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This HTML document consists of Chapter 1 of the book <I>Markov Random
Field Modeling in Computer Vision</I> (© Springer-Verlag 1995). The
rest of the book can be browsed at
<A HREF="http://www.ntu.ac.sg/~szli/MRF_Book/MRF_Book.html">
http://www.ntu.ac.sg/~szli/MRF_Book/MRF_Book.html></A> but which may slow.
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<b>
<font size="+3">M</font><font size="+2">arkov</font>
<font size="+3">R</font><font size="+2">andom</font>
<font size="+3">F</font><font size="+2">ield</font><BR>
<font size="+2"> Modeling in Computer Vision</font><BR>
<BR>
<A HREF="http://www.ntu.ac.sg/~szli/"><I>S.Z. Li</I></A>
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<BR><BR>
<P>
<P><IMG ALIGN=BOTTOM SRC="./book_icons/cover_pic.gif"><P>
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<BR><BR>
<H4><IMG SRC="./book_icons/springer.gif" ALT="" align=middle> © Springer-Verlag 1995</H4>
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</b>
<BR>
ISBN 0-387-70145-1 Spinger-Verlag New York Berlin Heidelberg Tokyo <br>
ISBN 3-540-70145-1 Spinger-Verlag Berlin Heidelberg New York Tokyo <br>
ISBN 4-431-70145-1 Spinger-Verlag Tokyo Berlin Heidelberg New York <br>
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<font size="+1">
``An excellent book --- very thorough and very
clearly written.''
<P>
--- <I> Stuart Geman</I>
<BR><P>
<HR>
<BR><P>
``I have found the book to be a very valuable reference. I am very
impressed by both the breath and depth of the coverage. This must
have been a truly monumental undertaking.''<BR>
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--- <I> Charles A. Bouman</I>
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<H3>Summary</H3>
<P>
Markov random field (MRF) theory provides a basis for modeling
contextual constraints in visual processing and interpretation. It
enables us to develop optimal vision algorithms systematically when used
with optimization principles. This book presents a comprehensive study
on the use of MRFs for solving computer vision problems. The book covers
the following parts essential to the subject: introduction to
fundamental theories, formulations of MRF vision models, MRF parameter
estimation, and optimization algorithms. Various vision models are
presented in a unified framework, including image restoration and
reconstruction, edge and region segmentation, texture, stereo and
motion, object matching and recognition, and pose estimation. This book
is an excellent reference for researchers working in computer vision,
image processing, statistical pattern recognition and applications of
MRFs. It is also suitable as a text for advanced courses in these areas.
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<BR> <HR> <BR>
<font size="+2"><B>Table of Contents</B></font>
<BR><BR>
<A NAME=tex2html501 HREF="Head-Tail/foreword.html"> Foreword by Anil K. Jain</A>
<BR>
<A NAME=tex2html502 HREF="Head-Tail/preface.html"> Preface</A>
<BR><BR>
<font size="+1">C</font>hapter <font size="+1"> 1.
