ngan, meier, chai - advanced video coding principles and techniques

Advanced Video Coding:
Principles and Techniques
Series Editor: J. Biemond, Delft University of Technology, The Netherlands
Volume 1
Volume 2
Volume 3
Volume 4
Volume 5
Volume 6
Volume 7
Three-Dimensional Object Recognition Systems
(edited by A.K. Jain and P.J. Flynn)
VLSI Implementations for Image Communications
(edited by P. Pirsch)
Digital Moving Pictures - Coding and Transmission on ATM Networks
(J P. Leduc)
Motion Analysis for Image Sequence Coding (G.Tziritas and C. Labit)
Wavelets in Image Communication (edited by M. Barlaud)
Subband Compression of Images: Principles and Examples
(T.A. Ramstad, S.O. Aase and J.H. Husey)
Advanced Video Coding: Principles and Techniques
(K.N. Ngan, T. Meier and D. Chai)
ADVANCES IN IMAGE COMMUNICATION 7
Advanced Video Coding:
Principles and Techniques
King N. Ngan, Thomas Meier and Douglas Chai
University of Western Australia,
Dept. of Electrical and Electronic Engineering,
Visual Communications Research Group,
Nedlands, Western Australia 6907

1999
Elsevier
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To Elena
To June
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Preface
The rapid advancement in computer and telecommunication technologies is
affecting every aspects of our daily lives. It is changing the way we interact
with each other, the way we conduct business and has profound impact on
the environment in which we live. Increasingly, we see the boundaries be-
tween computer, telecommunication and entertainment are blurring as the
three industries become more integrated with each other. Nowadays, one no
longer uses the computer solely as a computing tool, but often as a console
for video games, movies and increasingly as a telecommunication terminal
for fax, voice or videoconferencing. Similarly, the traditional telephone net-
work now supports a diverse range of applications such as video-on-demand,
videoconferencing, Internet, etc.
One of the main driving forces behind the explosion in information traffic
across the globe is the ability to move large chunks of data over the exist-
ing telecommunication infrastructure. This is made possible largely due to

the tremendous progress achieved by researchers around the world in data
compression technology, in particular for video data. This means that for
the first time in human history, moving images can be transmitted over long
distances in real-time, i.e., the same time as the event unfolds over at the
sender's end.
Since the invention of image and video compression using DPCM (differ-
ential pulse-code-modulation), followed by transform coding, vector quanti-
zation, subband/wavelet coding, fractal coding, object-oreinted coding and
model-based coding, the technology has matured to a stage that various cod-
ing standards had been promulgated to enable interoperability of different
equipment manufacturers implementing the standards. This promotes the
adoption of the standards by the equipment manufacturers and popularizes
the use of the standards in consumer products.
JPEG is an image coding standard for compressing still images accord-
ing to a compression/quality trade-off. It is a popular standard for image
exchange over the Internet. For video, MPEG-1 caters for storage media
vii
viii
up to a bit rate of 1.5 Mbits/s; MPEG-2 is aimed at video transmission
of typically 4-10 Mbits/s but it alSo can go beyond that range to include
HDTV (high-definition TV) image~. At the lower end of the bit rate spec-
trum, there are H.261 for videoconmrencing applications at p x 64 Kbits/s,
where p = 1, 2, , 30; and H.263,~which can transmit at bit rates of less
than 64 Kbits/s, clearly aiming at the videophony market.
The standards above have a number of commonalities: firstly, they are
based on predictive/transform coder architecture, and secondly, they pro-
cess video images as rectangular frames. These place severe constraints as
demand for greater variety and access of video content increases. Multi-
media including sound, video, graphics, text, and animation is contained
in many of the information content encountered in daily life. Standards

