Part I Wavelet Transform: Theory and Implementation
1 The Wavelet Transform: A Surfing Guide
Akram Aldroubi
1.1 Introduction ................................................... 3
1.2 Notations ...................................................... 8
1.3 The Continuous Wavelet Transform .............................. 11
1.3.1 The Continuous Wavelet Transform of 1-D Signals ......... 11
1.3.2 Multidimensional Wavelet Transform ...................... 13
1.4 The Discrete Wavelet Transforms ............................... 15
1.4.1 The Dyadic Wavelet Transform ............................ 15
1.4.2 The Redundant Discrete Wavelet Transforms ............... 17
1.5 Multiresolutions and Wavelets ................................. 18
1.5.1 Multiresolution Approximations of L2 .................... 18
1.5.2 Orthogonal MRA-Type Wavelets ............................ 20
1.5.3 Semi-Orthogonal MRA-Type Wavelet Bases .................. 21
1.5.4 Bi-Orthogonal MRA-Type Wavelet Bases .................... 23
1.5.5 Local and Global Characterization of Functions in
Terms of Their Wavelet Coefficients ..................... 23
1.6 Special Bases of Scaling Functions ............................ 24
1.6.1 Interpolating Scaling Functions ......................... 25
1.6.2 Interpolating Wavelets .................................. 26
1.7 Applications and generalizations .............................. 27
1.7.1 Applications of the Wavelet Transform ................... 27
1.7.2 Generalizations of the Wavelet Transform ................ 28
1.8 Frame Representations ......................................... 28
2 A Practical Guide to the Implementation of the Wavelet Transform
Michael Unser
2.1 Introduction .................................................. 37
2.2 Basic Tools ................................................... 39
2.2.1 Scaling Functions and Multiresolution
Representations ......................................... 39
2.2.2 Inner Products Via Discrete Convolutions ................ 42
2.2.3 Boundary Conditions ..................................... 43
2.3 Wavelet Bases (Nonredundant Transform) ........................ 46
2.3.1 Fast Dyadic Wavelet Transform ........................... 46
2.3.2 Implementation Details .................................. 48
2.3.3 Extensions .............................................. 52
2.4 Dyadic Wavelet Frames ......................................... 52
2.5 Nondyadic Wavelet Analyses .................................... 57
2.5.1 Wavelet Representation .................................. 58
2.5.2 Fast Redundant Dyadic Wavelet Transform ................. 60
2.5.3 Fast Redundant Wavelet Transform with Integer
Scales .................................................. 61
2.5.4 Fast Redundant Wavelet Transform
(Arbitrary Scales) ....................................... 62
2.5.5 Fast Redundant Morlet or Gabor Wavelet Transform ........ 65
2.6 Conclusion .................................................... 67
Part II Wavelets in Medical Imaging and Tomography
3 An Application of Wavelet Shrinkage to Tomography
Eric D. Kolaczyk
3.1 Introduction .................................................. 77
3.1.1 Tomography .............................................. 77
3.1.2 Why Wavelets? ........................................... 78
3.1.3 Wavelet Shrinkage and the Proposed Method ............... 79
3.2 Inversion ..................................................... 80
3.2.1 Direct Data Vs. Indirect Data ........................... 80
3.2.2 The Wavelet-Vaguelette Decomposition .................... 81
3.2.3 Efficient Expressions for the Radon Vaguelette
Coefficients ............................................. 82
3.2.4 Calculation of the Radon Vaguelette Coefficient ......... 83
3.3 Denoising Using Wavelet Shrinkage ............................. 84
3.3.1 Wavelet Shrinkage with Direct Data ...................... 85
3.3.2 Wavelet Shrinkage with Tomographic Data ................. 85
3.3.3 The Proposed Reconstruction Method ...................... 86
3.4 A Short Comparative Study ..................................... 87
3.5 Discussion .................................................... 89
4 Wavelet Denoising of Functional MRI Data
Michael Hilton, Todd Ogden, David Hattery, Guinevere Eden,
and Bjorn Jawerth
4.1 Functional MRI and Brain Mapping .............................. 93
4.2 Image Acquisition ............................................. 95
4.3 fMRI Time Series Analysis ..................................... 96
4.3.1 The Hemodynamic Response Function ....................... 97
4.4 Wavelet Denoising of Signals .................................. 98
4.4.1 Data Analytic Thresholding ............................. 100
4.5 Experimental Results ......................................... 104
4.5.1 Data Set Descriptions .................................. 104
4.5.2 Analysis Technique ..................................... 105
4.5.3 Denoising Results ...................................... 108
4.6 Conclusions .................................................. 111
4.7 Acknowledgment ............................................... 112
5 Statistical Analysis of Image Differences by Wavelet
