S.C. Zhu and D. Mumford (2006) A Stochastic Grammar of Images

S.C. Zhu and D. Mumford (2006) A Stochastic Grammar of Images
R
Foundations and Trends
in
Computer Graphics and Vision
Vol. 2, No. 4 (2006) 259–362
c 2007 S.-C. Zhu and D. Mumford
DOI: 10.1561/0600000018
A Stochastic Grammar of Images
Song-Chun Zhu1,∗ and David Mumford2
1
2
University of California, Los Angeles USA, [email protected]
Brown University, USA, David [email protected]
Abstract
This exploratory paper quests for a stochastic and context sensitive
grammar of images. The grammar should achieve the following four
objectives and thus serves as a unified framework of representation,
learning, and recognition for a large number of object categories. (i) The
grammar represents both the hierarchical decompositions from scenes,
to objects, parts, primitives and pixels by terminal and non-terminal
nodes and the contexts for spatial and functional relations by horizontal links between the nodes. It formulates each object category as the
set of all possible valid configurations produced by the grammar. (ii)
The grammar is embodied in a simple And–Or graph representation
where each Or-node points to alternative sub-configurations and an
And-node is decomposed into a number of components. This representation supports recursive top-down/bottom-up procedures for image
parsing under the Bayesian framework and make it convenient to scale
up in complexity. Given an input image, the image parsing task constructs a most probable parse graph on-the-fly as the output interpretation and this parse graph is a subgraph of the And–Or graph after
* Song-Chun
Zhu is also affiliated with the Lotus Hill Research Institute, China.
making choice on the Or-nodes. (iii) A probabilistic model is defined
on this And–Or graph representation to account for the natural occurrence frequency of objects and parts as well as their relations. This
model is learned from a relatively small training set per category and
then sampled to synthesize a large number of configurations to cover
novel object instances in the test set. This generalization capability
is mostly missing in discriminative machine learning methods and can
largely improve recognition performance in experiments. (iv) To fill the
well-known semantic gap between symbols and raw signals, the grammar includes a series of visual dictionaries and organizes them through
graph composition. At the bottom-level the dictionary is a set of image
primitives each having a number of anchor points with open bonds to
link with other primitives. These primitives can be combined to form
larger and larger graph structures for parts and objects. The ambiguities in inferring local primitives shall be resolved through top-down
computation using larger structures. Finally these primitives forms a
primal sketch representation which will generate the input image with
every pixels explained. The proposal grammar integrates three prominent representations in the literature: stochastic grammars for composition, Markov (or graphical) models for contexts, and sparse coding
with primitives (wavelets). It also combines the structure-based and
appearance based methods in the vision literature. Finally the paper
presents three case studies to illustrate the proposed grammar.
1
Introduction
1.1
The Hibernation and Resurgence of Image Grammars
Understanding the contents of images has always been the core problem in computer vision with early work dated back to Fu [22], Riseman
[33], Ohta and Kanade [54, 55] in the 1960–1970s. By analogy to natural
language understanding, the task of image parsing [72], as Figure 1.1
illustrates, is to compute a parse graph as the most probable interpretation of an input image. This parse graph includes a tree structured decomposition for the contents of the scene, from scene labels, to
objects, parts, primitives, so that all pixels are explained, and a number of spatial and functional relations between nodes for contexts at all
levels of the hierarchy.
People who worked on image parsing in the 1960–1970s were, obviously, ahead of their time. In Kanade’s own words, they had only 64K
memory to work with at that time. Indeed, his paper with Ohta [55]
was merely 4-page long! The image parsing efforts and structured methods encountered overwhelming difficulties in the 1970s and since then
entered a hibernation state for a quarter of a century. The syntactic
and grammar work have been mostly studied in the backstage as we
261
262 Introduction
a football match scene
sports field
spectator
person
point process
face
curve groups
texture
persons
texture
text
color region
texture
Fig. 1.1 Illustrating the task of image parsing. The parse graph includes a tree structured
decomposition in vertical arrows and a number of spatial and functional relations in horizontal arrows. From [72].
shall review in later section. These difficulties remain challenging even
today.
Problem 1: There is an enormous amount of visual knowledge about
the real world scenes that has to be represented in the computer in order
to make robust inference. For example, there are at least 3, 000 object
categories1 and many categories have wide intra-category structural
variations. The key questions are: how does one define an object category, say a car or a jacket? and how does one represent these categories
in a consistent framework?
The visual knowledge is behind our vivid dreams and imaginations
as well as the top-down computation. It was known that there are far
more downward fibers than upward fibers in the visual pathways of
primate animals. For example, it is reported in [65] that only 5%–10%
of the input to the geniculate relay cells derives from the retina. The
1 This
number comes from Biederman who adopted a method used by pollsters. Take an
English dictionary, open some pages at random, and count the number of nouns which
are object categories at a page and then times the number of pages of the dictionary
proportionally.
1.1 The Hibernation and Resurgence of Image Grammars
263
rest derives from local inhibitory inputs and descending inputs from
layer 6 of the visual cortex. The weakness in knowledge representation
and top-down inference is, in our opinion, the main obstacle in the road
toward robust and large scale vision systems.
Problem 2: The computational complexity is huge.2 A simple glance
of Figure 1.1 reveals that an input image may contain a large number
of objects. Human vision is known [70] to simultaneously activate the
computation at all levels from scene classification to edge detection —
all occurs in a very short time ≤400 ms, and to adopt multiple visual
routines [76] to achieve robust computation. In contrast, most pattern recognition or machine learning algorithms are feedforward and
computer vision systems rarely possess enough visual knowledge for
reasoning.
