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Considerable attention from the international scientific community is currently focused on the wide ranging applications of wavelets. For the first time, the field's leading experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. Wavelets in Medicine and Biology provides accessible, detailed, and comprehensive guidelines for all those interested in learning about wavelets and their applications to biomedical problems.
Contents
Part I Wavelet Transform: Theory and Implementation -- 1 The Wavelet Transform: A Surfing Guide /Akram Aldroubi -- 1.1 Introduction -- 1.2 Notations -- 1.3 The Continuous Wavelet Transform -- 1.3.1 The Continuous Wavelet Transform of 1-D Signals -- 1.3.2 Multidimensional Wavelet Transform -- 1.4 The Discrete Wavelet Transforms -- 1.4.1 The Dyadic Wavelet Transform -- 1.4.2 The Redundant Discrete Wavelet Transforms . -- 1.5 Multi resolutions and Wavelets -- 1.5.1 Multiresolution Approximations of L2 -- 1.5.2 Orthogonal MRA-Type Wavelets -- 1.5.3 Semi-Orthogonal MRA-Type Wavelet Bases -- 1.5.4 Bi-Orthogonal MRA-Type Wavelet Bases -- 1.5.5 Local and Global Characterization of Functions in Terms of Their Wavelet Coefficients -- 1.6 Special Bases of Scaling Functions -- 1.6.1 Interpolating Scaling Functions -- 1.6.2 Interpolating Wavelets -- 1.7 Applications and generalizations -- 1.7.1 Applications of the Wavelet Transform -- 1.7.2 Generalizations of the Wavelet Transform -- 1.8 Frame Representations -- 2 A Practical Guide to the Implementation of the Wavelet Transform /Michael Unser -- 2.1 Introduction -- 2.2 Basic Tools -- 2.2.1 Scaling Functions and Multiresolution Representations -- 2.2.2 Inner Products Via Discrete Convolutions -- 2.2.3 Boundary Conditions -- 2.3 Wavelet Bases (Nonredundant Transform) -- 2.3.1 Fast Dyadic Wavelet Transform -- 2.3.2 Implementation Details -- 2.3.3 Extensions -- 2.4 Dyadic Wavelet Frames -- 2.5 Nondyadic Wavelet Analyses -- 2.5.1 Wavelet Representation -- 2.5.2 Fast Redundant Dyadic Wavelet Transform -- 2.5.3 Fast Redundant Wavelet Transform with Integer Scales -- 2.5.4 Fast Redundant Wavelet Transform (Arbitrary Scales) -- 2.5.5 Fast Redundant Morlet or Gabor Wavelet Transform -- 2.6 Conclusion -- Part II Wavelets in Medical Imaging and Tomography -- 3 An Application of Wavelet Shrinkage to Tomography /Eric D. Kolaczyk -- 3.1 Introduction -- 3.1.1 Tomography -- 3.1.2 Why Wavelets? -- 3.1.3 Wavelet Shrinkage and the Proposed Method -- 3.2 Inversion -- 3.2.1 Direct Data Vs. Indirect Data -- 3.2.2 The Wavelet-Vaguelette Decomposition -- 3.2.3 Efficient Expressions for the Radon Vaguelette Coefficients -- 3.2.4 Calculation of the Radon Vaguelette Coefficient. -- 3.3 Denoising Using Wavelet Shrinkage -- 3.3.1 Wavelet Shrinkage with Direct Data -- 3.3.2 Wavelet Shrinkage with Tomographic Data -- 3.3.3 The Proposed Reconstruction Method -- 3.4 A Short Comparative Study -- 3.5 Discussion -- 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 -- 4.2 Image Acquisition -- 4.3 fMRI Time Series Analysis -- 4.3.1 The Hemodynamic Response Function -- 4.4 Wavelet Denoising of Signals -- 4.4.1 Data Analytic Thresholding -- 4.5 Experimental Results -- 4.5.1 Data Set Descriptions -- 4.5.2 Analysis Technique -- 4.5.3 Denoising Results -- 4.6 Conclusions -- 4.7 Acknowledgment -- 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 -- 5.2 Wavelet Transform -- 5.3 Correlation of Wavelet Coefficients -- 5.4 Statistical Tests -- 5.5 Experimental Results -- 5.5.1 Functional Magnetic Resonance Images -- 5.5.2 Positron Emission Tomography Images -- 5.6 Discussion -- 6 Feature Extraction in Digital Mammography -- R. A. DeVore, B. Lucier, and Z. Yang -- 6.1 Introduction -- 6.2 Mammograms as Digitized Images -- 6.2.1 Characteristics of Mammographic Images -- 6.3 Compression and Noise Removal -- 6.4 Some Issues in Compression Algorithms -- 6.4.1 Choice of Wavelet Basis -- 6.4.2 Choice of Metric -- 6.4.3 Level of Compression -- 6.5 Algorithms -- 6.6 Examples -- 7 Multiscale Contrast Enhancement and Denoising in Digital Radiographs /Jian Fan and Andrew Laine -- 7.1 Introduction -- 