Time Frequency and Wavelets in Biomedical Signal Processing

by Akay, Metin

• ISBN13: 9780780311473
• ISBN10: 0780311477
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-11-07
• Publisher: Wiley-IEEE Press

Summary

Brimming with top articles from experts in signal processing and biomedical engineering, Time Frequency and Wavelets in Biomedical Signal Processing introduces time-frequency, time-scale, wavelet transform methods, and their applications in biomedical signal processing. This edited volume incorporates the most recent developments in the field to illustrate thoroughly how the use of these time-frequency methods is currently improving the quality of medical diagnosis, including technologies for assessing pulmonary and respiratory conditions, EEGs, hearing aids, MRIs, mammograms, X rays, evoked potential signals analysis, neural networks applications, among other topics.Time Frequency and Wavelets in Biomedical Signal Processing will be of particular interest to signal processing engineers, biomedical engineers, and medical researchers.Topics covered include: Time-frequency analysis methods and biomedical applications Wavelets, wavelet packets, and matching pursuits and biomedical applications Wavelets and medical imaging Wavelets, neural networks, and fractals

Author Biography

About the Editor Metin Akay is IEEE Press Series Editor for the IEEE Press Series in Biomedical Engineering, and a member of the IEEE Engineering in Medicine and Biology Society Publication Committee. Dr. Akay has authored Biomedical Signal Processing (Academic Press, 1994); Detection and Estimation of Biomedical Signals (Academic Press, 1996); and coauthored the most recent edition of Theory and Design of Biomedical Instruments (Academic Press, 1991). He has published a number of technical papers in the areas of noninvasive detection of coronary artery disease, early human development, and control of breathing. In addition, Dr. Akay holds two U.S. patents and has given several keynote/plenary and invited talks at international conferences, workshops, and symposiums in these areas.

Table of Contents

List of Contributors	xxiii	(4)
Preface	xxvii
PART I TIME-FREQUENCY ANALYSIS METHODS WITH BIOMEDICAL APPLICATIONS	1	(208)
Chapter 1 Recent Advances in Time-Frequency Representations: Some Theoretical Foundation	3	(42)
William J. Williams
1.1. Introduction	3	(5)
1.2. The Reduced Interference Distribution	8	(12)
1.2.1 Ambiguity Function Relationships	8	(3)
1.2.2 The Exponential Distribution	11	(2)
1.2.3 Zhao-Atlas-Marks	13	(1)
1.2.4 Kernel Selection for RID	13	(3)
1.2.5 Design Procedures for Effective RID Kernels	16	(2)
1.2.6 Limitations of RID	18	(2)
1.3. Additional Distributions with Designed or Adaptive Kernels	20	(5)
1.3.1 Fixed Kernel Designs	20	(1)
1.3.2 Distributions with Adaptive Kernels	21	(1)
1.3.3 Some Adaptive RID Results	22	(3)
1.4. Noise Considerations	25	(1)
1.5. Discrete Formulations and Fast Algorithms	26	(7)
1.5.1 Discrete Realizations	26	(3)
1.5.2 Binomial Time-Frequency Distribution Results	29	(2)
1.5.3 Fast Algorithms Using Spectrogram Decompositions	31	(2)
1.6. Time-Varying Filtering and Synthesis	33	(4)
1.7. Analysis Window Comparisons: Wavelets and Cohen's Class	37	(2)
1.8. Conclusions	39	(1)
Acknowledgments	39	(1)
References	39	(6)
Chapter 2 Biological Applications and Interpretations of Time-Frequency Signal Analysis	45	(28)
William J. Williams
2.1. Introduction	45	(2)
2.2. Cohen's Class of Distributions	47	(13)
2.2.1 Electrophysiological Signals and Epilepsy	48	(8)
2.2.2 The Importance of Invariance in EEG Representation	56	(1)
2.2.3 Event Related Potentials	56	(3)
2.2.4 Other Electrophysiological Results	59	(1)
2.3. Bioacoustics Examples	60	(8)
2.3.1 Temporomandibular Joint Sounds	60	(5)
2.3.2 Animal Sounds	65	(2)
2.3.3 Heart and Muscle Sounds	67	(1)
2.4. Conclusions	68	(1)
Acknowledgments	68	(1)
References	69	(4)
Chapter 3 The Application of Advanced Time-Frequency Analysis Techniques to Doppler Ultrasound	73	(28)