The key questions are: how does one achieve robust computation
that can be scaled to thousands of categories? and how does one coordinate these bottom-up and top-down procedures? To achieve scalable
computation, the vision algorithm must be based on simple procedures
and structures that are common to all categories.
Problem 3: The most obvious reason that sent the image parsing
work to dormant status was the so-called semantic gap between the
raw pixels and the symbolic token representation in early syntactic and
structured methods. That is, one cannot reliably compute the symbols
from raw images. This has motivated the shift of focus to appearance
based methods in the past 20 years, such as PCA [75], AAM [12], and
appearance based recognition [51], image pyramids [69] and wavelets
[15], and machine learning methods [21, 63, 78] in the past decade.
Though the appearance based methods and machine learning algorithms have made remarkable progress, they have intrinsic problems
that could be complemented by structure based methods. For example,
they require too many training examples due to the lack the compositional and generative structures. They are often over-fit to specific
training set and can hardly generalize to novel instances or configurations especially for categories that have large intra-class variations.
2 The
NP-completeness is no longer an appropriate measure of complexity, because even
many simplified vision problems are known to be NP-hard.
264 Introduction
After all these developments, the recent vision literature has
observed a pleasing trend for returning to the grammatical and compositional methods, for example, the work in the groups of Ahuja [71],
Geman [27, 36], Dickinson [14, 40], Pollak [79], Buhmann [57] and Zhu
[9, 32, 44, 59, 72, 74, 85, 86]. The return of grammar is in response to
the limitations of the appearance based and machine learning methods
when they are scaled up.
The return of grammar is powered by progresses in several aspects,
which were not available in the 1970s. (i) A consistent mathematical and statistical framework to integrate various image models, such
as Markov (graphical) models [90], sparse coding [56], and stochastic context free grammar [10]. (ii) More realistic appearance models
for the image primitives to connect the symbols to pixels. (iii) More
powerful algorithms including discriminative classification and generative methods, such as the Data-Driven Markov China Monte Carlo
(DDMCMC) [73]. (iv) Huge number of realistic training and testing
images [87].
1.2
Objectives
This exploratory paper will review the issues and recent progress in
developing image grammars, and introduce a stochastic and context
sensitive grammar as a unified framework for representation, learning,
and recognition. This framework integrates many existing models and
algorithms in the literature and addresses the problems raised in the
previous subsection. This image grammar should achieve the following
four objectives.
Objective 1: A common framework for visual knowledge representation and object categorization. Grammars, studied mostly in language
[1, 26], are known for their expressive power in generating a very large
set of configurations or instances, i.e., their language, by composing
a relatively much smaller set of words, i.e., shared and reusable elements, using production rules. Hierarchic and structural composition
is the key concept behind grammars in contrast to enumerating all
possible configurations.
1.2 Objectives
265
In this paper, we embody the image grammar in an And–Or
graph representation3 where each Or-node points to alternative subconfigurations and an And-node is decomposed into a number of
components. This And–Or graph represents both the hierarchical
decompositions from scenes, to objects, parts, primitives and pixels
by terminal and non-terminal nodes and the contexts for spatial and
functional relations by horizontal links between the nodes. It is an alternate way of representing production rules and it contains all possible
parse trees. Then we will define a probabilistic model for the And–Or
graph which can be learned from examples using maximum likelihood
estimation. Therefore, all the structural and contextual information
are represented in the And–Or graph (and equivalently the grammar).
This also resolve the object categorization problem. We can define each
object category as the set of all valid configurations which are produced
by the grammar, with its probability learned to reproduce natural frequency of instances occurring in the observed ensemble.
As we will show in later section, this probability model integrates
popular generative models, such as sparse coding (wavelet coding) and
stochastic context free grammars (SCFG), with descriptive models,
such as Markov random fields and graphical models. The former represents the generative hierarchy for reconfigurability while the latter
models context.
Objective 2: Scalable and recursive top-down/bottom-up computation. The And–Or graph representation has recursive structures with
two types of nodes. It can be easily scalable up in the number of nodes
and object categories. For example, suppose an Or-node represents an
object, say car, it then has a number of children nodes for different
views (front, side, back etc.) of cars. By adding a new child node, we
can augment to new views. This representation supports recursive topdown/bottom-up procedures for image parsing and make it convenient
to scale up in complexity.
Figure 1.2 shows a parsing graph under construction at a time step.
This simple grammar is one of our case study in later section uses one
3 The
And–Or graph was previously used by Pearl in [58] for heuristic searches. In our work,
we use it in a very different purpose and should not be confused with Pearl’s work.
266 Introduction
S
scene
objects
parse graph
G
mesh rule
r3
A
B cube rule
r6
C nest rule
r4
rectangular
surfaces
top-down
proposals
configuration
C
bottom-up
proposals
edge map
image
I
Fig. 1.2 Illustrating the recursive bottom-up/top-down computation processes in image
parsing. The detection of rectangles (in red) instantiates some non-terminal nodes shown as
upward arrows. They in turn activate graph grammar rules for grouping larger structures
in nodes A, B, and C, respectively. These rules generate top-down prediction of rectangles (in blue). The predictions are validated from the image under the Bayesian posterior
probability. Modified from [59].