have to evolve to integrate and code the multimedia content. The concept
of video as a sequence of rectangular frames displayed in time is outdated
since video nowadays can be captured in different locations and composed as
a composite scene. Furthermore, video can be mixed with graphics and an-
imation to form a new video, and so on. The new paradigm is to view video
content as audiovisual object which
and composed in whatever way an
MPEG-4 is the emerging stanc
tent. It defines a syntax for a set c
content-based interactivity, compre
does not specify how the video con
as an entity can be coded, manipulated
application requires.
lard for the coding of multimedia con-
,f content-based functionalities, namely,
ssion and universal access. However, it
tent is to be generated. The process of
video generation is difficult and under active research. One simple way is to
capture the visual objects separately , as it is done in TV weather reports,
where the weather reporter stands in front of a weather map captured sepa-
rately and then composed together yith the reporter. The problem is this is
not always possible as in the case mj outdoor live broadcasts. Therefore, au-
tomatic segmentation has to be employed to generate the visual content in
real-time for encoding. Visual content is segmented as semantically mean-
ingful object known as video objec I plane. The video object plane is then
tracked making use of the tempora I ~ correlation between frames so that its
location is known in subsequent frames. Encoding can then be carried out
using MPEG-4. " L
This book addresses the more ~dvanced topics in video coding not in-
cluded in most of the video codingbooks in the market. The focus of the

book is on coding of arbitrarily shaped visual objects and its associated
topics. |
It is organized into six chapters:Image and Video
Segmentation (Chap-
ter 1), Face Segmentation (Chapter" 2), Foreground/Background Coding
ix
(Chapter 3), Model-based Coding (Chapter 4), Video Object Plane Ex-
traction and Tracking (Chapter 5), and MPEG-4 Video Coding Standard
(Chapter 6).
Chapter 1 deals with image and video segmentation. It begins with
a review of Bayesian inference and Markov random fields, which are used
in the various techniques discussed throughout the chapter. An important
component of many segmentation algorithms is edge detection. Hence, an
overview of some edge detection techniques is given. The next section deals
with low level image segmentation involving morphological operations and
Bayesian approaches. Motion is one of the key parameters used in video
segmentation and its representation is introduced in Section 1.4. Motion
estimation and some of its associated problems like occlusion are dealt with
in the following section. In the last section, video segmentation based on
motion information is discussed in detail.
Chapter 2 focuses on the specific problem of face segmentation and its
applications in videoconferencing. The chapter begins by defining the face
segmentation problem followed by a discussion of the various approaches
along with a literature review. The next section discusses a particular face
segmentation algorithm based on a skin color map. Results showed that this
particular approach is capable of segmenting facial images regardless of the
facial color and it presents a fast and reliable method for face segmentation
suitable for real-time applications. The face segmentation information is
exploited in a video coding scheme to be described in the next chapter where
the facial region is coded with a higher image quality than the background

region.
Chapter 3 describes the foreground/background (F/B) coding scheme
where the facial region (the foreground) is coded with more bits than the
background region. The objective is to achieve an improvement in the
perceptual quality of the region of interest, i.e., the face, in the encoded
image. The F/B coding algorithm is integrated into the H.261 coder with
full compatibility, and into the H.263 coder with slight modifications of
its syntax. Rate control in the foreground and background regions is also
investigated using the concept of joint bit assignment. Lastly, the MPEG-4
coding standard in the context of foreground/background coding scheme is
studied.
As mentioned above, multimedia content can contain synthetic objects
or objects which can be represented by synthetic models. One such model
is the 3-D wire-frame model (WFM) consisting of 500 triangles commonly
used to model human head and body. Model-based coding is the technique
used to code the synthetic wire-frame models. Chapter 4 describes the pro-
cedure involved in model-based coding for a human head. In model-based
coding, the most difficult problem is the automatic location of the object
in the image. The object location is crucial for accurate fitting of the 3-D
WFM onto the physical object to be coded. The techniques employed for
automatic facial feature contours extraction are active contours (or snakes)
for face profile and eyebrow extraction, and deformable templates for eye
and mouth extraction. For synthesis of the facial image sequence, head mo-
tion parameters and facial expression parameters need to be estimated. At
the decoder, the facial image sequence is synthesized using the facial struc-
ture deformation method which deforms the structure of the 3-D WFM to
stimulate facial expressions. Facial expressions can be represented by 44 ac-
tion units and the deformation of the WFM is done through the movement
of vertices according to the deformation rules defined by the action units.
Facial texture is then updated to improve the quality of the synthesized

