The authors of this book take a very pragmatic approach in the selection and the order of the material they present. After a brief Chapter 0 discussing what is meant by Pattern Theory, the Chapters 1 through 6 deal progressively with topics ranging from Coding to Music to Character and Image recognition. By posing concrete, real-life problems, the authors take the reader through an eclectic tour of mathematics.
Almost all branches of the undergraduate mathematics curriculum appear at one point or another in the book. Probability and Statistics dominate with Bayesian probability, Markov Chains and Stochastic Processes. Riemannian geometry, Lie Groups and Fluid Mechanics tools are employed in the analysis of images. By going to infinite dimensions, aspects of Functional Analysis such as Distributions, Fourier Transforms and Sobolev spaces, are discussed. Differential Equations appear as Euler-Lagrange equations for the extremals in variational problems. There are numerous algorithms spread throughout the book. Some of these algorithms serve as simulations, and some are pieces of code used in image processing.
Each chapter ends with a number of exercises. Some of the exercises require manipulating relatively large datasets, and this is made available through the book’s website: http://www.dam.brown.edu/ptg/MDbook/index.html
There are 241 references ranging from technical journal articles and PhD dissertations, to classical textbooks.
The magic of this book is the mastery with which real-life problems are interweaved with serious mathematics and small bursts of flavor contained in things like Card Shuffling, Shannon’s n-gram approximations, and comparison of Kandinsky and Pollock paintings with pictures obtained from random wavelet models. The book is peppered with such little gems.
The first author, David Mumford, is a name well known to every mathematician. He is a member of the Pattern Theory Group which was founded in 1972 by Ulf Grenander at Brown University. The book evolved from lectures delivered by the first author, and covers topics taught by both authors. This is a book worth reading by anyone interested in mathematics.
Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.