Up: Table of Contents
Previous: List of Notation
 Adaptive interaction function
 Discontinuities in MRFs, Defining the DA , Defining the DA , Robust M Estimator, Redefinition of M
 Adaptive potential function
 Defining the DA , Robust M Estimator, Redefinition of M
 AIF
 see Adaptive interaction function
 Akaike information criterion
 Estimating the Number
 AMestimator
 Problems with M , AM Estimator
 Annealing
 AM Estimator
 Analog network
 Graduated NonConvexity
 Anisotropic diffusion
 Discontinuities in MRFs
 Annealing labeling
 Local vs. Global , Annealing Labeling for
 APF
 see Adaptive potential function
 Autobinomial
 AutoModels, Hierarchical GRF Model, MRF Texture Modeling
 Autologistic
 AutoModels, Maximum Likelihood, PseudoLikelihood, Coding Method , Simultaneous Restoration and
 Autonormal
 AutoModels, Hierarchical GRF Model, MRF Texture Modeling, Maximum Likelihood, Simultaneous Segmentation and , Simultaneous Segmentation and
 interaction coefficients
 AutoModels
 interaction matrix
 AutoModels
 Band of convexity
 Defining the DA
 Bayes estimation
 Bayes Estimation
 Binary (bilateral) relation
 Relational Structure Representation
 Canonical potential
 Normalized and Canonical
 Clique
 Neighborhood System and
 for irregular sites
 Neighborhood System and
 for regular sites
 Neighborhood System and
 type of
 Neighborhood System and
 Clique potential
 Gibbs Random Fields, AutoModels
 for MLL
 MultiLevel Logistic Model
 quadratic
 MRF Prior for
 Clique potential
 for autonormal model
 AutoModels
 Closeness term
 Regularization
 Coding method
 Coding Method , Iterated Conditional Modes
 Coloring
 The Labeling Problem
 Combinatorial minimization
 comparison
 Experimental Comparison
 Conditional probability
 Gibbs Random Fields
 Configuration
 The Labeling Problem, Markov Random Fields
 Configuration space
 The Labeling Problem
 size of
 The Labeling Problem
 Constrained minimization
 Constrained Minimization
 Contextual constraint
 Introduction , AutoModels, Work in Relational
 Contextual constraints
 Labeling with Contextual
 Contextual constraint
 Labeling with Contextual
 Continuation
 AM Estimator
 Correctness
 Role of Energy
 Coupled MRF
 see Markov random field, coupled
 Cross validation
 Cross Validation
 Cross validation
 Solving the Euler
 DA
 see Discontinuity adaptive model
 Debugging
 Model Debugging
 Discontinuities
 Discontinuities in MRFs, Discontinuities in MRFs
 Discontinuity
 adaptivity
 Defining the DA
 Discontinuity adaptive model
 Discontinuities in MRFs
 convex
 Convex DA and
 definition of
 Defining the DA
 Edge detection
 Edge Detection
 using line process
 Edge Labeling using
 Edge detection
 Forbidden edge patterns
 Forbidden Edge Patterns
 thresholding
 Edge Labeling using
 Effective energy
 Mean Field Annealing
 Effective potential
 Line Process Model
 EM
 see Expectationmaximization
 Energy
 Gibbs Random Fields
 order of
 AutoModels, Regularization Solution
 order of
 AutoModels, Regularization Solution
 Energy minimization
 MAPMRF Labeling
 seeMinimization
 Energy Minimization
 Euler equation
 Discontinuities in MRFs, Discontinuities in MRFs, The Discontinuity Adaptive , Defining the DA , Defining the DA , Defining the DA
 solution of
 Solving the Euler
 Euler equation
 Defining the DA
 Expectationmaximization
 ExpectationMaximization
 Fixedpoint equation
 Rotation Angle Estimation, Classical Minimization with
 Fixedpoint iteration
 Classical Minimization with , Classical Minimization with , Classical Minimization with , Classical Minimization with , Iterative Updating Equations, Mean Field Annealing
 Fuzzy assignment
 Representation of Continuous
 Gaussian MRF (GMRF)
 see Autonormal
 Genetic algorithm
 Genetic Algorithms
 Gibbs distribution
 Line Process Model
 sampling
 Gibbs Random Fields
 Gibbs random field
 Gibbs Random Fields
 hierarchical
 Observation Models, MRF Texture Modeling
 hierarchical
 Observation Models, MRF Texture Modeling
 isotropic
 Gibbs Random Fields
 Gibbs random field
 hierarchical
 Hierarchical GRF Model, Simultaneous Segmentation and
