The Maximum Entropy Method
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1 Nailong Wu The Maximum Entropy Method With 53 Figures Springer
2 1. Introduction What is the Maximum Entropy Method Definition of Entropy Rationale of the Maximum Entropy Method Present and Future Research 9 2. Maximum Entropy Method MEMl and Its Application in Spectral Analysis Definition and Expressions of Entropy HI Approach Approach Discussion Formulation and Solution Formulation Solution Discussion Equivalents and Signal Model ACF Extension Subject to the Nonnegativity Constraint Principle of MCE AR Process (Signal Model) Bayesian Method Wiener Filter and Approximation Theoretic Approach Algorithms and Numerical Example (Given ACF) Levinson's Recursion for 1-D Noiseless Data Lim-Malik Algorithm for 2-D Noiseless Data Wernecke-D'Addario Algorithm for 2-D Noisy Data Numerical Example Algorithms and Numerical Example (Given Time Series) Burg Algorithm Marple Algorithm Other Fast Algorithms Numerical Example 91
3 X 2.6 Order Selection FPE Criterion AIC Criterion Other Criteria Summary Maximum Entropy Method MEM2 and Its Application in Image Restoration Definition and Expressions of Entropy HI MLM Direct Definition Method Discussion Formulation and Implicit Solution Formulation Implicit Solution Iterative Algorithm Discussion Explicit Solution Explicit Solution Discussion Examples Equivalents and Signal Model ACF Extension Subject to the Nonnegativity Constraint Principle of MCE Exponential Process (Signal Model) Bayesian Method MLM R-X Procedure Statements of the MEM2 Problem R-X Procedure Example Algorithms and Numerical Examples (I) Frieden Algorithm Gull-Daniell Algorithm Revised GD Algorithm Simplified Newton-Raphson Algorithm Numerical Example Algorithms and Numerical Examples (II) Skilling-Bryan Algorithm Differential Equation Approach Algorithms and Numerical Examples (III) MEM/MemSys5 Package MEM Task in IRAF Restoration with Variable Resolution 184
4 3.8.4 Numerical Examples Other Algorithms Analysis and Comparison of the Maximum Entropy Method Generalized MEM Formulation of GMEM "Entropy" Expressions in GMEM Properties of GMEM Expressions of Entropy Solution's Properties Existence Uniqueness Consistency Statistical Properties Resolution Enhancement and Data Extension (Experimental Results) Examples Resolvability in 1-D Spectral Estimation Resolvability in 2-D Spectral Estimation Superresolution and Spectral Line Splitting Resolution Enhancement and Data Extension (Theoretical Analysis) Data Extension in MEM1 and MEM Resolution Enhancement of MEM1 and MEM MEM1 and MEM2 Spectra at Low SNR Line Splitting of MEM Peak Location and Relative Power Estimation (Experimental Results) Peak Location (Given ACF) Peak Location (Given Time Series) Relative Power Estimation (Given ACF) Summary and Comments Peak Location and Relative Power Estimation (Theoretical Analysis) Interference Between Peaks Causes Peak Shifting Explanation of the Peak Shifting in MEM1 Spectra Relative Power Estimation for MEM Summary for Sects Comments on the Three Schools of Thought on MEM Applications of the Maximum Entropy Method in Mathematics and Physics Solution of Moment Problems General Theory 252 XI
5 XII Numerical Methods Noisy Moment Problems Numerical Examples Solution of Integral Equations Conversion of Integral Equations to Moment Problems Solution of Moment Problems by MEM Numerical Examples Discussion Solution of Partial Differential Equations Theory Numerical Example Discussion Predictive Statistical Mechanics Formulation and Solution Useful Formulae Distributions of Particles Among Energy Levels Boltzmann Distribution Fermi-Dirac and Bose-Einstein Distributions Classical Statistical Ensembles Micro canonical Ensemble Canonical Ensemble Grand Canonical Ensemble Quantum Statistical Ensembles Microcanonical Ensemble Canonical Ensemble Grand Canonical Ensemble 303 Appendices 305 A. Cepstral Analysis 305 A.I Cepstral Analysis System 305 A.2 I/O Relationship 306 A.3 Properties of the Complex Cepstrum 307 A.4 I/O Relationship for Minimum-Phase Input 309 B. Image Restoration 311 B.I Image Formation 311 B.2 Image Restoration 313 B.3 Relationship Between Image Restoration and Spectral Estimation 314 References 317 Index 323
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