1.2 Objectives
267
primitive: rectangular surfaces projected onto the image plane. The
grammar rules represents various organization, such as alignments of
the rectangles in mesh, linear, nesting, cubic structures. In the kitchen
scene, the four rectangles (in red) accepted through bottom-up process
and they activate the production rules represented by the non-terminal
nodes A, B, and C, respectively. Which then predict a number of candidates (in blue) in top-down search. The solid upward arrows show
the bottom-up binding, while the downward arrows show the top-down
prediction. As the ROC curves in Figure 9.5 shows in later section, the
top-down prediction largely improves the recognition rate of the rectangles, as certain rectangles can only be hallucinated through top-down
process due to occlusion and severe image degradation.
Given an input image, the image parsing task constructs a most
probable parse graph on-the-fly as the output interpretation and this
parse graph is a subgraph of the And–Or graph after making choices
on the Or-nodes.
As we shall discuss in later section, the computational algorithm
maintains the same data structures for each of the And-nodes and
Or-nodes in the And–Or graph and adopt the same computational
procedure: (i) bottom-up detecting and binding using a cascade of features; and (ii) top-down on-line template composition and matching.
To implement the system, we only need to write one common class (in
C++ programming) for all the nodes, and different objects and parts
are realized as instances of this class. These nodes use different bottomup features/tests and the top-down templates during the computational
process. The features and templates are learned off-line through training images and loaded into the instances of the C++ class during the
computational process. This recursive algorithm has the potential to
be implemented in a massively parallel machine where each unit has
the same data structures and functions described above.
Objective 3: Small sample learning and generalization. The probabilistic model defined on this And–Or graph representation can be
learned from a relatively small training set per category and then sampled through Monte Carlo simulation to synthesize a large number of
configurations. This is in fact an extension to the traditional texture
synthesis experiment by the minimax entropy principle [90], where new
268 Introduction
texture samples are synthesized which are different from the observed
texture but are perceptually equivalent to the observed texture. The
minimax entropy learning scheme is extended to the And–Or graph
models in [59], which can generate novel configurations through composition to cover unforeseen object instances in the test set. This generalization capability is mostly missed in discriminative machine learning
methods.
In the experiments reported in [44, 59], they seek for the minimum number of distinct training samples needed for each category,
usually in the range of 20–50. They prune some redundant examples which can be derived through other examples by composition.
Then they found that the generated samples can largely improve the
object recognition performance. For example, a 15% recognition rate is
reported in [44].
Objective 4: Mapping the visual vocabulary to fill the semantic gap.
To fill the well-known semantic gap between symbols and pixels, the
grammar includes a series of visual dictionaries for visual concepts at
all levels. There are two key observations for these dictionaries.
1. The elements of the dictionaries are organized through graph
composition. At the bottom-level the dictionary is a set of
image primitives each having a number of anchor points in a
small graph with open bonds to link with other primitives.
These primitives can be combined to form larger and larger
graph structures for parts and objects, in a way similar to
Lego pieces that kids play with.4
2. Vision is distinct from other sensors, like speech in the aspect
that objects can appear at arbitrary scales. As a result, the
instances of each node can occur at any sizes. The nonterminal nodes at all levels of the And–Or graph can terminate directly as image primitives. Thus one has to account
for the transitions between instances of the same node over
scales. This is the topics studied in the perceptual scale space
theory [80].
4 Note
that Lego pieces are well designed to have standardized teeth to fit each other, this
is not true in the image primitives. The latter are more flexible.
1.3 Overview of the Image Grammar
269
Though there are variations in the literature for what the low level
primitives should be, the differences are really minor between what
people called textons, texels, primitives, patches, and fragments. The
ambiguities in inferring these local primitives shall be resolved through
top-down computation using larger structures.
Finally the primitives are connected to form a primal sketch graph
representation [31] which will generate the input image with every pixels explained. This closes the semantic gap.
1.3
Overview of the Image Grammar
In this subsection, we overview the basic concepts in the image grammar. We divided it into two parts: (i) representation and data structures, (ii) Image annotation dataset to learn the grammar, and the
learning and computing issues.
1.3.1
Overview of the Representational Concepts
and Data Structures
We use Figure 1.3 as an example to review the representational concepts
in the following:
1. An And–Or graph. Figure 1.3(a) shows a simple example of
an And–Or graph. An And–Or graph includes three types
of nodes: And-nodes (solid circles), Or-nodes (dashed circles), and terminal nodes (squares). An And-node represents
a decomposition of an entity into its parts. It corresponds to
the grammar rules, for example,
A → BCD,
H → N O.
The horizontal links between the children of an And-node
represent relations and constraints. The Or-nodes act as
“switches” for alternative sub-structures, and stands for
labels of classification at various levels, such as scene category, object classes, and parts etc. It corresponds to production rules like,
B → E | F,
C → G | H | I.
270 Introduction
(b) parse graph 1
(a) And-Or graph
(c) parse graph 2
A
A
and-node
or-node
A
leaf node
B
C
B
D
E
F
E
L
H
G
M
I
J
D
B
C
D
H
J
F
I
I
9
9'
O
N
K
S
U
6
8
L
M
P
R
2
4
O
N
1
R
S
T
U
2
3
4 5 6
7
8
(d) configuration 1
9
10
<B,C
11
>
1
<B,C
>
6
8
(e) configuration 2
4
<C,D
>
10
>
<C,D
2
<B,C>
9
<C,D>
Q
10
<L,M>
P
<N,O>
1
C
9'
Fig. 1.3 Illustrating the And–Or graph representation. (a) An And–Or graph embodies the
grammar productions rules and contexts. It contains many parse graphs, one of which is
shown in bold arrows. (b) and (c) are two distinct parse graphs by selecting the switches at
related Or-nodes. (d) and (e) are two graphical configurations produced by the two parse
graphs, respectively. The links of these configurations are inherited from the And–Or graph
relations. Modified from [59].
