Wavelet Toolbox™ User's Guide R2022b.

1214 Pages·2022·19.807 MB·English

by  Michel Misiti, Yves Misiti, Georges Oppenheim, Jean-Michel Poggi

Wavelet Toolbox™
User's Guide
Michel Misiti
Yves Misiti
Georges Oppenheim
Jean-Michel Poggi
R2022b
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: [phone]
The MathWorks, Inc.
1 Apple Hill Drive
Natick, MA 01760-2098
Wavelet Toolbox™ User's Guide
© COPYRIGHT 1997–2022 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. By accepting delivery of the Program or Documentation, the government
hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer
software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014.
Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain
to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and
Documentation by the federal government (or other entity acquiring for or through the federal government) and shall
supersede any conflicting contractual terms or conditions. If this License fails to meet the government's needs or is
inconsistent in any respect with federal procurement law, the government agrees to return the Program and
Documentation, unused, to The MathWorks, Inc.
Trademarks
MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See
www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be
trademarks or registered trademarks of their respective holders.
Patents
MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for
more information.
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)
September 2016 Online only Revised for Version 4.17 (Release 2016b)
March 2017 Online only Revised for Version 4.18 (Release 2017a)
September 2017 Online only Revised for Version 4.19 (Release 2017b)
March 2018 Online only Revised for Version 5.0 (Release 2018a)
September 2018 Online only Revised for Version 5.1 (Release 2018b)
March 2019 Online only Revised for Version 5.2 (Release 2019a)
September 2019 Online only Revised for Version 5.3 (Release 2019b)
March 2020 Online only Revised for Version 5.4 (Release 2020a)
September 2020 Online only Revised for Version 5.5 (Release 2020b)
March 2021 Online only Revised for Version 5.6 (Release 2021a)
September 2021 Online only Revised for Version 6.0 (Release 2021b)
March 2022 Online only Revised for Version 6.1 (Release 2022a)
September 2022 Online only Revised for Version 6.2 (Release 2022b)