S. Lawrence Marple, Jr.
Tom Brotherton
Doug Jones
3.1. Introduction	73	(14)
3.1.1 Adaptive Quadratic Time-Frequency Representations	76	(5)
3.1.2 The Wavelet Transform Time-Frequency Representation	81	(2)
3.1.3 Model-Based Approaches	83	(4)
3.2. Doppler Ultrasound Data Processing Results	87	(7)
3.2.1 The Short-Time Fourier Transform (STFT)	87	(1)
3.2.2 Generalized Wigner-Ville and Complex Ambiguity Functions	87	(1)
3.2.3 The Adaptive Optimal Kernel (AOK) TFR	87	(7)
3.2.4 The Adaptive Cone Kernel (ACK) Distribution	94	(1)
3.2.5 The Wavelet Transform Time-Frequency Representation	94	(1)
3.2.6 Model-Based Approaches: Signal Subspace Enhancement Linear Prediction for Extended Data STFT	94	(1)
3.3. Conclusions	94	(5)
References	99	(2)
Chapter 4 Analysis of ECG Late Potentials Using Time-Frequency Methods	101	(16)
Hartmut Dickhaus
Hartmut Heinrich
4.1. Introduction	101	(1)
4.2. Methods	102	(4)
4.2.1 Data Acquisition and Preprocessing	102	(1)
4.2.2 Comparison of Time-Frequency Representations by Simulated ECG Test Signals	103	(3)
4.3. Application of Time-Frequency Transformations to Clinical ECG Data	106	(6)
4.3.1 Evaluation of Time-Frequency Representations	106	(4)
4.3.2 Parameter Optimization for Classification Purposes	110	(2)
4.4. Conclusion	112	(2)
Acknowledgment	114	(1)
References	114	(3)
Chapter 5 Time-Frequency Distributions Applied to Uterine EMG: Characterization and Assessment	117	(30)
Jacques Duchene
Dominique Devedeux
5.1. Introduction	117	(4)
5.2. Time-Frequency Distributions	121	(6)
5.2.1 The Parametric Approach: AR Modeling	121	(1)
5.2.2 Cohen's Class Distributions	122	(2)
5.2.3 Signal-Dependent Optimal Kernel	124	(1)
5.2.4 Reassignment Procedure	125	(2)
5.3. Criteria for Determining the Representation Quality	127	(1)
5.3.1 Back to the Initial Problem: Modulation Extraction	127	(1)
5.3.2 Criteria Definition	127	(1)
5.4. Results	128	(11)
5.4.1 Method Validation	128	(7)
5.4.2 Results on the Comparison Between Representations	135	(1)
5.4.3 Robustness and Selectivity	136	(3)
5.4.4 Toward a Possible Final Choice	139	(1)
5.5. Some Examples on Real Signals	139	(3)
5.6. Conclusion	142	(1)
References	143	(4)
Chapter 6 Time-Frequency Analyses of the Electrogastrogram	147	(36)
Zhiyue Lin
Jiande Z. Chen
6.1. Introduction	147	(3)
6.2. Electrogastrography	150	(2)
6.2.1 Myoelectrical Activities in the Stomach	150	(1)
6.2.2 Electrogastrogram (EGG)	151	(1)
6.3. Short-Time Fourier Transform and Spectrogram	152	(4)
6.3.1 Advantages and Limitations	152	(2)
6.3.2 Applications	154	(2)
6.4. Exponential Distribution	156	(2)
6.4.1 Advantages and Limitations	156	(1)
6.4.2 Applications	156	(2)
6.5. Adaptive Arma Modeling	158	(6)
6.5.1 Definition and Implementation	158	(3)
6.5.2 Advantages and Limitations	161	(1)
6.5.3 Applications	161	(3)
6.6. Performance Comparison	164	(9)
6.6.1 Simulation Results	164	(7)