Due to this recursive definition, one may merge the And–
Or graphs for many objects or scene categories into a larger
graph. In theory, all scene and object categories can be represented by one huge And–Or graph, as it is the case for natural
language. The nodes in an And–Or graph may share common
parts, for example, both cars and trucks have rubber wheels
as parts, and both clock and pictures have frames.
2. A parse graph, as shown in Figure 1.1, is a hierarchic generative interpretation of a specific image. A parse graph is
augmented from a parse tree, mostly used in natural or programming language by adding a number of relations, shown
as side links, among the nodes. A parse graph is derived
from the And–Or graph by selecting the switches or classification labels at related Or-nodes. Figures 1.3(b) and 1.3(c)
1.3 Overview of the Image Grammar
271
are two instances of the parse graph from the And–Or graph
in Figure 1.3(a). The part shared by two node may have different instances, for example, node I is a child of both nodes
C and D. Thus we have two instances for node 9.
3. A configuration is a planar attribute graph formed by linking the open bonds of the primitives in the image plane.
Figures 1.3(d) and 1.3(e) are two configurations produced
by the parse graphs in Figures 1.3(b) and 1.3(c), respectively. Intuitively, when the parse graph collapses, it produces a planar configuration. A configuration inherits the
relations from its ancestor nodes, and can be viewed as a
Markov networks (or deformable templates [19]) with reconfigurable neighborhood. We introduce a mixed random field
model [20] to represent the configurations. The mixed random field extends conventional Markov random field models
by allowing address variables and handles non-local connections caused by occlusions. In this generative model, a configuration corresponds to a primal sketch graph [31].
4. The visual vocabulary. Due to scaling property, the terminal nodes could appear at all levels of the And–Or graph.
Each terminal node takes instances from certain set. The set
is called a dictionary and contains image patches of various
complexities. The elements in the set may be indexed by
variables such as its type, geometric transformations, deformations, appearance changes etc. Each patch is augmented
with anchor points and open bond to connect with other
patches.
5. The language of a grammar is the set of all possible valid
configurations produced by the grammar. In stochastic grammar, each configuration is associated with a probability. As
the And–Or graph is directed and recursive, the sub-graph
underneath any node A can be considered a sub-grammar for
the concept represented by node A. Thus a sub-language for
node A is the set of all valid configurations produced by the
And–Or graph rooted at A. For example, if A is an object category, say a car, then this sub-language defines all the valid
272 Introduction
configurations of car. In an exiting case, the sub-language of
a terminal node contains only the atomic configurations and
thus is called a dictionary.
In comparison, an element in a dictionary is an atomic structure and
an element in a language is a composite structure (or configuration)
made of a number of atomic structures. A configuration of node A in
zoomed-out view loses its resolution and details, and becomes an atomic
element in the dictionary of node A. For example, a car viewed in close
distance is a configuration consisting of many parts and primitives.
But in far distance, a car is represented by a small image patch as a
whole and is not decomposable. This is a special property of the image
grammar. The perceptual transition over scales is studied in [80, 84].
1.3.2
Overview of the Dataset and Learning
Now we briefly overview the learning and computing issues with
stochastic image grammars.
A foremost question that one may ask is: how do you build this
grammar and where is the dataset? Collecting the dataset for learning
and training is perhaps more challenging than the learning task itself.
Although fully automated learning is most ideal, for example, let
a computer program watch Disney cartoon or Hollywood movies and
hope it figures out all the object categories and relations. But purely
unsupervised learning is less practical for learning the structured compositional models at present for two reasons. (i) Visual learning must be
guided by objectives and purposes of vision, not purely based on statistical information. Ideally one has to integrate this automatic learning
process with autonomous robot and AI reasoning at the higher level.
Before the robotics and AI systems are ready, we should guide the
learning process with some human supervision. For example, what are
important structures and what are decorative stuff. (ii) In almost all
the unsupervised learning methods, the trainers still have to select their
data carefully to contrast the involved concepts. For example, to learn
the concept that a car has doors, we must select images of cars with
doors both open and closed. Otherwise the concept of door cannot be
learned.
1.3 Overview of the Image Grammar
273
We propose to learn the image grammar in a semi-automatic way.
We shall start with a supervised learning with manually annotated
images and objects to produce the parse graphs. We use this dataset
to initiate the process and then shift to weakly supervised learning.
This initial dataset is still very large if we target thousands of object
categories.
To make the large scale grammar learning framework practical, the
first author founded an independent non-profit research institute which
started to operate in the summer of 2005.5 It has a full time annotation
team for parsing the image structures and a development team for the
annotation tools and database construction. Each image or object is
parsed, semi-automatically, into a parse graph where the relations are
specified and objects are names using the wordnet standard. Figure 1.4
lists an inventory of the current ground truth dataset parsed at LHI.
It has now over 500, 000 images (or video frames) parsed, covering 280
object categories. Figure 1.5 shows two examples — the parse trees
of cat and car. For clarity we only show the parse trees with naming
of the nodes. Beyond the object parsing, there are many scene images
annotated with the objects and their spatial relations labeled. As stated
in a report [87], this ground truth annotation is aimed at broader scope
and more hierarchic structures than other datasets collected in various
groups, such as Berkeley [4, 50], Caltech [16, 29], and MIT [62].
