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Pattern Theory

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Table of Contents

Preface Notation What Is Pattern Theory? The Manifesto of Pattern Theory The Basic Types of Patterns Bayesian Probability Theory: Pattern Analysis and Pattern Synthesis English Text and Markov Chains Basics I: Entropy and Information Measuring the n-gram Approximation with Entropy Markov Chains and the n-gram Models Words Word Boundaries via Dynamic Programming and Maximum Likelihood Machine Translation via Bayes' Theorem Exercises Music and Piece wise Gaussian Models Basics III: Gaussian Distributions Basics IV: Fourier Analysis Gaussian Models for Single Musical Notes Discontinuities in One-Dimensional Signals The Geometric Model for Notes via Poisson Processes Related Models Exercises Character Recognition and Syntactic Grouping Finding Salient Contours in Images Stochastic Models of Contours The Medial Axis for Planar Shapes Gestalt Laws and Grouping Principles Grammatical Formalisms Exercises Contents Image Texture, Segmentation and Gibbs Models Basics IX: Gibbs Fields (u + v)-Models for Image Segmentation Sampling Gibbs Fields Deterministic Algorithms to Approximate the Mode of a Gibbs Field Texture Models Synthesizing Texture via Exponential Models Texture Segmentation Exercises Faces and Flexible Templates Modeling Lighting Variations Modeling Geometric Variations by Elasticity Basics XI: Manifolds, Lie Groups, and Lie Algebras Modeling Geometric Variations by Metrics on Diff Comparing Elastic and Riemannian Energies Empirical Data on Deformations of Faces The Full Face Model Appendix: Geodesics in Diff and Landmark Space Exercises Natural Scenes and their Multiscale Analysis High Kurtosis in the Image Domain Scale Invariance in the Discrete and Continuous Setting The Continuous and Discrete Gaussian Pyramids Wavelets and the "Local" Structure of Images Distributions Are Needed Basics XIII: Gaussian Measures on Function Spaces The Scale -Rotation- and Translation-Invariant Gaussian Distribution Mode lII: Images Made Up of Independent Objects Further Models Appendix: A Stability Property of the Discrete Gaussian Pyramid Exercises Bibliography Index

About the Author

David Mumford is a professor emeritus of applied mathematics at Brown University. His contributions to mathematics fundamentally changed algebraic geometry, including his development of geometric invariant theory and his study of the moduli space of curves. In addition, Dr. Mumford's work in computer vision and pattern theory introduced new mathematical tools and models from analysis and differential geometry. He has been the recipient of many prestigious awards, including U.S. National Medal of Science (2010), the Wolf Foundation Prize in Mathematics (2008), the Steele Prize for Mathematical Exposition (2007), the Shaw Prize in Mathematical Sciences (2006), a MacArthur Foundation Fellowship (1987-1992), and the Fields Medal (1974). Agnes Desolneux is a researcher at CNRS/Universite Paris Descartes. A former student of David Mumford's, she earned her Ph.D. in applied mathematics from CMLA, ENS Cachan. Dr. Desolneux's research interests include statistical image analysis, Gestalt theory, mathematical modeling of visual perception, and medical imaging.


The book comes with a large number of exercises and problems, some requiring computer programming. Thanks to these, it can be used as a textbook to support a quite original course that could be offered by a department of applied mathematics, computer science or electrical engineering. In fact, this excellent book targets and deserves a broad readership. It will provide precious and interesting material to anyone who would like to discover pattern theory and how it traverses across geometry, probability and signal processing. --Laurent Younes, Mathematical Reviews, Issue 2011m ! a masterpiece. It is one of the best books I have ever read. ! What singles out this outstanding book is an extremely original subject development. ! This book is so exciting. It is a detective fiction. It is an inquiry into 'real-world signals.' In contrast to most detective stories, the beauty of the style is exceptional and meets the standards of the best writers. Art and beauty are present everywhere in this marvelous book. ! The overall organisation of the book is also marvelous. ! The authors are leaders in signal and image processing and this book is based on their extremely innovative research. Reading this book is like entering David Mumford's office and beginning a friendly and informal scientific discussion with him and Agnes. That is a good approximation to paradise. --Yves Meyer, EMS Newsletter, September 2011 Pattern Theory covers six classic attempts at modeling signals from the human and natural world: natural language (written), music, character recognition, texture modeling, face recognition, and natural scenes. These applications, appealing to students and researchers alike, include fourteen 'crash courses' giving all the needed basics, exercises, and numerical simulations. ... a complete pedagogic tool at master or first-year graduate level. I endorse the publication of Pattern Theory, and will actually use it and recommend it to other researchers. --Jean-Michel Morel, CMLA This book is fascinating. It develops a statistic approach to finding the patterns in the signals generated by the world. The style is lucid. I'm reminded of Mumford's exposition of Theta functions and Abelian varieties in his Tata lectures. The exposition is thorough. The authors provide the necessary mathematical tools allowing scientists to pursue an exciting subject. I've been running a seminar at MIT entitled 'New Opportunities for the Interactions of Mathematics and Other Disciplines' because I'm convinced that mathematics will move in surprising new directions. Pattern Theory, a decade's effort, is a prime example. --I.M. Singer, Institute Professor, MIT

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