Pattern theory : from representation to inference / / Ulf Grenander and Michael I. Miller

• Record Nr.: UNINA9910966108003321
• Autore: Grenander Ulf
• Pubbl/distr/stampa: Oxford ; ; New York, : Oxford University Press, 2007
• ISBN: 0-19-152311-9; 1-4294-7014-3; 0-19-191656-0; 1-280-75385-4
• Edizione: [1st ed.]
• Descrizione fisica: xii, 596 p. : ill. (some col.)
• Altri autori (Persone): MillerMichael I
• Disciplina: 511.33
• Soggetti: Pattern perception; Pattern recognition systems
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references (p. 563-579) and index.
• Nota di contenuto: Intro -- Contents -- 1 Introduction -- 1.1 Organization -- 2 The Bayes Paradigm, Estimation and Information Measures -- 2.1 Bayes Posterior Distribution -- 2.1.1 Minimum Risk Estimation -- 2.1.2 Information Measures -- 2.2 Mathematical Preliminaries -- 2.2.1 Probability Spaces, Random Variables, Distributions, Densities, and Expectation -- 2.2.2 Transformations of Variables -- 2.2.3 The Multivariate Normal Distribution -- 2.2.4 Characteristic Function -- 2.3 Minimum Risk Hypothesis Testing on Discrete Spaces -- 2.3.1 Minimum Probability of Error via Maximum A Posteriori Hypothesis Testing -- 2.3.2 Neyman-Pearson and the Optimality of the Likelihood Ratio Test -- 2.4 Minimum Mean-Squared Error Risk Estimation in Vector Spaces -- 2.4.1 Normed Linear and Hilbert Spaces -- 2.4.2 Least-Squares Estimation -- 2.4.3 Conditional Mean Estimation and Gaussian Processes -- 2.5 The Fisher Information of Estimators -- 2.6 Maximum-Likelihood and its consistency -- 2.6.1 Consistency via Uniform Convergence of Empirical Log-likelihood -- 2.6.2 Asymptotic Normality and &amp -- #8730 -- n Convergence Rate of the MLE -- 2.7 Complete-Incomplete Data Problems and the EM Algorithm -- 2.8 Hypothesis Testing and Model Complexity -- 2.8.1 Model-Order ; Estimation and the d/2 log Sample-Size Complexity -- 2.8.2 The Gaussian Case is Special -- 2.8.3 Model Complexity and the Gaussian Case -- 2.9 Building Probability Models via the Principle of Maximum Entropy -- 2.9.1 Principle of Maximum Entropy -- 2.9.2 Maximum Entropy Models -- 2.9.3 Conditional Distributions are Maximum Entropy -- 3 Probabilistic Directed Acyclic Graphs and Their Entropies -- 3.1 Directed Acyclic Graphs (DAGs) -- 3.2 Probabilities on Directed Acyclic Graphs (PDAGs) -- 3.3 Finite State Markov Chains -- 3.4 Multi-type Branching Processes -- 3.4.1 The Branching Matrix -- 3.4.2 The Moment-Generating Function.; 3.5 Extinction for Finite-State Markov Chains and Branching Processes -- 3.5.1 Extinction in Markov Chains -- 3.5.2 Extinction in Branching Processes -- 3.6 Entropies of Directed Acyclic Graphs -- 3.7 Combinatorics of Independent, Identically Distributed Strings via the Aymptotic Equipartition Theorem -- 3.8 Entropy and Combinatorics of Markov Chains -- 3.9 Entropies of Branching Processes -- 3.9.1 Tree Structure of Multi-Type Branching Processes -- 3.9.2 Entropies of Sub-Critical, Critical, and Super-Critical Processes -- 3.9.3 Typical Trees and the Equipartition Theorem -- 3.10 Formal Languages and Stochastic Grammars -- 3.11 DAGs for Natural Language Modelling -- 3.11.1 Markov Chains and m-Grams -- 3.11.2 Context-Free Models -- 3.11.3 Hierarchical Directed Acyclic Graph Model -- 3.12 EM Algorithms for Parameter Estimation in Hidden Markov Models -- 3.12.1 MAP Decoding of the Hidden State Sequence -- 3.12.2 ML Estimation of HMM parameters