Wavelet Toolbox™ User's Guide Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi R2016a How to Contact MathWorks Latest news: www.mathworks.com Sales and services: www.mathworks.com/sales_and_services User community: www.mathworks.com/matlabcentral Technical support: www.mathworks.com/support/contact_us Phone: 508-647-7000 The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Wavelet Toolbox™ User's Guide © COPYRIGHT 1997–2016 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. 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Revision History March 1997 First printing New for Version 1.0 September 2000 Second printing Revised for Version 2.0 (Release 12) June 2001 Online only Revised for Version 2.1 (Release 12.1) July 2002 Online only Revised for Version 2.2 (Release 13) June 2004 Online only Revised for Version 3.0 (Release 14) July 2004 Third printing Revised for Version 3.0 October 2004 Online only Revised for Version 3.0.1 (Release 14SP1) March 2005 Online only Revised for Version 3.0.2 (Release 14SP2) June 2005 Fourth printing Minor revision for Version 3.0.2 September 2005 Online only Minor revision for Version 3.0.3 (Release R14SP3) March 2006 Online only Minor revision for Version 3.0.4 (Release 2006a) September 2006 Online only Revised for Version 3.1 (Release 2006b) March 2007 Online only Revised for Version 4.0 (Release 2007a) September 2007 Online only Revised for Version 4.1 (Release 2007b) October 2007 Fifth printing Revised for Version 4.1 March 2008 Online only Revised for Version 4.2 (Release 2008a) October 2008 Online only Revised for Version 4.3 (Release 2008b) March 2009 Online only Revised for Version 4.4 (Release 2009a) September 2009 Online only Minor revision for Version 4.4.1 (Release 2009b) March 2010 Online only Revised for Version 4.5 (Release 2010a) September 2010 Online only Revised for Version 4.6 (Release 2010b) April 2011 Online only Revised for Version 4.7 (Release 2011a) September 2011 Online only Revised for Version 4.8 (Release 2011b) March 2012 Online only Revised for Version 4.9 (Release 2012a) September 2012 Online only Revised for Version 4.10 (Release 2012b) March 2013 Online only Revised for Version 4.11 (Release 2013a) September 2013 Online only Revised for Version 4.12 (Release 2013b) March 2014 Online only Revised for Version 4.13 (Release 2014a) October 2014 Online only Revised for Version 4.14 (Release 2014b) March 2015 Online only Revised for Version 4.14.1 (Release 2015a) September 2015 Online only Revised for Version 4.15 (Release 2015b) March 2016 Online only Revised for Version 4.16 (Release 2016a) Contents Acknowledgments Wavelets, Scaling Functions, and Conjugate Quadrature Mirror Filters 1 Wavelet Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Daubechies Wavelets: dbN . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6 Symlet Wavelets: symN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Coiflet Wavelets: coifN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Biorthogonal Wavelet Pairs: biorNr.Nd . . . . . . . . . . . . . . . . . 1-8 Meyer Wavelet: meyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 Gaussian Derivatives Family: gaus . . . . . . . . . . . . . . . . . . . 1-12 Mexican Hat Wavelet: mexh . . . . . . . . . . . . . . . . . . . . . . . . 1-13 Morlet Wavelet: morl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14 Additional Real Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 Complex Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 Wavelet Families and Associated Properties — I . . . . . . . . . 1-19 Wavelet Families and Associated Properties — II . . . . . . . . 1-21 Adding Your Own Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 Preparing to Add a New Wavelet Family . . . . . . . . . . . . . . 1-23 Adding a New Wavelet Family . . . . . . . . . . . . . . . . . . . . . . 1-28 After Adding a New Wavelet Family . . . . . . . . . . . . . . . . . . 1-35 Lifting Method for Constructing Wavelets . . . . . . . . . . . . . . 1-37 Lifting Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-38 Polyphase Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 1-39 Split, Predict, and Update . . . . . . . . . . . . . . . . . . . . . . . . . . 1-40 Haar Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . 1-41 Bior2.2 Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . 1-42 v Lifting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-43 Primal Lifting from Haar . . . . . . . . . . . . . . . . . . . . . . . . . . 1-45 Integer-to-Integer Wavelet Transform . . . . . . . . . . . . . . . . . 1-46 Orthogonal and Biorthogonal Filter Banks . . . . . . . . . . . . . 1-48 Lifting a Filter Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-60 Adding a Quadrature Mirror Filter . . . . . . . . . . . . . . . . . . . . 