With this annotated dataset, we can construct the And–Or graph
for object and scene categories and learn the probability model on the
And–Or graphs. These learning steps are guided by a minimax entropy
learning scheme [90] and maximum likelihood estimation. It is divided
into three parts:
1. Learning the probabilities at the Or-node so that the configurations generated account for the natural co-occurrence
frequency. This is typical in stochastic context free grammars [10].
2. Learning and pursuing the Markov models on the horizontal
links and relations to account for the spatial relations, as well
5 It
is called the Lotus Hill Research Institute (LHI) in China (www.lotushill.org).
activity
cat
pig
horse
tiger
cattle
bear
panda
orangutang
kangaroo
zebra
...
land mammal
indoor
other
manmade
25,449 images
146,835 POs
marine
insert
plant
bird
flower
fruit
mountain/hill
crocodile
crane
bass
ant
body of water
turtle
eagle
butterfly
shark
...
ibis
dolpin cockroach forg
parrot
crab
dragonfly
trout
snak
flamingo goldfish mayfly
...
owl
shrimp scorpion
pigeon
octopus tick
...
...
robin
duck
hen
...
other
generic
object
battleship
television
cannon
lamp
microwave helicopter
tank
camera
rifle
ceiling fan
ambulance sword
...
telepnone
cell phone
mp3
air-condition
...
weapon
chinese
vehicle furniture electronic
airplane table
car
chair
bed
bus
bicycle bench
couch
SUV
...
truck
motorcycle
cruise ship
ambulance
...
1,854 images
46,419 POs
english
text
636,748 images
3,927,130 POs
1,271 images
14,784 POs
age
pose
expression
face
Database
587,391 frames
3,121,798POs
food
flag
container
computer
tools
music instrument
stationery
...
other
cartoon
movie clips
surveillance
video clips
video
4,798 images
156,665 POs
attribute curve
graphlet
weak boundary
...
low-middle
level vision
others
Fig. 1.4 Inventory of the current human annotated image database from Lotus Hill Research Institute for learning and testing. From [87].
A large set of human annotated images and video ground truth is available at the website www.imageparsing.com.
outdoor
natural
aerial 1,625 images
image 117,215 POs
business
parking
airport
sports
street
bathroom shopping residential
parking bedroom meeting industry
corridor dinner
intersection
harbor
marina
lecture
highway hall
kitchen
school
forest
office
landscape
livingroom
animal
rural
cityview
seashore
images
scene 14,360
323,414 POs
PO means a parsed object node in the database
Inventory of the annotated image database by Nov.06
274 Introduction
1.3 Overview of the Image Grammar
275
Fig. 1.5 Two examples of the parse trees (cat and car) in the Lotus Hill Research Institute
image corpus. From [87].
276 Introduction
as consistency of appearance between nodes in the And–Or
graphs. This is similar to the learning of Markov random
fields [90], except that we are dealing with a dynamic graphical configuration instead of a fixed neighborhood.
3. Learning the And–Or graph structures and dictionaries. The
terminal nodes are learned through clustering and the nonterminal nodes are learned through binding. We only briefly
discuss this issue in this paper as the current literature has
not made significant progress in this part.
The proposed stochastic context sensitive grammar (SCSG) combines the reconfigurability of SCFG with the contextual constraints of
graphical (MRF) models, and has the following properties: (a) Compositional power for representing large intra-class structural variations.
The grammar can generate a huge number of configurations (i.e., its
language) for scenes and objects by composing a relatively much smaller
vocabulary. All are represented in graphical configurations. The language of the grammar is the set of all valid configurations of a category, such as furniture, clothes, vehicles, etc. Thus it has enormous
expressive power. (b) Recursive structures for scalable computing. The
grammar is embodied into an And–Or graph which has recursive structure. The latter is easy to scale in terms of increasing the number of
object categories or augmenting more levels (e.g., scene nodes). Consequently the inference algorithms is also recursively defined. We only
need to write general top-down and bottom-up functions for a common And–Or node, and re-use the code for all nodes in the And–Or
graph. (c) Small sample for effective learning. Due to explicit composition and part-sharing between categories, the state spaces for all object
categories are decomposed into products of subspaces of lower dimensions for the vocabulary and relations. Thus we need relatively smaller
number of training examples (20–100 instances) for each category. In
recent experiments (see Figure 2.6), we can sample the learned object
model to generate novel object configurations for generalization, and
observe remarkable (over 15% improvement in object category) recognition tasks.
1.3 Overview of the Image Grammar
277
The rest of the paper is organized in the following way. We first discuss in Chapter 2 the background of stochastic grammar, its formulation, the new issues of image grammar in contrast to language grammar,
and previous work on image grammar. Then we present the grammar
and And–Or graph representation in Chapters 3–6 sequentially: the
visual grammar, the relations and configurations, the parse graphs,
and finally the And–Or graph. The learning algorithm and results are
discussed in Chapter 7, which is followed by the top-down/bottom-up
inference algorithm in Chapter 8, and three case studies in Chapter 9.
Finally, we raise a number of unsolved problems in Chapter 10 to conclude the paper.
2
Background
2.1
The Origin of Grammars
The origin of grammar in real-world signals, either language or vision,
is that certain parts of a signal s tend to occur together more frequently than by chance. Such co-occurring elements can be grouped
together forming a higher order part of the signal and this process can
be repeated to form increasingly larger parts. Because of their higher
probability, these parts are found to re-occur in other similar signals,
so they form a vocabulary of “reusable” parts. A basic statistical measure, which indicates whether something is a good part, is a quantity
which measures, in bits, the strength of binding of two parts s|A and
s|B of the signal s:
p(s|A∪B )
.
