Language: English
Pages: 934
Preface......Page 15
About the Authors......Page 17
Notes by Yves Meyer......Page 18
Notes by Ingrid Daubechies......Page 20
Acknowledgments......Page 21
What Is the Wavelet Toolbox?......Page 22
All Users......Page 23
Caution......Page 24
For More Background......Page 25
Windows Fonts......Page 26
Mouse Compatibility......Page 27
Typographical Conventions......Page 28
Related Products......Page 29
Wavelets: A New Tool for Signal Analysis......Page 31
Time Aspects......Page 33
Wavelet Decomposition as a Whole......Page 34
Fourier Analysis......Page 35
Short-Time Fourier Analysis......Page 36
Wavelet Analysis......Page 37
What Can Wavelet Analysis Do?......Page 38
Number of Dimensions......Page 39
The Continuous Wavelet Transform......Page 40
Scaling......Page 41
Five Easy Steps to a Continuous Wavelet Transform......Page 42
The Scale of Nature......Page 45
What’s Continuous About the Continuous Wavelet Transform?......Page 47
One-Stage Filtering: Approximations and Details......Page 48
Number of Levels......Page 51
Wavelet Reconstruction......Page 52
Reconstructing Approximations and Details......Page 53
Relationship of Filters to Wavelet Shapes......Page 55
The Scaling Function......Page 57
Multistep Decomposition and Reconstruction......Page 58
Wavelet Packet Analysis......Page 59
History of Wavelets......Page 61
An Introduction to the Wavelet Families......Page 62
Daubechies......Page 63
Biorthogonal......Page 64
Symlets......Page 65
Mexican Hat......Page 66
Complex Wavelets......Page 67
Using Wavelets......Page 69
One-Dimensional Continuous Wavelet Analysis......Page 72
Continuous Analysis Using the Command Line......Page 73
Continuous Analysis Using the Graphical Interface......Page 76
Loading Signals into the Continuous Wavelet 1-D Tool......Page 85
Saving Wavelet Coefficients......Page 86
One-Dimensional Complex Continuous Wavelet Analysis......Page 87
Complex Continuous Analysis Using the Command Line......Page 88
Complex Continuous Analysis Using the Graphical Interface......Page 90
Importing and Exporting Information from the Graphical Interface......Page 95
Decomposition Structure Utilities......Page 96
De-noising and Compression......Page 97
One-Dimensional Analysis Using the Command Line......Page 98
One-Dimensional Analysis Using the Graphical Interface......Page 107
Selecting Different Views of the Decomposition......Page 114
Saving Information to Disk......Page 125
Saving Synthesized Signals......Page 126
Saving Discrete Wavelet Transform Coefficients......Page 128
Saving Decompositions......Page 129
Loading Signals......Page 130
Loading Decompositions......Page 131
Decomposition Structure Utilities......Page 133
De-noising and Compression......Page 134
Two-Dimensional Analysis Using the Command Line......Page 135
Two-Dimensional Analysis Using the Graphical Interface......Page 142
Using Square Mode Features......Page 146
Using Tree Mode Features......Page 147
Zooming in on Detail......Page 148
Importing and Exporting Information from the Graphical Interface......Page 151
Saving Synthesized Images......Page 152
Saving Discrete Wavelet Transform Coefficients......Page 153
Saving Decompositions......Page 154
Loading Information into the Wavelet 2-D Tool......Page 155
Loading Images......Page 156
Loading Discrete Wavelet Transform Coefficients......Page 157
Loading Decompositions......Page 158
Indexed Images......Page 160
Wavelet Decomposition of Indexed Images......Page 161
Other Images......Page 162
Example 1 - Converting Color Indexed Images......Page 163
Example 2 - Converting an RGB tif-Image.......Page 165
Synthesis-Reconstruction Functions......Page 167
One-Dimensional Analysis Using the Command Line......Page 168
One-Dimensional Analysis for De-Noising Using the Graphical Interface......Page 177
Importing and Exporting Information from the Graphical Interface......Page 182
Synthesis-Reconstruction Function......Page 184
Two-Dimensional Analysis Using the Command Line......Page 185
Two-Dimensional Analysis for De-Noising Using the Graphical Interface......Page 194
Importing and Exporting Information from the Graphical Interface......Page 199
One-Dimensional Estimation Using the GUI for Equally Spaced Observations (Fixed Design)......Page 201
One-Dimensional Estimation Using the GUI for Randomly Spaced Observations (Stochastic Design)......Page 207
Saving Function......Page 209
Loading Data......Page 210
One-Dimensional Estimation Using the Graphical Interface......Page 211