via EM Forward/Backward Algorithm -- 3.13 EM Algorithms for Parameter Estimation in Natural Language Models -- 3.13.1 EM Algorithm for Context-Free Chomsky Normal Form -- 3.13.2 General Context-Free Grammars and the Trellis Algorithm of Kupiec -- 4 Markov Random Fields on Undirected Graphs -- 4.1 Undirected Graphs -- 4.2 Markov Random Fields -- 4.3 Gibbs Random Fields -- 4.4 The Splitting Property of Gibbs Distributions -- 4.5 Bayesian Texture Segmentation: The log-Normalizer Problem -- 4.5.1 The Gibbs Partition Function Problem -- 4.6 Maximum-Entropy Texture Representation -- 4.6.1 Empirical Maximum Entropy Texture Coding -- 4.7 Stationary Gibbs Random Fields -- 4.7.1 The Dobrushin/Lanford/Ruelle Definition -- 4.7.2 Gibbs Distributions Exhibit Multiple Laws with the Same Interactions (Phase Transitions): The Ising Model at Low Temperature -- 4.8 1D Random Fields are Markov Chains.; 4.9 Markov Chains Have a Unique Gibbs Distribution -- 4.10 Entropy of Stationary Gibbs Fields -- 5 Gaussian Random Fields on Undirected Graphs -- 5.1 Gaussian Random Fields -- 5.2 Difference Operators and Adjoints -- 5.3 Gaussian Fields Induced via Difference Operators -- 5.4 Stationary Gaussian Processes on Z[sup(d)] and their Spectrum -- 5.5 Cyclo-Stationary Gaussian Processes and their Spectrum -- 5.6 The log-Determinant Covariance and the Asymptotic Normalizer -- 5.6.1 Asymptotics of the Gaussian processes and their Covariance -- 5.6.2 The Asymptotic Covariance and log-Normalizer -- 5.7 The Entropy Rates of the Stationary Process -- 5.7.1 Burg's Maximum Entropy Auto-regressive Processes on Z[sup(d)] -- 5.8 Generalized Auto-Regressive Image Modelling via Maximum-Likelihood Estimation -- 5.8.1 Anisotropic Textures -- 6 The Canonical Representations of General Pattern Theory -- 6.1 The Generators, Configurations, and Regularity of Patterns -- 6.2 The Generators of Formal Languages and Grammars -- 6.3 Graph Transformations -- 6.4 The Canonical Representation of Patterns: DAGs, MRFs, Gaussian Random Fields -- 6.4.1 Directed Acyclic Graphs -- 6.4.2 Markov Random Fields -- 6.4.3 Gaussian Random Fields: Generators induced via difference operators -- 7 Matrix ; Group Actions Transforming Patterns -- 7.1 Groups Transforming Configurations -- 7.1.1 Similarity Groups -- 7.1.2 Group Actions Defining Equivalence -- 7.1.3 Groups Actions on Generators and Deformable Templates -- 7.2 The Matrix Groups -- 7.2.1 Linear Matrix and Affine Groups of Transformation -- 7.2.2 Matrix groups acting on R[sup(d)] -- 7.3 Transformations Constructed from Products of Groups -- 7.4 Random Regularity on the Similarities -- 7.5 Curves as Submanifolds and the Frenet Frame -- 7.6 2D Surfaces in R[sup(3)] and the Shape Operator -- 7.6.1 The Shape Operator.; 7.7 Fitting Quadratic Charts and Curvatures on Surfaces -- 7.7.1 Gaussian and Mean Curvature -- 7.7.2 Second Order Quadratic Charts -- 7.7.3 Isosurface Algorithm -- 7.8 Ridge Curves and Crest Lines -- 7.8.1 Definition of Sulcus, Gyrus, and Geodesic Curves on Triangulated Graphs -- 7.8.2 Dynamic Programming -- 7.9 Bijections and Smooth Mappings for Coordinatizing Manifolds via Local Coordinates -- 8 Manifolds, Active Models, and Deformable Templates -- 8.1 Manifolds as Generators, Tangent Spaces, and Vector Fields -- 8.1.1 Manifolds -- 8.1.2 Tangent Spaces -- 8.1.3 Vector Fields on M -- 8.1.4 Curves and the Tangent Space -- 8.2 Smooth Mappings, the Jacobian, and