images.
Chapter 5 addresses the extraction of video object planes (VOPs) and
their tracking thereafter. An intrinsic problem of video object plane extrac-
tion is that objects of interest are not homogeneous with respect to low-level
features such as color, intensity, or optical flow. Hence, conventional seg-
mentation techniques will fail to obtain semantically meaningful partitions.
The most important cue exploited by most of the VOP extraction algo-
rithms is motion. In this chapter, an algorithm which makes use of motion
information in successive frames to perform a separation of foreground ob-
jects from the background and to track them subsequently is described in
detail. The main hypothesis underlying this approach is the existence of
a dominant global motion that can be assigned to the background. Areas
in the frame that do not follow this background motion then indicate the
presence of independently moving physical objects which can be character-
ized by a motion that is different from the dominant global motion. The
algorithm consists of the following stages: global motion estimation, ob-
ject motion detection, model initialization, object tracking, model update
and VOP extraction. Two versions of the algorithm are presented where
the main difference is in the object motion detection stage. Version I uses
morphological motion filtering whilst Version II employs change detection
masks to detect the object motion. Results will be shown to illustrate the
effectiveness of the algorithm.
The last chapter of the book, Chapter 6, contains a description of the
MPEG-4 standard. It begins with an explanation of the MPEG-4 devel-
opment process, followed by a brief description of the salient features of
MPEG-4 and an outline of the technical description. Coding of audio ob-
xi
jects including natural sound and synthesized sound coding is detailed in
Section 6.5. The next section containing the main part of the chapter, Cod-
ing of Natural Textures, Images And Video, is extracted from the MPEG-4

Video Verification Model 11. This section gives a succinct explanation of
the various techniques employed in the coding of natural images and video
including shape coding, motion estimation and compensation, prediction,
texture coding, scalable coding, sprite coding and still image coding. The
following section gives an overview of the coding of synthetic objects. The
approach adopted here is similar to that described in Chapter 4. In order
to handle video transmission in error-prone environment such as the mobile
channels, MPEG-4 has incorporated error resilience functionality into the
standard. The last section of the chapter describes the error resilient tech-
niques used in MPEG-4 for video transmission over mobile communication
networks.
King N. Ngan
Thomas Meier
Douglas Chai
June 1999
Acknowledgments
The authors would ike to thank Professor K. Aizawa of University of
Tokyo, Japan, for the use of the "Makeface" 3-D wireframe synthesis soft-
ware package, from which some of the images in Chapter 4 are obtained.
Xll
This Page Intentionally Left Blank
Table of Contents
Preface vii
Acknowledgments
xi
1
Image and Video Segmentation
1
1.1 Bayesian Inference and MRF's 2
1.1.1 MAP Estimation 3

1.1.2 Markov Random Fields (MRFs) 4
1.1.3 Numerical Approximations 7
1.2 Edge Detection 15
1.2.1 Gradient Operators: Sobel, Prewitt, Frei-Chen 16
1.2.2 Canny Operator 17
1.3 Image Segmentation 20
1.3.1 Morphological Segmentation 22
1.3.2 Bayesian Segmentation 28
1.4 Motion 32
1.4.1 Real Motion and Apparent Motion 33
1.4.2 The Optical Flow Constraint (OFC) 34
1.4.3 Non-parametric Motion Field Representation 35
1.4.4 Parametric Motion Field Representation 36
1.4.5 The Occlusion Problem 40
1.5 Motion Estimation 41
1.5.1 Gradient-based Methods 42
1.5.2 Block-based Techniques 44
1.5.3 Pixel-recursive Algorithms 46
1.5.4 Bayesian Approaches 47
1.6 Motion Segmentation 49
1.6.1 3-D Segmentation 50
1.6.2 Segmentation Based on Motion Information Only 52
1.6.3 Spatio-Temporal Segmentation 54
xiii
xiv TABLE OF CONTENTS
1.6.4 Joint Motion Estimation and Segmentation 56
References 60
2 Face Segmentation 69
2.1 Face Segmentation Problem 69
2.2 Various Approaches 70