(2.1)
log2
(p(s|A ) · p(s|B )
Two parts of a signal are bound if the probability of their co-occurrence
is significantly greater than the probability if their occurence was
independent. The classic example which goes back to Laplace is the
sequence of 14 letters “CONSTANTINOPLE”: these occur much more
frequently in normal text than in random sequences of the 26 letters
278
2.1 The Origin of Grammars
279
S
S
A
B
(a)
B
A
(b)
Fig. 2.1 (a) Two parallel lines form a reusable part containing as its constituents the two
lines, (b) A T-junction is another reusable part formed from two lines.
in which the letters are chosen independently, even with their standard
frequencies. In this example, the composite part is a word, its constituents are letters. A more elaborate example from vision is shown
in Figure 2.1. On the left, this illustrates how nearby lines tend to be
parallel more often than at other mutual orientations, hence a pair of
parallel lines forms a reusable part. On the right, we see how another
frequent configuration is when the two lines are roughly perpendicular
and touch forming a “T-junction.”
The set of reusable parts that one identifies in some class of signals,
e.g., in images, is called the vocabulary for this class of signals. Each
such reusable part has a name or label. In language, a noun phrase,
whose label is “NP” is a common reusable part, an element of the
linguistic vocabulary. In vision, a face is a clear candidate for such a very
high-level reusable part. The set of such parts which one encounters in
analyzing statistically a specific signal is called the parse graph of the
signal. Abstractly, one first associates to a signal s : D → I the set of
subsets {Ai } of D such that s|Ai is a reusable part. Then these subsets
are made into the vertices or nodes Ai of the parse graph. In the
graph, the proper inclusion of one subset in another, Ai Aj , is shown
by a “vertical” directed edge Aj → Ai . For simplicity, we prune
redundant edges in this graph, adding edges only when Ai Aj and
there is no Ak such that Ai Ak Aj .
In the ideal situation, the parse graph is a tree with the whole signal
at the top and the domain D (the letters of the text or the pixels of the
image) at the bottom. Moreover, each node Ai should be the disjoint
union of its children, the parts {Aj |Aj Ai }. This is the case for the
280 Background
simple parse trees of Figure 2.1 or in most sentences, such as the ones
shown below in Figure 2.6.
2.2
The Traditional Formulation of Grammar
The formal idea of grammars goes back to Panini’s Sanskit grammar in
the first millenium BCE, but its modern formalization can be attributed
to Chomsky [11]. Here one finds the definition making a grammar into
a 4-tuple G = (VN , VT , R, S), where VN is a finite set of non-terminal
nodes, VT a finite set of terminal nodes, S ∈ VN is a start symbol at
the root, and R is a set of production rules,
R = {γ : α → β}.
(2.2)
One requires that α, β ∈ (VN ∪ VT )+ are strings of terminal or nonterminal symbols, with α including at least a non-terminal symbol.1
Chomsky classified languages into four types according to the form of
their production rules. A type 3 grammar has rules A → aB or A → a,
where a ∈ VT and A, B ∈ VN . It is also called a finite state or regular
grammar. A type 2 grammar has rules A → β and is called a context
free grammar. A type 1 grammar is context sensitive with rules ξAη →
ξβη where a non-terminal node A is rewritten by β in the context of
two strings ξ and η. The type 0 grammar is called a phrase structure
or free grammar with no constraint on α and β.
The set of all possible strings of terminals ω derived from a grammar
G is called its language, denoted by
R∗
(2.3)
L(G) = ω : S =⇒ ω, ω ∈ VT∗ .
R∗ means a sequence of production rules deriving ω from S, i.e.,
S
γ1 ,γ2 ,...,γn(ω)
=⇒
ω
(2.4)
If the grammar is of type 1, 2, or 3, then given a sequence of rules generating the terminal string ω, we obtain a parse tree for ω, denoted by
pt(ω) = (γ1 , γ2 , . . . , γn(ω) ),
1V ∗
(2.5)
means a string consisting of n ≥ 0 symbols from V , and V + means a string with n ≥ 1
symbols from V .
2.2 The Traditional Formulation of Grammar
281
if each production rule creates one node labeled by its head A and
a set of vertical arrows between A and each symbol in the string β.
To relate this to the general setup of the previous section, note that
each node has a set of ultimate descendents in the string ω. This is to
be a reusable part. If we give this part the label A ∈ VN , we see that
the tree can equally well be generated by taking these parts as nodes
and putting in vertical arrows when one part contains another with
no intermediate part. Thus the standard Chomskian formulation is a
special case of our general setup.
As is illustrated in Figure 2.4, the virtue of the grammar lies in its
expressive power of generating a very large set of valid sentences (or
strings), i.e., its language, through a relatively much smaller vocabulary VT , VN and production rules R. Generally speaking, the following
inequality is often true in practice,
|L(G)| |Vn |, |VT |, |R|.
(2.6)
In images, VT can be pixels, but here we will find it more convenient
to make it correspond to a simple set of local structures in the image,
textons, and other image primitives [30, 31]. Then VN will be reusable
parts and objects in the image, and a production rule A → β is a template which enables you to expand A. Then the L(G) will be the set
of all valid object configurations, i.e., scenes. The grammar rules represent both structural regularity and flexibility. The structural regularity
is enforced by the template which decomposes an entity A, such as
object into certain elements in β. The structural flexibility is reflected
by the fact that each structure A has many alternative decompositions.