Diffeomorphisms -- 8.2.1 Smooth Mappings and the Jacobian -- 8.2.2 The Jacobian and Local Diffeomorphic Properties -- 8.3 Matrix Groups are Diffeomorphisms which are a Smooth Manifold -- 8.3.1 Diffeomorphisms -- 8.3.2 Matrix Group Actions are Diffeomorphisms on the Background Space -- 8.3.3 The Matrix Groups are Smooth Manifolds (Lie Groups) -- 8.4 Active Models and Deformable Templates as Immersions -- 8.4.1 Snakes and Active Contours -- 8.4.2 Deforming Closed Contours in the Plane -- 8.4.3 Normal Deformable Surfaces -- 8.5 Activating Shapes in Deformable Models -- 8.5.1 Likelihood of Shapes Partitioning Image -- 8.5.2 A General Calculus for Shape Activation -- 8.5.3 Active Closed Contours in R[sup(2)] -- 8.5.4 Active Unclosed Snakes and Roads -- 8.5.5 Normal Deformation of Circles and Spheres -- 8.5.6 Active Deformable Spheres -- 8.6 Level Set Active Contour Models -- 8.7 Gaussian Random Field Models for Active Shapes -- 9 Second Order and Gaussian Fields -- 9.1 Second Order Processes (SOP) and the Hilbert Space of Random Variables -- 9.1.1 Measurability, Separability, Continuity -- 9.1.2 Hilbert space of random variables -- 9.1.3 Covariance and Second Order Properties.; 9.1.4 Quadratic Mean Continuity and Integration -- 9.2 Orthogonal Process Representations on Bounded Domains -- 9.2.1 Compact Operators and Covariances -- 9.2.2 Orthogonal Representations for Random Processes and Fields -- 9.2.3 Stationary Periodic Processes and Fields on Bounded Domains -- 9.3 Gaussian Fields on the Continuum -- 9.4 Sobolev Spaces, Green's Functions, and Reproducing Kernel Hilbert Spaces -- 9.4.1 Reproducing Kernel Hilbert Spaces -- 9.4.2 Sobolev Normed Spaces -- 9.4.3 Relation to Green's Functions -- 9.4.4 Gradient and Laplacian Induced Green's Kernels -- 9.5 Gaussian Processes Induced via Linear Differential Operators -- 9.6 Gaussian Fields in the Unit Cube -- 9.6.1 Maximum Likelihood Estimation of the Fields: Generalized ARMA Modelling -- 9.6.2 Small Deformation Vector Fields Models in the Plane and Cube -- 9.7 Discrete Lattices and Reachability of Cyclo-Stationary Spectra -- 9.8 Stationary Processes on the Sphere -- 9.8.1 Laplacian Operator Induced Gaussian Fields on the Sphere -- 9.9 Gaussian Random Fields on an Arbitrary Smooth Surface -- 9.9.1 Laplace-Beltrami Operator with Neumann Boundary Conditions -- 9.9.2 Smoothing an Arbitrary Function on Manifolds by Orthonormal Bases of the Laplace-Beltrami Operator -- 9.10 Sample Path Properties and Continuity -- 9.11 Gaussian Random Fields as Prior Distributions in Point Process Image Reconstruction -- 9.11.1 The Need for ; Regularization in Image Reconstruction -- 9.11.2 Smoothness and Gaussian Priors -- 9.11.3 Good's Roughness as a Gaussian Prior -- 9.11.4 Exponential Spline Smoothing via Good's Roughness -- 9.12 Non-Compact Operators and Orthogonal Representations -- 9.12.1 Cramer Decomposition for Stationary Processes -- 9.12.2 Orthogonal Scale Representation -- 10 Metrics Spaces for the Matrix Groups -- 10.1 Riemannian Manifolds as Metric Spaces.; 10.1.1 Metric Spaces and Smooth Manifolds.
• Sommario/riassunto: A comprehensive overview of the challenges in signal, data and pattern analysis in speech recognition, computational linguistics, image analysis and computer vision. Includes numerous exercises, an extensive bibliography, and additional resources -- extended proofs, selected solutions and examples -- on a companion website.