Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration (eBook)

Preface 5
Contents 8
Maximizing Adaptivity in Hierarchical Topological Models Using Cancellation Trees 10
1 Introduction 10
2 Morse–Smale Complex 12
3 Cancellation Forest 14
4 Hierarchy 17
5 Results 21
6 Conclusions and Future Research 26
References 26
The Toporrery: Computation and Presentation of Multiresolution Topology 28
1 Introduction 28
2 Multiresolution Contour Trees 31
3 Hierarchical Morse Functions 36
4 Multiresolution Contour Trees 39
5 Output Sensitive Visualization 44
6 Conclusions 47
Isocontour Based Visualization of Time-Varying Scalar Fields 50
1 Introduction 50
2 Isocontour Extraction 52
3 Topological Structures for Supporting Visualization 57
4 Conclusions 72
References 74
DeBruijn Counting for Visualization Algorithms 78
1 Introduction 78
2 History and Literature 80
3 Permuting the Colors: The Power Group 83
4 Summary 96
References 96
Topological Methods for Visualizing Vortical Flows 98
1 Introduction 98
2 Related Work 100
3 Theoretical Aspects of (Parametric) Topology 100
4 Topology Tracking 104
5 Planar Topology Tracking for Volume Exploration 109
6 Results 111
7 Conclusion 115
References 116
Stability and Computation of Medial Axes: A State-of-the-Art Report 117
1 Introduction 117
2 Medial Axis and Skeleton 118
3 Finiteness Properties 119
4 Homotopy Equivalence 120
5 Instability and Semi-continuity 121
6 Stability Under C2-Perturbations 122
7 Exact Computation of Medial Axes 122
8 Approximation Paradigm for Medial Axes 123
9 Punctured Euclidean Spaces 124
10 Voronoi Graph and Medial Axis 124
11 Pruning in the Presence of Noise 126
12 Stability of the .-Medial Axis 128
13 What Now? 129
References 131
Local Geodesic Parametrization: An Ant’s Perspective 134
1 Introduction 134
2 The Algorithm 136
3 Implementation 139
4 Sample Application 142
5 Conclusion and Future Work 143
References 144
Tensor-Fields Visualization Using a Fabric-like Texture Applied to Arbitrary Two-dimensional Surfaces 145
1 Introduction 145
2 Related Work 146
3 Mathematical Background and Notation 147
4 MethodOverview 148
5 Metric Definition 149
6 Surface Definition and Tensor Projection 150
7 Texture Generation 151
8 Results and Conclusions 157
References 160
Flow Visualization via Partial Differential Equations 162
1 Introduction 162
2 Review of Texture Based Flow Visualization 164
3 A Brief Introduction to Scale Space Methods in Image Processing 166
4 A Flow Aligned Differential Operator 168
5 Anisotropic Diffusion for Stationary Flow 168
6 Transport and Diffusion for Non-stationary Flow Fields 174
7 Continuous Clustering via Anisotropic Phase Separation 178
8 Remarks on the Finite Element Implementation 183
9 Clustering Based on Hierarchical Decomposition of a Differential Operator 184
10 Conclusions 191
References 192
Iterative Twofold Line Integral Convolution for Texture-Based Vector Field Visualization 195
1 Introduction 195
2 Previous Work 196
3 Continuous Twofold Convolution Along Straightened Lines 197
4 Discretization and Sampling 200
5 Extended Scenarios 202
6 Discussion of Costs and Quality 204
7 Implementation and Results 205
8 Conclusion 211
References 214
Constructing 3D Elliptical Gaussians for Irregular Data 216
1 Introduction 216
2 Previous Work 218
3 Creating 3D Elliptical Gaussian 219
4 Evaluation 223
5 Rendering 224
6 Implementation and Results 224
7 Conclusion and Future Work 227
References 228
From Sphere Packing to the Theory of Optimal Lattice Sampling 229
1 Introduction 229
2 Multidimensional Sampling Theory 230
3 The Optimal Lattice Sampling 235
4 The BCC Lattice 236
5 Reconstruction Filters 238
6 Implementation 245
7 Results and Discussion 247
8 Conclusion 253
9 Future Research 254
References 256
Reducing Interpolation Artifacts by Globally Fairing Contours 258
1 Introduction 258
2 Related Work 259
3 Fairing Contours of a Scalar Field 260
4 Numerical Examples 266
5 Conclusions and Future Work 269
References 270
Time- and Space-Efficient Error Calculation for Multiresolution Direct Volume Rendering 271
1 Introduction 271
2 Previous Work 272
3 Error Estimation 274
4 Multiresolution Representation 279
5 Rendering 279
6 Results and Discussion 280
7 Conclusions and Future Work 282
References 283
Massive Data Visualization: A Survey 284
1 Introduction 284
2 Driving Problems 285
3 How Do We Explore Massive Data? 287
4 Simplification Methods 288
5 Multiresolution Methods 290
6 External Memory Methods 292
7 Visual Scalability 294
8 Conclusions 296
References 296
Compression and Occlusion Culling for Fast Isosurface Extraction from Massive Datasets 302
1 Introduction 302
2 Related Work 304
3 Mesh Refinement and Preprocessing 306
4 Data Compression 308
5 Occlusion Culling 311
6 Results 313
7 Conclusions and Future Work 318
References 319
Volume Visualization of Multiple Alignment of Large Genomic DNA 323
1 Introduction 323
2 Our Approach 326
3 Implementation 330
4 Results 332
5 Conclusions 338
References 338
Model-Based Visualization: Computing Perceptually Optimal Visualizations 341
1 Introduction 341
2 Approach 342
3 Discussion 346
References 348