Representing Spherical Functions with Rhombic Dodecahedron Ng Lai Sze A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Philosophy in Computer Science and Engineering ©The Chinese University of Hong Kong April 2006 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. 1 IC 2 b M aTjl) ^>4IBRARY system/-；^ Abstract of thesis entitled: Representing Spherical Functions with Rhombic Dodecahedron Submitted by Ng Lai Sze for the degree of Master of Philosophy at The Chinese University of Hong Kong in April 2006 In this thesis, we present a novel method to efficiently represent datasets over the 2D spherical domain based on rhombic dodecahedron. We propose a new subdivision strategy, a family of skew great circles, for parameterization of the spherical domain. The sampling pattern produced over the sphere surface does not only have low discrepancy, low stretch efficiency and low area stan- dard deviation, but also allows fast location of sample points in real time. The data querying process of skew great circle subdivision strategy can be done through an analytical equation whereas recursive search is required for exist- ing recursive subdivision strategy. This can speed up the data sampling and retrieval process. The effectiveness of this spherical data representation has been experimented with environment maps, shadow maps and high dynamic range data. The low discrepancy sampling pattern results in good rendering quality for environment mapping application and shadow mapping. The fast data query technique ensures real time environment mapping to be practi- cal. The nice structure of the subdivision strategy allows the construction of spherical quadrilateral-based “quadtree", which is used for adaptive and deterministic sampling of static and dynamic sequence of high dynamic range environment maps. i 論文摘要 在這份論文，我們提出一個新穎的方法有效地以菱形十二面體代表在球面領域上 的資料°我們提出了一個新的細分方法——扭曲大圓系列(family of skew great circles)，爲球面領域參數化。所生產的取點樣本不僅有均句取樣、低舒展效率和 低面積標準偏差，而且能夠迅速地找到樣本點。利用扭曲大圓系列，只需通過分 析等式便可得到資料點的位置。若利用遞迴細分方法，需要遞迴尋找直到最底層 才能找到資料點的位置。我們建議的細分方法能加速資料取樣過程和資料檢索過 程。我們將所提出的球面資料表示法用於環境映射(environment mapping)、陰影貼 圖(shadow mapping)和高動態範圍資料(high dynamic range data)作爲試驗。將均句取 樣的球面資料表示法運用在環境映射方面，會有高質素的影像描給結果。利用快 速的資料詢問的技術，即時環境映射便可以做到。除此之外，細分方法的結構允 許球面四邊形四分樹的建立。此四分樹可用於靜態和動態的高動態範圍資料環境 貼圖的取樣。 ii Acknowledgement I would like to express my sincere thanks to my supervisor Professor Tien-Tsin Wong for his kindly guidance, support and help throughout this dissertation. I would like to express my special thanks and appreciation to Professor Chi- Wing Fu and Professor Chi-Sing Leung for their valuable comments, advice and help throughout the research project. I would like to thanks Laurent Thion, the author of the web site Ecliptique (http://ecliptique.com/)’ for the panorama data (the statue of liberty in Paris). I would also like to thanks Paul Debevec for the HDR environment maps. Finally, I would like to extend my thanks to all of my colleagues including Albert, Alex, Brian, Cow, Food, Gordon, Louis, Oscar, Royce, Table, Wai and Wyman for their warmest and greatest support. iii Contents Abstract i Acknowledgement iii 1 Introduction 1 2 Spherical Data Representation 4 3 Rhombic Dodecahedron 7 3.1 Introduction 7 3.2 Platonic Solids, Archimedean Solids and Dual Polyhedron . 8 3.2.1 Platonic Solids 8 3.2.2 Archimedean Solids 10 3.2.3 Dual Polyhedron 13 3.3 Rhombic Dodecahedron 16 3.3.1 Basic Property of Rhombic Dodecahedron 16 3.3.2 Construction of Rhombic Dodecahedron 16 3.3.3 Advantages of Rhombic Dodecahedron 16 3.4 Summary 19 4 Subdivision Scheme 21 4.1 Introduction 21 4.2 Motivation 22 iv 4.3 Great Circle Subdivision 22 4.3.1 Normal Space Analysis 23 4.4 Small Circle Subdivision 25 4.5 Skew Great Circle Subdivision 27 4.6 Analysis 28 4.6.1 Sampling Uniformity 29 4.6.2 Area Uniformity 32 4.6.3 Stretch Measurement 35 4.6.4 Query Efficiency 39 4.7 Summary 40 5 Data Querying and Indexing 49 5.1 Introduction ^ 5.2 Location of base polygon 43 5.2.1 General Method 43 5.2.2 Tailored Table Look Up Method 45 5.3 Location of the subdivided area ^^ 5.3.1 On Deriving the Indexing Equation 50 54 5.4 Summary 56 6 Environment Mapping 56 6.1 Introduction 6.2 Related Work 57 6.3 Methodology ^^ 6.4 Data Preparation ^^ r* rv 6.4.1 Re-sampling of Data on Sphere 6.4.2 Preparation of Texture 65 6.5 Reflection and Refraction by environment mapping 68 6.5.1 Location and Retrieval of Data 68 V 6.5.2 Cg Implementation 70 6.6 Experiments 76 6.6.1 Experiment Setup 76 6.6.2 Experiment Result and Analysis 78 6.7 Summary 89 7 Shadow Mapping 92 