Matlab - Wavelet Toolbox User's Guide

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