6.6.2 Clinical Applications	171	(2)
6.7. Conclusions	173	(4)
Acknowledgments	177	(1)
References	177	(6)
Chapter 7 Recent Advances in Time-Frequency and Time-Scale Methods	183	(26)
Claudia Mello
Metin Akay
7.1. Introduction	183	(1)
7.1.1 Notation	184	(1)
7.2. Fourier Representation	184	(3)
7.3. Cohen's Class Operators	187	(5)
7.4. Wavelets: Frames, Multiresolution Approximation, and Beyond	192	(8)
7.5. More Transformations	200	(2)
7.6. Conclusions	202	(1)
Acknowledgments	202	(1)
References	202	(7)
PART II WAVELETS, WAVELET PACKETS, AND MATCHING PURSUITS WITH BIOMEDICAL APPLICATIONS	209	(214)
Chapter 8 Fast Algorithms for Wavelet Transform Computation	211	(32)
Olivier Rioul
Pierre Duhamel
8.1. Introduction	211	(3)
8.1.1 Classification of Wavelet Transforms	211	(2)
8.1.2 Note on the Choice of the Wavelet	213	(1)
8.2. Multiresolution and Two-Scale Equations	214	(2)
8.2.1 Multiresolution Spaces	214	(1)
8.2.2 Examples	215	(1)
8.2.3 Two-Scale Equations	215	(1)
8.3. The Initial Signal Approximation	216	(2)
8.3.1 Remarks on Initialization and Sampling	217	(1)
8.4. The Discrete Wavelet Transform (DWT)	218	(1)
8.5. The DWT For WS Computation	219	(4)
8.5.1 WS Computation: Mallat and Shensa Algorithm	219	(2)
8.5.2 The Wavelet Approximation	221	(1)
8.5.3 Using the Inverse DWT to Compute the Inverse WS (IWS)	222	(1)
8.6. The DWT For CWT Computation	223	(4)
8.6.1 Finer Sampling in Scale	223	(2)
8.6.2 Finer Sampling in Time: Modified Shensa and "a trous" Algorithms	225	(1)
8.6.3 A slightly Different Building Block	225	(2)
8.6.4 Inner Product Implementation of the CWT	227	(1)
8.7. Efficient Implementations of the DWT	227	(3)
8.7.1 Preliminaries	228	(1)
8.7.2 Reorganization of the Computations	228	(2)
8.8. Faster DWT Algorithms	230	(6)
8.8.1 An FFT-Based DWT Algorithm	231	(2)
8.8.2 A Generalization: The Vetterli Algorithm	233	(1)
8.8.3 DWT Algorithms for Short Filters	234	(1)
8.8.4 Other Considerations	235	(1)
8.8.5 Faster CWT Algorithms	236	(1)
8.9. Other Algorithms for CWT Computation	236	(4)
8.9.1 Reproducing Kernels	236	(1)
8.9.2 Algorithms Using Splines	236	(3)
8.9.3 Mellin-Transform-Based Algorithms	239	(1)
8.10. Conclusion	240	(1)
References	240	(3)
Chapter 9 Analysis of Cellular Vibrations in the Living Cochlea Using the Continuous Wavelet Transform and the Short-Time Fourier Transform	243	(28)
M. C. Teich, C. Heneghan
S. M. Khanna
9.1. Introduction	243	(1)
9.2. Methods	244	(1)
9.3. Theory	245	(13)
9.3.1 The Continuous-Time Fourier Transform	245	(1)
9.3.2 The Short-Time Fourier Transform	245	(3)
9.3.3 The Continuous Wavelet Transform	248	(2)
9.3.4 Wavelet Bases	250	(3)
9.3.5 STFT and CWT Implementation	253	(5)
9.4. Results	258	(8)
9.5. Discussion	266	(1)
9.6. Conclusion	267	(1)
Acknowledgments	267	(1)
References	267	(4)
Chapter 10 Iterative Processing Method Using Gabor Wavelets and the Wavelet Transform for the Analysis of Phonocardiogram Signals	271	(34)