In this paper, we will find it convenient to describe the entire
grammar by one universal And–Or tree, which contains all parsings
as subtrees. In this tree, the Or-nodes are labeled by VN ∪ VT and
the And-nodes are labeled by production rules R. We generate this
tree recursively, starting by taking start symbol as a root which is an
Or-node. We proceed as follows: wherever we have an Or-node with
non-terminal label A, we consider all rules which have A on the left
and create children which are And-nodes labeled by the corresponding
rules. These in turn expand to a set of Or-nodes labeled by the symbols
on the right of the rule. An Or-node labeled by a non-terminal does
282 Background
And−Or tree
and
S
or
r2
r1
a
leaf
b
S
r2
r1
A parsing tree pt(abb)
a
S
b
Fig. 2.2 A very simple grammar, its universal And–Or tree and a specific parse tree in
shadow.
not expand further. Clearly, all specific parse trees will be contained
in the universal And–Or tree by selecting specific children for each Ornode reached when descending the tree. This tree is often infinite. An
example is shown in Figure 2.2.
A vision example of an And–Or tree, using the reusable parts in
Figure 2.1, is shown in Figure 2.3. A, B, C are non-terminal nodes and
A
Or-node
And-node
leaf-node
C
B
a
b
c
c
Fig. 2.3 An example of binding elements a, b, c into a larger structures A in two alternative
ways, represented by an And–Or tree.
2.2 The Traditional Formulation of Grammar
283
a, b, c are terminal or leaf nodes. B, C are the two ambiguous ways to
interpret A. B represents an occlusion configuration with two layers
while C represents a butting/alignment configuration at one layer. The
node A in Figure 2.3 is a frequently observed local structure in natural
images when a long bar (e.g., a tree trunk) occludes a surface boundary
(e.g., a fence).
The expressive power of an And–Or tree is illustrated in Figure 2.4.
On the left is an And-node A which has two components B and C.
Both B and C are Or-nodes with three alternatives shown by the six
leaf nodes. The 6 leaf nodes can compose a set of configurations for
node A, which is called the “language” of A – denoted by L(A). Some
of the valid configurations are shown at the bottom. The power of composition is crucial for representing visual concepts which have varying
structures, for example, if A is an object category, such as car or chair,
then L(A) is a set of valid designs of cars or chairs. The expressive
power of the And–Or tree rooted at A is reflected in the ratio of the
total number of configurations that it can compose over the number of
nodes in the And–Or tree. For example, Figure 2.4(b) shows two levels of And-nodes and two levels of Or-nodes. Both have branch factor
Or-node
A
And-node
leaf-node
C
B
a
b
d
c
L(A) ={
f
e
... }
(a)
(b)
Fig. 2.4 (a) An And-node A is composed of two Or-nodes B and C, each of which includes
three alternative leaf nodes. The 6 leaf nodes can compose a set of configurations for node A,
which is called the “language” of A. (b) An And–Or tree (5-level branch number = 3)
with 10 And-nodes, 30 Or-nodes, and 81 leaf nodes, can produce 312 = 531, 441 possible
configurations.
284 Background
b = 3. This tree has a total of 10 And-nodes, 30 Or-nodes, and 81 leaf
nodes, the number of possible structures is (3 × 33 )3 = 531, 441, though
some structures may be repeated.
In Section 2.6, we shall discuss three major differences between
vision grammars and language grammars.
2.3
Overlapping Reusable Parts
As mentioned, in good cases, there are no overlapping reusable parts in
the base signal and each part is the disjoint union of its children. But
this need not be the case. If two reusable parts do overlap, typically
this leads to parse structures with a diamond in them, Figure 2.5 is
an example. Many sentences, for example, are ambiguous and admit
two reasonable parses. If there exists a string ω ∈ L(G) that has more
than one parse tree, then G is said to be an ambiguous grammar.
For example, Figure 2.6 shows two parse trees for a classic ambiguous
sentence (discussed in [26]). Note that in the first parse, the reusable
part “saw the man” is singled out as a verb phrase or VP; in the
second, one finds instead the noun phrase (NP) “the man with the
telescope.” Thus the base sentence has two distinct reusable parts
which overlap in “the man.” Fixing a specific parse eliminates this
complication. In context, the sentence is always spoken with only one
of these meanings, so one parse is right, one is wrong, one reusable
part is accepted, one is rejected. If we reject one, the remaining
parts do not overlap.
A
B
a
C
b
c
Fig. 2.5 Parts sharing and the diamond structure in And–Or graphs.
2.3 Overlapping Reusable Parts
S
S
VP
NP
VP
NP
VP
NP
V
PP
PP
NP
V
NP
P
NP
Det
Det
I
saw the
285
N
man
Det
N
N
with the telescope
P
NP
Det
I
saw the
man
N
with the telescope
Fig. 2.6 An example of ambiguous sentence with two parse trees. The non-terminal nodes
S, V, NP, VP denotes sentence, verbal, noun phrase, and verbal phrase, respectively. Note
that if the two parses are merged, we obtain a graph, not a tree, with a “diamond” in it as
above.
The above is, however, only the simplest case where reusable parts
overlap. Taking vision, there seem to occur an overlap in four ways.
1. Ambiguous scenes where distinct parses suggest themselves.
2. High level patterns which incorporate multiple partial
patterns.
3. “Joints” between two high level parts where some sharing of
pixels or edges occurs.
4. Occlusion where a background object is completed behind a
foreground object, so the two objects overlap.
