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