Mustafa Matalgah
Jerome Knopp
Salah Mawagdeh
10.1. Introduction	271	(2)
10.2. Theoretical Background	273	(2)
10.2.1 The Fourier Transform and the STFT	273	(1)
10.2.2 The Wigner Distribution	273	(1)
10.2.3 The Wavelet Transform	274	(1)
10.3. Combined Wavelet-Fourier Transform	275	(3)
10.3.1 Theorem and Proof	275	(3)
10.4. Computer Simulation and Real Data	278	(11)
10.4.1 The Fourier Transform	279	(1)
10.4.2 The Short-Time Fourier Transform	279	(1)
10.4.3 The Wigner Distribution	279	(1)
10.4.4 The Wavelet Transform	279	(6)
10.4.5 Iterative Processing Method	285	(4)
10.5. Discussion and Conclusion	289	(2)
References	301	(4)
Chapter 11 Wavelet Feature Extraction from Neurophysiological Signals	305	(18)
Mingui Sun
Robert J. Sclabassi
11.1. Introduction	305	(2)
11.2. Wavelet Transforms	307	(1)
11.3. Signal-to-Noise Ratio	307	(1)
11.4. Wavelet Spectral Division	308	(1)
11.5. Variance	309	(2)
11.6. Spectral Features in the Wavelet Extrema and Zero-Crossings	311	(2)
11.7. Computation	313	(2)
11.8. Experimental Results	315	(3)
11.9. Discussion	318	(1)
Acknowledgments	318	(1)
Appendix: Expected Number of Zero Crossings	318	(2)
References	320	(3)
Chapter 12 Experiments with Adapted Wavelet De-Noising for Medical Signals and Images	323	(24)
Ronald R. Coifman
Mladen Victor Wickerhauser
12.1. Time and Frequency Analysis	323	(1)
12.2. Example Libraries of Waveforms	324	(7)
12.3. Choosing the "Best Basis"	331	(1)
12.4. Compression	332	(1)
12.5. Adapted Waveform "De-Noising"	332	(5)
12.6. Experiments with SNR Improvement	337	(1)
12.6.1 Procedure	337	(1)
12.6.2 Results	338	(1)
12.7. Conclusion	338	(7)
12.A INSTRUCTIONS AND SAMPLE OUTPUT FOR THE PROGRAM "DENOISE"	342	(1)
12.A.1 Summary of the Algorithm	342	(1)
12.A.2 Manual Page	342	(2)
12.A.3 Output from denoise-i4-m9-t0.2 sine+8db.asc	344	(1)
References	345	(2)
Chapter 13. Speech Enhancement for Hearing Aids	347	(20)
Janet C. Rutledge
13.1. Introduction	347	(1)
13.2. Background	348	(5)
13.2.1 Hearing Impairments	348	(2)
13.2.2 Hearing Loss Compensation Techniques	350	(2)
13.2.3 Noise Reduction	352	(1)
13.2.4 Motivation for Using Wavelets	353	(1)
13.3. Wavelet-Based Compression	353	(5)
13.3.1 Comparison with Multiband Filter Compression	355	(3)
13.4. Wavelet-Based Noise Reduction	358	(5)
13.4.1 Simultaneous Compression and De-Noising	358	(1)
13.4.2 Adaptive Multi-band MDL	359	(2)
13.4.3 Preliminary Results	361	(1)
13.4.4 Discussion	361	(2)
13.5. Concluding Remarks	363	(1)
References	364	(3)
Chapter 14 From Continuous Wavelet Transform to Wavelet Packets: Application to the Estimation of Pulmonary Microvascular Pressure	367	(22)
Mohsine Karrakchou
Murat Kunt
14.1. Introduction	367	(1)
14.2. Wavelet Packets	368	(4)
14.2.1 The Best-Basis Method	370	(1)
14.2.2 Criteria for the Selection of the Best-Basis	370	(2)
