Tutorials on Multiresolution in Geometric Modelling

by Iske, Armin; Quak, Ewald; Floater, Michael S.

• ISBN13: 9783540436393
• ISBN10: 3540436391
• Format: Hardcover
• Copyright: 2002-06-01
• Publisher: Springer-Verlag New York Inc

Summary

Multiresolution methods in geometric modelling are concerned with the generation, representation, and manipulation of geometric objects at several levels of detail. Applications include fast visualization and rendering as well as coding, compression and digital transmission of 3D geometric objects.This book is based on thirteen tutorials presented during the European Summer School "Principles of Multiresolution in Geometric Modelling", held at the Munich University of Technology, Germany, during August 22-30, 2001. The book covers: subdivision; wavelets; scattered data modelling; and coding and data structures.The tutorials are designed to be introductory in character, and include supporting exercises. Other supplementary material and software can be downloaded from the Web Site www.ma.tum.de/primus 2001/.

Table of Contents

Part I. Subdivision
Subdivision of Box-Splines	3	(22)
Malcolm Sabin
B-Splines	4	(4)
Univariate Box-Splines	8	(4)
Bivariate Box-Splines	12	(5)
Freedom of Topology	17	(4)
Programming Subdivision Schemes	21	(1)
Bibliographic Notes	22	(3)
References	22	(3)
Interpolatory Subdivision Schemes	25	(26)
Nira Dyn
The Univariate Stationary Case	25	(10)
Non-Stationary Univariate Interpolatory Schemes, Exact for Exponentials	35	(2)
Tensor-Product Interpolatory Schemes for Surfaces	37	(2)
The Butterfly Scheme - An Interpolatory Subdivision Scheme on Triangulations	39	(4)
Bibliographic Notes	43	(8)
References	43	(2)
Appendix: Solutions of Selected Exercises	45	(6)
Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials	51	(18)
Nira Dyn
Introduction	51	(1)
Analysis of Univariate Schemes	52	(6)
Analysis of Bivariate Schemes with Factorizable Symbols	58	(5)
Bibliographic Notes	63	(6)
References	64	(1)
Appendix: Solutions of Selected Exercises	65	(4)
Eigenanalysis and Artifacts of Subdivision Curves and Surfaces	69	(24)
Malcolm Sabin
Analysis of Curves	69	(10)
Analysis of Subdivision Surfaces	79	(10)
Bibliographic Notes	89	(4)
References	90	(3)
Nonlinear Subdivision Schemes: Applications to Image Processing	93	(8)
Albert Cohen
Basarab Matei
Extended Abstract	93	(4)
References	97	(4)
Part II. Wavelets
Nonuniform B-Splines and B-Wavelets	101	(46)
Ewald Quak
Introduction	101	(1)
B-Splines on Nonuniform Knot Sequences	102	(10)
Refinement of B-Splines	112	(8)
A Nonuniform Spline Multiresolution Analysis	120	(5)
An Example: Multiresolution Analysis based on Piecewise Linear Splines	125	(17)
Nonuniform B-Wavelets	142	(5)
References	145	(2)
BLaC Wavelets and Non-Nested Wavelets	147	(18)
Georges-Pierre Bonneau
Introduction	147	(1)
Multiresolution Analysis with Non-Nested Approximating Spaces	147	(4)
BLaC Wavelets	151	(6)
Multiresolution Analysis on Hierarchical Triangulations	157	(8)
References	162	(3)
Multiresolution on the Sphere	165	(40)
Matthias Conrad
Jurgen Prestin
Introduction	165	(1)
Preliminaries	166	(7)
Spherical Harmonics	173	(18)
Spherical Basis Functions	191	(14)
References	200	(5)
Part III. Scattered Data Modelling
Scattered Data Modelling Using Radial Basis Functions	205	(38)
Armin Iske
Introduction	205	(2)
Interpolation	207	(3)
Optimal Recovery	210	(3)
Pointwise Optimality	213	(3)
Uncertainty Principle	216	(3)
Optimal Point Sampling	219	(5)
Least Squares Approximation	224	(3)
Multilevel Approximation Schemes	227	(1)
Scattered Data Filtering	228	(15)
References	233	(3)
Appendix: Exercises	236	(7)
Scattered Data Fitting with Bivariate Splines	243	(44)
Frank Zeilfelder
Introduction	243	(3)
Splines	246	(5)
Bernstein-Bezier Techniques	251	(4)
Dimension of Bivariate Spline Spaces	255	(4)
Interpolation Methods for Bivariate Splines	259	(11)
A Direct Approximation Method for Bivariate Splines	270	(17)
References	276	(8)
Appendix: Exercises	284	(3)
Parameterization of Triangulations and Unorganized Points	287	(32)
Michael Floater
Kai Hormann
Introduction	287	(1)
Parameterization of Simple Triangulations	288	(3)
Parameterization by Linear Methods	291	(2)
Choosing the Weights	293	(3)
Parameterization by Weighted Least Squares	296	(2)
Parameterization by Non-Linear Methods	298	(7)
Remeshing, Subdivision Connectivity, and Multiresolution	305	(3)
Parameterization of Unorganized Points	308	(3)
Parameterization of Manifold Triangulations	311	(8)
References	314	(5)
Part IV. Coding and Data Structures
Simplification and Compression of 3D Meshes	319	(44)
Craig Gotsman
Stefan Gumhold
Leif Kobbelt
Introduction	319	(4)
Mesh Decimation Techniques	323	(11)
Connectivity Coding	334	(12)
Compression of Geometric Data	346	(17)
References	356	(4)
Appendix: Exercises	360	(3)
Multiresolution Mesh Representation: Models and Data Structures	363	(56)
Leila De Floriani
Paola Magillo
Introduction	363	(2)
Background Notions	365	(4)
A Reference Framework for Multiresolution Meshes	369	(6)
Classification of Multiresolution Meshes	375	(12)
Level-Of-Detail (LOD) Queries on a Multiresolution Mesh	387	(4)
General-Purpose Data Structures for Multiresolution Meshes	391	(3)
Data Structures for Nested Models	394	(3)
Data Structures for Non-Nested Models	397	(13)
Concluding Remarks	410	(9)
References	411	(6)
Appendix: Exercises	417	(2)
Index	419