Wavelet Toolbox For Use with MATLAB® Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi Computation Visualization Programming User’s Guide Version 1 How to Contact The MathWorks: ☎ 508-647-7000 Phone 508-647-7001 Fax FAX The MathWorks, Inc. Mail ✉ 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web ftp.mathworks.com Anonymous FTP server comp.soft-sys.matlab Newsgroup @ [email protected] Technical support [email protected] Product enhancement suggestions [email protected] Bug reports [email protected] Documentation error reports [email protected] Subscribing user registration [email protected] Order status, license renewals, passcodes [email protected] Sales, pricing, and general information Wavlet Toolbox User’s Guide COPYRIGHT 1996 - 1997 by The MathWorks, Inc. All Rights Reserved. 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 repro- duced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the software on behalf of any unit or agency of the U. S. Government, the following shall apply: (a) for units of the Department of Defense: RESTRICTED RIGHTS LEGEND: Use, duplication, or disclosure by the Government is subject to restric- tions as set forth in subparagraph (c)(1)(ii) of the Rights in Technical Data and Computer Software Clause at DFARS 252.227-7013. (b) for any other unit or agency: NOTICE - Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction and disclosure are as set forth in Clause 52.227-19(c)(2) of the FAR. Contractor/manufacturer is The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760-1500. MATLAB, Simulink, Handle Graphics, and Real-Time Workshop are registered trademarks and Stateflow and Target Language Compiler are trademarks of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders. Printing History: March 1996 First printing Contents Preface About the Authors . xv Acknowledgments . xvi What is the Wavelet Toolbox? . xvii How to Use This Guide . xviii For More Background . xix Installation . xx System Recommendations . xx Platform-Specific Details . xx Windows Fonts . xx Other Platforms Fonts . xxi Mouse Compatibility . xxi Typographical Conventions . xxii Wavelets: A New Tool for Signal Analysis 1 Fourier Analysis . 1-3 Short-Time Fourier Analysis . 1-4 Wavelet Analysis . 1-5 What Can Wavelet Analysis Do? . 1-5 iv What is Wavelet Analysis? . 1-7 Number of Dimensions . 1-7 The Continuous Wavelet Transform . 1-8 Scaling . 1-9 Shifting . 1-10 Five Easy Steps to a Continuous Wavelet Transform . 1-10 Scale and Frequency . 1-13 The Scale of Nature . 1-13 What’s Continuous About the Continuous Wavelet Transform? . 1-15 The Discrete Wavelet Transform . 1-16 One-Stage Filtering: Approximations and Details . 1-16 Multiple-Level Decomposition . 1-19 Number of Levels . 1-19 Wavelet Reconstruction . 1-20 Reconstruction Filters . 1-21 Reconstructing Approximations and Details . 1-21 Relationship of Filters to Wavelet Shapes . 1-23 The Scaling Function . 1-25 Multistep Decomposition and Reconstruction . 1-25 Wavelet Packet Analysis . 1-27 History of Wavelets . 1-29 An Introduction to the Wavelet Families . 1-30 Haar . 1-31 Daubechies . 1-31 Biorthogonal . 1-32 Coiflets . 1-33 Symlets . 1-33 Morlet . 1-34 Mexican Hat . 1-34 Meyer . 1-35 v Contents Using Wavelets 2 Continuous Wavelet Analysis (One-Dimensional) . 2-3 Continuous Analysis Using the Command Line . 2-3 Continuous Analysis Using the Graphical Interface . 2-7 Importing and Exporting Information from the Graphical Interface . 2-11 Loading Signals into the Continuous Wavelet 1-D Tool . 2-11 Saving Wavelet Coefficients . 2-12 One-Dimensional Discrete Wavelet Analysis . 2-13 Analysis Decomposition Functions: . 2-13 Synthesis Reconstruction Functions: . 2-13 Decomposition Structure Utilities: Analysis Decomposition Functions: . 2-14 One-Dimensional Analysis Using the Command Line . 2-15 One-Dimensional Analysis Using the Graphical Interface . 2-22 Importing and Exporting Information from the Graphical Interface . 2-38 Saving Information to the Disk . 2-38 Loading Information into the Wavelet 1-D Tool . 2-40 Two-Dimensional Discrete Wavelet Analysis . 2-43 Analysis-Decomposition Functions: . 2-43 Synthesis-Reconstruction Functions: . 2-43 Decomposition Structure Utilities: . 2-43 De-noising and Compression: . 2-44 Two-Dimensional Analysis Using the Command Line . 2-44 Two-Dimensional Analysis Using the Graphical Interface . 2-52 Importing and Exporting Information from the Graphical Interface . 2-59 Saving Information to the Disk . 2-59 Loading Information into the Wavelet 2-D Tool . 2-62 Working with Indexed Images . 2-66 Understanding Images in MATLAB . 2-66 Indexed Images . 2-66 Wavelet Decomposition of Indexed Images . 2-68 How Decompositions Are Displayed . 2-71 vi Wavelet Applications 3 Detecting Discontinuities and Breakdown Points I . 3-3 Discussion . 3-4 Guidelines for Detecting Discontinuities . 3-4 Detecting Discontinuities and Breakdown Points II . 3-6 Discussion . 3-7 Detecting Long-Term Evolution . 3-8 Discussion . 3-9 Detecting Self-Similarity . 3-10 Wavelet Coefficients and Self-Similarity . 3-10 Discussion . 3-11 Identifying Pure Frequencies . 3-12 Discussion . 3-12 Suppressing Signals . 3-15 Discussion . 3-16 Vanishing Moments . 3-17 De-Noising Signals . 3-18 Discussion . 3-18 Compressing Signals . 3-21 Discussion . 3-22 Wavelets in Action: Examples and Case Studies 4 Illustrated Examples . 4-3 Advice to the Reader . 4-6 About Further Exploration . 4-7 Example #1: A Sum of Sines . 4-8 vii Contents Example #2: A Frequency Breakdown . 4-10 Example #3: Uniform White Noise . ..