Wavelets - Caltech Multi-Res Modeling Group

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Table of Contents Preamble – Alain Fournier 1 1 Prolegomenon :::::::::::::::::::::::::::::::::::::::: 1 I Introduction – Alain Fournier 5 1 Scale ::::::::::::::::::::::::::::::::::::::::::::: 5 1.1 Image pyramids ::::::::::::::::::::::::::::::::::: 5 2 Frequency :::::::::::::::::::::::::::::::::::::::::: 7 3 The Walsh transform ::::::::::::::::::::::::::::::::::::: 8 4 Windowed Fourier transforms :::::::::::::::::::::::::::::::: 10 5 Relative Frequency Analysis ::::::::::::::::::::::::::::::::: 12 6 Continuous Wavelet Transform ::::::::::::::::::::::::::::::: 12 7 From Continuous to Discrete and Back ::::::::::::::::::::::::::: 13 7.1 Haar Transform ::::::::::::::::::::::::::::::::::: 13 7.2 Image Pyramids Revisited :::::::::::::::::::::::::::::: 14 7.3 Dyadic Wavelet Transforms ::::::::::::::::::::::::::::: 15 7.4 Discrete Wavelet Transform ::::::::::::::::::::::::::::: 16 7.5 Multiresolution Analysis ::::::::::::::::::::::::::::::: 17 7.6 Constructing Wavelets :::::::::::::::::::::::::::::::: 17 7.7 Matrix Notation ::::::::::::::::::::::::::::::::::: 19 7.8 Multiscale Edge Detection :::::::::::::::::::::::::::::: 19 8 Multi-dimensional Wavelets ::::::::::::::::::::::::::::::::: 20 8.1 Standard Decomposition ::::::::::::::::::::::::::::::: 20 8.2 Non-Standard Decomposition :::::::::::::::::::::::::::: 20 8.3 Quincunx Scheme :::::::::::::::::::::::::::::::::: 21 9 Applications of Wavelets in Graphics :::::::::::::::::::::::::::: 21 9.1 Signal Compression ::::::::::::::::::::::::::::::::: 21 9.2 Modelling of Curves and Surfaces :::::::::::::::::::::::::: 33 9.3 Radiosity Computations ::::::::::::::::::::::::::::::: 33 10 Other Applications :::::::::::::::::::::::::::::::::::::: 33
Page 1: f[n] = a[n] 0.8 0.6 0.4 0.2 0 -0.2
Page 5: Lecturers Michael F. Cohen Microsof
Page 9 and 10: 5 Multiresolution Analysis ::::::::
Page 11 and 12: 4.3 Algorithms ::::::::::::::::::::
Page 13 and 14: 1 Prolegomenon N : Preamble These a
Page 15 and 16: PREAMBLE 3 be a disparate collectio
Page 17 and 18: 1 Scale 1.1 Image pyramids I: Intro
Page 19 and 20: INTRODUCTION 7 If we look at the op
Page 21 and 22: and the function can be reconstruct
Page 23 and 24: Heisenberg inequality that bounds t
Page 25 and 26: We can reconstruct the signal as: Z
Page 27 and 28: 7.3 Dyadic Wavelet Transforms INTRO
Page 29 and 30: 7.5 Multiresolution Analysis INTROD
Page 31 and 32: 7.7 Matrix Notation INTRODUCTION 19
Page 33 and 34: and three wavelet functions, (x; y)
Page 35 and 36: 0.0 0.0 Figure I.10: Signal 2 analy
Page 37 and 38: Haar 0 Haar 1 Haar 2 Haar 3 Haar 4
Page 39 and 40: f[n] = a[n] f[n] = a[n] H G a[n/2]
Page 41 and 42: 0 0 (x) 1 1 (y) 0 0 (x) 1 0 (y) 0 0
Page 43 and 44: 30 25 20 15 10 5 0 -5 -10 -15 -20 2
Page 45 and 46: INTRODUCTION 33 We now go through t
Page 47 and 48: 300 250 200 150 100 50 -10 -20 -30
Page 49 and 50: 1 Introduction II: Multiresolution
Page 51 and 52: MULTIRESOLUTION AND WAVELETS 39 ori
Page 53 and 54: 1.3 Example of wavelet decompositio
Page 55 and 56: MULTIRESOLUTION AND WAVELETS 43 fun
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MULTIRESOLUTION AND WAVELETS 45 ort
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a : a(t) = (x, a): uf (a) = hf; ai
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1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 MU
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MULTIRESOLUTION AND WAVELETS 51 The
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Z 6= 0: MULTIRESOLUTION AND WAVELET
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Z h 00; 0ni = j ˆj2e in! Z 2 d! =
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Then the product 1 p 2 1Y 1 m0( ! )
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MULTIRESOLUTION AND WAVELETS 59 and
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3.7 Some shortcomings of compactly
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4.1.1 Examples MULTIRESOLUTION AND
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and the following “analyzing” a
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1 Introduction III: Building Your O
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1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0
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BUILDING YOUR OWN WAVELETS AT HOME
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constant average interpolation cons
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1.0 0.0 -1.0 1.0 0.0 -1.0 0.0 0.2 0
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scaling functions ˜'j;k, so that w