Importing and Exporting Information from the Graphical Interface......Page 216
One-Dimensional Variance Adaptive Thresholding of Wavelet Coefficients......Page 218
One-Dimensional Local Thresholding for De-noising Using the Graphical Interface......Page 219
Modifying Interval Dependent Thresholds.......Page 224
Importing and Exporting Information from the Graphical Interface......Page 226
One-Dimensional Selection of Wavelet Coefficients Using the Graphical Interface......Page 228
Selecting Biggest Coefficients Globally.......Page 231
Selecting Coefficients Manually......Page 233
Selecting Coefficients Automatically.......Page 235
Two-Dimensional Selection of Wavelet Coefficients Using the Graphical Interface......Page 237
Selecting Biggest Coefficients Globally.......Page 240
Selecting Biggest Coefficients by Level.......Page 241
Selecting Coefficients Automatically.......Page 242
One-Dimensional Extension Using the Graphical Interface......Page 244
Extending Signal for SWT.......Page 248
Truncating Signal.......Page 251
Importing and Exporting Information from the Graphical Interface......Page 252
Two-Dimensional Extension Using the Graphical Interface......Page 253
Importing and Exporting Information from the Graphical Interface......Page 256
Wavelet Applications......Page 257
Detecting Discontinuities and Breakdown Points I......Page 259
Guidelines for Detecting Discontinuities......Page 260
Detecting Discontinuities and Breakdown Points II......Page 262
Discussion......Page 263
Detecting Long-Term Evolution......Page 264
Discussion......Page 265
Wavelet Coefficients and Self-Similarity......Page 266
Discussion......Page 267
Discussion......Page 268
Suppressing Signals......Page 271
Discussion......Page 272
Vanishing Moments......Page 273
Discussion......Page 274
De-Noising Images......Page 277
Discussion......Page 278
Discussion......Page 282
Fast Multiplication of Large Matrices......Page 284
Example 1: Effective Fast Matrix Multiplication......Page 285
Example 2: Ineffective Fast Matrix Multiplication......Page 287
Wavelets in Action: Examples and Case Studies......Page 289
Illustrated Examples......Page 291
Advice to the Reader......Page 294
Tip 4......Page 296
Example 1: A Sum of Sines......Page 297
Example 2: A Frequency Breakdown......Page 299
Example 3: Uniform White Noise......Page 301
Example 4: Colored AR(3) Noise......Page 303
Example 5: Polynomial + White Noise......Page 305
Example 6: A Step Signal......Page 307
Example 7: Two Proximal Discontinuities......Page 309
Example 8: A Second-Derivative Discontinuity......Page 311
Example 9: A Ramp + White Noise......Page 313
Example 10: A Ramp + Colored Noise......Page 315
Example 11: A Sine + White Noise......Page 317
Example 12: A Triangle + A Sine......Page 319
Example 13: A Triangle + A Sine + Noise......Page 321
Example 14: A Real Electricity Consumption Signal......Page 323
Data and the External Information......Page 325
Analysis of the Midday Period......Page 327
Analysis of the End of the Night Period......Page 328
Identify the Sensor Failure......Page 331
Suppress the Noise......Page 332
Identify Patterns in the Details......Page 334
Locate and Suppress Outlying Values......Page 335
Study Missing Data......Page 336
Using Wavelet Packets......Page 337
Synthesis-Reconstruction Functions......Page 339
De-Noising and Compression......Page 340
Starting the Wavelet Packet 1-D Tool......Page 343
Loading a Signal......Page 344
Analyzing a Signal......Page 345
Computing the Best Tree......Page 346
Selecting a Threshold for Compression......Page 348
Compressing a Signal......Page 350
Starting the Wavelet Packet 1-D Tool......Page 351
Loading a Signal......Page 352
Analyzing a Signal......Page 353
Computing the Best Tree and Performing De-Noising......Page 354
Two-Dimensional Wavelet Packet Analysis......Page 357
Starting the Wavelet Packet 2-D Tool......Page 358
Loading an Image......Page 359
Analyzing an Image......Page 360
Compressing an Image Using Wavelet Packets......Page 361
Saving Synthesized Signals......Page 365
Saving Synthesized Images......Page 366
Saving One-Dimensional Decomposition Structures......Page 367
Saving Two-Dimensional Decomposition Structures......Page 368
Loading Signals......Page 369
Loading Images......Page 370
Loading Wavelet Packet Decomposition Structures......Page 371
Advanced Concepts......Page 373
Mathematical Conventions......Page 374
Wavelet Decomposition: A Hierarchical Organization......Page 377
Wavelet Shapes......Page 378