Contents
Acknowledgments

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Wavelets, Scaling Functions, and Conjugate Quadrature
Mirror Filters
1
Wavelet Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
Daubechies Wavelets: dbN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Symlet Wavelets: symN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Coiflet Wavelets: coifN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Biorthogonal Wavelet Pairs: biorNr.Nd . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Meyer Wavelet: meyr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8
Gaussian Derivatives Family: gaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9
Mexican Hat Wavelet: mexh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Morlet Wavelet: morl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10
Additional Real Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
Complex Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
Wavelet Families and Associated Properties — I . . . . . . . . . . . . . . . . . . . . 1-16
Wavelet Families and Associated Properties — II . . . . . . . . . . . . . . . . . . . 1-17
Lifting Method for Constructing Wavelets . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Lifting Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19
Polyphase Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21
Split, Predict, and Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21
Haar Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-22
Bior2.2 Wavelet Via Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23
Add Lifting Step To Haar Lifting Scheme . . . . . . . . . . . . . . . . . . . . . . . . 1-24
Integer-to-Integer Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26
Orthogonal and Biorthogonal Filter Banks . . . . . . . . . . . . . . . . . . . . . . . . 1-28
Scaling Function and Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-37
Lifting a Filter Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-40
Add Quadrature Mirror and Biorthogonal Wavelet Filters . . . . . . . . . . . 1-44
Least Asymmetric Wavelet and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52
v
Continuous Wavelet Analysis
2
Using Wavelet Time-Frequency Analyzer App . . . . . . . . . . . . . . . . . . . . . . . 2-2
1-D Continuous Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Morse Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10
What Are Morse Wavelets? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10
Morse Wavelet Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10
Effect of Parameter Values on Morse Wavelet Shape . . . . . . . . . . . . . . . . 2-11
Relationship Between Analytic Morse Wavelet and Analytic Signal . . . . . 2-13
Comparison of Analytic Wavelet Transform and Analytic Signal Coefficients
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14
Recommended Morse Wavelet Settings for the CWT . . . . . . . . . . . . . . . . 2-18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18
Boundary Effects and the Cone of Influence . . . . . . . . . . . . . . . . . . . . . . . 2-20
Time-Frequency Analysis and Continuous Wavelet Transform . . . . . . . . 2-29
Continuous Wavelet Analysis of Modulated Signals . . . . . . . . . . . . . . . . . 2-41
Remove Time-Localized Frequency Components . . . . . . . . . . . . . . . . . . . 2-44
Time-Varying Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-49
Continuous Wavelet Analysis of Cusp Signal . . . . . . . . . . . . . . . . . . . . . . 2-53
Two-Dimensional CWT of Noisy Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 2-56
2-D Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64
Discrete Wavelet Analysis
3
Critically Sampled and Oversampled Wavelet Filter Banks . . . . . . . . . . . . 3-2
Double-Density Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3
Dual-Tree Complex Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
Dual-Tree Double-Density Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . 3-7
1-D Decimated Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Decomposition Structure Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
1-D Analysis Using the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10
1-D Analysis Using the Wavelet Analyzer App . . . . . . . . . . . . . . . . . . . . . 3-16
Importing and Exporting Information from the Wavelet Analyzer App . . . 3-27
vi Contents
Fast Wavelet Transform (FWT) Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 3-34
Filters Used to Calculate the DWT and IDWT . . . . . . . . . . . . . . . . . . . . . 3-34
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-36
Why Does Such an Algorithm Exist? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-40
1-D Wavelet Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-43
2-D Wavelet Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-44
Border Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-45
Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-45
Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-46
Arbitrary Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-49
Image Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-51
Nondecimated Discrete Stationary Wavelet Transforms (SWTs) . . . . . . . 3-55
ε-Decimated DWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-55
How to Calculate the ε-Decimated DWT: SWT . . . . . . . . . . . . . . . . . . . . . 3-55
Inverse Discrete Stationary Wavelet Transform (ISWT) . . . . . . . . . . . . . . 3-58
More About SWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-59
1-D Stationary Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
1-D Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-60
Wavelet Changepoint Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-67
Scale-Localized Volatility and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 3-78
R Wave Detection in the ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-87
Wavelet Cross-Correlation for Lead-Lag Analysis . . . . . . . . . . . . . . . . . . . 3-96
1-D Multisignal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-107
1-D Multisignal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-107
2-D Discrete Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-114
Analysis-Decomposition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-114
Synthesis-Reconstruction Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-114
Decomposition Structure Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-114
Denoising and Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-114
Wavelet Image Analysis and Compression . . . . . . . . . . . . . . . . . . . . . . . 3-115
2-D Stationary Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-121
Analysis-Decomposition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-121
Synthesis-Reconstruction Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-121
2-D Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-121
Shearlet Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-128
Shearlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-128
Transform Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-129
Dual-Tree Complex Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 3-131
vii
Analytic Wavelets Using the Dual-Tree Wavelet Transform . . . . . . . . . . 3-159