 hierarchical
 Hierarchical GRF Model, Simultaneous Segmentation and
 homogeneous
 Gibbs Random Fields
 Gibbs sampler
 MRF Texture Modeling, Random Sampling Algorithms
 Gibbs distribution
 Gibbs Random Fields
 sampling
 MRF Texture Modeling
 Global minimum
 Minimization  Local
 multiple
 Minimization  Local
 unique
 Minimization  Local
 Global optimization
 annealing
 Minimization  Global
 performance comparison
 Annealing
 GNC
 see Graduated nonconvexity
 Goodness of fit
 Reduction of Nonzero
 Graduated nonconvexity
 DiscontinuityAdaptivity Model and , AM Estimator, Annealing, Graduated NonConvexity
 Graduated nonconvexity
 Minimization  Global , Annealing
 Graph matching
 Relational Structure Representation
 HammersleyClifford Theorem
 MarkovGibbs Equivalence
 Hard constraint
 Work in Relational
 HCF
 seeHighest confidence first
 Highest Confidence First
 Heuristics
 Use of Heuristics
 Hierarchical MRF model
 Hierarchical GRF Model, MRF Texture Modeling
 Highest confidence first
 Highest Confidence First
 Homogeneous
 Gibbs Random Fields, AutoModels
 Hopfield method
 Hopfield Method
 Hopfield network
 Line Process Model , RL using LagrangeHopfield , Minimization  Global
 Identical independent distribution
 Surface Reconstruction, Iterated Conditional Modes
 Illposed problem
 Regularization, Regularization Solution, SmoothnessRegularization and
 Image restoration
 piecewise constant
 Piecewise Constant Restoration
 piecewise continuous
 Piecewise Continuous Restoration
 Instability
 Instability
 Integral limit method
 Mean Field Annealing
 Intensity constancy
 Variational Approach
 Interaction function
 Defining the DA
 Ising model
 AutoModels
 Ising model
 generalized
 MultiLevel Logistic Model
 Label set
 continuity
 Sites and Labels, Neighborhood System and
 continuity
 Sites and Labels, Neighborhood System and
 continuous
 Sites and Labels
 discrete
 Sites and Labels
 real
 Sites and Labels
 Labeling assignment
 Representation of Continuous
 feasibility
 Representation of Continuous
 unambiguity
 Representation of Continuous
 Labeling of sites
 The Labeling Problem
 Labeling problem
 Sites and Labels, The Labeling Problem, Markov Random Fields
 categories LP1  LP4
 Labeling Problems in
 categorization
 Labeling Problems in
 under contextual constraint
 Labeling with Contextual
 with parameter estimation
 Unsupervised Estimation with
 Lagrange function
 Lagrange Multipliers
 augmented
 Lagrange Multipliers
 Lagrange multiplier method
 Lagrange Multipliers
 LagrangeHopfield Method
 RL using LagrangeHopfield
 Lagrangian multiplier
 RL using LagrangeHopfield
 Least squares
 OptimizationBased Vision, Robust M Estimator, AM Estimator, Rotation Angle Estimation, Rotation Angle Estimation, Rotation Angle Estimation, Rotation Angle Estimation, Rotation Angle Estimation, Rotation Angle Estimation, Least Squares Fit, Least Squares Fit, Reduction of Nonzero
 Least squares
 Least Squares Fit
 Likelihood function
 Bayes Estimation, Posterior Probability and
 Likelihood function
 Observation Models
 Line process
 Edge Labeling using , Discontinuities in MRFs, Line Process Model , Work in Relational
 approximation of
 Line Process Model
 elimination of
 Line Process Model
 potential function
 Line Process Model
 Line process
 MRF Prior for , Edge Labeling using
 elimination of
 Edge Labeling using
 Local minimum
 Minimization  Local
 M estimator
 annealing algorithm
 AM Estimator
 stabilized
 Redefinition of M
 Mestimator
 (, Robust M Estimator, )
 problems with
 Problems with M
 redefinition of
 Redefinition of M
 MAPMRF framework
 Summary of MAPMRF
 MAPMRF framework
 Introduction
 Mapping
 from scene to model objects
 Matching to Multiple
 from sites to labels
 The Labeling Problem
 involving NULL label
 Relational Structure Representation
 morphic
 Relational Structure Representation
 structural
 Relational Structure Representation, Criteria for Parameter , Posterior Energy
 structural
 Relational Structure Representation, Criteria for Parameter , Posterior Energy
 structural
