Wavelets in medicine and biology
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Wavelets in medicine and biology / ed. by Aldroubi A., Unser M. - Boca Raton: CRC, 1996. - 616 p. - ISBN 0-8493-9483-X.
 
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  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


Вверх Wavelets in medicine and biology / ed. by Aldroubi A., Unser M. - Boca Raton: CRC, 1996. - 616 p. - ISBN 0-8493-9483-X.

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