1-67 Continuous Wavelet Analysis 2 1-D Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . 2-2 Command Line Continuous Wavelet Analysis . . . . . . . . . . . . 2-4 Continuous Analysis Using the Graphical Interface . . . . . . . 2-8 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18 Time-Frequency Analysis of Modulated Signals . . . . . . . . . 2-20 Remove Time-Localized Frequency Components . . . . . . . . 2-25 Time-Varying Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-31 One-Dimensional Complex Continuous Wavelet Analysis . 2-35 Complex Continuous Analysis Using the Command Line . . . 2-36 Complex Continuous Analysis Using the Graphical Interface 2-38 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Interactive DFT-Based Continuous Wavelet Analysis . . . . . 2-43 Manual Selection of CWT Coefficients . . . . . . . . . . . . . . . . . 2-48 Two-Dimensional CWT of Noisy Pattern . . . . . . . . . . . . . . . 2-52 2-D Continuous Wavelet Transform App . . . . . . . . . . . . . . . 2-61 2-D Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . 2-61 2-D CWT App Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-62 vi Contents Discrete Wavelet Analysis 3 Critically Sampled and Oversampled Wavelet Filter Banks . 3-2 Double-Density Wavelet Transform . . . . . . . . . . . . . . . . . . . . 3-3 Dual-Tree Complex Wavelet Transform . . . . . . . . . . . . . . . . 3-6 Dual-Tree Double-Density Wavelet Transforms . . . . . . . . . . 3-10 1-D Decimated Wavelet Transforms . . . . . . . . . . . . . . . . . . . 3-12 Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . 3-12 Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . . 3-12 Decomposition Structure Utilities . . . . . . . . . . . . . . . . . . . . 3-12 Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . . . 3-12 One-Dimensional Analysis Using the Command Line . . . . . 3-14 One-Dimensional Analysis Using the Graphical Interface . . 3-22 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-36 Fast Wavelet Transform (FWT) Algorithm . . . . . . . . . . . . . . 3-45 Filters Used to Calculate the DWT and IDWT . . . . . . . . . . 3-45 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-48 Why Does Such an Algorithm Exist? . . . . . . . . . . . . . . . . . . 3-53 One-Dimensional Wavelet Capabilities . . . . . . . . . . . . . . . . 3-57 Two-Dimensional Wavelet Capabilities . . . . . . . . . . . . . . . . 3-58 Border Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59 Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59 Nondecimated Discrete Stationary Wavelet Transforms (SWTs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-68 -Decimated DWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-68 How to Calculate the -Decimated DWT: SWT . . . . . . . . . . . 3-69 Inverse Discrete Stationary Wavelet Transform (ISWT) . . . 3-73 More About SWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-73 One-Dimensional Nondecimated Discrete Stationary Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-74 Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . 3-74 Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . . 3-74 One-Dimensional Analysis Using the Command Line . . . . . 3-75 Interactive 1-D Stationary Wavelet Transform Denoising . . 3-84 Importing and Exporting from the GUI . . . . . . . . . . . . . . . . 3-88 vii Wavelet Changepoint Detection . . . . . . . . . . . . . . . . . . . . . . . 3-90 Scale-Localized Volatility and Correlation . . . . . . . . . . . . . 3-105 R Wave Detection in the ECG . . . . . . . . . . . . . . . . . . . . . . . 3-116 Wavelet Cross-Correlation for Lead-Lag Analysis . . . . . . . 3-126 One-Dimensional Multisignal Analysis . . . . . . . . . . . . . . . . 3-138 One-Dimensional Multisignal Analysis — Command Line . 3-138 Interactive One-Dimensional Multisignal Analysis . . . . . . 3-147 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-180 Two-Dimensional Discrete Wavelet Analysis . . . . . . . . . . . 3-187 Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . 3-187 Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . 3-187 Decomposition Structure Utilities . . . . . . . . . . . . . . . . . . . 3-187 Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . . 3-188 Two-Dimensional Analysis — Command Line . . . . . . . . . . 3-188 Interactive Two-Dimensional Wavelet Analysis . . . . . . . . . 3-196 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-205 Two-Dimensional Discrete Stationary Wavelet Analysis . 3-214 Analysis-Decomposition Function . . . . . . . . . . . . . . . . . . . 3-214 Synthesis-Reconstruction Function . . . . . . . . . . . . . . . . . . 3-214 Two-Dimensional Analysis Using the Command Line . . . . 3-214 Interactive 2-D Stationary Wavelet Transform Denoising . 