Chapter 1 Introduction Motivation Approach Significance of Research Overview of Material... 5

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1 ACKNOWLEDGEMENT First and foremost, I would like to thank my thesis advisor, Dr. A. Lynn Abbott, for his invaluable guidance and patience throughout my entire study at Virginia Tech. Beyond gaining academic knowledge, I am also indebted to him for his examples in life. I would like to thank my friends in Blacksburg, Virginia and in Boulder, Colorado for their warmhearted friendship throughout my life in the United States. First special thanks go to Somkiat Appipattanavis, Dr. Chalor Jarusukthiruk and Kenichiro Shimada for the good memories and second special thanks go to Anbu Subramanian and Erol Sarigul, who were always quick to help and provide information. I also will never forget Jay, Michele and Ellen Lester for our weekly meetings and studies. I cannot thank my father and mother enough for their love and support. Thanks also go to my brothers, sisters and sister-in-law for their encouragement. I also would like to thank my special aunt, Vichulada Leerujikul, who has been advising and guiding me through my high school and college study years. III

2 TABLE OF CONTENTS Chapter 1 Introduction Motivation Approach Significance of Research Overview of Material Chapter 2 Discrete Wavelet Transform Fourier Transform Wavelet Transform Separability Property Multiresolution Analysis Signal Decomposition Requirements for Multiresolution Choice of Wavelet Chapter 3 Background and Related Work Input and Output Data Gray-Scale Image Triangulation Delaunay and Data-Dependent Triangulations Gouraud Shading Error Metric Graph Theory IV

3 Chapter 4 Wavelet-Based Initial Triangulation Overview of the Initial Triangulation Basic and Variant Templates SML Initial Triangulation Vertex Compatibility and Variant Templates Complete Set Chapter 5 Local Operators High-Level Algorithm SML High-Level Algorithm New Approach Region Selection Candidate Selection Neighborhood Assignment Mesh Simplification Vertex Removal Operation Edge Collapse Operation Image Enhancement Edge Swap Operation Wavelet Coefficient Attraction Operation Shape-Preserving Split Operation Chapter 6 Result and Performance Analysis Memory Requirement and Time Complexity Overview Memory Requirement Time Complexity Test Image V

4 6.3 Results Mesh Regularity Comparison Level-of-Detail (LOD) Result Time Complexity Result Mesh Reduction Result Multiresolution Mesh Generation Applications Region-of-Interest (ROI) Refinement Eye Detection Segmentation Chapter 7 Conclusion and Future Work Summary and Conclusion Future Works Bibliography Appendix I Data Structure VI

5 LIST OF FIGURES Figure 1.1 Approximation using triangulation at different resolution levels Figure 1.2 Segmentation of the peppers image Figure 1.3 Example of eye detection Figure 2.1 Example of Fourier transforms of two-dimensional signals Figure 2.2 Separability property of the wavelet transform Figure 2.3 Discrete wavelet transform at different levels Figure 2.4 Filter banks used for decomposition and reconstruction Figure 2.5 Discrete wavelet signals Figure 2.6 Space and subspaces represented in multiresolution Figure 2.7 Approximations at different levels Figure 2.8 Magnitude response and its absolute magnitude response Figure 2.9 Shifting comparison between orthogonal and biorthogonal wavelets Figure 3.1 Regular sample grid Figure 3.2 Approximation of an image using regularly sampled triangular mesh Figure 3.3 Example of a Voronoi diagram and its Delaunay triangulation Figure 3.4 Concept of Delaunay triangulation Figure 3.5 Comparison of Delaunay triangulation and non-delaunay triangulation Figure 3.6 Gouraud shading Figure 3.7 Mach band phenomenon Figure 3.8 Error calculation for the Gouraud shading reconstructed image Figure 3.9 General and simple planar graphs Figure 3.10 Equivalent graph Figure 4.1 Construction of basic templates Figure 4.2 Nine fundamental forms for variant construction Figure 4.3 Wavelet coefficients for initial triangulation and its comparison Figure 4.4 Dual templates Figure 4.5 Demonstration for the directionality property for the discontinuity Figure 4.6 Directionality and slope of discontinuity Figure 4.7 Basic template assignment Figure 4.8 Vertex incompatibility VII