14.3. Estimation of Pulmonary Capillary Pressure	372	(4)
14.3.1 The Clinical Importance of Effective Pulmonary Capillary Pressure	372	(1)
14.3.2 Arterial Occlusion (AO)	373	(2)
14.3.3 Limitations of the Arterial Occlusion to Apneic Transients	375	(1)
14.4. How Wavelets Can Help To Solve The Problem	376	(6)
14.4.1 Classical Finite Impulse Response Adaptive Filtering	376	(2)
14.4.2 Fundamentals of Adaptive Filtering in Subbands	378	(1)
14.4.3 The Decomposition of Mutual Wavelet Packets	379	(2)
14.4.4 Implementation Scheme	381	(1)
14.4.5 Experimental Results	381	(1)
14.5. Conclusion	382	(1)
References	383	(6)
Chapter 15 In Pursuit of Time-Frequency Representation of Brain Signals	389	(8)
P. J. Durka
K. J. Blinowska
15.1. Introduction	389	(1)
15.2. Application of the Wavelet Transform to Evoked-Potential Analysis	390	(8)
15.2.1 Method	390	(2)
15.2.2 Application to EP Analysis	392	(2)
15.2.3 Discussion	394	(4)
15.3. Matching Pursuit Method and its Applications	398	(6)
15.3.1 Method	398	(2)
15.3.2 Results and Discussion	400	(4)
15.4. Conclusion	404	(1)
Acknowledgments	405	(1)
References	405	(2)
Chapter 16 EEG Spike Directors Based on Different Decompositions: A Comparative Study	407	(16)
L. Senhadji
J. J. Bellanger
G. Carrault
16.1. Introduction	407	(2)
16.2. Problem Statement	409	(1)
16.3. Description of the Test T(1)	410	(1)
16.4. Variations of S(1)	411	(2)
16.4.1 Detectors Built Without Using the Spike Waveform	411	(1)
16.4.2 Detectors Based on Objective Knowledge on P(1) (Other Than Their Time Duration)	412	(1)
16.5. Experimentation and Performance Evaluation	413	(1)
16.6. Results and Discussion	414	(5)
16.7. Conclusion	419	(1)
References	420	(3)
PART III WAVELETS AND MEDICAL IMAGING	423	(220)
Chapter 17 A Discrete Dyadic Wavelet Transform for Multidimensional Feature Analysis	425	(26)
Iztok Koren
Andrew Laine
17.1. Introduction	425	(1)
17.2. One-Dimensional Discrete Dyadic Wavelet Transform	426	(11)
17.2.1 Wavelet Transform	426	(4)
17.2.2 Implementation	430	(6)
17.2.3 Remarks	436	(1)
17.3. Multidimensional Discrete Dyadic Wavelet Transform	437	(5)
17.3.1 Wavelet Transform	437	(3)
17.3.2 Implementation	440	(1)
17.3.3 Remarks	441	(1)
17.4. Applications	442	(6)
17.4.1 Contrast Enhancement in Digital Mammography	442	(3)
17.4.2 Edge Detection in Echocardiographic Image Sequences	445	(2)
17.4.3 Remarks	447	(1)
17.5. Conclusion	448	(1)
Acknowledgment	448	(1)
References	448	(3)
Chapter 18 Hexagonal QMF Banks and Wavelets	451	(22)
Sergio Schuler
Andrew Laine
18.1. Introduction	451	(1)
18.2. Hexagonal Sampling System	451	(13)
18.2.1 Hexagonal Systems	452	(2)
18.2.2 Up-Sampling and Down-Sampling in Hexagonal Systems	454	(3)
18.2.3 Analysis Synthesis Filter Banks in Hexagonal Systems	457	(3)
18.2.4 Redundant Analysis Synthesis Filter Banks in Hexagonal Systems	460	(2)
18.2.5 The Discrete Fourier Transform in Hexagonal Systems	462	(2)
