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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010341723 | RC78.7.R4 S57 2015 | Open Access Book | Book | Searching... |
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Summary
Summary
Statistical Computing in Nuclear Imaging introduces aspects of Bayesian computing in nuclear imaging. The book provides an introduction to Bayesian statistics and concepts and is highly focused on the computational aspects of Bayesian data analysis of photon-limited data acquired in tomographic measurements.
Basic statistical concepts, elements of decision theory, and counting statistics, including models of photon-limited data and Poisson approximations, are discussed in the first chapters. Monte Carlo methods and Markov chains in posterior analysis are discussed next along with an introduction to nuclear imaging and applications such as PET and SPECT.
The final chapter includes illustrative examples of statistical computing, based on Poisson-multinomial statistics. Examples include calculation of Bayes factors and risks as well as Bayesian decision making and hypothesis testing. Appendices cover probability distributions, elements of set theory, multinomial distribution of single-voxel imaging, and derivations of sampling distribution ratios. C++ code used in the final chapter is also provided.
The text can be used as a textbook that provides an introduction to Bayesian statistics and advanced computing in medical imaging for physicists, mathematicians, engineers, and computer scientists. It is also a valuable resource for a wide spectrum of practitioners of nuclear imaging data analysis, including seasoned scientists and researchers who have not been exposed to Bayesian paradigms.
Author Notes
Arkadiusz Sitek is an associate physicist at Massachusetts General Hospital in Boston and an assistant professor at Harvard Medical School. He received his doctorate from the University of British Columbia in Canada and since 2001 has worked as a nuclear imaging scientist in the Lawrence Berkeley National Laboratory, Beth Israel Medical Center, and Brigham and Women's Hospital before joining Massachusetts General Hospital. He has authored more than 100 scientific journal and proceedings papers, book chapters, and patents, and served as a principal investigator on nuclear imaging research projects. Dr. Sitek is a practitioner of the Bayesian school of thought and a member of the International Society for Bayesian Analysis.
Table of Contents
List of Figures | p. xi |
List of Tables | p. xvi |
About the Series | p. xviii |
Preface | p. xxi |
About the Author | p. xxiii |
Chapter 1 Basic statistical concepts | p. 1 |
1.1 Introduction | p. 1 |
1.2 Before- and after-the-experiment concepts | p. 2 |
1.3 Definition of probability | p. 6 |
1.3.1 Countable and uncountable quantities | p. 8 |
1.4 Joint and conditional probabilities | p. 10 |
1.5 Statistical model | p. 13 |
1.6 Likelihood | p. 17 |
1.7 Pre-posterior and posterior | p. 19 |
1.7.1 Reduction of pre-posterior to posterior | p. 19 |
1.7.2 Posterior through Bayes theorem | p. 19 |
1.7.3 Prior selection | p. 20 |
1.7.4 Examples | p. 22 |
1.7.5 Designs of experiments | p. 23 |
1.8 Extension to multi- dimensions | p. 25 |
1.8.1 Chain rule and marginalization | p. 26 |
1.8.2 Nuisance quantities | p. 27 |
1.9 Unconditional and conditional independence | p. 29 |
1.10 Summary | p. 34 |
Chapter 2 Elements of decision theory | p. 37 |
2.1 Introduction | p. 37 |
2.2 Loss function and expected loss | p. 39 |
2.3 After-the-experiment decision making | p. 42 |
2.3.1 Point estimation | p. 43 |
2.3.2 Interval estimation | p. 48 |
2.3.3 Multiple-alternative decisions | p. 50 |
2.3.4 Binary hypothesis testing/detection | p. 52 |
2.4 Before-the-experiment decision making | p. 56 |
2.4.1 Bayes risk | p. 58 |
2.4.2 Other methods | p. 62 |
2.5 Robustness of the analysis | p. 64 |
Chapter 3 Counting statistics | p. 67 |
3.1 Introduction to statistical models | p. 67 |