Wavelets and Associated Families......Page 380
Wavelet Transforms: Continuous and Discrete......Page 385
Local and Global Analysis......Page 387
Details and Approximations......Page 388
Filters Used to Calculate the DWT and IDWT......Page 392
Algorithms......Page 396
Why Does Such an Algorithm Exist?......Page 401
Synthesis-reconstruction capabilities......Page 405
Basic two-dimensional objects......Page 406
Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding......Page 408
Zero-Padding......Page 412
Smooth Padding......Page 413
Original Image......Page 414
Symmetric Extension......Page 415
Smooth Padding......Page 416
e-Decimated DWT......Page 417
How to Calculate the e-Decimated DWT: SWT......Page 418
Step 0 (Original Data)......Page 419
Step 3......Page 420
More About SWT......Page 423
Why Are Wavelets Useful for Space-Saving Coding?......Page 424
What About the Regularity of a Wavelet y?......Page 425
Are Wavelets Useful in Fields Other Than Signal or Image Processing?......Page 426
Is It Easy to Build a New Wavelet?......Page 427
What Is the Link Between Wavelet and Fourier Analysis?......Page 428
How to Connect Scale to Frequency?......Page 429
Wavelet Families: Additional Discussion......Page 434
Daubechies Wavelets: dbN......Page 435
dbN......Page 436
Symlet Wavelets: symN......Page 437
Coiflet Wavelets: coifN......Page 438
Biorthogonal Wavelet Pairs: biorNr.Nd......Page 439
Meyer Wavelet: meyr......Page 441
Battle-Lemarie Wavelets......Page 442
Mexican Hat Wavelet: mexh......Page 443
Morlet Wavelet: morl......Page 444
Gaussian Derivatives Family: gaus......Page 445
FIR Based Approximation of the Meyer Wavelet: dmey......Page 446
Complex Gaussian Wavelets: cgau......Page 447
Complex Morlet Wavelets: cmor......Page 448
Complex Frequency B-Spline Wavelets: fbsp......Page 449
Complex Shannon Wavelets: shan......Page 450
Summary of Wavelet Families and Associated Properties (Part 1)......Page 452
Summary of Wavelet Families and Associated Properties (Part 2)......Page 454
Suppressing Signals......Page 456
Noise Processing......Page 459
The Basic One-Dimensional Model......Page 461
De-Noising Procedure Principles......Page 462
Soft or Hard Thresholding?......Page 463
Threshold Selection Rules......Page 464
Dealing with Unscaled Noise and Nonwhite Noise......Page 466
De-Noising in Action......Page 467
Extension to Image De-Noising......Page 470
One-Dimensional Variance Adaptive Thresholding of Wavelet Coefficients......Page 471
Data Compression......Page 474
Density Estimation......Page 477
How to Perform Wavelet-Based Density Estimation......Page 478
A More Technical Viewpoint......Page 479
Regression Applications......Page 482
How to Perform Wavelet-Based Regression Estimation?......Page 483
A More Technical Viewpoint......Page 484
One-Dimensional DWT and SWT De-Noising......Page 485
Two-Dimensional DWT and SWT De-Noising......Page 486
One-Dimensional Wavelet Packet De-Noising......Page 487
Density Estimation......Page 488
Scarce High, Medium, and Low......Page 489
Penalized High, Medium, and Low......Page 490
Fixed Form......Page 491
Global, and By level 1, 2, 3......Page 492
From Wavelets to Wavelet Packets: Decomposing the Details......Page 493
How to Obtain and Explain These Graphs?......Page 494
Example 2: Analyzing a Chirp Signal......Page 497
Building Wavelet Packets......Page 498
Wavelet Packet Atoms......Page 500
Organizing the Wavelet Packets......Page 503
Choosing the Optimal Decomposition......Page 504
Example 2: Minimum-Entropy Decomposition......Page 506
Some Interesting Subtrees......Page 509
Wavelet Packets 2-D Decomposition Structure......Page 510
Wavelet Packets for Compression and De-Noising......Page 511
References......Page 512
Adding Your Own Wavelets......Page 517
Determine the Wavelet Type......Page 519
Define the Orders of Wavelets Within the Given Family......Page 520
Type 1 (Orthogonal with FIR Filter)......Page 521
Type 4 or Type 5 (No FIR Filter; No Scale Function)......Page 522
Define the Effective Support......Page 523
Example 1......Page 524
Example 2......Page 528
After Adding a New Wavelet Family......Page 532
Function Reference......Page 533
Functions - By Category......Page 534
Wavelet Families......Page 535
Discrete Wavelets: Two-Dimensional......Page 537
Discrete Stationary Wavelet Transform Algorithms......Page 539
Tree Management Utilities......Page 540
Obsolete Functions......Page 542
Functions - Alphabetical List......Page 543
allnodes......Page 547
appcoef......Page 549
