Spherical Sampling

This book presents, in a consistent and unified overview, results and developments in the field of today ́s spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling. Preface 6 About the Authors 9 Contents 11 Chapter 1 Introduction 16 1.1 Motivation and Justification 16 1.2 Spherical Signals, Spherical Harmonics, and Pseudodifferential Operators 18 1.3 Signal Band and Space Limitation 22 1.4 Uncertainty Principle and Zonal Kernel Functions 23 1.5 Spline and Wavelet Sampling 29 1.6 Goal and Layout of the Work 36 Part I Preparatory Material 45 Chapter 2 Basics and Settings 46 2.1 Three-Dimensional Cartesian Framework 47 2.2 Kelvin Transform 54 2.3 Two-Dimensional Spherical Framework 56 2.4 Stereographic Projection 72 Part II Function Systems 74 Chapter 3 Spherical Harmonics 75 3.1 Spherical Harmonics and Their Essential Properties 76 3.2 Shannon Kernels and Means 83 3.3 Bernstein Kernels and Means 86 3.4 Abel-Poisson Kernels and Means 89 3.5 Latitude-Longitude Generated Spherical Harmonics 93 3.6 Spectral Signal-to-Noise Ratio 100 3.7 Circular Harmonics 106 3.8 Inner/Outer Harmonics 110 Chapter 4 Zonal Functions 116 4.1 Bandlimited/Spacelimited Functions 117 4.2 Radial Basis Functions, Zonal Kernel Functions 118 4.3 Singular Integral Kernels and Approximate Identities 121 4.4 Uncertainty Principle: Space Versus Frequency Localization 123 4.5 Localization of Representative Zonal Kernels 129 Chapter 5 Slepian Functions: Basics and Settings 142 5.1 Classical One-Dimensional Concept 143 5.2 Spatial Concentration of Bandlimited Functions 145 5.3 Spectral Concentration of Bandlimited Functions 148 5.4 Significant Eigenvalues 149 5.5 Slepian Function Approximation 151 5.6 Sparse Slepian Function Expansions 153 5.7 Generalized Slepian Functions 154 Part III Plane Involved Stereographic Sampling 157 Chapter 6 Stereographic Shannon-Type Sampling 158 6.1 1D-Preparatory Sampling 159 6.2 Bivariate Lattice Point Identities 168 6.3 Over- and Undersampling 173 6.4 Bivariate Shannon-Type Sampling 179 6.5 Stereographically Projected Bivariate Shannon-Type Sampling 182 Chapter 7 Plane Based Scaling and Wavelet Functions 184 7.1 Mapping the Plane to the Sphere 185 7.2 From Bivariate to Spherical Wavelets 188 7.3 Numerical Test Example 189 7.4 Algorithmic Aspects 192 Part IV Plane Involved Polar Coordinate Sampling 195 Chapter 8 Sampling Based on Bivariate Fourier Coefficient Integration 196 8.1 Periodic Integration Revisited 197 8.2 Gauss-Legendre Integration Revisited 205 8.3 Sampling Based on Latitude-Longitude Grids 214 8.4 Sparse Recovery of Truncated Spherical Harmonic Expansions 222 8.5 Sufficient Conditions for Sampling and Interpolation 224 Chapter 9 Orthogonal Zonal, Tesseral, and Sectorial Wavelet Reconstruction 230 9.1 Index Sets 232 9.2 Zonal, Tesseral, and Sectorial Spherical Harmonics 233 9.3 Sobolev Spaces and Reproducing Kernel Hilbert Spaces 235 9.4 Orthogonal Shannon-Type Wavelets 237 9.5 Horizontal Shannon-Type Wavelets 244 9.6 Vertical Shannon-Type Wavelets 245 Chapter 10 Biorthogonal Finite-Cap-Element Multiscale Tree Sampling 253 10.1 Block Grids 255 10.2 Scaling Functions 259 10.3 Wavelet Functions 262 10.4 Multiresolution Analysis and Sampling 263 10.5 Tree Algorithm 266 Part V Sphere Intrinsic Frequency Limited Sampling 270 Chapter 11 Spherical Harmonics Interpolatory Sampling 271 11.1 Unisolvent Point Systems 272 11.2 Shannon Lagrangians 274 11.3 Beltrami Green’s Functions and Associated Integral Formulas 275 11.4 Interpolating Splines, Best Approximation, and Beltrami Spline Sampling 281 11.5 Non-Bandlimited Remainder Term Estimates 290 11.6 Interpolation Operators and Lebesgue Function 296 11.7 Lebesgue Constants 297 11.8 Combined Polynomial Sampling and Spline Interpolation 302 Chapter 12 Bandlimited Multiscale Tree Sampling 305 12.1 Exact Spherical Harmonics Integration 305 12.2 Shannon Reconstruction Sampling 307 12.3 Shannon Tree Sampling 309 12.4 Finite-Dimensional Sobolev Space Settings 313 12.5 Bernstein Wavelet Reconstruction 314 12.6 Bernstein Tree Sampling 317 Part VI Sphere Intrinsic Frequency Versus Space Sampling 319 Chapter 13 RKHS Framework and Spline Sampling 320 13.1 Sobolev Space Settings 321 13.2 Pseudodifferential Operators 323 13.3 RKHS Framework 327 13.4 Finite-Dimensional Spline Interpolation and Sampling 354 13.5 Spline-Inversion of Pseudodifferential Operators 365 13.6 Combined Interpolation and Smoothing 368 13.7 Closure of Spline Spaces 373 13.8 Infinite-Dimensional Spline Interpolation and Sampling 376 13.9 Multiscale Spline Sampling 379 Chapter 14 Orthogonal/Non-OrthogonalWavelet Approximations and Tree Sampling 387 14.1 Admissible Generators 388 14.2 Scale-Discrete Scaling Functions 390 14.3 Multiresolution Analysis 391 14.4 Scale-Discrete Wavelet Functions 392 14.5 Non-Bandlimited Examples 396 14.6 Bandlimited Examples 399 14.7 Sampling by Exact Fully Discrete Wavelet Transform 404 14.8 Tree Sampling 408 Part VII Sphere Intrinsic Spacelimited Sampling 411 Chapter 15 Non-Orthogonal Finite-Cap-Element Multiscale Sampling 412 15.1 Low Discrepancy Method 413 15.2 Abel-Poisson-Type Scaling Functions 417 15.3 Haar-Type Scaling