<A NAME=tex2html503 HREF="Chapter_1/node3.html#SECTION00300000000000000000"> Introduction</A>
</font>
<UL>
<LI>1.1 <A NAME=tex2html504 HREF="Chapter_1/node4.html#SECTION00310000000000000000"> Visual Labeling</A>
<UL>
<LI>1.1.1 <A NAME=tex2html505 HREF="Chapter_1/node5.html#SECTION00311000000000000000"> Sites and Labels</A>
<LI>1.1.2 <A NAME=tex2html506 HREF="Chapter_1/node6.html#SECTION00312000000000000000"> The Labeling Problem</A>
<LI>1.1.3 <A NAME=tex2html507 HREF="Chapter_1/node7.html#SECTION00313000000000000000"> Labeling Problems in Vision</A>
<LI>1.1.4 <A NAME=tex2html508 HREF="Chapter_1/node8.html#SECTION00314000000000000000"> Labeling with Contextual Constraints</A>
</UL>
<LI>1.2 <A NAME=tex2html509 HREF="Chapter_1/node9.html#SECTION00320000000000000000"> Markov Random Fields and Gibbs Distributions</A>
<UL>
<LI>1.2.1 <A NAME=tex2html510 HREF="Chapter_1/node10.html#SECTION00321000000000000000"> Neighborhood System and Cliques</A>
<LI>1.2.2 <A NAME=tex2html511 HREF="Chapter_1/node11.html#SECTION00322000000000000000"> Markov Random Fields</A>
<LI>1.2.3 <A NAME=tex2html512 HREF="Chapter_1/node12.html#SECTION00323000000000000000"> Gibbs Random Fields</A>
<LI>1.2.4 <A NAME=tex2html513 HREF="Chapter_1/node13.html#SECTION00324000000000000000">
<A NAME=tex2html513 HREF="Chapter_1/node13.html#SECTION00324000000000000000"> Markov-Gibbs Equivalence</A>
<LI>1.2.5 <A NAME=tex2html514 HREF="Chapter_1/node14.html#SECTION00325000000000000000"> Normalized and Canonical Forms</A>
</UL>
<LI>1.3 <A NAME=tex2html515 HREF="Chapter_1/node15.html#SECTION00330000000000000000"> Useful MRF Models</A>
<UL>
<LI>1.3.1 <A NAME=tex2html516 HREF="Chapter_1/node16.html#SECTION00331000000000000000"> Auto-Models</A>
<LI>1.3.2 <A NAME=tex2html517 HREF="Chapter_1/node17.html#SECTION00332000000000000000"> Multi-Level Logistic Model</A>
<LI>1.3.3 <A NAME=tex2html518 HREF="Chapter_1/node18.html#SECTION00333000000000000000"> The Smoothness Prior</A>
<LI>1.3.4 <A NAME=tex2html519 HREF="Chapter_1/node19.html#SECTION00334000000000000000"> Hierarchical GRF Model</A>
</UL>
<LI>1.4 <A NAME=tex2html520 HREF="Chapter_1/node20.html#SECTION00340000000000000000"> Optimization-Based Vision</A>
<UL>
<LI>1.4.1 <A NAME=tex2html521 HREF="Chapter_1/node21.html#SECTION00341000000000000000"> Research Issues</A>
<LI>1.4.2 <A NAME=tex2html522 HREF="Chapter_1/node22.html#SECTION00342000000000000000"> Role of Energy Functions</A>
<LI>1.4.3 <A NAME=tex2html523 HREF="Chapter_1/node23.html#SECTION00343000000000000000"> Formulation of Objective Functions</A>
<LI>1.4.4 <A NAME=tex2html524 HREF="Chapter_1/node24.html#SECTION00344000000000000000"> Optimality Criteria</A>
</UL>
<LI>1.5 <A NAME=tex2html525 HREF="Chapter_1/node25.html#SECTION00350000000000000000"> Bayes Labeling of MRFs</A>
<UL>
<LI>1.5.1 <A NAME=tex2html526 HREF="Chapter_1/node26.html#SECTION00351000000000000000"> Bayes Estimation</A>
<LI>1.5.2 <A NAME=tex2html527 HREF="Chapter_1/node27.html#SECTION00352000000000000000"> MAP-MRF Labeling</A>
<LI>1.5.3 <A NAME=tex2html528 HREF="Chapter_1/node28.html#SECTION00353000000000000000"> Regularization</A>
<LI>1.5.4 <A NAME=tex2html529 HREF="Chapter_1/node29.html#SECTION00354000000000000000"> Summary of MAP-MRF Approach</A>
</UL>
</UL>
</td>
<td colspan=2 width=420 align=left valign=top>
<BR><BR>
<font size="+1">C</font>hapter <font size="+1"> 2.
