Ng Lai Sze. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 135-140). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Spherical Data Representation --- p.4 / Chapter 3 --- Rhombic Dodecahedron --- p.7 / Chapter 3.1 --- Introduction --- p.7 / Chapter 3.2 --- "Platonic Solids, Archimedean Solids and Dual Polyhedron" --- p.8 / Chapter 3.2.1 --- Platonic Solids --- p.8 / Chapter 3.2.2 --- Archimedean Solids --- p.10 / Chapter 3.2.3 --- Dual Polyhedron --- p.13 / Chapter 3.3 --- Rhombic Dodecahedron --- p.16 / Chapter 3.3.1 --- Basic Property of Rhombic Dodecahedron --- p.16 / Chapter 3.3.2 --- Construction of Rhombic Dodecahedron --- p.16 / Chapter 3.3.3 --- Advantages of Rhombic Dodecahedron --- p.16 / Chapter 3.4 --- Summary --- p.19 / Chapter 4 --- Subdivision Scheme --- p.21 / Chapter 4.1 --- Introduction --- p.21 / Chapter 4.2 --- Motivation --- p.22 / Chapter 4.3 --- Great Circle Subdivision --- p.22 / Chapter 4.3.1 --- Normal Space Analysis --- p.23 / Chapter 4.4 --- Small Circle Subdivision --- p.25 / Chapter 4.5 --- Skew Great Circle Subdivision --- p.27 / Chapter 4.6 --- Analysis --- p.28 / Chapter 4.6.1 --- Sampling Uniformity --- p.29 / Chapter 4.6.2 --- Area Uniformity --- p.32 / Chapter 4.6.3 --- Stretch Measurement --- p.35 / Chapter 4.6.4 --- Query Efficiency --- p.39 / Chapter 4.7 --- Summary --- p.40 / Chapter 5 --- Data Querying and Indexing --- p.42 / Chapter 5.1 --- Introduction --- p.42 / Chapter 5.2 --- Location of base polygon --- p.43 / Chapter 5.2.1 --- General Method --- p.43 / Chapter 5.2.2 --- Tailored Table Look Up Method --- p.45 / Chapter 5.3 --- Location of the subdivided area --- p.49 / Chapter 5.3.1 --- On Deriving the Indexing Equation --- p.50 / Chapter 5.4 --- Summary --- p.54 / Chapter 6 --- Environment Mapping --- p.56 / Chapter 6.1 --- Introduction --- p.56 / Chapter 6.2 --- Related Work --- p.57 / Chapter 6.3 --- Methodology --- p.58 / Chapter 6.4 --- Data Preparation --- p.59 / Chapter 6.4.1 --- Re-sampling of Data on Sphere --- p.60 / Chapter 6.4.2 --- Preparation of Texture --- p.65 / Chapter 6.5 --- Reflection and Refraction by environment mapping --- p.68 / Chapter 6.5.1 --- Location and Retrieval of Data --- p.68 / Chapter 6.5.2 --- Cg Implementation --- p.70 / Chapter 6.6 --- Experiments --- p.76 / Chapter 6.6.1 --- Experiment Setup --- p.76 / Chapter 6.6.2 --- Experiment Result and Analysis --- p.78 / Chapter 6.7 --- Summary --- p.89 / Chapter 7 --- Shadow Mapping --- p.92 / Chapter 7.1 --- Introduction --- p.92 / Chapter 7.2 --- Related Work --- p.93 / Chapter 7.3 --- Methodology --- p.95 / Chapter 7.4 --- Data Preparation --- p.97 / Chapter 7.5 --- Shadow Determination and Scene Illumination --- p.98 / Chapter 7.6 --- Experiments --- p.100 / Chapter 7.6.1 --- Experiment Setup --- p.100 / Chapter 7.6.2 --- Experiment Result and Analysis --- p.101 / Chapter 7.7 --- Summary --- p.107 / Chapter 8 --- Dynamic HDR Environment Sequences Sampling --- p.110 / Chapter 8.1 --- Introduction --- p.110 / Chapter 8.2 --- Related Work on HDR Distant Environment Map Sampling --- p.112 / Chapter 8.3 --- Static Sampling by Spherical Quad-Quad Tree --- p.114 / Chapter 8.3.1 --- Importance Metric --- p.117 / Chapter 8.4 --- Dynamic Sampling by Spherical Quad-Quad Tree --- p.121 / Chapter 8.5 --- Experiments --- p.125 / Chapter 8.5.1 --- Static Sampling --- p.125 / Chapter 8.5.2 --- Dynamic Sampling --- p.126 / Chapter 8.6 --- Summary --- p.132 / Chapter 9 --- Conclusion --- p.133 / Bibliography --- p.135

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Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2008. / Includes bibliographical references (leaves 35-36). Also available in electronic version.

This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on Rn, as described in the following two paragraphs respectively. Let Hn be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K, Hn) is a Gelfand pair. Also assume that the K-action on Cn is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair (K, Hn). For the special case K = U(n), this was proved by Geller, giving a formula for the Weyl transform of a function f of the type f = Pg, where g is a radial function, and P a bigraded solid U(n)-harmonic polynomial. Using our general Hecke-Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on Hn that are invariant under the action of K and the left action of Hn . We consider convolution equations of the type f * T = g, where f, g ε Lp(Rn) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T , we show that f is compactly supported, provided g is.