Forty years ago, in 1957, the Principle of Maximum Entropy was first intro­ duced by Jaynes into the field of statistical mechanics. Since that seminal publication, this principle has been adopted in many areas of science and technology beyond its initial application. It is now found in spectral analysis, image restoration and a number of branches ofmathematics and physics, and has become better known as the Maximum Entropy Method (MEM). Today MEM is a powerful means to deal with ill-posed problems, and much research work is devoted to it. My own research in the area ofMEM started in 1980, when I was a grad­ uate student in the Department of Electrical Engineering at the University of Sydney, Australia. This research work was the basis of my Ph.D. the­ sis, The Maximum Entropy Method and Its Application in Radio Astronomy, completed in 1985. As well as continuing my research in MEM after graduation, I taught a course of the same name at the Graduate School, Chinese Academy of Sciences, Beijingfrom 1987to 1990. Delivering the course was theimpetus for developing a structured approach to the understanding of MEM and writing hundreds of pages of lecture notes.

1. Introduction.- 1.1 What is the Maximum Entropy Method.- 1.2 Definition of Entropy.- 1.3 Rationale of the Maximum Entropy Method.- 1.4 Present and Future Research.- 2. Maximum Entropy Method MEM1 and Its Application in Spectral Analysis.- 2.1 Definition and Expressions of Entropy H1.- 2.2 Formulation and Solution.- 2.3 Equivalents and Signal Model.- 2.4 Algorithms and Numerical Example (Given ACF).- 2.5 Algorithms and Numerical Example (Given Time Series).- 2.6 Order Selection.- 3. Maximum Entropy Method MEM2 and Its Application in Image Restoration.- 3.1 Definition and Expressions of Entropy H2.- 3.2 Formulation and Implicit Solution.- 3.3 Explicit Solution.- 3.4 Equivalents and Signal Model.- 3.5 R - ? Procedure.- 3.6 Algorithms and Numerical Examples (I).- 3.7 Algorithms and Numerical Examples (II).- 3.8 Algorithms and Numerical Examples (III).- 4. Analysis and Comparison of the Maximum Entropy Method.- 4.1 Generalized MEM.- 4.2 Expressions of Entropy.- 4.3 Solution's Properties.- 4.4 Resolution Enhancement and Data Extension (Experimental Results).- 4.5 Resolution Enhancement and Data Extension (Theoretical Analysis).- 4.6 Peak Location and Relative Power Estimation (Experimental Results).- 4.7 Peak Location and Relative Power Estimation (Theoretical Analysis).- 4.8 Comments on the Three Schools of Thought on MEM.- 5. Applications of the Maximum Entropy Method in Mathematics and Physics.- 5.1 Solution of Moment Problems.- 5.2 Solution of Integral Equations.- 5.3 Solution of Partial Differential Equations.- 5.4 Predictive Statistical Mechanics.- 5.5 Distributions of Particles Among Energy Levels.- 5.6 Classical Statistical Ensembles.- 5.7 Quantum Statistical Ensembles.- Appendices.- A. Cepstral Analysis.- A.1 Cepstral Analysis System.- A.2 I/O Relationship.-A.3 Properties of the Complex Cepstrum.- A.4 I/O Relationship for Minimum-Phase Input.- B. Image Restoration.- B.1 Image Formation.- B.2 Image Restoration.- B.3 Relationship Between Image Restoration and Spectral Estimation.- References.