Low Level MRF Models
<UL>
</font>
<LI>2.1 Observation Models
<LI>2.2 Image Restoration and Reconstruction
<UL>
<LI>2.2.1 MRF Priors for Image Surfaces
<UL>
<LI> MRF Prior for Piecewise Constant Surfaces
<LI> MRF Prior for Piecewise Continuous Surfaces
</UL>
<LI>2.2.2 Piecewise Constant Restoration
<UL>
<LI> Deriving Posterior Energy
<LI> Energy Minimization
</UL>
<LI>2.2.3 Piecewise Continuous Restoration
<UL>
<LI> Deriving Posterior Energy
<LI> Energy Minimization
</UL>
<LI>2.2.4 Surface Reconstruction
<UL>
<LI> Regularization Solution
</UL>
</UL>
<LI>2.2 Edge Detection
<UL>
<LI>2.2.1 Edge Labeling using Line Process
<LI>2.2.2 Forbidden Edge Patterns
</UL>
<LI>2.3 Texture Synthesis and Analysis
<UL>
<LI>2.3.1 MRF Texture Modeling
<LI>2.3.2 Texture Segmentation
</UL>
<LI>2.4 Optical Flow
<UL>
<LI>2.4.1 Variational Approach
<LI>2.4.2 Flow Discontinuities
</UL>
</UL>
</tr>
</td>
<tr>
<td colspan=2 width=420 align=left valign=top>
<BR><BR>
<font size="+1">C</font>hapter <font size="+1"> 3.
Discontinuities in MRFs
</font>
<UL>
<LI>3.1 Smoothness, Regularization and Discontinuities
<UL>
<LI>3.1.1 Regularization and Discontinuities
<UL>
<LI> Standard Regularization
<LI> Line Process Model and Its Approximations
</UL>
<LI>3.1.2 Other Regularization Models
</UL>
<LI>3.2 The Discontinuity Adaptive MRF Model
<UL>
<LI>3.2.1 Defining the DA Model
<LI>3.2.2 Relations with Previous Models
<LI>3.2.3 Discrete Data and 2D Cases
<LI>3.2.4 Solution Stability
</UL>
<LI>3.3 Computation of DA Solutions
<UL>
<LI>3.3.1 Solving the Euler Equation
<LI>3.3.2 Experimental Results
</UL>
<LI>3.4 Conclusion
</UL>
<td width=420 colspan=2 valign=top align=left>
<font size="+1">C</font>hapter <font size="+1"> 4.
Discontinuity-Adaptivity Model and Robust Estimation
</font>
<UL>
<LI>4.1 The DA Prior and Robust Statistics
<UL>
<LI>4.1.1 Robust M Estimator
<LI>4.1.2 Problems with M Estimator
<LI>4.1.3 Redefinition of M Estimator
<LI>4.1.4 AM Estimator
<LI>4.1.5 Convex DA and M-Estimation Models
</UL>
<LI>4.2 Experimental Comparison
<UL>
<LI>4.2.1 Location Estimation
<LI>4.2.2 Rotation Angle Estimation
</UL>
</UL>
</tr>
</td>
<tr>
<td colspan=2 valign=top width=420 align=left>
<font size="+1">C</font>hapter <font size="+1"> 5.
High Level MRF Models
</font>
<UL>
<LI>5.1 Matching under Relational Constraints
</font>
<UL>
<LI>5.1.1 Relational Structure Representation
<LI>5.1.2 Work in Relational Matching
</UL>
<LI>5.2 MRF-Based Matching
<UL>
<LI>5.2.1 Posterior Probability and Energy
<LI>5.2.2 Matching to Multiple Objects
<LI>5.2.3 Experiments
<UL>
<LI> Matching Objects of Point Patterns
<LI> Matching Objects of Line Patterns
<LI> Matching Curved Objects under Similarity Transformations
</UL>
<LI>5.2.4 Extensions
<UL>
<LI> Incorporating Higher Constraints
<LI> Coupled MRFs for Matching of Different Features
<LI> Relationships with Low Level MRF Models
</UL>
</UL>
<LI>5.3 Pose Computation
<UL>
<LI>5.3.1 Pose Clustering and Estimation
<LI>5.3.2 Simultaneous Matching and Pose
<LI>5.3.3 Discussion
</UL>
</UL>
<td align=left valign=top width=420 colspan=2>
<font size="+1">C</font>hapter <font size="+1"> 6.