7.1 Introduction 92 7.2 Related Work 93 7.3 Methodology 95 7.4 Data Preparation 97 7.5 Shadow Determination and Scene Illumination 98 7.6 Experiments 100 7.6.1 Experiment Setup 100 7.6.2 Experiment Result and Analysis 101 7.7 Summary 107 8 Dynamic HDR Environment Sequences Sampling 110 8.1 Introduction 110 8.2 Related Work on HDR Distant Environment Map Sampling • 112 8.3 Static Sampling by Spherical Quad-Quad Tree 114 8.3.1 Importance Metric 117 8.4 Dynamic Sampling by Spherical Quad-Quad Tree 121 8.5 Experiments 125 8.5.1 Static Sampling 125 8.5.2 Dynamic Sampling 126 8.6 Summary 132 9 Conclusion 133 vi Bibliography 135 vii List of Figures 3.1 Platonic solids model 9 3.2 Archimedean solids model 11 3.3 Dual polyhedron construction 14 3.4 Construction of rhombic dodecahedron 17 3.5 Spherical models based on recursive subdivision method . 18 4.1 Great circle subdivision scheme 23 4.2 Normal space and normal curve for great circle subdivision . 24 4.3 Approximating recursive subdivision by great circles 26 4.4 Point set associated with recursive subdisivion 26 4.5 The curve stitching pattern around the Pole 27 4.6 Discrepancy plotting for different geometry models and differ- ent subdivision schemes 30 4.7 Discrepancy plotting for some selected geometry model and sampling pattern 31 4.8 Plot of area standard deviation against number of subdivided area for different geometry models and different subdivision schemes 33 4.9 Magnified the last part of the plot of area standard deviation . 34 4.10 Projection of vector onto tangent plane 36 4.11 Stretch measurement for different models 38 viii 5.1 Icosahedron model and rhombic dodecahedron model 44 5.2 Front view and back view of rhombic dodecahedron 45 5.3 Relationship between query point and the normal curve .... 51 5.4 Sphere cutting off at small circle plane 52 6.1 Relationship between normal, reflected vector and viewing vector 60 6.2 Sampling points located at grid point and sampling points lo- cated at the middle of the subdivided quad 61 6.3 Discrepancy comparison for rhombic dodecahedron with differ- ent subdivision methods and different kinds of distribution of sample points 62 6.4 PSNR plotting against number of jitter sampling for data set lib2 and iceplaceday 64 6.5 Sampling of data set lib2 on sphere 66 6.6 Sampled lib2 in rhombic texture format 67 6.7 Frame rate plotting for data set lib2 and lib2(bilinear) .... 79 6.8 Frame rate plotting for data set Milky Way and Milky Way (bilinear) 80 6.9 Frame rate plotting for data set icepalaceday and icepalace- day (bilinear) 81 6.10 PSNR plotting for data set milky way and lib2 82 6.11 PSNR plotting for data set land shallow and land ice cloud . 83 6.12 PSNR plotting for data set icepalaceday 84 6.13 PSNR plotting for data set milky way and lib2 with bilinear interpolation 85 6.14 PSNR plotting for data set land shallow and land ice cloud with bilinear interpolation 86 6.15 PSNR plotting for data set icepalaceday with bilinear interpo- lation 87 ix 6.16 One of the control images for data set lib2 89 6.17 Magnification of two regions in a rendered result for data set lib2 90 7.1 PSNR plotting for data set 1 and data set 2 103 7.2 PSNR plotting for data set 3 and data set 4 104 7.3 PSNR plotting for data set 5 and data set 6 105 7.4 PSNR plotting for data set 7 and data set 8 106 7.5 One of the control images for data set 6 108 7.6 Magnification of two regions in the rendered result for data set 6109 8.1 Fish view look and the corresponding quad-quad tree 116 8.2 Subdivision steps of the spherical quad-quad tree based on im- portance metric 119 8.3 Possible subdivision of a rhombic face 120 8.4 Sampling pattern on environment maps 122 8.5 Merge and split operation for sampling of HDR dynamic se- quence of environment maps 124 8.6 Static HDR environment map with different sampling methods 127 8.7 Top: HDR environment Grace Cathedral with fire. Boxed re- gion is zoom in and placed below each corresponding rendering result in next figure. Bottom: An illustration of dragon under the synthetic fire flame 129 8.8 Dynamic HDR environment map sequence sampling experiment result 130 V List of Tables 3.1 Basic property of Platonic solids 9 3.2 Basic property of Archimedean solids 12 3.3 Archimedean solids and its dual polyhedron 15 3.4 Platonic solids and its dual polyhedron 15 5.1 Great circles definition and its associated normal vector calcu- lation 46 5.2 Sign of the dot product of the query point and the six great circles' normal when the query point locates within the face .