Wavelets in Medicine and Biology (eBook, ePUB)

Redaktion: Aldroubi, Akram; Unser, Michael

• Format: ePub

Produktbeschreibung

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.

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Produktdetails

• Verlag: Taylor & Francis
• Seitenzahl: 632
• Erscheinungstermin: 22. November 2017
• Englisch
• ISBN-13: 9781351404716
• Artikelnr.: 50403440

Autorenporträt

Aldroubi, Akram Unser, Michael

Inhaltsangabe

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