Decomposition
Urs E. Ruttimann, Michael Unser, Philippe Thevenaz, Chulhee Lee,
Daniel Rio, and Daniel W. Hommer
5.1 Introduction ................................................. 115
5.2 Wavelet Transform ............................................ 119
5.3 Correlation of Wavelet Coefficients .......................... 123
5.4 Statistical Tests ............................................ 128
5.5 Experimental Results ......................................... 132
5.5.1 Functional Magnetic Resonance Images ................... 132
5.5.2 Positron Emission Tomography Images .................... 137
5.6 Discussion ................................................... 139
6 Feature Extraction in Digital Mammography
R.A.DeVore, B.Lucier, and Z.Yang
6.1 Introduction ................................................. 145
6.2 Mammograms as Digitized Images ............................... 146
6.2.1 Characteristics of Mammographic Images ................. 148
6.3 Compression and Noise Removal ................................ 149
6.4 Some Issues in Compression Algorithms ........................ 152
6.4.1 Choice of Wavelet Basis ................................ 152
6.4.2 Choice of Metric ....................................... 153
6.4.3 Level of Compression ................................... 153
6.5 Algorithms ................................................... 153
6.6 Examples ..................................................... 156
7 Multiscale Contrast Enhancement and Denoising in Digital
Radiographs
Jian Fan and Andrew Laine
7.1 Introduction ................................................. 163
7.2 One-Dimensional Wavelet Transform ............................ 165
7.2.1 General Structure and Channel Characteristics .......... 165
7.2.2 Two Possible Filters ................................... 168
7.3 Linear Enhancement and Unsharp Masking ....................... 170
7.3.1 Review of Unsharp Masking .............................. 170
7.3.2 Inclusion of Unsharp Masking within RDWT Frame-
Work ................................................... 171
7.4 Nonlinear Enhancement ........................................ 173
7.4.1 Minimum Constraint for an
Enhancement Function ................................... 173
7.4.2 Filter Selection ....................................... 173
7.4.3 A Nonlinear Enhancement Function ....................... 174
7.5 Combined Denoising and Enhancement ........................... 176
7.5.1 Incorporating Wavelet Shrinkage into
Enhancement ............................................ 177
7.5.2 Threshold Estimation for Denoising ..................... 179
7.6 Two-Dimensional Extension .................................... 180
7.7 Experimental Results and Comparisons ......................... 180
7.8 Conclusion ................................................... 183
7.9 Acknowledgment ............................................... 187
8 Using Wavelets to Suppress Noise in Biomedical Images
Maurits Malfait
8.1 Introduction ................................................. 192
8.2 Overview of Wavelet-Based Noise Suppression .................. 193
8.2.1 Wavelet Shrinkage ...................................... 193
8.2.2 Correlating Coefficients Between Wavelet
Levels ................................................. 194
8.2.3 Smoothness Measure from Wavelet Extrema ................ 195
8.2.4 Example ................................................ 195
8.3 Introducing an A Priori Model ................................ 196
8.3.1 Motivation ............................................. 196
8.3.2 Basic Idea and Notation ................................ 198
8.3.3 Bayesian Method ........................................ 199
8.3.4 The Conditional Probability ............................ 200
8.3.5 The A Priori Probability ............................... 201
8.3.6 Coefficient Manipulation ............................... 201
8.4 Results for Biomedical Images ................................ 202
9 Wavelet Transform and Tomography: Continuous and
Discrete Approaches
F.Peyrin and M.Zaim
9.1 Introduction ................................................. 210
9.2 Basis of Tomography ......................................... 211
9.2.1 Problem Position ....................................... 211
9.2.2 Reconstruction Methods: Transform Methods .............. 212
9.2.3 Series Expansion Methods ............................... 213
9.3 Continuous Wavelet Decomposition ............................. 214
9.3.1 Continuous Wavelet Decomposition of
Projections ............................................ 214
9.3.2 Continuous Wavelet Decomposition of the Image .......... 216
9.4 Discrete Wavelet Decomposition ............................... 219
9.4.1 1-D DWT of the Projections ............................. 220
9.4.2 2-D Discrete WT of the Image ........................... 223
9.5 Conclusion ................................................... 225
9.5.1 Acknowledgments ........................................ 225
9.6 Appendix 1 ................................................... 226
10 Wavelets and Local Tomography
Carlos A. Berenstein and David F. Walnut