3.2 Fundamental statistical law | p. 69 |
3.3 General models of photon-limited data | p. 71 |
3.3.1 Binomial statistics of nuclear decay | p. 71 |
3.3.2 Multinomial statistics of detection | p. 72 |
3.3.3 Statistics of complete data | p. 77 |
3.3.4 Poisson-multinomial distribution of nuclear data | p. 84 |
3.4 Poisson approximation | p. 88 |
3.4.1 Poisson statistics of nuclear decay | p. 88 |
3.4.2 Poisson approximation of nuclear data | p. 90 |
3.5 Normal distribution approximation | p. 93 |
3.5.1 Approximation of binomial law | p. 94 |
3.5.2 Central limit theorem | p. 95 |
Chapter 4 Monte Carlo methods in posterior analysis | p. 99 |
4.1 Monte Carlo approximations of distributions | p. 99 |
4.1.1 Continuous distributions | p. 99 |
4.1.2 Discrete distributions | p. 104 |
4.2 Monte Carlo integrations | p. 107 |
4.3 Monte Carlo summations | p. 110 |
4.4 Markov chains | p. 111 |
4.4.1 Markov processes | p. 113 |
4.4.2 Detailed balance | p. 114 |
4.4.3 Design of Markov chain | p. 116 |
4.4.4 Metropolis-Hastings sampler | p. 118 |
4.4.5 Equilibrium | p. 120 |
4.4.6 Resampling methods (bootstrap) | p. 126 |
Chapter 5 Basics of nuclear imaging | p. 129 |
5.1 Nuclear radiation | p. 130 |
5.1.1 Basics of nuclear physics | p. 130 |
5.1.1.1 Atoms and chemical reactions | p. 130 |
5.1.1.2 Nucleus and nuclear reactions | p. 131 |
5.1.1.3 Types of nuclear decay | p. 133 |
5.1.2 Interaction of radiation with matter | p. 136 |
5.1.2.1 Inelastic scattering | p. 137 |
5.1.2.2 Photoelectric effect | p. 138 |
5.1.2.3 Photon attenuation | p. 138 |
5.2 Radiation detection in nuclear imaging | p. 141 |
5.2.1 Semiconductor detectors | p. 142 |
5.2.2 Scintillation detectors | p. 143 |
5.2.2.1 Photomultiplier tubes | p. 144 |
5.2.2.2 Solid-state photomultipliers | p. 145 |
5.3 Nuclear imaging | p. 147 |
5.3.1 Photon-limited data | p. 150 |
5.3.2 Region of response (ROR) | p. 152 |
5.3.3 Imaging with gamma camera | p. 153 |
5.3.3.1 Gamma camera | p. 153 |
5.3.3.2 SPECT | p. 157 |
5.3.4 Positron emission tomography (PET) | p. 159 |
5.3.4.1 PET nuclear imaging scanner | p. 159 |
5.3.4.2 Coincidence detection | p. 161 |
5.3.4.3 ROR for PET and TOF-PET | p. 162 |
5.3.4.4 Quantitation of PET | p. 165 |
5.3.5 Compton imaging | p. 166 |
5.4 Dynamic imaging and kinetic modeling | p. 168 |
5.4.1 Compartmental model | p. 169 |
5.4.2 Dynamic measurements | p. 171 |
5.5 Apphcations of nuclear imaging | p. 173 |
5.5.1 Clinical applications | p. 173 |
5.5.2 Other applications | p. 174 |
Chapter 6 Statistical computing | p. 179 |
6.1 Computing using Poisson-multinomial distribution (PMD) | p. 179 |
6.1.1 Sampling the posterior | p. 180 |
6.1.2 Computationally efficient priors | p. 182 |
6.1.3 Generation of Markov chain | p. 186 |
6.1.4 Metropolis-Hastings algorithm | p. 187 |
6.1.5 Origin ensemble algorithms | p. 190 |
6.2 Examples of statistical computing | p. 193 |
6.2.1 Simple tomographic system (STS) | p. 194 |
6.2.2 Image reconstruction | p. 195 |
6.2.3 Bayes factors | p. 198 |
6.2.4 Evaluation of data quality | p. 200 |
6.2.5 Detection-Bayesian decision making | p. 203 |
6.2.6 Bayes risk | p. 205 |
Appendix A Probability distributions | p. 209 |
A.1 Univariate distributions | p. 209 |
A.1.1 Binomial distribution | p. 209 |
A.1.2 Gamma distribution | p. 209 |
A.1.3 Negative binomial distribution | p. 209 |
A.1.4 Poisson-binomial distribution | p. 210 |
A.1.5 Poisson distribution | p. 210 |
A.1.6 Uniform distribution | p. 210 |
A.1.7 Univariate normal distribution | p. 210 |
A.2 Multivariate distributions | p. 211 |
A.2.1 Multinomial distribution | p. 211 |
A.2.2 Multivariate normal distribution | p. 211 |
A.2.3 Poisson-multinomial distribution | p. 211 |
Appendix B Elements of set theory | p. 213 |
Appendix C Multinomial distribution of single-voxel imaging | p. 217 |
Appendix D Derivations of sampling distribution ratios | p. 221 |
Appendix E Equation (6.11) | p. 223 |
Appendix F C++ OE code for STS | p. 225 |
References | p. 231 |