Wavelets - Caltech Multi-Res Modeling Group

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V Curves and Surfaces – Leena-Maija Reissell, Tony D. DeRose, Michael Lounsbery 123 1 Wavelet representation for curves (L-M. Reissell) ::::::::::::::::::::::123 1.1 Introduction :::::::::::::::::::::::::::::::::::::123 1.2 Parametric wavelet decomposition notation :::::::::::::::::::::125 1.3 Basic properties of the wavelet decomposition of curves ::::::::::::::126 1.4 Choosing a wavelet :::::::::::::::::::::::::::::::::127 2 Wavelets with interpolating scaling functions ::::::::::::::::::::::::128 2.1 The construction :::::::::::::::::::::::::::::::::::129 2.2 The pseudocoiflet family P2N ::::::::::::::::::::::::::::131 2.3 Examples ::::::::::::::::::::::::::::::::::::::132 3 Applications :::::::::::::::::::::::::::::::::::::::::134 3.1 Adaptive scaling coefficient representation :::::::::::::::::::::134 3.2 Example of adaptive approximation using scaling coefficients ::::::::::::135 3.3 Finding smooth sections of surfaces; natural terrain path planning ::::::::::137 3.4 Error estimation :::::::::::::::::::::::::::::::::::137 4 Conclusion ::::::::::::::::::::::::::::::::::::::::::140 5 Multiresolution Analysis for Surfaces of Arbitrary Topological Type (T. DeRose) ::::::141 5.1 Introduction :::::::::::::::::::::::::::::::::::::141 5.2 A preview of the method :::::::::::::::::::::::::::::::142 5.3 Our view of multiresolution analysis :::::::::::::::::::::::::143 5.4 Nested linear spaces through subdivision ::::::::::::::::::::::143 5.5 Inner products over subdivision surfaces :::::::::::::::::::::::147 5.6 Multiresolution analysis based on subdivision ::::::::::::::::::::148 5.7 Examples ::::::::::::::::::::::::::::::::::::::151 5.8 Summary ::::::::::::::::::::::::::::::::::::::152 VI Wavelet Radiosity – Peter Schröder 155 1 Introduction :::::::::::::::::::::::::::::::::::::::::155 1.1 A Note on Dimensionality ::::::::::::::::::::::::::::::156 2 Galerkin Methods ::::::::::::::::::::::::::::::::::::::156 2.1 The Radiosity Equation :::::::::::::::::::::::::::::::156 2.2 Projections ::::::::::::::::::::::::::::::::::::::158 3 Linear Operators in Wavelet Bases ::::::::::::::::::::::::::::::160 3.1 Standard Basis ::::::::::::::::::::::::::::::::::::161 3.2 Vanishing Moments :::::::::::::::::::::::::::::::::162 3.3 Integral Operators and Sparsity :::::::::::::::::::::::::::163 3.4 Non-Standard Basis :::::::::::::::::::::::::::::::::164 4 Wavelet Radiosity ::::::::::::::::::::::::::::::::::::::166 4.1 Galerkin Radiosity ::::::::::::::::::::::::::::::::::166 4.2 Hierarchical Radiosity ::::::::::::::::::::::::::::::::167
4.3 Algorithms ::::::::::::::::::::::::::::::::::::::168 4.4 O(n) Sparsity ::::::::::::::::::::::::::::::::::::174 5 Issues and Directions :::::::::::::::::::::::::::::::::::::176 5.1 Tree Wavelets ::::::::::::::::::::::::::::::::::::176 5.2 Visibility :::::::::::::::::::::::::::::::::::::::177 5.3 Radiance :::::::::::::::::::::::::::::::::::::::178 5.4 Clustering ::::::::::::::::::::::::::::::::::::::179 6 Conclusion ::::::::::::::::::::::::::::::::::::::::::179 VII More Applications – Michael F. Cohen, Wim Sweldens, Alain Fournier 183 1 Hierarchical Spacetime Control of Linked Figures (M. F. Cohen) ::::::::::::::183 1.1 Introduction :::::::::::::::::::::::::::::::::::::183 1.2 System overview :::::::::::::::::::::::::::::::::::185 1.3 Wavelets :::::::::::::::::::::::::::::::::::::::185 1.4 Implementation :::::::::::::::::::::::::::::::::::188 1.5 Results ::::::::::::::::::::::::::::::::::::::::189 1.6 Conclusion ::::::::::::::::::::::::::::::::::::::190 2 Variational Geometric Modeling with Wavelets (M. F. Cohen) ::::::::::::::::190 2.1 Abstract :::::::::::::::::::::::::::::::::::::::190 2.2 Introduction :::::::::::::::::::::::::::::::::::::191 2.3 Geometric Modeling with Wavelets :::::::::::::::::::::::::192 2.4 Variational Modeling ::::::::::::::::::::::::::::::::194 2.5 Conclusion ::::::::::::::::::::::::::::::::::::::199 3 Wavelets and Integral and Differential Equations (W. Sweldens) ::::::::::::::200 3.1 Integral equations ::::::::::::::::::::::::::::::::::200 3.2 Differential equations ::::::::::::::::::::::::::::::::201 4 Light Flux Representations (A. Fournier) ::::::::::::::::::::::::::202 5 Fractals and Wavelets (Alain Fournier) :::::::::::::::::::::::::::204 5.1 Fractal Models ::::::::::::::::::::::::::::::::::::204 5.2 Fractional Brownian Motion :::::::::::::::::::::::::::::204 5.3 Stochastic Interpolation to Approximate fBm ::::::::::::::::::::205 5.4 Generalized Stochastic Subdivision :::::::::::::::::::::::::205 5.5 Wavelet Synthesis of fBm ::::::::::::::::::::::::::::::206 VIII Pointers and Conclusions 211 1 Sources for Wavelets :::::::::::::::::::::::::::::::::::::211 2 Code for Wavelets ::::::::::::::::::::::::::::::::::::::211 3 Conclusions :::::::::::::::::::::::::::::::::::::::::212