18.3. Implementation	464	(7)
18.3.1 Image Support in Hexagonal Systems	464	(3)
18.3.2 Multiresolution Representations in Hexagonal Systems	467	(2)
18.3.3 Overcomplete Multiresolution Representations in Hexagonal Systems	469	(2)
Acknowledgment	471	(1)
References	472	(1)
Chapter 19 Inversion of the Radon Transform under Wavelet Constraints	473	(26)
Berkman Sahiner
Andrew E. Yagle
19.1. Introduction	473	(1)
19.2. Inverse Radon Transforms and Discrete Wavelet Transforms	474	(4)
19.2.1 The Inverse Radon Transform	474	(1)
19.2.2 The Discrete Wavelet Transform	475	(2)
19.2.3 The Unsubsampled Wavelet Transform	477	(1)
19.3. Filtering with Use of DWT Constraints	478	(8)
19.3.1 Problem Definition	480	(1)
19.3.2 Constraints on a Single Wavelet	480	(2)
19.3.3 Constraints on Several Sub-Wavelets	482	(1)
19.3.4 Examples and Discussion	483	(3)
19.4. Image Restoration with Use of UWT Constraints	486	(10)
19.4.1 Wavelet Interpretation of the Missing Angle Problem	487	(1)
19.4.2 Interpolation of Low-Resolution Missing Data	488	(2)
19.4.3 Summary of the Algorithm	490	(1)
19.4.4 Numerical Examples	491	(5)
19.5. Conclusion	496	(1)
References	496	(3)
Chapter 20 Wavelets Applied to Mammograms	499	(20)
Walter B. Richardson, Jr.
20.1. Introduction	499	(1)
20.2. Wavelets and Multiresolution Analysis	500	(4)
20.3. Data Compression and Teleradiology	504	(4)
20.4. Feature Enhancement and Classification	508	(2)
20.5. Wavelets, Fractals, and Texture	510	(2)
20.6. De-Noising	512	(3)
20.7. Discussion and Conclusions	515	(1)
Acknowledgments	516	(1)
References	516	(3)
Chapter 21 Hybrid Wavelet Transform for Image Enhancement for Computer-Assisted Diagnosis and Telemedicine Applications	519	(14)
Laurence P. Clarke
Wei Qian
Maria Kallergi
Priya Venugopal
Robert A. Clark
21.1. Introduction	519	(2)
21.2. Design of a Hybrid Filter	521	(4)
21.2.1 Introduction	521	(1)
21.2.2 Hybrid Filter Architecture	521	(1)
21.2.3 Adaptive Multistage Nonlinear Filtering	522	(2)
21.2.4 Wavelet Decomposition and Reconstruction	524	(1)
21.3. Experimental Results	525	(5)
21.3.1 Influence of Preprocessing for a Hybrid Filter	525	(2)
21.3.2 Influence of Sensor Resolution	527	(1)
21.3.3 Influence of Linear Versus Order Statistic Operator	528	(2)
21.4. Conclusion	530	(1)
References	531	(2)
Chapter 22 Medical Image Enhancement Using Wavelet Transform and Arithmetic Coding	533	(16)
Pongskorn Saipetch
Bruce K. T. Ho
Ramesh K. Panwar
Marco Ma
22.1. Introduction	533	(1)
22.2. Wavelet Transform	534	(3)
22.2.1 Wavelet Transform of Images	534	(3)
22.3. Quantization	537	(1)
22.4. Arithmetic Coding	538	(1)
22.5. Experiments	539	(2)
22.6. Results	541	(4)
22.6.1 Lossly Compression	541	(3)
22.6.2 Lossless Compression	544	(1)
22.7. Conclusions	545	(1)
References	546	(3)
Chapter 23 Adapted Wavelet Encoding in Functional Magnetic Resonance Imaging	549	(56)
Dennis M. Healy, Jr.