3-222 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-225 Three-Dimensional Discrete Wavelet Analysis . . . . . . . . . . 3-227 Performing Three-Dimensional Analysis Using the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-227 Performing Three-Dimensional Analysis Using the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-227 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-234 Dual-Tree Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . 3-238 Analytic Wavelets Using the Dual-Tree Wavelet Transform 3-260 viii Contents Wavelet Packets 4 About Wavelet Packet Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4-2 One-Dimensional Wavelet Packet Analysis . . . . . . . . . . . . . . 4-6 Starting the Wavelet Packet 1-D Tool . . . . . . . . . . . . . . . . . . 4-6 Loading a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 Analyzing a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Computing the Best Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Compressing a Signal Using Wavelet Packets . . . . . . . . . . . 4-10 De-Noising a Signal Using Wavelet Packets . . . . . . . . . . . . 4-13 Two-Dimensional Wavelet Packet Analysis . . . . . . . . . . . . . 4-20 Starting the Wavelet Packet 2-D Tool . . . . . . . . . . . . . . . . . 4-20 Compressing an Image Using Wavelet Packets . . . . . . . . . . 4-23 Importing and Exporting from Graphical Tools . . . . . . . . . 4-27 Saving Information to Disk . . . . . . . . . . . . . . . . . . . . . . . . . 4-27 Loading Information into the Graphical Tools . . . . . . . . . . . 4-30 Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34 From Wavelets to Wavelet Packets . . . . . . . . . . . . . . . . . . . 4-34 Wavelet Packets in Action: An Introduction . . . . . . . . . . . . 4-35 Building Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . 4-38 Wavelet Packet Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-41 Organizing the Wavelet Packets . . . . . . . . . . . . . . . . . . . . . 4-42 Choosing the Optimal Decomposition . . . . . . . . . . . . . . . . . 4-44 Some Interesting Subtrees . . . . . . . . . . . . . . . . . . . . . . . . . 4-49 Wavelet Packets 2-D Decomposition Structure . . . . . . . . . . 4-52 Wavelet Packets for Compression and Denoising . . . . . . . . . 4-52 Introduction to Object-Oriented Features . . . . . . . . . . . . . . 4-53 Objects in the Wavelet Toolbox Software . . . . . . . . . . . . . . . 4-54 Examples Using Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-55 plot and wpviewcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-55 drawtree and readtree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-59 Change Terminal Node Coefficients . . . . . . . . . . . . . . . . . . 4-61 Thresholding Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . 4-63 ix Description of Objects in the Wavelet Toolbox Software . . 4-67 WTBO Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-67 NTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-68 Private . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-68 DTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-68 WPTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-70 Advanced Use of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-73 Building a Wavelet Tree Object (WTREE) . . . . . . . . . . . . . . 4-73 Building a Right Wavelet Tree Object (RWVTREE) . . . . . . . 4-74 Building a Wavelet Tree Object (WVTREE) . . . . . . . . . . . . . 4-75 Building a Wavelet Tree Object (EDWTTREE) . . . . . . . . . . 4-77 Denoising, Nonparametric Function Estimation, and Compression 5 Wavelet Denoising and Nonparametric Function Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Threshold Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Soft or Hard Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 Dealing with Unscaled Noise and Nonwhite Noise . . . . . . . . 5-7 Denoising in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Extension to Image Denoising . . . . . . . . . . . . . . . . . . . . . . . 5-11 One-Dimensional Wavelet Variance Adaptive Thresholding . 5-12 Wavelet Denoising Analysis Measurements . . . . . . . . . . . . . 5-14 Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-15 Translation Invariant Wavelet Denoising with Cycle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 1-D Cycle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-23 2-D Cycle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26 One-Dimensional Adaptive Thresholding of Wavelet Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-30 One-Dimensional Interactive Local Thresholding . . . . . . . . 5-30 Importing and Exporting Information from the Graphical Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-38 x Contents