10.1 Introduction ................................................ 231
10.2 Background and Notation ..................................... 233
10.3 Why Wavelets? ............................................... 235
10.3.1 The Nonlocality of the Radon Transform ............... 235
10.3.2 Wavelets, Vanishing Moments, Λ-Tomography ............ 236
10.4 Wavelet Inversion of the Radon Transform .................... 237
10.4.1 The Continuous Wavelet Transform ..................... 237
10.4.2 The Semi-Continuous Wavelet Transform ................ 239
10.4.3 The Discrete Wavelet Transform ....................... 241
10.5 Wavelet Localization of Radon Transform ..................... 249
10.6 Conclusions ................................................. 251
10.7 Appendix: Proofs of Theorems ................................ 251
10.8 Acknowledgments ............................................. 258
11 Optimal Time-Frequency Projections for Localized Tomography
Tim Olson
11.1 Introduction ................................................ 263
11.1.1 Historical Notes ..................................... 263
11.1.2 Prior Work ........................................... 264
11.1.3 Organization ......................................... 266
11.2 Algorithmic Goals ........................................... 266
11.3 Background .................................................. 266
11.3.1 The Radon Transform .................................. 266
11.3.2 Basic Fourier Analysis ............................... 270
11.4 Reconstruction Techniques ................................... 271
11.4.1 Fourier Reconstruction ............................... 271
11.4.2 Filtered Backprojection .............................. 273
11.4.3 Nonlocality of the Radon Inversion ................... 273
11.4.4 Visualization via the Sinogram ....................... 276
11.4.5 Comparison to Local Tomography ....................... 277
11.5 Localization ................................................ 278
11.5.1 Utilizing Functions with Zero Moments ................ 278
11.5.2 How Many Frequency Windows? .......................... 278
11.5.3 High Frequency Computation ........................... 279
11.5.4 Low Frequency Computation ............................ 280
11.5.5 The Algorithm ........................................ 281
11.6 Numerical Results ........................................... 282
11.7 Optimality .................................................. 283
11.7.1 Minimization of Nonlocal Data ........................ 287
11.8 Conclusion .................................................. 288
11.9 Appendix: Error Analysis .................................... 289
11.9.1 Aliasing Error Analysis .............................. 289
11.9.2 Truncation Error Analysis ............................ 291
11.10 Local Cosine and Sine Bases ................................ 292
11.11 Acknowledgments ............................................ 295
12 Adapted Wavelet Techniques for Encoding Magnetic
Resonance Images
Dennis M. Healy, Jr. and John B. Weaver
12.1 Introduction ................................................ 298
12.2 Encoding in Magnetic Resonance Imaging ...................... 299
12.2.1 Nuclear Magnetic Resonance ........................... 300
12.2.2 Imaging .............................................. 303
12.2.3 Imaging Time and Signal-to-Noise Ratio ............... 311
12.3 Adapted Waveform Encoding in MRI ............................ 314
12.3.1 MRI Encoding with a Basis ............................ 315
12.3.2 Figures of Merit in Adapted Waveform Encoding ........ 324
12.3.3 Choosing a Basis for Encoding ........................ 329
12.3.4 Implementation of Adapted Waveform Encoding .......... 330
12.4 Reduced Imaging Times ....................................... 334
12.4.1 Adapted Waveform Encoding with K-L Bases ............. 335
12.4.2 Approximate K-L Bases ................................ 339
12.4.3 Approximate Karhunen-Loeve Encoding .................. 342
12.5 Conclusions ................................................. 346
12.6 Acknowledgments ............................................. 347
Part III Wavelets and Biomedical Signal Processing
13 Sleep Images Using the Wavelet Transform to Process
Polysomnographic Signals
Richard Sartene, Laurent Poupard, Jean-Louis Bernard and
Jean-Christophe Wallet
13.1 Introduction ................................................ 355
13.2 Sleep Polygraphy ............................................ 357
13.2.1 Signals .............................................. 357
13.2.2 Sleep Architecture (Figure 13.3) ..................... 360
13.2.3 Sleep and Cardiorespiratory Activity [5, 9] .......... 362
13.3 The Wavelet Transform-Practical Use ......................... 365
13.3.1 Practical Considerations ............................. 365
13.3.2 Validation of the Modulation Laws (FM-AM) ............ 367
13.4 Application of the Wavelet Transform ........................ 370
13.5 Cardiorespiratory Variations ................................ 376
13.6 Interaction Between Two Systems ............................. 377