2.2.1 Shape Analysis 71
2.2.2 Motion Analysis 72
2.2.3 Statistical Analysis 72
2.2.4 Color Analysis 73
2.3 Applications 74
2.3.1 Coding Area of Interest with Better Quality 74
2.3.2 Content-based Representation and MPEG-4 76
2.3.3 3D Human Face Model Fitting 76
2.3.4 Image Enhancement 76
2.3.5 Face Recognition, Classification and Identification . . 76
2.3.6 Face Tracking 78
2.3.7 Facial Expression Study 78
2.3.8 Multimedia Database Indexing 78
2.4 Modeling of Human Skin Color 79
2.4.1 Color Space 80
2.4.2 Limitations of Color Segmentation 84
2.5 Skin Color Map Approach 85
2.5.1 Face Segmentation Algorithm 85
2.5.2 Stage One- Color Segmentation 87
2.5.3 Stage Two- Density Regularization 90
2.5.4 Stage Three- Luminance Regularization 92
2.5.5 Stage Four- Geometric Correction 93
2.5.6 Stage Five- Contour Extraction 94
2.5.7 Experimental Results 95
References 107
3 Foreground/Background Coding 113
3.1 Introduction 113
3.2 Related Works 116
3.3 Foreground and Background Regions 122
3.4 Content-based Bit Allocation 123

3.4.1 Maximum Bit Transfer 123
3.4.2 Joint Bit Assignment 127
3.5 Content-based Rate Control 131
TABLE OF CONTENTS
xv
3.6 H.261FB Approach 132
3.6.1 H.261 Video Coding System 133
3.6.2 Reference Model 8 137
3.6.3 Implementation of the H.261FB Coder 139
3.6.4 Experimental Results 145
3.7 H.263FB Approach 165
3.7.1 Implementation of the H.263FB Coder 165
3.7.2 Experimental Results 167
3.8 Towards MPEG-4 Video Coding : 171
3.8.1 MPEG-4 Coder 171
3.8.2 Summary ~ . . 180
References 181
4 Model-Based Coding 183
4.1 Introduction 183
4.1.1 2-D Model-Based Approaches 183
4.1.2 3-D Model-Based Approaches ~. 184
4.1.3 Applications of 3-D Model-Based Coding , 186
4.2 3-D Human Facial Modeling 187
4.2.1 Modeling A Person's Face 188
4.3 Facial Feature Contours Extraction ,. 193
4.3.1 Rough Contour Location Finding , 196
4.3.2 Image Processing 198
4.3.3 Features Extraction Using Active Contour Models . . 204
4.3.4 Features Extraction Using Deformable Templates . . . 210
4.3.5 Nose Feature Points Extraction Using Geometrical

Properties 218
4.4 WFM Fitting and Adaptation 220
4.4.1 Head Model Adjustment 220
4.4.2 Eye Model Adjustment 223
4.4.3 Eyebrow Model Adjustment 225
4.4.4 Mouth Model Adjustment 225
4.5 Analysis of Facial Image Sequences 227
4.5.1 Estimation of Head Motion Parameters 231
4.5.2 Estimation of Facial Expression Parameters 233
4.5.3 High Precision Estimation by Iteration 234
4.6 Synthesis of Facial Image Sequences 234
4.6.1 Facial Structure Deformation Method 235
4.7 Update of 3-D Facial Model 237
4.7.1 Update of Texture Information 239
xvi TABLE OF CONTENTS
4.7.2 Update of Depth Information 242
4.7.3 Transmission Bit Rates 243
References 245
5 VOP Extraction and Tracking
251
5.1 Video Object Plane Extraction Techniques 251
5.2 Outline of VOP Extraction Algorithm 258
5.3 Version I: Morphological Motion Filtering 260
5.3.1 Global Motion Estimation 261
5.3.2 Object Motion Detection Using Morphological Mo-
tion Filtering 265
5.3.3 Model Initialization 277
5.3.4 Object Tracking Using the Hausdorff Distance 277
5.3.5 Model Update 284
5.3.6 VOP Extraction 288

5.3.7 Results 294
5.4 Version II: Change Detection Masks 297
5.4.1 Object Motion Detection Using CDM 298
5.4.2 Model Initialization 300
5.4.3 Model Update 301
5.4.4 Background Filter 301
5.4.5 Results 304
References 310
6 MPEG-4 Standard 315
6.1 Introduction 315
6.2 MPEG-4 Development Process 315
6.3 Features of the MPEG-4 Standard [2] 316
6.3.1 Coded Representation of Primitive AVOs 317
6.3.2 Composition of AVOs 318
6.3.3 Description, Synchronization and Delivery of Stream-
ing Data for AVOs 318
6.3.4 Interaction with AVOs 321
6.3.5 Identification of Intellectual Property 321
6.4 Technical Description of the MPEG-4 Standard 321
6.4.1 DMIF 322
6.4.2 Demultiplexing, Sychronization and Buffer Manage-
ment 324
6.4.3 Syntax Description 326
6.5 Coding of Audio Objects 326
TABLE OF CONTENTS xvii
6.5.1 Natural Sound 326
6.5.2 Synthesized Sound 328
6.6 Coding of Natural Visual Objects 329
6.6.1 Video Object Plane (VOP) 329
6.6.2 The Encoder 331

