Statistics of Medical Imaging (Chapman & Hall/CRC Interdisciplinary Statistics)

Description

Statistical investigation into technology not only provides a better understanding of the intrinsic features of the technology (analysis), but also leads to an improved design of the technology (synthesis). Physical principles and mathematical procedures of medical imaging technologies have been extensively studied during past decades. However, less work has been done on the statistical aspects of these techniques. Statistics of Medical Imaging fills this gap and provides a theoretical framework for statistical investigation into medical imaging technologies. Features Describes physical principles and mathematical procedures of two medical imaging techniques: X-ray CT and MRIPresents statistical properties of imaging data (measurements) at each stage in the imaging processes of X-ray CT and MRIDemonstrates image reconstruction as a transform from a set of random variables (imaging data) to another set of random variables (image data)Presents statistical properties of image data (pixel intensities) at three levels: a single pixel, any two pixels, and a group of pixels (a region)Provides two stochastic models for X-ray CT and MR image in terms of their statistics and two model-based statistical image analysis methodsEvaluates statistical image analysis methods in terms of their detection, estimation, and classification performancesIndicates that X-ray CT, MRI, PET and SPECT belong to a category of imaging: the non-diffraction computed tomography Rather than offering detailed descriptions of statistics of basic imaging protocols of X-ray CT and MRI, this book provides a method to conduct similar statistical investigations into more complicated imaging protocols. b11413-1 Contents Preface b11413-2 1. Introduction 1.1 Data Flow and Statistics 1.2 Imaging and Image Statistics 1.3 Statistical Image Analysis 1.4 Motivation and Organization 1.4.1 Motivation 1.4.2 Organization b11413-3 2. X-Ray CT Physics and Mathematics 2.1 Introduction 2.2 Photon Emission, Attenuation, and Detection 2.2.1 Emission 2.2.2 Attenuation 2.2.3 Detection 2.3 Attenuation Coefficient 2.3.1 Linear Attenuation Coefficient 2.3.2 Relative Linear Attenuation Coefficient 2.4 Projections 2.4.1 Projection 2.4.2 Parallel and Divergent Projections 2.5 Mathematical Foundation of Image Reconstruction 2.5.1 Fourier Slice Theorem 2.5.2 Inverse Radon Transform 2.6 Image Reconstruction 2.6.1 Convolution Method 2.6.2 Computational Implementation 2.7 Appendices 2.7.1 Appendix 2A 2.7.2 Appendix 2B 2.7.3 Appendix 2C Problems References b11413-4 3. MRI Physics and Mathematics 3.1 Introduction 3.1.1 History 3.1.2 Overview 3.2 Nuclear Spin and Magnetic Moment 3.3 Alignment and Precession 3.3.1 Alignment 3.3.2 Precession 3.4 Macroscopic Magnetization 3.4.1 Macroscopic Magnetization 3.4.2 Thermal Equilibrium Macroscopic Magnetization 3.5 Resonance and Relaxation 3.5.1 Resonance 3.5.2 Relaxation 3.6 Bloch Eq. and Its Solution 3.6.1 Homogeneous Sample and Uniform Magnetic Field 3.6.2 Complex Representation 3.6.3 Heterogeneous Sample and Nonuniform Magnetic Field 3.7 Excitation 3.7.1 Nonselective Excitation 3.7.2 Selective Excitation 3.8 Induction 3.8.1 Signal Detection 3.8.2 Signal Demodulation 3.8.3 Spatial Localization 3.9 k-Space and k-Space Sample 3.9.1 Concepts 3.9.2 Sampling Protocols 3.9.3 Sampling Requirements 3.10 Image Reconstruction 3.10.1 Fourier Transform 3.10.2 Projection Reconstruction 3.11 Echo Signal 3.11.1 T*2 Decay 3.11.2 Echoes 3.11.3 T2 Decay 3.12 Appendices 3.12.1 Appendix 3A 3.12.2 Appendix 3B 3.12.3 Appendix 3C 3.12.4 Appendix 3D Problems References b11413-5 4. Non-Diffraction Computed Tomography 4.1 Introduction 4.2 Interaction between EM Wave and Object 4.3 Inverse Scattering Problem 4.3.1 Relationship between Incident and Scattered Waves 4.3.2 Inverse Scattering Problem Solutions 4.4 Non-Diffraction Computed Tomography 4.4.1 X-Ray Computed Tomography 4.4.2 Magnetic Resonance Imaging 4.4.3 Emission Computed Tomography 4.5 Appendix 4.5.1 Appendix 4A Problems References