Pattern theory from representation to inference

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 -...

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Detalles Bibliográficos
Autor principal:	Grenander, Ulf (-)
Otros Autores:	Miller, Michael I.
Formato:	Libro electrónico
Idioma:	Inglés
Publicado:	Oxford ; New York : Oxford University Press 2007.
Edición:	1st ed
Materias:	Pattern perception. Pattern recognition systems.
Ver en Biblioteca Universitat Ramon Llull:	https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009798390306719

Tabla de Contenidos:

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.

Pattern theory from representation to inference

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