Douglas W. Warner
John B. Weaver
23.1. Parsimonious Representations of Images	551	(3)
23.2. Standard MRI and Fourier Transforms	554	(2)
23.3. Alternatives to the Fourier Basis	556	(4)
23.4. Finding Approximate K-L Bases	560	(2)
23.5. Adapted Waveform Encoding in MRI	562	(4)
23.5.1 Wavelet Encoding	563	(2)
23.5.2 More General Bases for Encoding	565	(1)
23.5.3 Choosing a Basis for Fast MRI Encoding	565	(1)
23.6. K-L Bases in MRI	566	(4)
23.6.1 K-L Waveform Encoding	566	(1)
23.6.2 Simulation Results	567	(2)
23.6.3 Implementation and Practical Limitations of K-L Encoding	569	(1)
23.7. Approximate K-L Bases in MRI	570	(7)
23.7.1 Approximate K-L Waveform Encoding	571	(1)
23.7.2 Application to Dynamical Imaging	571	(5)
23.7.3 Two-Dimensional Approximate K-L Encoding	576	(1)
23.8. Conclusion	577	(2)
Appendix A: Encoding in MRI	579	(20)
A.1 Nuclear Magnetic Resonance	581	(3)
A.2 Imaging	584	(7)
A.3 Imaging Time and SNR	591	(3)
A.4 Adapted Waveform Encoding in MRI	594	(1)
A.5 MRI Encoding with a Basis	594	(3)
A.6 MR Phosphorus Spectroscopy	597	(2)
References	599	(6)
Chapter 24 A Tutorial Overview of a Stabilization Algorithm for Limited-Angle Tomography	605	(18)
Tom Olson
24.1. Introduction	605	(1)
24.2. Background and Definitions	606	(3)
24.2.1 The Radon Transform	606	(1)
24.2.2 Tomography and Limited-Angle Tomography	607	(2)
24.2.3 Physical Motivation and Prior Work	609	(1)
24.3. Limitations of the Singular Value Decomposition	609	(2)
24.3.1 Unbounded Inverses and Approximate Identities	609	(1)
24.3.2 Uncorrelated, Exact Bases versus Induced Correlations and Redundant Bases	610	(1)
24.3.3 Decreasing Signal-to-Noise Ratio	611	(1)
24.4. Mollification Methods	611	(5)
24.4.1 Szego's Theory for Finite Toeplitz Operators	611	(1)
24.4.2 Limited-Angle Spectra	612	(2)
24.4.3 Uncertainty Principles and Signal Recovery	614	(1)
24.4.4 Nonlinear Constraints, Induced Correlations, and POCS	615	(1)
24.5. The Algorithm	616	(1)
24.6. Numerical Results	617	(3)
24.7. Conclusion	620	(1)
Acknowledgment	621	(1)
References	621	(2)
Chapter 25 Wavelet Compression of Medical Images	623	(20)
Armando Manduca
25.1. Introduction	623	(1)
25.2. Discrete Wavelet Transforms	624	(2)
25.3. Image Compression with Wavelets	626	(9)
25.3.1 Implementation	628	(1)
25.3.2 Set Partitioning in Hierachical Trees	629	(1)
25.3.3 Sample Compressions	630	(2)
25.3.4 Compression of 3-D Images	632	(2)
25.3.5 Preserving Arbitrary Regions	634	(1)
25.4. Discussion	635	(5)
25.4.1 Comparisons with JPEG	636	(2)
25.4.2 Human Visual System Response	638	(1)
25.4.3 Medical Acceptance of Lossy Compression	638	(1)
25.4.4 Related Advanced Techniques	639	(1)
References	640	(3)
PART IV WAVELETS, NEURAL NETWORKS, AND FRACTALS	643	(86)
Chapter 26 Single Side Scaling Wavelet Frame and Neural Network	645	(24)
Qinghua Zhang
26.1. A Short Introduction to Neural Networks	645	(2)
26.2. Wavelet Series and Wavelet Network	647	(1)
26.3. Double Side Scaling Wavelet Frames	648	(3)
26.3.1 A Sufficient Condition	649	(1)
26.3.2 Radial Case	650	(1)
26.4. Single Side Scaling Wavelet Frame	651	(3)