 Relational Structure Representation, Criteria for Parameter , Posterior Energy
 under weak constraint
 Relational Structure Representation
 with continuous labels
 The Labeling Problem
 with discrete labels
 The Labeling Problem
 Markov process
 Markov Random Fields
 Markov random field
 Labeling with Contextual , Markov Random Fields
 coupled
 Markov Random Fields
 homogeneous
 Markov Random Fields, Markov Random Fields
 homogeneous
 Markov Random Fields, Markov Random Fields
 isotropic
 Markov Random Fields
 Markovianity
 Markov Random Fields
 positivity
 Markov Random Fields
 MarkovGibbs equivalence
 MarkovGibbs Equivalence
 Markov random field
 coupled
 Edge Labeling using
 Markovianity
 Markov Random Fields
 Maximum a posteriori
 Introduction , Optimality Criteria, Bayes Estimation
 Maximum a posteriori marginal
 Optimality Criteria
 Maximum entropy
 Optimality Criteria
 Maximum likelihood
 Bayes Estimation, Summary of MAPMRF , Maximum Likelihood
 Maximum likelihood
 Optimality Criteria
 Mean field
 Mean Field Approximations
 annealing
 Mean Field Annealing
 approximation
 Discontinuities in MRFs, Line Process Model , Mean Field Approximations
 approximation
 Discontinuities in MRFs, Line Process Model , Mean Field Approximations
 approximation
 Discontinuities in MRFs, Line Process Model , Mean Field Approximations
 approximation
 Discontinuities in MRFs, Line Process Model , Mean Field Approximations
 Mean field annealing
 Experimental Comparison
 Mean field
 annealing
 Minimization  Global
 Metropolis algorithm
 MRF Texture Modeling
 Metropolis algorithm
 Random Sampling Algorithms
 Minimization
 Energy Minimization, Energy Minimization, Forbidden Edge Patterns, Texture Segmentation, Discontinuities in MRFs, SmoothnessRegularization and , Line Process Model , Solving the Euler , Rotation Angle Estimation, Rotation Angle Estimation, Experiments, Discussion, Simultaneous Segmentation and
 constrained
 Forbidden Edge Patterns, Forbidden Edge Patterns, Relaxation Labeling
 constrained
 Forbidden Edge Patterns, Forbidden Edge Patterns, Relaxation Labeling
 constrained
 Forbidden Edge Patterns, Forbidden Edge Patterns, Relaxation Labeling
 global methods
 Minimization  Global
 local methods
 Minimization  Local
 Minimum description length
 Optimality Criteria, Estimating the Number
 ML
 see Maximum likelihood
 MLL
 see Multilevel logistic, Dynamic Programming
 Modeling
 geometric
 Research Issues
 photometric
 Research Issues
 Monte Carlo method
 Random Sampling Algorithms
 Morphism
 Relational Structure Representation
 of relational structures
 Relational Structure Representation
 MRFGRF equivalence
 see MarkovGibbs equivalence
 Multilevel logistic
 MultiLevel Logistic Model, Simultaneous Restoration and
 conditional probability of
 MultiLevel Logistic Model
 multiplesite clique potential
 MultiLevel Logistic Model
 pairsite clique potential
 MultiLevel Logistic Model
 singlesite clique potential
 MultiLevel Logistic Model
 Multiresolution computation
 Multiresolution Methods
 Neighbor set
 Neighborhood System and
 Neighborhood
 nearest
 Labeling with Contextual , Neighborhood System and
 nearest
 Labeling with Contextual , Neighborhood System and
 shape of
 Neighborhood System and
 Neighborhood system
 Neighborhood System and , Relational Structure Representation, Relational Structure Representation, Posterior Probability and , Pose Clustering and , Simultaneous Matching and
 nearest
 Neighborhood System and
 order of
 Neighborhood System and
 Neighborhood system
 Sites and Labels
 4neighborhood
 Neighborhood System and
 8neighborhood
 Neighborhood System and
 Normalized clique potential
 Normalized and Canonical , Reduction of Nonzero , Reduction of Nonzero
 Object recognition
 (, ), (, )
 Objective function
 Introduction
 Observation model
 Observation Models
 Optical flow
 Optical Flow
 Optimizationbased approach
 OptimizationBased Vision
 Optimizationbased approach
 Labeling Problems in
 Ordering
 of labels
 Sites and Labels
 of sites
 Sites and Labels
 Outlier
 DiscontinuityAdaptivity Model and
 P.d.f.