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P j+1 subsample P j P j BUILDING YO
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that BUILDING YOUR OWN WAVELETS AT
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The wavelet coefficients can be fou
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1.0 0.0 -1.0 1.0 0.0 -1.0 0.0 0.2 0
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2.0 1.0 0.0 -1.0 -2.0 0.0 0.2 0.4 0
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9.3 Weighted Inner Products BUILDIN
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11 Outlook BUILDING YOUR OWN WAVELE
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IV: 1 Wavelets and signal compressi
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original signal forward transform W
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WAVELETS, SIGNAL COMPRESSION AND IM
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1 Wavelet representation for curves
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CURVES AND SURFACES 125 Wavelets ca
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CURVES AND SURFACES 127 The proofs
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- The first 2N moments for and ˜ v
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CURVES AND SURFACES 131 The biortho
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1.5 1 0.5 0 -0.5 scaling function w
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CURVES AND SURFACES 135 linearly in
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3.3 Finding smooth sections of surf
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The total error at level i is now b
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CURVES AND SURFACES 141 5 Multireso
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CURVES AND SURFACES 143 reflects a
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M 3 0 1 2 P 0 = A 0 B 0 O 0 N 0 I 0
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CURVES AND SURFACES 147 The notatio
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5.6.2 Computation of wavelets Figur
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into a lower resolution part in V j
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There are numerous areas for future
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1 Introduction VI: Wavelet Radiosit
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x WAVELET RADIOSITY: WAVELET METHOD
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WAVELET RADIOSITY: WAVELET METHODS
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4.3.4 Oracle WAVELET RADIOSITY: WAV
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φ 1 i, 2j φ 2 i, 2j φ 1 i, 2j+1
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Siggraph ’95 Course Notes: #26 Wa
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1 Hierarchical Spacetime Control of
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Hierarchical Spacetime Constraints
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MORE APPLICATIONS 187 it is underst
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1.5 Results MORE APPLICATIONS 189 A
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2.2 Introduction MORE APPLICATIONS
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MORE APPLICATIONS 193 different res
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MORE APPLICATIONS 195 Where H is th
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Error Error 1.0 0.8 0.6 0.4 0.2 0.0
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2.5 Conclusion MORE APPLICATIONS 19
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MORE APPLICATIONS 201 of a wavelet
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MORE APPLICATIONS 203 gation in par
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5.3 Stochastic Interpolation to App
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MORE APPLICATIONS 207 If we conduct
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1 Sources for Wavelets VIII: Pointe
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Bibliography [1] AHARONI, G.,AVERBU
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BIBLIOGRAPHY 215 [31] CHUI, C.K.,AN
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BIBLIOGRAPHY 217 [67] DONOHO, D.L.,
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BIBLIOGRAPHY 219 [103] HECKBERT, P.
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BIBLIOGRAPHY 221 [135] MALLAT, S. G
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BIBLIOGRAPHY 223 [171] SIEGEL, R.,A
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pyramid scheme, 38, 40 quantization
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