6 Figure 4.9 Neighbor checking for vertex compatibility Figure 4.10 Solution to vertex incompatibility Figure 4.11 Example of final initial triangulation Figure 4.12 Complete set of the templates and their generating isomorphism functions Figure 5.1 Flow diagram of SML algorithm Figure 5.2 Flow diagram of the new algorithm Figure 5.3 Neighborhood assignment Figure 5.4 Triangle subdivision Figure 5.5 Gauss mapping Figure 5.6 Biased Gauss mapping and autocorrelation representation Figure 5.7 Edge collapse example Figure 5.8 Top view of an edge collapse operation Figure 5.9 Undesirable condition for edge collapse Figure 5.10 Undesirable result of edge collapse Figure 5.11 Short edge collapse Figure 5.12 Triangulation violation caused by edge collapse Figure 5.13 Effect of edge collapse on convex polygon Figure 5.14 Effect of edge collapse on concave polygon Figure 5.15 Delaunay triangulation decision Figure 5.16 Error-based edge swap Figure 5.17 Limitation of edge swap Figure 5.18 Wavelet coefficient attraction Figure 5.19 Templates for wavelet coefficient attraction Figure 5.20 Triangular mesh violation after vertex translation Figure 5.21 Triangle split Figure 5.22 Directional split based on wavelet coefficient Figure 6.1 Test Images Figure 6.2 Figure 6.3 Figure 6.4 PSNR comparison on the Lena and peppers images at different values of mesh regularity factor, α The number of triangles needed to represent the Lena and peppers images at different values of α Processing times of the Lena and peppers images at different values of the mesh regularity factor, α VIII

7 Figure 6.5 Reconstructed image comparison on pepper image Figure 6.6 Figure 6.7 Level-of-detail (LOD) analysis on PSNR comparison at different values of ω max Level-of-detail (LOD) analysis on the number of triangles comparison at different values of ω max Figure 6.8 Elaine results from level-of-detail experiment Figure 6.9 Element multiplication from the Lena image Figure 6.10 Time comparison based on the number of triangles Figure 6.11 Comparison of the number of triangles at different vertex removal threshold, δ v Figure 6.12 Comparison of PSNR on Lena and Elaine images at different vertex removal threshold, δ v, from 0.00 to Figure 6.13 Reconstructed Image of Lena Figure 6.14 Reconstructed Image of Peppers Figure 6.15 Reconstructed Image of Mandrill Figure 6.16 Reconstructed image of Elaine Figure 6.17 Reconstructed Image of Moon Figure 6.18 Reconstructed Image of Goldhill Figure 6.19 Discrete wavelet transform results Figure 6.20 PSNR comparison on all images at different levels Figure 6.21 Number of triangles comparison on all images at different levels Figure 6.22 Histogram of the test images Figure 6.23 Masking for the Lena and pepper images Figure 6.24 Triangulation refinement based on masking shown in Figure Figure 6.25 Triangulation refinement of two different peppers Figure 6.26 Figure 6.27 Diagram of eye detection algorithm using triangular mesh representation as input Results of eye detection on the Lena and Elaine image after relational screening Figure 6.28 Comparison of the eye templates Figure 6.29 Results of the eye detection Figure 6.30 Segmentation of peppers image Figure A1 Data structure of the triangular mesh Figure A2 Hierarchical ring data structure Figure A3 Half-edge data structure IX

8 LIST OF TABLES Table 6.1 Table 6.2 Memory requirements for the vertex, loop, half-edge and face linked-lists, wavelet coefficients and input image of the algorithm Time complexity of the triangle selection, edge swap, edge collapse, vertex removal and triangle split operations used by the algorithm Table 6.3 Statistics information of the test images Table 6.4 Vertex removal comparison as a function of vertex removal threshold, δ v Table 6.5 Results obtained from Lena image from level 6 to Table 6.6 Results obtained from pepper image from level 6 to Table 6.7 Results obtained from Mandrill image from level 6 to Table 6.8 Results obtained from Elaine image from level 6 to Table 6.9 Results obtained from moon image from level 6 to Table 6.10 Results obtained from Goldhill image from level 6 to X