13.7 Conclusion-Perspectives ..................................... 380
13.8 Acknowledgments ............................................. 381
14 Estimating the Fractal Exponent of Point Processes in
Biological Systems Using Wavelet- and Fourier-Transform Methods
Malvin C. Teich, Conor Heneghan, Steven B. Lowen and
Robert G. Turcott
14.1 Introduction ................................................ 383
14.1.1 Mathematical Descriptions of Stochastic Point
Processes ............................................ 384
14.1.2 Fractal Stochastic Point Processes (FSPPs)
Exhibit Scaling ...................................... 385
14.1.3 The Standard Fractal Renewal Process ................. 386
14.1.4 Examples of Fractal Stochastic Point Processes
in Nature ............................................ 387
14.2 Estimating the Fractal Exponent ............................. 388
14.2.1 Coincidence Rate ..................................... 389
14.2.2 Power Spectral Density ............................... 389
14.2.3 Fano Factor .......................................... 390
14.2.4 Allan Factor ......................................... 392
14.2.5 Haar-Basis Representation of the Fano and Allan
Factors .............................................. 393
14.2.6 Wavelet-Based Fano and Allan Factors ................. 397
14.3 Comparison of Techniques .................................... 402
14.4 Discussion .................................................. 405
14.5 Conclusion .................................................. 408
14.6 Acknowledgments ............................................. 408
15 Point Processes, Long-Range Dependence and Wavelets
Patrice Abry and Patrick Flandrin
15.1 Motivation .................................................. 413
15.1.1 Long-Range Dependence ................................ 413
15.1.2 Point Processes ...................................... 414
15.1.3 Long-Range Dependent Point Processes ................. 415
15.1.4 Fano Factor .......................................... 415
15.1.5 Wavelet Analysis ..................................... 415
15.2 The Standard Fano Factor .................................... 416
15.2.1 Some Definitions ..................................... 416
15.2.2 Poisson Process: Theme and Variations ................ 416
15.2.3 A Long-Dependent Poisson Process ..................... 417
15.2.4 Main Limitations ..................................... 418
15.3 The Wavelet-Based Fano Factor ............................... 419
15.3.1 The Multiresolution Point of View .................... 419
15.3.2 Unbiased Estimation of the Long Range-
Dependence Parameter: A Key Feature .................. 422
15.3.3 Reduction of the Range of the Dependence:
Another Key Feature .................................. 424
15.3.4 Fano Factor, Allan Variance and Wavelets ............. 426
15.3.5 Choosing the Number of Vanishing Moments N ........... 427
15.4 Practical Issues ............................................ 428
15.5 Fano Factor and Spectral Estimation ......................... 430
15.6 An Example: Spiketrain of an Auditory-Nerve Response ........ 432
15.7 Conclusion .................................................. 434
16 Continuous Wavelet Transform: ECG Recognition Based on Phase
and Modulus Representations and Hidden Markov Models
Lotfi Senhadji, Laurent Thoraval and Guy Carrault
16.1 Introduction ................................................ 439
16.2 Properties of Square Modulus and Phase ...................... 441
16.2.1 Square Modulus Approximation ......................... 441
16.2.2 Phase Behavior ....................................... 442
16.3 Illustration on Signals ..................................... 445
16.3.1 Results on Simulated Data ............................ 445
16.3.2 Results on Real Data ................................. 447
16.4 Cardiac Beat Recognition .................................... 448
16.5 Results ..................................................... 458
16.6 Conclusion .................................................. 460
16.7 Appendix .................................................... 460
17 Interference Canceling in Biomedical Systems: The Mutual
Wavelet Packets Approach
Mohsine Karrakchou and Murat Kunt
17.1 Introduction ................................................ 465
17.2 Pulmonary Capillary Pressure: A Short Review ................ 466
17.2.1 Clinical Relevance ................................... 466
17.2.2 In Vivo Estimation: The Occlusion Techniques ......... 468
17.2.3 Limitations in Patients .............................. 471
17.3 Basics of Interference Canceling ............................ 472
17.3.1 Classical FIR Adaptive Filtering ..................... 472
17.4 Multirate Adaptive Filtering ................................ 475
17.4.1 Fundamentals of Adaptive Filtering in
Subbands ............................................. 476
17.5 Wavelet Packets ............................................. 477
17.5.1 The Best Basis Method ................................ 478
17.6 Mutual Wavelet Packet Decomposition ......................... 480
17.6.1 Introductory Comments ................................ 480