26.4.1 A Sufficient Condition for Single Side Scaling Wayelet Frame	651	(2)
26.4.2 Radial Case	653	(1)
26.4.3 Some Practical Considerations	653	(1)
26.5. Combining Wavelet and Neural Network	654	(7)
26.5.1 Modeling Nonlinear Systems	654	(1)
26.5.2 Sparse Data and Thinned Wavelet Frame	655	(1)
26.5.3 Regression Analysis Applied to Wavelets	656	(2)
26.5.4 The Network Size	658	(1)
26.5.5 Additional Optimization	659	(1)
26.5.6 Implementation of the Wavelet Network	659	(1)
26.5.7 Numerical Example	660	(1)
26.6. Conclusion	661	(1)
Appendix A: Proof of Theorem 3	662	(2)
Appendix B: Proof of Theorem 4	664	(2)
Appendix C: Some Comments on Theorem 4	666	(1)
References	666	(3)
Chapter 27 Analysis of Evoked Potentials Using Wavelet Networks	669	(16)
Hartmut Heinrich
Hartmut Dickhaus
27.1. Introduction	669	(1)
27.2. Wavelet Networks	670	(8)
27.2.1 Basic Method	670	(3)
27.2.2 Constraints for a Uniform WN Parameterization	673	(1)
27.2.3 Advanced WN Learning Algorithm	674	(4)
27.3. Wavelet Nets Applied to EP Signals	678	(4)
27.3.1 Clinical and Methodical Background	678	(1)
27.3.2 Data Acquisition and Preprocessing	679	(1)
27.3.3 Parameterization and Discrimination by Means of WN Parameters	679	(3)
27.4. Conclusion	682	(1)
References	683	(2)
Chapter 28 Self-Organizing Wavelet-Based Neural Networks	685	(18)
Kunikazu Kobayashi
28.1. Introduction	685	(2)
28.2. Preliminaries	687	(2)
28.2.1 Wavelet Transform	687	(1)
28.2.2 Inversion Formula	687	(1)
28.2.3 Windows	688	(1)
28.3. Network Expression	689	(1)
28.4. Function Approximation and Network Optimization	689	(5)
28.4.1 Function Approximation Problem	690	(1)
28.4.2 Self-Organization of Networks	691	(2)
28.4.3 Minimization of Errors	693	(1)
28.5. Computer Simulations	694	(7)
28.5.1 Simulation I	695	(2)
28.5.2 Simulation II	697	(1)
28.5.3 Simulation III	698	(3)
28.6. Conclusion	701	(1)
References	701	(2)
Chapter 29 On Wavelets and Fractal Processes	703	(16)
Patrick Flandrin
29.1. Introduction	703	(1)
29.2. Fractal Processes	704	(1)
29.3. Wavelets and Fractional Brownian Motion	705	(4)
29.3.1 The Fractional Brownian Motion Model	705	(1)
29.3.2 Wavelet Analysis of fBm	705	(1)
29.3.3 Wavelet Estimation of the Hurst Exponent	706	(2)
29.3.4 Some Further Remarks on Wavelets and fBm	708	(1)
29.4. Wavelets and Point Processes	709	(2)
29.4.1 Some Models	709	(1)
29.4.2 A Wavelet-Based Fano Factor	710	(1)
29.5. Further Comments and Extensions	711	(3)
29.5.1 On Implementation	711	(1)
29.5.2 On Time-Dependent Fractal Processes	712	(1)
29.5.3 On Multifractal Processes	713	(1)
29.6. Conclusion	714	(1)
Acknowledgment	715	(1)
References	715	(4)
Chapter 30 Fractal Analysis of Heart Rate Variability	719	(10)
Russell Fischer
Metin Akay
30.1. Introduction	719	(1)
30.2. The fBm Model	720	(1)
30.3. The Autocorrelation Function for DFGN	720	(1)
30.4. The Probability Density Function for DFGN	721	(1)
30.5. A Maximum Likelihood Estimator for DFGN	721	(1)
30.6. PSD Estimators for fBm and DFGN	722	(1)
30.7. A Wavelet Estimator for DFGN	723	(2)
30.8. The Heart Rate Variability Signal	725	(2)
References	727	(2)
Index	729	(10)
Editor's Biography	739