6.6.3 Shape Coding 332
6.6.4 Motion Estimation and Compensation 338
6.6.5 Texture Coding 352
6.6.6 Prediction and Coding of B-VOPs 368
6.6.7 Generalized Scalable Coding 373
6.6.8 Sprite Coding 378
6.6.9 Still Image Texture Coding 386
6.7 Coding of Synthetic Objects 391
6.7.1 Facial Animation 391
6.7.2 Body Animation 393
6.7.3 2-D Animated Meshes 393
6.8 Error Resilience 395
6.8.1 Resynchronization 395
6.8.2 Data Recovery 396
6.8.3 Error Concealment 396
6.8.4 Modes of Operation 397
6.8.5 Error Resilience Encoding Tools 398
References 400
Index 401
This Page Intentionally Left Blank
Chapter 1
Image and Video
Segmentation
Segmentation plays a crucial role in second-generation image and video
coding schemes, as well as in content-based video coding. It is one of the
most difficult tasks in image processing, and it often determines the eventual
success or failure of a system.
Broadly speaking, segmentation seeks to subdivide images into regions of
similar attribute. Some of the most fundamental attributes are luminance,
color, and optical flow. They result in a so-called low-level segmentation,

because the partitions consist of primitive regions that usually do not have
a one-to-one correspondence with physical objects.
Sometimes, images must be divided into physical objects so that each
region constitutes a semantically meaningful entity. This higher-level seg-
mentation is generally more difficult, and it requires contextual information
or some form of artificial intelligence. Compared to low-level segmentation,
far less research has been undertaken in this field.
Both low-level and higher-level segmentation are becoming increasingly
important in image and video coding. The level at which the partitioning
is carried out depends on the application. So-called second generation cod-
ing schemes [1, 2] employ fairly sophisticated source models that take into
account the characteristics of the human visual system. Images are first
partitioned into regions of similar intensity, color, or motion characteristics.
Each region is then separately and efficiently encoded, leading to less arti-
facts than systems based on the discrete cosine transform (DCT) [3, 4, 5].
The second-generation approach has initiated the development of a signifi-
cant number of segmentation and coding algorithms [6, 7, 8, 9, 10], which
are based on a low-level segmentation.
2 CHAPTER 1. IMAGE AND VIDEO SEGMENTATION
The new video coding standard MPEG-4 [11, 12], on the other hand,
targets more than just large coding gains. To provide new functionali-
ties for future multimedia applications, such as content-based interactivity
and content-based scalability, it introduces a content-based representation.
Scenes are treated as compositions of several semantically meaningful ob-
jects, which are separately encoded and decoded. Obviously, MPEG-4 re-
quires a prior decomposition of the scene into physical objects or so-called
video object planes (VOPs). This corresponds to a higher-level partition.
As opposed to the intensity or motion-based segmentation for the second-
generation techniques, there does not exist a low-level feature that can be
utilized for grouping pixels into semantically meaningful objects. As a con-

sequence, VOP segmentation is generally far more difficult than low-level
segmentation. Furthermore, VOP extraction for content-based interactivity
functionalities is an unforgiving task. Even small errors in the contour can
render a VOP useless for such applications.
This chapter starts with a review of Bayesian inference and Markov
random fields (MRFs), which will be needed throughout this chapter. A
brief discussion of edge detection is given in Section 1.2, and Section 1.3
deals with low-level still image segmentation. The remaining three sections
are devoted to video segmentation. First, an introduction to motion and
motion estimation is given in Sections 1.4 and 1.5, before video segmentation
techniques are examined in Sections 1.6 and 5.1. For a review of VOP
segmentation algorithms, we refer the reader to Chapter 5.
1.1
Bayesian Inference and Markov Random Fields
Bayesian inference is among the most popular and powerful tools in image
processing and computer vision [13, 14, 15]. The basis of Bayesian tech-
niques is the famous inversion formula
p(xlo)_ P(OIX)P(X).
(1.1)
P(O)
Although equation (1.1) is trivial to derive using the axioms of probability
theory, it represents a major concept. To understand this better, let X
denote an unknown parameter and 0 an observation that provides some
information about X. In the context of decision making, X and 0 are
sometimes referred to as hypothesis and evidence, respectively.
P(XIO )
can now be viewed as the likelihood of the unknown parameter
X, given the observation O. The inversion formula (1.1) enables us to
express
P(XIO )

