Index

Gibbs Random Fields, Auto-Models

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

for MLL: Multi-Level Logistic Model
quadratic: MRF Prior for

Clique potential

for auto-normal model: Auto-Models

Closeness term

Regularization

Coding method

Coding Method , Iterated Conditional Modes

Coloring

The Labeling Problem

Combinatorial minimization

comparison: Experimental Comparison

Conditional probability

The Labeling Problem, Markov Random Fields

Configuration

Configuration space

The Labeling Problem

size of: The Labeling Problem

Constrained minimization

Constrained Minimization

Contextual constraint

Introduction , Auto-Models, 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 Expectation-maximization

Energy

Discontinuities in MRFs, Discontinuities in MRFs, The Discontinuity Adaptive , Defining the DA , Defining the DA , Defining the DA

order of: Auto-Models, Regularization Solution
order of: Auto-Models, Regularization Solution

Energy minimization

MAP-MRF Labeling

seeMinimization: Energy Minimization

Euler equation

solution of: Solving the Euler

Euler equation

Defining the DA

Expectation-maximization

Expectation-Maximization

Fixed-point equation

Rotation Angle Estimation, Classical Minimization with

Fixed-point 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 Auto-normal

Genetic algorithm

Genetic Algorithms

Gibbs distribution

Line Process Model

sampling: Gibbs Random Fields

Gibbs random field

MRF Texture Modeling, Random Sampling Algorithms

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

Gibbs distribution

Discontinuity-Adaptivity Model and , AM Estimator, Annealing, Graduated Non-Convexity

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 non-convexity

Goodness of fit

Reduction of Nonzero

Graduated non-convexity

Graduated non-convexity

Minimization -- Global , Annealing

Graph matching

Hierarchical GRF Model, MRF Texture Modeling

Hammersley-Clifford Theorem

Markov-Gibbs Equivalence

Hard constraint

Work in Relational

HCF

seeHighest confidence first: Highest Confidence First

Heuristics

Use of Heuristics

Hierarchical MRF model

Highest confidence first

Highest Confidence First

Homogeneous

Gibbs Random Fields, Auto-Models

Hopfield method

Hopfield Method

Hopfield network

Line Process Model , RL using Lagrange-Hopfield , Minimization -- Global

Identical independent distribution

Surface Reconstruction, Iterated Conditional Modes

Ill-posed 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

Auto-Models

Ising model

generalized: Multi-Level 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

Lagrange-Hopfield Method

RL using Lagrange-Hopfield

Lagrangian multiplier

RL using Lagrange-Hopfield

Least squares

Optimization-Based 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

M-estimator

(, Robust M Estimator, )

problems with: Problems with M
redefinition of: Redefinition of M

MAP-MRF framework

Summary of MAP-MRF

MAP-MRF 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

Labeling with Contextual , Markov Random Fields

Markov random field

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

Markov-Gibbs equivalence

Markov-Gibbs Equivalence

Markov random field

coupled: Edge Labeling using

Markovianity

Introduction , Optimality Criteria, Bayes Estimation

Maximum a posteriori

Maximum a posteriori marginal

Optimality Criteria

Maximum entropy

Optimality Criteria

Maximum likelihood

Bayes Estimation, Summary of MAP-MRF , 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 Multi-level logistic, Dynamic Programming

Modeling

geometric: Research Issues
photometric: Research Issues

Monte Carlo method

Random Sampling Algorithms

Morphism

Multi-Level Logistic Model, Simultaneous Restoration and

of relational structures: Relational Structure Representation

MRF-GRF equivalence

see Markov-Gibbs equivalence

Multi-level logistic

conditional probability of: Multi-Level Logistic Model
multiple-site clique potential: Multi-Level Logistic Model
pair-site clique potential: Multi-Level Logistic Model
single-site clique potential: Multi-Level Logistic Model

Multi-resolution computation

Multi-resolution 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

4-neighborhood: Neighborhood System and
8-neighborhood: 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

Optimization-based approach

Optimization-Based Vision

Optimization-based approach

Labeling Problems in

Ordering

of labels: Sites and Labels
of sites: Sites and Labels

Outlier

Discontinuity-Adaptivity 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

Forbidden Edge Patterns, Penalty Functions

Penalty function method

Perceptual organization

Edge Detection

Pose estimation

(, )

Positivity

Regularization and Discontinuities, Defining the DA , Defining the DA

Potential function

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

Probability distribution function

Probability distribution function

Simultaneous Matching and , Pseudo-Likelihood

Pseudo-likelihood

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, Work in Relational

Relational matching

Relational structure

Work in Relational , Relaxation Labeling, RL using Lagrange-Hopfield

matching of: Relational Structure Representation

Relaxation Labeling

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 M-estimator

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

Single-site 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 Multi-level logistic

Surface interpolation

Surface Reconstruction

Surface reconstruction

Surface Reconstruction

Temperature

Texture

modeling: MRF Texture Modeling
segmentation: Simultaneous Segmentation and

Texture segmentation

Texture Segmentation

Unary property

Relational Structure Representation, Relational Structure Representation, Work in Relational

Weak constraint

Weak membrane

Deriving Posterior Energy

Weak morphism