Multifractal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-162
Time-Frequency Gallery
4
Time-Frequency Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2
Short-Time Fourier Transform (Spectrogram) . . . . . . . . . . . . . . . . . . . . . . 4-3
Continuous Wavelet Transform (Scalogram) . . . . . . . . . . . . . . . . . . . . . . . 4-8
Wigner-Ville Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10
Reassignment and Synchrosqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12
Constant-Q Gabor Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18
Data-Adaptive Methods and Multiresolution Analysis . . . . . . . . . . . . . . . 4-19
Wavelet Packets
5
About Wavelet Packet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2
Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
From Wavelets to Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5
Wavelet Packets in Action: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5-6
Building Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8
Wavelet Packet Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11
Organizing the Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12
Choosing the Optimal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-13
Some Interesting Subtrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16
Wavelet Packets 2-D Decomposition Structure . . . . . . . . . . . . . . . . . . . . 5-19
Wavelet Packets for Compression and Denoising . . . . . . . . . . . . . . . . . . . 5-19
Introduction to Object-Oriented Features . . . . . . . . . . . . . . . . . . . . . . . . . 5-20
Objects in the Wavelet Toolbox Software . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21
Examples Using Wavelet Packet Tree Objects . . . . . . . . . . . . . . . . . . . . . . 5-22
plot and wpviewcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-22
Change Terminal Node Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-24
Thresholding Wavelet Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25
Description of Objects in the Wavelet Toolbox Software . . . . . . . . . . . . . 5-28
WTBO Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
NTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28
Private . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29
DTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-29
WPTREE Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-30
Build Wavelet Tree Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33
Building a Wavelet Tree Object (WTREE) . . . . . . . . . . . . . . . . . . . . . . . . 5-33
viii Contents
Working With Wavelet Tree Objects (WTREE) . . . . . . . . . . . . . . . . . . . . . 5-33
Building a Right Wavelet Tree Object (RWVTREE) . . . . . . . . . . . . . . . . . 5-40
Working With Right Wavelet Tree Objects (RWVTREE) . . . . . . . . . . . . . . 5-41
Building a Wavelet Tree Object (WVTREE) . . . . . . . . . . . . . . . . . . . . . . . 5-46
Working With Wavelet Tree Objects (WVTREE) . . . . . . . . . . . . . . . . . . . . 5-47
Building a Wavelet Tree Object (EDWTTREE) . . . . . . . . . . . . . . . . . . . . . 5-53
Working With Wavelet Tree Object (EDWTTREE) . . . . . . . . . . . . . . . . . . 5-54
Denoising, Nonparametric Function Estimation, and
Compression
6
Wavelet Denoising and Nonparametric Function Estimation . . . . . . . . . . 6-2
Denoising Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3
Soft or Hard Thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5
Dealing with Unscaled Noise and Nonwhite Noise . . . . . . . . . . . . . . . . . . 6-6
Wavelet Denoising in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7
Extension to Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11
1-D Wavelet Variance Adaptive Thresholding . . . . . . . . . . . . . . . . . . . . . 6-13
Wavelet Denoising Analysis Measurements . . . . . . . . . . . . . . . . . . . . . . . 6-16
Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18
Denoise a Signal with the Wavelet Signal Denoiser . . . . . . . . . . . . . . . . . 6-26
Translation Invariant Wavelet Denoising with Cycle Spinning . . . . . . . . 6-37
1-D Cycle Spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-37
Multivariate Wavelet Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-40
Multivariate Wavelet Denoising — Command Line . . . . . . . . . . . . . . . . . 6-40
Wavelet Multiscale Principal Components Analysis . . . . . . . . . . . . . . . . . 6-45
Wavelet Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-48
Compression Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-50
Wavelet Compression for Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-51
Effects of Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-51
True Compression Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-53
Quantitative and Perceptual Quality Measures . . . . . . . . . . . . . . . . . . . . 6-54
More Information on True Compression . . . . . . . . . . . . . . . . . . . . . . . . . 6-55
2-D Wavelet Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-56
Compression by Global Thresholding and Huffman Encoding . . . . . . . . . 6-56
Uncompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-57
Compression by Progressive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-58
Handling Truecolor Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-61
ix
Matching Pursuit
7
Matching Pursuit Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2
Redundant Dictionaries and Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2
Nonlinear Approximation in Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2
Basic Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3
Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5
Weak Orthogonal Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5
Matching Pursuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6
Sensing Dictionary Creation and Visualization . . . . . . . . . . . . . . . . . . . . . 7-6
Orthogonal Matching Pursuit on 1-D Signal . . . . . . . . . . . . . . . . . . . . . . 7-10
Electricity Consumption Analysis Using Matching Pursuit . . . . . . . . . . . . 7-13
Code Generation from MATLAB Support in Wavelet Toolbox
8
Code Generation Support, Usage Notes, and Limitations . . . . . . . . . . . . . 8-2
Denoise Signal Using Generated C Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5
Generate Code to Denoise a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9
CUDA Code from CWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11
Special Topics
9
Wavelet Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2
Wavelet Scattering Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2
Invariance Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3
Quality Factors and Filter Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6
In Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8
Wavelet Scattering Invariance Scale and Oversampling . . . . . . . . . . . . . 9-10
Empirical Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-14
Tunable Q-factor Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19
Frequency-Domain Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19
TQWT Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19
Redundancy and Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-20
Example: MRA of Audio Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-22
x Contents