Wavelet Toolbox User's Guide - University of …

1 Wavelet ToolboxComputationVisualizationProgrammi ngUser s GuideVersion 1 Michel MisitiYves MisitiGeorges OppenheimJean-Michel PoggiFor Use with MATLAB How to Contact The MathWorks:508-647-7000 Phone508-647-7001 FaxThe MathWorks, Prime Park WayNatick, MA 01760-1500 FTP Technical Product enhancement Bug Documentation error Subscribing user Order status, license renewals, Sales, pricing, and general informationWavlet Toolbox user s Guide COPYRIGHT 1996 - 1997 by The MathWorks, Inc. All Rights 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, GOVERNMENT: If Licensee is acquiring the software on behalf of any unit or agency of the U.

2 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 (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 (c)(2) of the FAR. Contractor/manufacturer is The MathWorks Inc., 24 Prime Park Way, Natick, MA , Simulink, Handle Graphics, and Real-Time Workshop are registered trademarks and Stateflow and Target Language Compiler are trademarks of The MathWorks, product or brand names are trademarks or registered trademarks of their respective History: March 1996 First printing%FAX)@ivContentsPrefaceAbout the Authors.

3 XvAcknowledgments .. xviWhat is the Wavelet Toolbox ? .. xviiHow to Use This Guide .. xviiiFor More Background .. xixInstallation .. xxSystem Recommendations .. xxPlatform-Specific Details .. xxWindows Fonts .. xxOther Platforms Fonts .. xxiMouse Compatibility .. xxiTypographical Conventions .. xxii1 Wavelets: A New Tool for Signal AnalysisFourier Analysis .. 1-3 Short-Time Fourier Analysis .. 1-4 Wavelet Analysis .. 1-5 What Can Wavelet Analysis Do? .. 1-5vContentsWhat 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.

4 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 .. 1-29An 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 .. 1-35vi2 Using WaveletsContinuous 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.

5 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: .. 2-43De-noising and Compression: .. 2-44 Two-Dimensional Analysis Using the Command Line.

6 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 .. 2-71viiContents3 Wavelet ApplicationsDetecting 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.

7 3-12 Discussion .. 3-12 Suppressing Signals .. 3-15 Discussion .. 3-16 Vanishing Moments .. 3-17De-Noising Signals .. 3-18 Discussion .. 3-18 Compressing Signals .. 3-21 Discussion .. 3-224 Wavelets in Action: Examples and Case StudiesIllustrated Examples .. 4-3 Advice to the Reader .. 4-6 About Further Exploration .. 4-7 Example #1: A Sum of Sines .. 4-8viiiExample #2: A Frequency Breakdown .. 4-10 Example #3: Uniform White Noise .. 4-12 Example #4: Colored AR(3) Noise .. 4-14 Example #5: Polynomial + White Noise .. 4-16 Example #6: A Step Signal .. 4-18 Example #7: Two Proximal Discontinuities .. 4-20 Example #8: A Second-Derivative Discontinuity .. 4-22 Example #9: A Ramp + White Noise .. 4-24 Example #10: A Ramp + Colored Noise .. 4-26 Example #11: A Sine + White Noise .. 4-28 Example #12: A Triangle + A Sine.

8 4-30 Example #13: A Triangle + A Sine + Noise .. 4-32 Example #14: A Real Electricity Consumption Signal .. 4-34A Case Study: An Electrical Signal .. 4-36 Data and the External Information .. 4-36 Analysis of the Midday Period .. 4-38 Analysis of the End of the Night Period .. 4-39 Suggestions for Further Analysis .. 4-42 Identify the Sensor Failure .. 4-42 Suppress the Noise .. 4-43 Identify Patterns in the Details .. 4-44 Locate and Suppress Outlying Values .. 4-46 Study Missing Data .. 4-47 Fast Multiplication of Large Matrices .. 4-48 Example 1: Effective Fast Matrix Multiplication .. 4-49 Example 2: Ineffective Fast Matrix Multiplication .. 4-515 Using Wavelet PacketsAbout Wavelet Packet Analysis .. 5-3 One-Dimensional Wavelet Packet Analysis .. 5-6De-Noising a Signal Using Wavelet Packet .. 5-14ixContentsTwo-Dimensional Wavelet Packet Analysis.

9 5-19 Importing and Exporting from Graphical Tools .. 5-26 Saving Information to the Disk .. 5-26 Saving Synthesized Signals .. 5-26 Saving Synthesized Images .. 5-27 Saving One-Dimensional Decomposition Structures .. 5-27 Saving Two-Dimensional Decomposition Structures .. 5-28 Loading Information into the Graphical Tools .. 5-28 Loading Signals .. 5-29 Loading Images .. 5-29 Loading Wavelet Packet Decomposition Structures .. 5-306 Advanced ConceptsMathematical Conventions .. 6-2 General Concepts .. 6-5 Wavelets: A New Tool for Signal Analysis .. 6-5 Wavelet Decomposition: A Hierarchical Organization .. 6-5 Finer and Coarser Resolutions .. 6-6 Wavelet Shapes .. 6-6 Wavelets and Associated Families .. 6-8 Wavelets on a Regular Discrete Grid .. 6-13 Wavelet Transforms: Continuous and Discrete .. 6-14 Local and Global Analysis.

10 6-16 Synthesis: An Inverse Transform .. 6-17 Details and Approximations .. 6-18 The Fast Wavelet Transform (FWT) Algorithm .. 6-21 Filters Used to Calculate the DWT and IDWT .. 6-21 Algorithms .. 6-24 Why Does Such an Algorithm Exist? .. 6-29xOne-Dimensional Wavelet Capabilities .. 6-34 Two-Dimensional Wavelet Capabilities .. 6-40 Dealing with Border Distortion .. 6-46 Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding .. 6-46 Periodized Wavelet Transform .. 6-55 Frequently Asked Questions .. 6-56 Continuous or Discrete Analysis? .. 6-56 Why Are Wavelets Useful for Space-Saving Coding? .. 6-56 Why Do All Wavelets Have Zero Average and SometimesSeveral Vanishing Moments? .. 6-57 What About the Regularity of a Wavelet ? .. 6-57 Are Wavelets Useful in Fields Other Than Signal or Image Processing? .. 6-58 What Functions Are Candidates to Be a Wavelet ?