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E-raamat: Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces

  • Formaat: 479 pages
  • Ilmumisaeg: 23-Aug-2010
  • Kirjastus: CRC Press Inc
  • ISBN-13: 9781000687293
  • Formaat - EPUB+DRM
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  • Formaat: 479 pages
  • Ilmumisaeg: 23-Aug-2010
  • Kirjastus: CRC Press Inc
  • ISBN-13: 9781000687293

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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.

Arvustused

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

List of Figures
xi
List of Tables
xv
Foreword xvii
Preface xix
Teaching and Reading Guides xxiii
1 Overview
1(36)
1.1 Curve representation and drawing
2(3)
1.2 Free-form parametric curves
5(6)
1.3 From subdivision to basis functions
11(5)
1.4 Wavelet subdivision and editing
16(13)
1.5 Surface subdivision
29(3)
1.6 Exercises
32(5)
2 Basis Functions for Curve Representation
37(38)
2.1 Refinability and scaling functions
39(7)
2.2 Generation of smooth basis functions
46(6)
2.3 Cardinal B-splines
52(4)
2.4 Stable bases for integer-shift spaces
56(6)
2.5 Splines and polynomial reproduction
62(5)
2.6 Exercises
67(8)
3 Curve Subdivision Schemes
75(58)
3.1 Subdivision matrices and stencils
76(9)
3.2 B-spline subdivision schemes
85(10)
3.3 Closed curve rendering
95(11)
3.4 Open curve rendering
106(23)
3.5 Exercises
129(4)
4 Basis Functions Generated by Subdivision Matrices
133(36)
4.1 Subdivision operators
134(4)
4.2 The up-sampling convolution operation
138(3)
4.3 Scaling functions from subdivision matrices
141(13)
4.4 Convergence of subdivision schemes
154(6)
4.5 Uniqueness and symmetry
160(3)
4.6 Exercises
163(6)
5 Quasi-Interpolation
169(36)
5.1 Sum-rule orders and discrete moments
170(3)
5.2 Representation of Polynomials
173(5)
5.3 Characterization of sum-rule orders
178(4)
5.4 Quasi-interpolants
182(16)
5.5 Exercises
198(7)
6 Convergence and Regularity Analysis
205(66)
6.1 Cascade operators
206(5)
6.2 Sufficient conditions for convergence
211(7)
6.3 Holder regularity
218(7)
6.4 Positive refinement sequences
225(8)
6.5 Convergence and regularity governed by two-scale symbols
233(11)
6.6 A one-parameter family
244(11)
6.7 Stability of the one-parameter family
255(5)
6.8 Exercises
260(11)
7 Algebraic Polynomial Identities
271(24)
7.1 Fundamental existence and uniqueness theorem
272(8)
7.2 Normalized binomial symbols
280(8)
7.3 Behavior on the unit circle in the complex plane
288(3)
7.4 Exercises
291(4)
8 Interpolatory Subdivision
295(44)
8.1 Scaling functions generated by interpolatory refinement sequences
296(6)
8.2 Convergence, regularity, and symmetry
302(10)
8.3 Rendering of closed and open interpolatory curves
312(10)
8.4 A one-parameter family of interpolatory subdivision operators
322(11)
8.5 Exercises
333(6)
9 Wavelets for Subdivision
339(72)
9.1 From scaling functions to synthesis wavelets
340(11)
9.2 Synthesis wavelets with prescribed vanishing moments
351(13)
9.3 Robust stability of synthesis wavelets
364(6)
9.4 Spline-wavelets
370(15)
9.5 Interpolation wavelets
385(16)
9.6 Wavelet subdivision and editing
401(5)
9.7 Exercises
406(5)
10 Surface Subdivision
411(34)
10.1 Control nets and net refinement
413(9)
10.2 Box splines as basis functions
422(5)
10.3 Surface subdivision masks and stencils
427(10)
10.4 Wavelet surface subdivision
437(5)
10.5 Exercises
442(3)
11 Epilogue
445(4)
Supplementary Readings 449(2)
Index 451
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. Chuis 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 Villierss research interests include computational mathematics, with an emphasis on wavelet and subdivision analysis.