7.2 One-Dimensional Wavelet Transform -- 7.2.1 General Structure and Channel Characteristics -- 7.2.2 Two Possible Filters -- 7.3 Linear Enhancement and Unsharp Masking -- 7.3.1 Review of Unsharp Masking -- 7.3.2 Inclusion of Unsharp Masking within RDWT Frame-Work -- 7.4 Nonlinear Enhancement -- 7.4.1 Minimum Constraint for an Enhancement Function -- 7.4.2 Filter Selection -- 7.4.3 A Nonlinear Enhancement Function -- 7.5 Combined Denoising and Enhancement -- 7.5.1 Incorporating Wavelet Shrinkage into Enhancement -- 7.5.2 Threshold Estimation for Denoising -- 7.6 Two-Dimensional Extension -- 7.7 Experimental Results and Comparisons -- 7.8 Conclusion -- 7.9 Acknowledgment -- 8 Using Wavelets to Suppress Noise in Biomedical Images /Maurits Malfait -- 8.1 Introduction -- 8.2 Overview of Wavelet-Based Noise Suppression -- 8.2.1 Wavelet Shrinkage -- 8.2.2 Correlating Coefficients Between Wavelet Levels -- 8.2.3 Smoothness Measure from Wavelet Extrema -- 8.2.4 Example -- 8.3 Introducing an A Priori Model -- 8.3.1 Motivation -- 8.3.2 Basic Idea and Notation -- 8.3.3 Bayesian Method -- 8.3.4 The Conditional Probability -- 8.3.5 The A Priori Probability -- 8.3.6 Coefficient Manipulation -- 8.4 Results for Biomedical Images -- 9 Wavelet Transform and Tomography: Continuous and Discrete Approaches /F. Peyrin and M. Zaim -- 9.1 Introduction -- 9.2 Basis of Tomography -- 9.2.1 Problem Position -- 9.2.2 Reconstruction Methods: Transform Methods . -- 9.2.3 Series Expansion Methods -- 9.3 Continuous Wavelet Decomposition -- 9.3.1 Continuous Wavelet Decomposition of Projections -- 9.3.2 Continuous Wavelet Decomposition of the Image -- 9.4 Discrete Wavelet Decomposition -- 9.4.1 1-D DWT of the Projections -- 9.4.2 2-D Discrete WT of the Image -- 9.5 Conclusion -- 9.5.1 Acknowledgments -- 9.6 Appendix 1 -- 10 Wavelets and Local Tomography /Carlos A. Berenstein and David F. Walnut -- 10.1 Introduction -- 10.2 Background and Notation -- 10.3 Why Wavelets? -- 10.3.1 The Nonlocality of the Radon Transform -- 10.3.2 Wavelets, Vanishing Moments, and A-Tomography -- 10.4 Wavelet Inversion of the Radon Transform -- 10.4.1 The Continuous Wavelet Transform -- 10.4.2 The Semi-Continuous Wavelet Transform -- 10.4.3 The Discrete Wavelet Transform -- 10.5 Wavelet Localization of Radon Transform -- 10.6 Conclusions -- 10.7 Appendix: Proofs of Theorems -- 10.8 Acknowledgments -- 11 Optimal Time-Frequency Projections for Localized Tomography /Tim Olson -- 11.1 Introduction -- 11.1.1 Historical Notes -- 11.1.2 Prior Work -- 11.1.3 Organization -- 11.2 Algorithmic Goals -- 11.3 Background -- 11.3.1 The Radon Transform -- 11.3.2 Basic Fourier Analysis -- 11.4 Reconstruction Techniques -- 11.4.1 Fourier Reconstruction -- 11.4.2 Filtered Back projection -- 11.4.3 Nonlocality of the Radon Inversion -- 11.4.4 Visualization via the Sinogram -- 11.4.5 Comparison to Local Tomography -- 11.5 Localization -- 11.5.1 Utilizing Functions with Zero Moments -- 11.5.2 How Many Frequency Windows? -- 11.5.3 High Frequency Computation -- 11.5.4 Low Frequency Computation -- 11.5.5 The Algorithm -- 11.6 Numerical Results -- 11.7 Optimality -- 11.7.1 Minimization of Nonlocal Data -- 11.8 Conclusion -- 11.9 Appendix: Error Analysis -- 11.9.1 Aliasing Error Analysis -- 11.9.2 Truncation Error Analysis -- 11.10 Local Cosine and Sine Bases -- 11.11 Acknowledgments -- 12 Adapted Wavelet Techniques for Encoding Magnetic Resonance Images /Dennis M. Healy, Jr. and John B. Weaver -- 12.1 Introduction -- 12.2 Encoding in Magnetic Resonance Imaging -- 12.2.1 Nuclear Magnetic Resonance -- 12.2.2 Imaging -- 12.2.3 Imaging Time and Signal-to-Noise Ratio -- 12.3 Adapted Waveform Encoding in MRI -- 12.3.1 MRI Encoding with a Basis -- 12.3.2 Figures of Merit in Adapted Waveform Encoding -- 12.3.3 Choosing a Basis for Encoding -- 12.3.4 Implementation of Adapted Waveform Encoding -- 12.4 Reduced Imaging Times -- 12.4.1 Adapted Waveform Encoding with K-L Bases . -- 12.4.2 Approximate K-L Bases -- 12.4.3 Approximate Karhunen-Loeve Encoding -- 12.5 Conclusions -- 12.6 Acknowledgments -- 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 --