17.6.2 The Mutual Wavelet Packets Decomposition ............. 480
17.6.3 Implementation Scheme ................................ 482
17.6.4 Algorithmic Complexity ............................... 482
17.6.5 Experimental Results ................................. 484
17.7 Conclusion .................................................. 486
18 Frame Signal Processing Applied to Bioelectric Data
John J. Benedetto
18.1 Introduction ................................................ 493
18.2 Notation .................................................... 494
18.3 The Theory of Frames ........................................ 494
18.3.1 Gabor and Wavelet Systems ............................ 494
18.3.2 Frames ............................................... 495
18.4 Frame Multiresolution Analysis (FMRA) ....................... 496
18.5 Noise reduction ............................................. 498
18.6 The Laplacian Method and Gaussian Frames .................... 501
18.7 An Interpretation of Spectral ECoG Data ..................... 503
18.8 Appendix .................................................... 508
19 Diagnosis of Coronary Artery Disease Using Wavelet - Based
Neural Networks
Metin Akay
19.1 Introduction ................................................ 513
19.2 Method ...................................................... 516
19.2.1 Fast Wavelet Transform ............................... 516
19.2.2 Fuzzy Min-Max Neural Networks ........................ 517
19.2.3 Patient Analysis ..................................... 518
19.3 Results ..................................................... 519
19.3.1 Feature Extraction ................................... 519
19.3.2 Network Output Representation ........................ 521
19.4 Conclusion .................................................. 522
19.5 Acknowledgment .............................................. 522
Part IV Wavelets and Mathematical Models in Biology
20 A Nonlinear Squeezing of the Continuous Wavelet
Transform Based on Auditory Nerve Models
Ingrid Daubechies and Stephane Maes
20.1 Introduction ................................................ 527
20.2 Cochlear Filtering .......................................... 528
20.3 Information Compression ..................................... 529
20.4 The Modulation Model for Speech ............................. 531
20.5 Squeezing the Continuous Wavelet Transform .................. 533
20.6 Short Discussion ............................................ 538
20.7 Results on Speech Signals ................................... 540
20.8 Acknowledgments ............................................. 544
21 The Application of Wavelet Transforms to Blood Flow
Velocimetry
Lora G. Weiss
21.1 Introduction ................................................ 547
21.2 1-D Measurement Devices ..................................... 549
21.3 1-D Velocimetry Methods ..................................... 551
21.3.1 Doppler Methods ...................................... 552
21.3.2 Time Domain Correlation Methods ...................... 554
21.3.3 Limitations .......................................... 555
21.3.4 Summary of Desirable Signal Characteristics .......... 557
21.4 Wideband / Wavelet Transform Processing ..................... 557
21.4.1 Wavelet Transform Processing ......................... 557
21.4.2 Parameter Estimation ................................. 560
21.4.3 Example .............................................. 562
21.5 Conclusions ................................................. 565
21.6 Acknowledgment .............................................. 566
21.7 Appendix .................................................... 567
22 Wavelet Models of Event - Related Potentials
Jonathan Raz and Bruce Turetsky
22.1 Introduction ................................................ 571
22.2 The Single Channel Wavelet Model ............................ 573
22.3 Application to Cat Potentials ............................... 575
22.4 The Topographic Wavelet Model ............................... 578
22.5 Application of Topographic Wavelet Model .................... 580
22.6 Discussion .................................................. 585
22.7 Acknowledgments ............................................. 586
23 Macromolecular Structure Computation Based on Energy
Function Approximation by Wavelets
Eberhard Schmitt
23.1 Introduction ................................................ 590
23.2 Domain and Function Decomposition ........................... 592
23.2.1 Reduction of the Degrees of Freedom .................. 592
23.2.2 Representation of the Energy Function ................ 595
23.3 Approximation by Wavelets ................................... 597
23.3.1 Local Approximation by Cubic B-Spline
Wavelets ............................................. 597
23.3.2 Global Approximation by Tensor Products .............. 598
23.4 Further Applications: Surface Representation ................ 602
23.5 Discussion .................................................. 603
Index ............................................................... 607
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