 1921." Includes bibliographical references. Also available on the Internet.

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published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Mechanical Engineering / Master / Master of Philosophy

The research discussed in this thesis is concerned with surface understanding from the viewpoint of recognition-oriented, scale-related processing based on surface curvatures and diffusion smoothing. Four problems below high level visual processing are investigated: 1) 3-dimensional data smoothing using a diffusion process; 2) Behaviour of shape features across multiple scales, 3) Surface segmentation over multiple scales; and 4) Symbolic description of surface features at multiple scales. In this thesis, the noisy data smoothing problem is treated mathematically as a boundary value problem of the diffusion equation instead of the well-known Gaussian convolution, In such a way, it provides a theoretical basis to uniformly interpret the interrelationships amongst diffusion smoothing, Gaussian smoothing, repeated averaging and spline smoothing. It also leads to solving the problem with a numerical scheme of unconditional stability, which efficiently reduces the computational complexity and preserves the signs of curvatures along the surface boundaries. Surface shapes are classified into eight types using the combinations of the signs of the Gaussian curvature K and mean curvature H, both of which change at different scale levels. Behaviour of surface shape features over multiple scale levels is discussed in terms of the stability of large shape features, the creation, remaining and fading of small shape features, the interaction between large and small features and the structure of behaviour of the nested shape features in the KH sign image. It provides a guidance for tracking the movement of shape features from fine to large scales and for setting up a surface shape description accordingly. A smoothed surface is partitioned into a set of regions based on curvature sign homogeneity. Surface segmentation is posed as a problem of approximating a surface up to the degree of Gaussian and mean curvature signs using the depth data alone How to obtain feasible solutions of this under-determined problem is discussed, which includes the surface curvature sign preservation, the reason that a sculptured surface can be segmented with the KH sign image alone and the selection of basis functions of surface fitting for obtaining the KH sign image or for region growing. A symbolic description of the segmented surface is set up at each scale level. It is composed of a dual graph and a geometrical property list for the segmented surface. The graph describes the adjacency and connectivity among different patches as the topological-invariant properties that allow some object's flexibility, whilst the geometrical property list is added to the graph as constraints that reduce uncertainty. With this organisation, a tower-like surface representation is obtained by tracking the movement of significant features of the segmented surface through different scale levels, from which a stable description can be extracted for inexact matching during object recognition.

by Chan Chun-hing. / Bibliography: leaves 66-68 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

by Ng Kwok Choi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1980. / Bibliography: leaves 39-43.

Yu Chengjie. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 65-67). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- A multiplicity estimate and applications --- p.5 / Chapter 2.1 --- A multiplicity estimate --- p.6 / Chapter 2.2 --- Sharp bounds for the dimensions of the spaces of holomorphic functions of polynomial growth --- p.14 / Chapter 2.3 --- Siegel's theorem on the fields of rational functions --- p.15 / Chapter 3 --- Quasi-embedding of complete Kahler manifolds --- p.21 / Chapter 3.1 --- The original map F0 --- p.21 / Chapter 3.2 --- Almost injectivity of F0 --- p.26 / Chapter 3.3 --- Almost surjectivity of F0 --- p.28 / Chapter 3.4 --- Weaker conditions for almost surjectivity --- p.41 / Chapter 3.5 --- Existence of quasi-embedding --- p.48 / Chapter 4 --- Desingularization of quasi-embeddings --- p.51 / Chapter 4.1 --- Normalization of a map with polynomial growth --- p.51 / Chapter 4.2 --- The method to desingularize a quasi-embedding --- p.54 / Chapter 4.3 --- The case of dimension two --- p.55 / Chapter 4.4 --- A uniformization theorem --- p.63 / Bibliography --- p.65