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48 L-M. REISSELL 2.4.1 Computing us
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50 L-M. REISSELL The Fourier transf
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52 L-M. REISSELL finite filters for
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54 L-M. REISSELL The last formulati
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56 L-M. REISSELL - m0(0) =1; m0( )=
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58 L-M. REISSELL The filter leads t
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60 L-M. REISSELL 1.5 1 0.5 0 -0.5 D
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62 L-M. REISSELL The last condition
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64 L-M. REISSELL 4.2 Approximation
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66 L-M. REISSELL 5.1 Orthogonalizat
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68 L-M. REISSELL This choice has th
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70 L-M. REISSELL 0.8 0.6 0.4 0.2 0
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72 WIM SWELDENS, PETER SCHRÖDER li
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74 WIM SWELDENS, PETER SCHRÖDER lo
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96 WIM SWELDENS, PETER SCHRÖDER P
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102 WIM SWELDENS, PETER SCHRÖDER 1
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104 WIM SWELDENS, PETER SCHRÖDER r
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106 WIM SWELDENS, PETER SCHRÖDER S
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108 W. SWELDENS 1. save storage, 2.
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110 W. SWELDENS bases by allowing d
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112 W. SWELDENS Frequency 0.015 0.0
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114 W. SWELDENS f( x, y) fLL ( x, y
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116 W. SWELDENS PSNR (dB) 42 40 38
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118 W. SWELDENS Encoder Decoder fi
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120 W. SWELDENS probing. The idea i
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122 W. SWELDENS filename descriptio
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124 L-M. REISSELL, T.DEROSE &MICHAE
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154 P. SCHRÖDER Siggraph ’95 Cou
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156 P. SCHRÖDER 1.1 A Note on Dime
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158 P. SCHRÖDER Once again we make
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160 P. SCHRÖDER E R I I E R E R E
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162 P. SCHRÖDER Haar basis (1 vani
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164 P. SCHRÖDER for some M>0, and
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166 P. SCHRÖDER G'i;'i recurse G'i
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168 P. SCHRÖDER GR, by going to hi
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170 P. SCHRÖDER As mentioned earli
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172 P. SCHRÖDER 2. the involved el
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174 P. SCHRÖDER a surface will be
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176 P. SCHRÖDER shape of Flatlets
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178 P. SCHRÖDER with discontinuity
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180 P. SCHRÖDER basis functions th
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182 M. F. COHEN, W.SWELDENS &A.FOUR
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208 and the correlation between lev
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210 Siggraph ’95 Course Notes: #2
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The latest edition of the popular N
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214 BIBLIOGRAPHY [13] BARTELS, R.H.
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216 BIBLIOGRAPHY [50] DAUBECHIES, I
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218 BIBLIOGRAPHY [85] FOURNIER, A.,
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220 BIBLIOGRAPHY [119] LEWIS, A.S.,
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222 BIBLIOGRAPHY [153] PEITGEN, H.O
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224 BIBLIOGRAPHY [189] VAN DE PANNE
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adaptive scaling coefficients, 134,
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