Chui / de Villiers Wavelet Subdivision Methods

GEMS for Rendering Curves and Surfaces
1. Auflage 2012
ISBN: 978-1-4398-1216-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 479 Seiten

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.

Zielgruppe

Advanced undergraduate and graduate students and researchers in computer science, mathematics, statistics, computer architecture, and engineering; computer graphics professionals.

Autoren/Hrsg.

Chui, Charles

de Villiers, Johan

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik EDV | Informatik Professionelle Anwendung
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

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.

Über Autor(innen)

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.