Functions 420 15.4 Sampling on Equidistributed Grids 425 15.5 Multiscale Signal-to-Noise Ratio 429 15.6 Finite-Cap-Element Multiscale Sampling 433 15.7 Haar Wavelets 435 15.8 Integration Using Large Equidistributed Data 437 Chapter 16 Non-Orthogonal Up Function Multiscale Tree Sampling 447 16.1 Locally Supported Kernels 448 16.2 Up Functions and Their Properties 456 16.3 Finite Truncations of Up Function Convolutions 457 16.4 Multiresolution in Terms of Up Functions 459 16.5 Wavelets in Terms of Up Functions 462 16.6 Decomposition and Reconstruction Schemes 465 Part VIII Applications 469 Chapter 17 Sampling Solutions of Inverse Pseudodifferential Equations 470 17.1 Ill-Posed Problems in Hilbert Space Framework 471 17.2 Rotation-Invariant Pseudodifferential Equations 477 17.3 Multiscale Solutions of Pseudodifferential Equations 479 17.4 Multiscale Signal-to-Noise Ratio 485 17.5 Tree Sampling 488 17.6 Scale Thesholding 490 Chapter 18 Sampling of Potential and Stream Functions 493 18.1 Space Mollified Green’s Functions 494 18.2 Multiscale Sampling of Surface Potential and Stream Functions 498 18.3 Geoidal Undulations from Deflections of the Vertical 502 18.4 Homogeneous Boundary Conditions 505 18.5 Dirichlet Boundary Conditions 509 18.6 Neumann Boundary Conditions 512 18.7 Dirichlet and Neumann Boundary Value Problem 514 18.8 Multiscale Cap-Sampling of Potential Functions from Surface Gradients 517 Part IX Final Remarks 521 Chapter 19 Applicabilities and Applications 522 19.1 From Univariate to Multivariate Sampling 522 19.2 From Sampling to Recovery Problems 524 19.3 Spherical Sampling in Geosystems Mathematics 526 19.4 Selective Publication List 526 Part X Appendix 529 Appendix Earth’s Gravitational Field: Ingredients, Observables, and Modeling 530 A.1 Gravitational, Centrifugal, and Gravity Acceleration 530 A.2 Disturbing Potential and Geoidal Undulations 533 A.3 Gravity Disturbance and Gravity Anomaly 541 A.4 Determination of the Disturbing Potential FromGravity Disturbances 542 A.5 Determination of the Disturbing Potential From Gravity Anomalies 544 A.6 Satellite Gravitational Tensor Field 546 A.7 Downward Continuation of Satellite-Gravity-Gradiometry (SGG)-Data 548 Bibliography 552 Index 584 Front Matter ....Pages i-xv Introduction (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 1-29 Front Matter ....Pages 31-31 Basics and Settings (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 33-60 Front Matter ....Pages 61-61 Spherical Harmonics (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 63-103 Zonal Functions (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 105-130 Slepian Functions: Basics and Settings (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 131-145 Front Matter ....Pages 147-147 Stereographic Shannon-Type Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 149-174 Plane Based Scaling and Wavelet Functions (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 175-185 Front Matter ....Pages 187-187 Sampling Based on Bivariate Fourier Coefficient Integration (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 189-222 Orthogonal Zonal, Tesseral, and Sectorial Wavelet Reconstruction (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 223-245 Biorthogonal Finite-Cap-Element Multiscale Tree Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 247-263 Front Matter ....Pages 265-265 Spherical Harmonics Interpolatory Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 267-300 Bandlimited Multiscale Tree Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 301-314 Front Matter ....Pages 315-315 RKHS Framework and Spline Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 317-383 Orthogonal/Non-Orthogonal Wavelet Approximations and Tree Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 385-408 Front Matter ....Pages 409-409 Non-Orthogonal Finite-Cap-Element Multiscale Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 411-445 Non-Orthogonal Up Function Multiscale Tree Sampling (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 447-468 Front Matter ....Pages 469-469 Sampling Solutions of Inverse Pseudodifferential Equations (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 471-493 Sampling of Potential and Stream Functions (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 495-522 Front Matter ....Pages 523-523 Applicabilities and Applications (Willi Freeden, M. Zuhair Nashed, Michael Schreiner)....Pages 525-531 Back Matter ....Pages 533-596 This book presents, in a consistent and unified overview, results and developments in the field of today ́s spherical sampling, particularly arising in mathematical geosciences. Although the book often refers to original contributions, the authors made them accessible to (graduate) students and scientists not only from mathematics but also from geosciences and geoengineering. Building a library of topics in spherical sampling theory it shows how advances in this theory lead to new discoveries in mathematical, geodetic, geophysical as well as other scientific branches like neuro-medicine. A must-to-read for everybody working in the area of spherical sampling.-- Provided by publisher