 see Probability density function
 Parameter estimation

 in high level vision
 (, )
 in low level vision
 (, )
 number of nonzero parameters
 Reduction of Nonzero
 while labeling
 (, )
 with labeled data
 Supervised Estimation with
 with unknown number of MRFs
 Estimating the Number
 with unlabeled data
 Unsupervised Estimation with
 Partition function
 Gibbs Random Fields, Line Process Model , MRF Parameter Estimation, Maximum Likelihood
 Pattern
 Gibbs Random Fields
 Penalty function method
 Forbidden Edge Patterns, Penalty Functions
 Perceptual organization
 Edge Detection
 Pose estimation
 (, )
 Positivity
 Markov Random Fields
 Potential function
 Regularization and Discontinuities, Defining the DA , Defining the DA
 Prior
 for piecewise constant surface
 MRF Prior for
 for piecewise continuous surface
 MRF Prior for
 for region
 MRF Prior for
 for surface
 MRF Priors for
 for texture
 Texture Segmentation
 smoothness
 The Smoothness Prior, Discontinuities in MRFs
 smoothness
 The Smoothness Prior, Discontinuities in MRFs
 Probability density function
 Markov Random Fields
 Probability distribution function
 Markov Random Fields
 Probability distribution function
 Markov Random Fields
 Pseudolikelihood
 Simultaneous Matching and , PseudoLikelihood
 Quadratic model
 truncated
 MRF Prior for
 Quaternion
 Pose Clustering and
 Random field
 The Labeling Problem, Markov Random Fields
 Region segmentation
 Piecewise Constant Restoration
 Regularization
 (, ), Deriving Posterior Energy, Regularization Solution, Discontinuities in MRFs, (, )
 quadratic
 Standard Regularization
 standard
 see Regularization,quadratic
 with line process
 Line Process Model
 Regularizer
 see Smoothness term
 Relational graph
 Relational Structure Representation
 Relational matching
 Relational Structure Representation, Work in Relational
 Relational structure
 Relational Structure Representation
 matching of
 Relational Structure Representation
 Relaxation Labeling
 Work in Relational , Relaxation Labeling, RL using LagrangeHopfield
 Relaxation labeling
 Relaxation Labeling
 Restoration with parameter estimation
 Simultaneous Restoration and
 RG
 see Relational graph
 RL
 see Relaxation labeling
 Robust estimation
 The DA Prior
 Robust M estimation
 see Mestimator
 RS
 see Relational structure
 Saddle point approximation
 Line Process Model , Mean Field Approximations, Mean Field Annealing
 Segmentation with parameter estimation
 Simultaneous Segmentation and
 Simulated annealing
 Experimental Comparison
 constrained
 Penalty Functions
 Singlesite clique potential
 Posterior Probability and
 Site
 image lattice
 Sites and Labels
 regular/irregular
 (, )
 regular/irregular
 (, )
 Smoothness
 The Smoothness Prior, Variational Approach, Discontinuities in MRFs, SmoothnessRegularization and
 complete
 MRF Prior for
 involving discontinuities
 Discontinuities in MRFs
 piecewise
 MRF Prior for
 Smoothness term
 Regularization
 discontinuity adaptive
 (, )
 discontinuity adaptive
 (, )
 membrane
 The Smoothness Prior
 plate
 The Smoothness Prior
 rod
 The Smoothness Prior
 string
 The Smoothness Prior
 Sparse data
 Surface Reconstruction
 Strauss process
 see Multilevel logistic
 Surface interpolation
 Surface Reconstruction
 Surface reconstruction
 Surface Reconstruction
 Temperature
 Gibbs Random Fields
 Texture
 modeling
 MRF Texture Modeling
 segmentation
 Simultaneous Segmentation and
 Texture segmentation
 Texture Segmentation
 Unary property
 Relational Structure Representation
 Weak constraint
 Relational Structure Representation, Relational Structure Representation, Work in Relational
 Weak membrane
 Deriving Posterior Energy
 Weak morphism
 Relational Structure Representation
 Weak string
 Deriving Posterior Energy