in terms of
P(OIX )
and
P(X).
In contrast to the
posterior
1.1. BAYESIAN INFERENCE AND MRF'S 3
probability P(XIO),
which is normally very difficult to establish,
P(OIX )
and the
prior probability P(X)
are intuitively easier to understand and can
usually be determined on a theoretical, experimental, or subjective basis [13,
14]. Bayes' theorem (1.1) can also be seen as an updating of the probability
of X from
P(X)
to
P(XIO )
after observing the evidence
O
[14].
1.1.1 MAP Estimation
Undoubtedly, the maximum a posteriori (MAP) estimator is the most im-
portant Bayesian tool. It aims at maximizing
P(XIO )
with respect to X,
which is equivalent to maximizing the numerator on the right-hand side
of (1.1), because
P(O)

does not depend on X. Hence, we can write
P(XIO) c~ P(OIX)P(X ).
(1.2)
For the purpose of a simplified notation, it is often more convenient to
minimize the negative logarithm of
P(X]O)
instead of maximizing
P(XIO )
directly. However, this has no effect on the outcome of the estimation. The
MAP estimate of X is now given by
XMAP
arg
n~x{P(OIX)P(X ) }
= arg n~n{- log
P(OIX) - log P(X)}.
(1.3)
From (1.3) it can be seen that the knowledge of two probability functions
is required. The likelihood
P(X)
contains the information that is available
a priori, that is, it describes our prior expectation on X before knowing O.
While it is often possible to determine
P(X)
from theoretical or experimen-
tal knowledge, subjective experience sometimes plays an important role. As
we will see later, Gibbs distributions are by far the most popular choice for
P(X)
in image processing, which means that X is assumed to be a sample
of a Markov random field (MRF).
The conditional probability

P(OIX),
on the other hand, defines how well
X explains the observation O and can therefore be viewed as an observation
model. It updates the a priori information contained in
P(X)
and is often
derived from theoretical or experimental knowledge. For example, assume
we wanted to recover the unknown original image X from a blurred image O.
The probability
P(OIX),
which describes the degradation process leading
to O, could be determined based on theoretical considerations. To this end,
a suitable mathematical model for blurring would be needed.
The major conceptual step introduced by Bayesian inference, besides the
inversion principle, is to model uncertainty about the unknown parameter X
4 CHAPTER 1. IMAGE AND VIDEO SEGMENTATION
by probabilities and combining them according to the axioms of probability
theory. Indeed, the language of probabilities has proven to be a powerful
tool to allow a quantitative treatment of uncertainty that conforms well
with human intuition. The resulting distribution
P(XIO),
after combining
prior knowledge and observations, is then the a posteriori belief in X and
forms the basis for inferences.
To summarize, by combining
P(X)
and
P(OIX )
the MAP estimator
incorporates both the a priori information on the unknown parameter X

that is available from knowledge and experience and the information brought
in by the observation O [16].
Estimation problems are frequently encountered in image processing and
computer vision. Applications include image and video segmentation [16,
17, 18, 19], where O represents an image or a video sequence and X is the
segmentation label field to be estimated. In image restoration [20, 21, 22], X
is the unknown original image we would like to recover and O the degraded
image. Bayesian inference is also popular in motion estimation [23, 24, 25,
26], with X denoting the unknown optical flow field and O containing two
or more frames of a video sequence. In all these examples, the unknown
parameter X is modeled by a random field.
1.1.2 Markov Random Fields (MRFs)
Without doubt the most important statistical signal models in image pro-
cessing and computer vision are based on Markov processes [27, 20, 28, 29].
Due to their ability to represent the spatial continuity that is inherent in
natural images, they have been successfully applied in various applications
to determine the prior distribution
P(X).
Examples of such Markov ran-
dom fields include region processes or label fields in segmentation prob-
lems [16, 17, 18, 30], models for texture or image intensity [20, 21, 30, 31],
and optical flow fields [23, 26].
First, some definitions will be introduced with focus on discrete 2-D
random fields. We denote by L- {(i,j)ll _< i_< M, 1 _<j _< N} afinite
M • N rectangular lattice of sites or pixels. A neighborhood system Af is
then defined as any collection of subsets Af/,j of L,
A/"- {Afi,jl(i,j) c L
and Af/,j C L},
(1.4)
such that for any pixel