MRF Parameter Estimation
</font>
<UL>
<LI>6.1 Supervised Estimation with Labeled Data
<UL>
<LI>6.1.1 Maximum Likelihood
<LI>6.1.2 Pseudo-Likelihood
<LI>6.1.3 Coding Method
<LI>6.1.4 Mean Field Approximations
<LI>6.1.5 Least Squares Fit
</UL>
<LI>6.2 Unsupervised Estimation with Unlabeled Data
<UL>
<LI>6.2.1 Simultaneous Restoration and Estimation
<LI>6.2.2 Simultaneous Segmentation and Estimation
<LI>6.2.3 Expectation-Maximization
<LI>6.2.4 Cross Validation
</UL>
<LI>6.3 Further Issues
<UL>
<LI>6.3.1 Estimating the Number of MRFs
<LI>6.3.2 Reduction of Nonzero Parameters
</UL>
</UL>
</tr>
</td>
<tr>
<td width=420 align=left valign=top colspan=2>
<font size="+1">C</font>hapter <font size="+1"> 7.
Parameter Estimation in Optimal Object Recognition
<P>
</font>
<UL>
<LI>7.1 Motivation
<LI>7.2 Theory of Parameter Estimation for Recognition
<UL>
<LI>7.2.1 Optimization-Based Object Recognition
<LI>7.2.2 Criteria for Parameter Estimation
<UL>
<LI> Correctness
<LI> Instability
<LI> Optimality
</UL>
<LI>7.2.3 Linear Classification Function
<LI>7.2.4 A Non-parametric Learning Algorithm
<LI>7.2.5 Reducing Search Space
</UL>
<LI>7.3 Application in MRF Object Recognition
<UL>
<LI>7.3.1 Posterior Energy
<LI>7.3.2 Energy in Linear Form
<LI>7.3.3 How Minimal Configuration Changes
<LI>7.3.4 Parametric Estimation under Gaussian Noise
</UL>
<LI>7.4 Experiments
<UL>
<LI>7.4.1 Recognition of Line Patterns
<LI>7.4.2 Recognition of Curved Objects
</UL>
<LI>7.5 Conclusion
</UL>
<td width=420 valign=top align=left colspan=2>
<font size="+1">C</font>hapter <font size="+1"> 8.
Minimization -- Local Methods
</font>
<UL>
<LI>8.1 Classical Minimization with Continuous Labels
<LI>8.2 Minimization with Discrete Labels
<UL>
<LI>8.2.1 Iterated Conditional Modes
<LI>8.2.2 Relaxation Labeling
<UL>
<LI> Representation of Continuous RL
<LI> Maximization Formulation
<LI> Iterative Updating Equations
<LI> Local vs. Global Solutions
</UL>
<LI>8.2.3 Highest Confidence First
<LI>8.2.4 Dynamic Programming
</UL>
<LI>8.3 Constrained Minimization
<UL>
<LI>8.3.1 Penalty Method
<LI>8.3.2 Lagrange Method
<LI>8.3.3 Hopfield Method
<LI>8.3.4 RL using Lagrange-Hopfield Method
</UL>
</UL>
</tr>
</td>
<tr>
<td align=left valign=top width=420 colspan=2>
<font size="+1">C</font>hapter <font size="+1"> 9.
Minimization -- Global Methods
</font>
<UL>
<LI>9.1 Simulated Annealing
<UL>
<LI>9.1.1 Random Sampling Algorithms
<LI>9.1.2 Annealing
</UL>
<LI>9.2 Mean Field Annealing
<LI>9.3 Graduated Non-Convexity
<UL>
<LI>9.3.1 Annealing Labeling for MAP-MRF Matching
</UL>
<LI>9.4 Genetic Algorithms
<LI>9.5 Experimental Comparison
<LI>9.6 Accelerating Computation
<UL>
<LI>9.6.1 Multi-resolution Methods
<LI>9.6.2 Use of Heuristics
</UL>
<LI>9.7 Model Debugging
</UL>
</UL>
<A NAME=tex2html661 HREF="Head-Tail/References.html">References<BR>
<A NAME=tex2html662 HREF="Head-Tail/node162.html#SECTION001300000000000000000"> List of Notation<BR>
<A NAME=tex2html663 HREF="Head-Tail/node163.html#SECTION001400000000000000000">Index<BR>
<BR><P>
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