SIGGRAPH '95 Course Notes a[n/4] a[n/8]H* a[1] H* a[n/2] d[1] H* G* G* H* d[n/8] G* Non-standard Haar Non-standard Flatlet 2 d[n/4] f[n] = a[n] d[n/2] G* Wavelets and their ApplicationsNon-standard Haar Non-standard Flatlet 2 in Computer Graphics AAA ... (a)B (b)B ... (c) Wavelet coefficients Wavelet coefficients Organizer: Alain Fournier University of British Columbia 0.8 3 scaling function scaling function wavelet wavelet 0.6 2 0.4 1 0.2 0 0 -0.2 -1 -0.4 -2 -0.6 -0.8 -3 0 50 100 150 200 250 300 350 400 0 100 200 300 400 500 600 700 Nothing to do with sea or anything else. Over and over it vanishes with the wave. ± Shinkichi Takahashi Lecturers Michael F. Cohen Microsoft Research One Microsoft Way Redmond, WA 98052 [email protected] Tony D. DeRose Department of Computer Science and Engineering FR-35 University of Washington Seattle, Washington 98195 [email protected] Alain Fournier Department of Computer Science University of British Columbia 2366 Main Mall Vancouver, British Columbia V6T 1Z4 [email protected] Michael Lounsbery Alias Research 219 S. Washington St. P.O. Box 4561 Seattle, WA 98104 [email protected] Leena-Maija Reissell Department of Computer Science University of British Columbia 2366 Main Mall Vancouver, British Columbia V6T 1Z4 [email protected] Peter SchroderÈ Department of Computer Science Le Conte 209F University of South Carolina Columbia, SC 29208 [email protected] Wim Sweldens Department of Mathematics University of South Carolina Columbia, SC 29208 [email protected] 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 II Multiresolution and Wavelets ± Leena-Maija Reissell 37 1 Introduction ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 37 1.1 A recipe for ®nding wavelet coef®cients :: ::: ::: ::: :: ::: ::: ::: : 37 1.2 Wavelet decomposition :: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 40 1.3 Example of wavelet decomposition : ::: ::: ::: ::: :: ::: ::: ::: : 41 1.4 From the continuous wavelet transform to more compact representations :: ::: : 42 2 Multiresolution: de®nition and basic consequences ::: ::: ::: :: ::: ::: ::: : 43 2.1 Wavelet spaces : ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 44 2.2 The re®nement equation :: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 46 2.3 Connection to ®ltering ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 46 2.4 Obtaining scaling functions by iterated ®ltering : ::: ::: :: ::: ::: ::: : 47 3 Requirements on ®lters for multiresolution : ::: ::: ::: ::: :: ::: ::: ::: : 52 3.1 Basic requirements for the scaling function ::: ::: ::: :: ::: ::: ::: : 52 3.2 Wavelet de®nition :: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 53 3.3 Orthonormality ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 54 3.4 Summary of necessary conditions for orthonormal multiresolution :: ::: ::: : 55 3.5 Suf®ciency of conditions : :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 56 3.6 Construction of compactly supported orthonormal wavelets :: ::: ::: ::: : 58 3.7 Some shortcomings of compactly supported orthonormal bases : ::: ::: ::: : 61 4 Approximation properties :: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 61 4.1 Approximation from multiresolution spaces ::: ::: ::: :: ::: ::: ::: : 61 4.2 Approximation using the largest wavelet coef®cients : ::: :: ::: ::: ::: : 64 4.3 Local regularity ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 64 5 Extensions of orthonormal wavelet bases :: ::: ::: ::: ::: :: ::: ::: ::: : 65 5.1 Orthogonalization :: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 66 5.2 Biorthogonal wavelets ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 66 5.3 Examples ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 68 5.4 Semiorthogonal wavelets : :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 68 5.5 Other extensions of wavelets :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 69 5.6 Wavelets on intervals ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 69 III Building Your Own Wavelets at Home ± Wim Sweldens, Peter SchrÈoder 71 1 Introduction ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 71 2 Interpolating Subdivision :: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 72 2.1 Algorithm ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 72 2.2 Formal Description* : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 74 3 Average-Interpolating Subdivision : :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 76 3.1 Algorithm ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 76 3.2 Formal Description* : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 79 4 Generalizations : ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 81 5 Multiresolution Analysis ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 83 5.1 Introduction :: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 83 5.2 Generalized Re®nement Relations :: ::: ::: ::: ::: :: ::: ::: ::: : 84 5.3 Formal Description* : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 84 5.4 Examples ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 85 5.5 Polynomial Reproduction : :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 85 5.6 Subdivision :: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 86 5.7 Coarsening ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 86 5.8 Examples ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 87 6 Second Generation Wavelets : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 88 6.1 Introducing Wavelets ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 88 6.2 Formal Description* : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 90 7 The Lifting Scheme :: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 91 7.1 Lifting and Interpolation: An Example :: ::: ::: ::: :: ::: ::: ::: : 91 7.2 Lifting: Formal Description* : ::: ::: ::: ::: ::: :: ::: ::: ::: : 93 7.3 Lifting and Interpolation: Formal description :: ::: ::: :: ::: ::: ::: : 94 7.4 Wavelets and Average-Interpolation: An Example ::: ::: :: ::: ::: ::: : 95 7.5 Wavelets and Average-Interpolation: Formal description* :: :: ::: ::: ::: : 98 8 Fast wavelet transform : ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: : 98 9Examples: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :100 9.1 Interpolation of Randomly Sampled Data : ::: ::: ::: :: ::: ::: ::: :100 9.2 Smoothing of Randomly Sampled Data :: ::: ::: ::: :: ::: ::: ::: :102 9.3 Weighted Inner Products :: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :103 10 Warning :: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :104 11 Outlook :: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :105 IV Wavelets, Signal Compression and Image Processing ± Wim Sweldens 107 1 Wavelets and signal compression : :: ::: ::: ::: ::: ::: :: ::: ::: ::: :107 1.1 The need for compression : :: ::: ::: ::: ::: ::: :: ::: ::: ::: :107 1.2 General idea :: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :108 1.3 Error measure : ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :110 1.4 Theory of wavelet compression ::: ::: ::: ::: ::: :: ::: ::: ::: :110 1.5 Image compression : ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :111 1.6 Video compression :: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :116 2 Wavelets and image processing :: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :117 2.1 General idea :: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :117 2.2 Multiscale edge detection and reconstruction :: ::: ::: :: ::: ::: ::: :117 2.3 Enhancement : ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :120 2.4 Others :: ::: ::: ::: :: ::: ::: ::: ::: ::: :: ::: ::: ::: :121 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 :::