Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces
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Features
- Focuses on subdivision convergence, regularity, and construction of scaling functions and synthesis wavelets
- Presents novel wavelet subdivision schemes developed by the authors
- Covers subdivision schemes for parametric curve rendering
- Extends curve subdivision to parametric surface rendering
- Provides guides to using the book for classroom teaching and self-study
- Requires very minimal prior mathematical background
- Lists advanced reading materials for further investigation
- Includes plenty of illustrative examples and over 300 end-of-chapter exercises
Summary
Prevalent in animation movies and interactive games, subdivision methods allow users to design and implement simple but efficient schemes for rendering curves and surfaces. Adding to the current subdivision toolbox, Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces introduces geometry editing and manipulation schemes (GEMS) and covers both subdivision and wavelet analysis for generating and editing parametric curves and surfaces of desirable geometric shapes. The authors develop a complete constructive theory and effective algorithms to derive synthesis wavelets with minimum support and any desirable order of vanishing moments, along with decomposition filters.
Through numerous examples, the book shows how to represent curves and construct convergent subdivision schemes. It comprehensively details subdivision schemes for parametric curve rendering, offering complete algorithms for implementation and theoretical development as well as detailed examples of the most commonly used schemes for rendering both open and closed curves. It also develops an existence and regularity theory for the interpolatory scaling function and extends cardinal B-splines to box splines for surface subdivision.
Keeping mathematical derivations at an elementary level without sacrificing mathematical rigor, this book shows how to apply bottom-up wavelet algorithms to curve and surface editing. It offers an accessible approach to subdivision methods that integrates the techniques and algorithms of bottom-up wavelets.
Table of Contents
OVERVIEW
Curve representation and drawing
Free-form parametric curves
From subdivision to basis functions
Wavelet subdivision and editing
Surface subdivision
BASIS FUNCTIONS FOR CURVE REPRESENTATION
Refinability and scaling functions
Generation of smooth basis functions
Cardinal B-splines
Stable bases for integer-shift spaces
Splines and polynomial reproduction
CURVE SUBDIVISION SCHEMES
Subdivision matrices and stencils
B-spline subdivision schemes
Closed curve rendering
Open curve rendering
BASIS FUNCTIONS GENERATED BY SUBDIVISION MATRICES
Subdivision operators
The up-sampling convolution operation
Scaling functions from subdivision matrices
Convergence of subdivision schemes
Uniqueness and symmetry
QUASI-INTERPOLATION
Sum-rule orders and discrete moments
Representation of polynomials
Characterization of sum-rule orders
Quasi-interpolants
CONVERGENCE AND REGULARITY ANALYSIS
Cascade operators
Sufficient conditions for convergence
Hölder regularity
Positive refinement sequences
Convergence and regularity governed by two-scale symbols
A one-parameter family
Stability of the one-parameter family
ALGEBRAIC POLYNOMIAL IDENTITIES
Fundamental existence and uniqueness theorem
Normalized binomial symbols
Behavior on the unit circle in the complex plane
INTERPOLATORY SUBDIVISION
Scaling functions generated by interpolatory refinement sequences
Convergence, regularity, and symmetry
Rendering of closed and open interpolatory curves
A one-parameter family of interpolatory subdivision operators
WAVELETS FOR SUBDIVISION
From scaling functions to synthesis wavelets
Synthesis wavelets with prescribed vanishing moments
Robust stability of synthesis wavelets
Spline-wavelets
Interpolation wavelets
Wavelet subdivision and editing
SURFACE SUBDIVISION
Control nets and net refinement
Box splines as basis functions
Surface subdivision masks and stencils
Wavelet surface subdivision
EPILOGUE
SUPPLEMENTARY READINGS
INDEX
Exercises appear at the end of each chapter.
Author Bio(s)
Charles Chui is a Curators’ Professor in the Department of Mathematics and Computer Science at the University of Missouri in St. Louis, and a consulting professor of statistics at Stanford University in California. Dr. Chui’s research interests encompass applied and computational mathematics, with an emphasis on splines, wavelets, mathematics of imaging, and fast algorithms.
Johan de Villiers is a professor in the Department of Mathematical Sciences, Mathematics Division at Stellenbosch University in South Africa. Dr. de Villiers’s research interests include computational mathematics, with an emphasis on wavelet and subdivision analysis.
Editorial Reviews
The monograph contains many examples, figures, and more than 300 exercises. It is friendly written for a broad readership and very convenient for students and researchers in applied mathematics and computer science. Doubtless, this nice book will stimulate further research in modeling of curves and surfaces with wavelet subdivision methods.
—Manfred Tasche, Zentralblatt MATH 1202
All topics are treated with great care, and a lot of effort is put into stating results and proofs with a very high precision and accuracy. This makes the book so self-contained that its list of references consists of only 24 items. This is exceptional for a monograph of 450 pages and quite clearly shows the intention of the authors and the approach they have taken for their book. … the book provides everything that is useful, for example, for classroom use: examples, exercises (even with marked difficulty levels), a carefully compiled index and even a very impressive reading guide. … Its extraordinary attention to detail makes it useful to undergraduate students or researchers who want to get familiar with the fundamental techniques of stationary subdivision, who want to see "how the machine works inside".
—Tomas Sauer, Mathematical Reviews, Issue 2011k
This book is the first writing that introduces and incorporates the wavelet component of the bottom-up subdivision scheme. A complete constructive theory, together with effective algorithms, is developed to derive such synthesis wavelets and analysis wavelet filters. The book contains a large collection of carefully prepared exercises and can be used both for classroom teaching and for self study. The authors have been in the forefront for advances in wavelets and wavelet subdivision methods and I congratulate them for writing such a comprehensive text.
—From the Foreword by Tom Lyche, University of Oslo, Norway
