Pattern Theory The Stochastic Analysis of Real-World Signals

Pattern theory is a distinctive approach to the analysis of all forms of real-world signals. At its core is the design of a large variety of probabilistic models whose samples reproduce the look and feel of the real signals, their patterns, and their variability. Bayesian statistical inference then allows you to apply these models in the analysis of new signals. This book treats the mathematical tools, the models themselves, and the computational algorithms for applying statistics to analyze six representative classes of signals of increasing complexity. The book covers patterns in text, sound, and images. Discussions of images include recognizing characters, textures, nature scenes, and human faces. The text includes onlineaccess to thematerials (data, code, etc.) needed for the exercises.

PrefaceNotationWhat Is Pattern Theory?The Manifesto of Pattern TheoryThe Basic Types of PatternsBayesian Probability Theory: Pattern Analysisand Pattern SynthesisEnglish Text and Markov ChainsBasics I: Entropy and InformationMeasuring the n-gram Approximation with EntropyMarkov Chains and the n-gram ModelsWordsWord Boundaries via Dynamic Programming and Maximum LikelihoodMachine Translation via Bayes’ TheoremExercisesMusic and Piece wise Gaussian ModelsBasics III: Gaussian DistributionsBasics IV: Fourier AnalysisGaussian Models for Single Musical NotesDiscontinuities in One-Dimensional SignalsThe Geometric Model for Notes via Poisson ProcessesRelated ModelsExercisesCharacter Recognition and Syntactic GroupingFinding Salient Contours in ImagesStochastic Models of ContoursThe Medial Axis for Planar ShapesGestalt Laws and Grouping PrinciplesGrammatical FormalismsExercisesContentsImage Texture, Segmentation and Gibbs ModelsBasics IX: Gibbs Fields(u + v)-Models for Image SegmentationSampling Gibbs FieldsDeterministic Algorithms to Approximate the Mode of a Gibbs FieldTexture ModelsSynthesizing Texture via Exponential ModelsTexture SegmentationExercisesFaces and Flexible TemplatesModeling Lighting VariationsModeling Geometric Variations by ElasticityBasics XI: Manifolds, Lie Groups, and Lie AlgebrasModeling Geometric Variations by Metrics on DiffComparing Elastic and Riemannian EnergiesEmpirical Data on Deformations of FacesThe Full Face ModelAppendix: Geodesics in Diff and Landmark SpaceExercisesNatural Scenes and their Multiscale AnalysisHigh Kurtosis in the Image DomainScale Invariance in the Discrete and Continuous SettingThe Continuous and Discrete Gaussian PyramidsWavelets and the "Local" Structure of ImagesDistributions Are NeededBasics XIII: Gaussian Measures on Function SpacesThe Scale -Rotation- and Translation-Invariant Gaussian DistributionMode lII: Images Made Up of Independent ObjectsFurther ModelsAppendix: A Stability Property of the DiscreteGaussian PyramidExercisesBibliographyIndex

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). Agnès Desolneux is a researcher at CNRS/Université 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.