(i, j)
1)
(i, j) Afi,j
and
2) (k, l) C - (i, j) e
(1.5)
1.1. BAYESIAN INFERENCE AND MRF'S 5
Figure 1.1" Eight-point neighborhood system: pixels belonging to the neigh-
borhood Af/,j of pixel (i, j) are marked in gray.
Generally speaking,
.hf/,j is
the set of neighbor pixels of (i, j).
A very popular neighborhood system is the one consisting of the eight
nearest pixels, as depicted in Fig. 1.1. The neighborhood Af/,j for this system
can be written as
Af/,j-{(i+h,j+v) I-l<h,v<land(h,v)r
(1.6)
whereby boundary pixels and the four corner pixels have only five and three
neighbors, respectively. The eight-point neighborhood system is also known
as the second-order neighborhood system. In contrast, the first-order system
is a four-point neighborhood system consisting of the horizontal and vertical
neighbor pixels only.
Now let X be a two-dimensional random field defined on L. Further, let
f~ denote the set of all possible realizations of X, the so-called sample or
configuration space. Then, X is a Markov random field (MRF) with respect
to Af if [20]
1)
P(X(i,j) IX(k,1),
all
(k,l) r (i,j))

= P(X(i,j) IX(k, 1), (k,l)C Hi,j)
2)
P(X - x) > O for all x E l2
(1.7)
for every (i, j) E L.
The first condition is the well-known Markovian property. It restricts
the statistical dependency of pixel (i, j) to its neighbors and thereby signif-
icantly reduces the complexity of the model. It is interesting to notice that
6 CHAPTER 1. IMAGE AND VIDEO SEGMENTATION
this condition is satisfied by any random field defined on a finite lattice if
the neighborhood is chosen large enough [29]. Such a neighborhood system
would, however, not benefit from a reduction in complexity like, for exam-
ple, a second-order system. The second condition in (1.7), the so-called
positivity condition, requires all realizations x E ~ of the MRF to have
positive probabilities. It is not always included into the definition of MRFs,
but it must be satisfied for the Hammersley-Clifford theorem below.
The definition (1.7) is not directly suitable to specify an MRF, but for-
tunately the Hammersley-Clifford theorem [27] greatly simplifies the speci-
fication. It states that a random field X is an MRF if and only if
P(X)
can
be written as a Gibbs distribution 1. That is,
1 (
P(X - x) - -2
-
1 )
-~U(x) , Vx e ft.
(1.8)
The Gibbs distribution was first used in physics and statistical mechanics.
Best known is the Ising Model, which was proposed to model the magnetic

properties of ferromagnetic materials [33].
Due to the analogy with physical systems,
U(x)
is called the energy
function and the constant T corresponds to temperature. For high temper-
atures T, the system is "melted" and all realizations x C ~ are more or
less equally probable. At low temperatures, on the other hand, the system
is forced to be in a state of low energy. Thus, in accordance with physical
systems, low energy levels correspond to a high likelihood and vice versa.
The so-called partition function Z is a normalizing constant and usually
does not have to be evaluated.
The energy function
U(x)
in (1.8) can be written as a sum of potential
functions
Vc(x):
U(x) - E Vc(x).
(1.9)
all cliques C
A clique C is defined as a subset C c L that contains either a single pixel
or several pixels that are all neighbors of each other. Note that the neigh-
borhood system Af determines exactly what types of cliques exist. For ex-
ample, all possible types of cliques for the eight-point neighborhood system
in Fig. 1.1 are illustrated in Fig. 1.2.
The clique potential
Vc(x)
in (1.9) represents the potential contributed
by clique C to the total energy
U(x)
and depends only on the pixels be-

longing to C. It follows that the energy function
U(x),
and therefore the
1sometimes called a Boltzmann-Gibbs distribution [32]