appcoef2......Page 551
bestlevt......Page 553
besttree......Page 555
biorfilt......Page 558
biorwavf......Page 563
centfrq......Page 564
cfs2wpt......Page 567
cgauwavf......Page 570
cmorwavf......Page 572
coifwavf......Page 574
cwt......Page 575
dbaux......Page 580
dbwavf......Page 583
ddencmp......Page 584
depo2ind......Page 587
detcoef......Page 589
detcoef2......Page 591
disp......Page 593
drawtree......Page 594
dtree......Page 596
dwt......Page 598
dwt2......Page 602
dwtmode......Page 606
dyaddown......Page 609
dyadup......Page 611
entrupd......Page 614
fbspwavf......Page 615
gauswavf......Page 617
get......Page 619
idwt......Page 621
idwt2......Page 625
ind2depo......Page 628
intwave......Page 630
isnode......Page 632
istnode......Page 634
iswt......Page 636
iswt2......Page 638
leaves......Page 640
mexihat......Page 644
meyer......Page 645
meyeraux......Page 648
morlet......Page 649
nodeasc......Page 650
nodedesc......Page 652
nodejoin......Page 655
nodepar......Page 657
nodesplt......Page 659
noleaves......Page 661
ntnode......Page 663
ntree......Page 664
orthfilt......Page 667
plot......Page 671
qmf......Page 676
rbiowavf......Page 678
read......Page 679
readtree......Page 682
scal2frq......Page 684
set......Page 694
shanwavf......Page 696
swt......Page 698
swt2......Page 702
symaux......Page 707
symwavf......Page 709
thselect......Page 710
tnodes......Page 712
treedpth......Page 714
treeord......Page 715
upcoef......Page 716
upcoef2......Page 720
upwlev......Page 722
upwlev2......Page 724
wavedec......Page 726
wavedec2......Page 730
wavedemo......Page 734
wavefun......Page 735
wavefun2......Page 739
waveinfo......Page 741
wavemenu......Page 743
wavemngr......Page 745
waverec......Page 757
waverec2......Page 758
wbmpen......Page 759
wcodemat......Page 763
wdcbm......Page 764
wdcbm2......Page 767
wden......Page 770
wdencmp......Page 775
wenergy......Page 783
wenergy2......Page 784
wentropy......Page 785
wextend......Page 789
wfilters......Page 793
wkeep......Page 796
wmaxlev......Page 798
wnoise......Page 800
wnoisest......Page 802
wp2wtree......Page 804
wpbmpen......Page 806
wpcoef......Page 810
wpcutree......Page 812
wpdec......Page 814
wpdec2......Page 817
wpdencmp......Page 820
wpfun......Page 826
wpjoin......Page 829
wprcoef......Page 831
wprec......Page 833
wprec2......Page 834
wpsplt......Page 835
wpthcoef......Page 837
wptree......Page 838
wpviewcf......Page 841
wrcoef......Page 843
wrcoef2......Page 845
wrev......Page 847
write......Page 848
wtbo......Page 850
wtbxmngr......Page 851
wthcoef......Page 853
wthcoef2......Page 854
wthresh......Page 855
wthrmngr......Page 856
De-noising using level dependent thresholds.......Page 857
Compression using level dependent thresholds.......Page 858
Compression using a global threshold.......Page 859
De-noising using a global threshold.......Page 860
wtreemgr......Page 861
wvarchg......Page 862
GUI Reference......Page 865
Connection of Plots......Page 867
Using the Mouse......Page 868
Translating Plots......Page 869
Controlling the Colormap......Page 870
Controlling the Number of Colors......Page 871
Controlling the Coloration Mode......Page 872
View Dynamical Visualization Tool Option......Page 874
File Menu Options......Page 875
Help Menu Options......Page 877
Using the View Axes Button......Page 878
Using the Interval-Dependent Threshold Settings Tool......Page 881
Continuous Wavelet Tool Features......Page 883
More Display Options......Page 884
Wavelet 2-D Tool Features......Page 886
Coefficients Coloration......Page 887
Node Action Functionality......Page 888
Wavelet Display Tool......Page 892
Wavelet Packet Display Tool......Page 893
Object-Oriented Programming......Page 895
Short Description of Objects in the Toolbox......Page 897
Example 1: plot and wpviewcf......Page 898
Example 2: drawtree and readtree......Page 901
Example 3: A Funny One......Page 903
Example 4: Thresholding Wavelet Packets......Page 905
Comments......Page 909
Private......Page 910
Fields Description......Page 911
Comments......Page 912
Fields Description......Page 913
Methods That Overload Those of DTREE Class......Page 914
Proper Methods of WPTREE Class.......Page 915
Example 1: Building a Wavelet Tree Object (WTREE)......Page 916
Example 2: Building a Right Wavelet Tree Object (RWVTREE)......Page 917
Running This Example......Page 918
Running This Example......Page 919
Example 4: Building a Wavelet Tree Object (EDWTTREE)......Page 920
Running This Example......Page 921
Index......Page 923