b11413-6 5. Statistics of X-Ray CT Imaging 5.1 Introduction 5.2 Statistics of Photon Measurements 5.2.1 Statistics of Signal Component 5.2.2 Statistics of Noise Component 5.2.3 Statistics of Photon Measurements 5.3 Statistics of Projections 5.3.1 Statistics of a Single Projection 5.3.2 Statistics of Two Projections 5.4 Statistical Interpretation of X-Ray CT Image Reconstruction 5.4.1 Signal Processing Paradigms 5.4.2 Statistical Interpretations 5.5 Appendices 5.5.1 Appendix 5A 5.5.2 Appendix 5B Problems References b11413-7 6. Statistics of X-Ray CT Image 6.1 Introduction 6.2 Statistics of the Intensity of a Single Pixel 6.2.1 Gaussianity 6.3 Statistics of the Intensities of Two Pixels 6.3.1 Spatially Asymptotic Independence 6.3.2 Exponential Correlation Coefficient 6.4 Statistics of the Intensities of a Group of Pixels 6.4.1 Stationarity 6.4.2 Ergodicity 6.5 Appendices 6.5.1 Appendix 6A Problems References b11413-8 7. Statistics of MR Imaging 7.1 Introduction 7.2 Statistics of Macroscopic Magnetizations 7.2.1 Statistics of Thermal Equilibrium Magnetization 7.2.2 Statistics of Transverse Precession Magnetizations 7.3 Statistics of MR Signals 7.3.1 Statistics of Signal Components of MR Signals 7.3.2 Statistics of the Noise Components of MR Signals 7.3.3 Statistics of MR Signals 7.4 Statistics of k-Space Samples 7.5 Statistical Interpretation of MR Image Reconstruction 7.5.1 Signal Processing Paradigms 7.5.2 Statistical Interpretations 7.6 Appendices 7.6.1 Appendix 7A 7.6.2 Appendix 7B 7.6.3 Appendix 7C Problems References b11413-9 8. Statistics of MR Imaging 8.1 Introduction 8.2 Statistics of the Intensity of a Single Pixel 8.2.1 Gaussianity 8.3 Statistics of the Intensities of Two Pixels 8.3.1 Spatially Asymptotic Independence 8.3.2 Exponential Correlation Coefficient 8.4 Statistics of the Intensities of a Group of Pixels 8.4.1 Stationarity 8.4.2 Ergodicity 8.4.3 Autocorrelaton and Spectral Density 8.5 Discussion and Remarks 8.5.1 Discussion 8.5.2 Remarks 8.6 Appendices 8.6.1 Appendix 8A 8.6.2 Appendix 8B 8.6.3 Appendix 8C 8.6.4 Appendix 8D Problems References b11413-10 9. Stochastic Image Models 9.1 Introduction 9.2 Stochastic Model I 9.2.1 Independent Finite Normal Mixture 9.2.2 Independence and Signal-to-Noise Ratio 9.3 Stochastic Model II 9.3.1 Markovianity 9.3.2 Correlated Finite Normal Mixture 9.4 Discussion 9.4.1 Mixture Models and Spatial Regularity 9.4.2 Mixture Models and Signal-to-Noise Ratio 9.4.3 Mixture Models and Hidden Markov Random Field 9.5 Appendix 9.5.1 Appendix 9A Problems References b11413-11 10. Statistical Image Analysis – I 10.1 Introduction 10.2 Detection of Number of Image Regions 10.2.1 Information Theoretic Criteria 10.3 Estimation of Image Parameters 10.3.1 Expectation-Maximization Method 10.3.2 Classification-Maximization Method 10.3.3 Discussion 10.4 Classification of Pixels 10.4.1 Bayesian Classifier 10.5 Statistical Image Analysis 10.5.1 Simulated Images 10.5.2 Physical Phantom Image 10.5.3 X-Ray CT Image 10.5.4 MR Image 10.6 Appendices 10.6.2 Appendix 10B 10.6.3 Appendix 10C Problems References b11413-12 11. Statistical Image Analysis – II 11.1 Introduction 11.2 Detection of the Number of Image Regions 11.2.1 Array Signal Processing Approach 11.2.2 Examples of Detection Results 11.2.3 Relationships to Other Related Methods 11.3 Estimation of Image Parameters 11.3.1 Extended Expectation-Maximization Method 11.3.2 Clique Potential Design 11.3.3 Energy Minimization 11.4 Classification of Pixels 11.4.1 Bayesian Classifier 11.5 Statistical Image Analysis 11.5.1 MR Images 11.6 Appendices 11.6.1 Appendix 11A 11.6.2 Appendix 11B 11.6.3 Appendix 11C Problems References b11413-13 12. Performance Evaluation of Image Analysis Methods 12.1 Introduction 12.2 Performance of the iFNM Model-Based Image Analysis Method 12.2.1 Detection Performance 12.2.2 Estimation Performance 12.2.3 Classification Performance 12.3 Performance of the cFNM Model-Based Image Analysis Method 12.4 Appendices 12.4.1 Appendix 12A 12.4.2 Appendix 12B Problems References b11413-14 Index A B C D E F G H I J K L M N P Q R S T U W X