A common cause of ambiguity in images is when there is an accidental match of color across the edge of an object. An example is shown in
Figure 2.7(a): the man’s face has similar color to the background and,
in fact, the segmenter decided the man had a pinnocio-like nose. The
true background and the false head with large nose overlap. As in the
linguistic examples, there is only “true” parse and the large nose part
should be rejected.
An example of the second is given by a square (or by many alphanumeric characters). A square may be broken up into two pairs of
parallel lines. A pair of parallel lines is a common reusable part in its
own right, so we may parse the square as having two child nodes, each
286 Background
(a)
(c)
(b)
(d)
Fig. 2.7 Four types of images in which “reusable parts” overlap. (a) The pinnocio nose
is a part of the background whose gray level is close to the face, so it can be grouped
with the face or the background. This algorithm chose the wrong parse. (b) The square
can be parsed in two different ways depending on which partial patterns are singled out.
Neither parse is wrong but the mid-level units overlap. (c) The two halves of a butt joint
have a common small edge. (d) The reconstructed complete sky, trees and field overlap
with the face.
such a pair. But the square is also built up from 4 line pairs meeting in a right angle. Such pairs of lines also form common reusable
parts. The two resulting parses are shown in Figure 2.7(b). One “solution” to this issue is to choose, once and for all, one of these as
the preferred parse for a square. In analyzing the image, both parses
may occur but, in order to give the whole the “square” label, one
parse is chosen and the other parts representing partial structures are
rejected.
“Joints” will be studied below: often two parts of the image are
combined in characteristic geometric ways. For example, two thin rectangles may butt against each other and then form a compound part.
But clearly, they share a small line segment which is common to both
2.4 Stochastic Grammar
287
their boundaries: see Figure 2.7(c). If the parsing begins at the pixel
level, such sharing between adjacent parts is almost inevitable. The
simplest way to restore the tree-like nature of the parse seems to be to
duplicate the overlapping part. For example, an edge is often part of
the structure on each side and it seems very natural to allocate to the
edge two nodes — the edge attached to side 1 and the edge attached
to side 2.
The most vision-specific case of overlap is caused by occlusion.
Occlusion is seen in virtually every image. It can be modeled by what
the second author has called the 2.1D sketch. Mentally, humans (and
presumably other visual animals) are quite aware that two complete
objects exist in space but that certain parts of the two objects project
to the same image pixels, with only one being visible. Here we consciously form duplicate image planes carrying the two objects: this is
crucial when we actually want to use our priors to reconstruct as much
as possible of the occluded object. It seems clear that the right parse
for such objects should add extra leaves at the bottom to represent
the occluded object. The new leaves carry colors, textures etc. extrapolated from the visible parts of the object. Their occluded boundaries
were what the gestalt school called amodal contours. The gestalt school
demonstrated that people often make very precise predictions for such
amodal contours.
Below we will assume that the reusable parts do not overlap so that
inclusion gives us a tree-like parse structure. This simplifies immensely
the computational algorithms. Future work may require dealing with
diamonds more carefully (REF Geman).
2.4
Stochastic Grammar
To connect with real-world signals, we must augment grammars with a
set of probabilities P as a fifth component. For example, in a stochastic
context free grammar (SCFG) — the most common stochastic grammar
in the literature, suppose A ∈ VN has a number of alternative rewriting
rules,
A → β1 | β2 | · · · | βn(A) ,
γi : A → βi .
(2.7)
288 Background
Each production rule is associated with a probability p(γi ) = p(A → βi )
such that:
n(A)
p(A → βi ) = 1.
(2.8)
i=1
This corresponds to what is called a random branching process in statistics [2]. Similarly a stochastic regular grammar corresponds to a Markov
chain process.
The probability of a parse tree is defined as the product,
n(ω)
p(pt(ω)) =
p(γj ).
(2.9)
j=1
The probability for a string (in language) or configuration (in image)
ω ∈ L(G) sums over the probabilities of all its possible parse trees.
p(pt(ω)).
(2.10)
p(ω) =
pt(ω)
Therefore a stochastic grammar G = (VN , VT , R, S, P) produces a probability distribution on its language
R∗
(2.11)
L(G) = (ω, p(ω)) : S =⇒ ω, ω ∈ VT∗ .
A stochastic grammar is said to be consistent if ω∈L(G) p(ω) = 1. This
is not necessarily true even when Equation (2.8) is satisfied for each
non-terminal node A ∈ VN . The complication is caused by cases when
there is a positive probability that the parse tree may not end in a
finite number of steps. For example, if we have a production rule that
expands A to AA or terminates to a, respectively,
A → AA | a with prob. ρ |(1 − ρ)
If ρ > 12 , then node A expands faster than it terminates, and it keeps
replicating. This poses some constraints for designing the set of probabilities P.
The set of probabilities P can be learned in a supervised way
from a set of observed parse trees {ptm , m = 1, 2, . . . , M } by maximum
2.5 Stochastic Grammar with Context
289
likelihood estimation,
M
P ∗ = arg max
p(pti ).
(2.12)
m=1
The solution is quite intuitive: the probability for each non-terminal
node A in (2.7) is
#(A → βi )
p(A → βi ) = n(A)
.
#(A
→
β
)
j
j=1
(2.13)
In the above equation, #(A → βi ) is the number of times a rule A → βi
is used in all the M parse trees. In an unsupervised learning case, when
the observation is a set of strings without parse trees, one can still
follow the ML-estimation above with an EM-algorithm. It was shown
in [10] that the ML-estimation of P can rule out infinite expansion and
produce a consistent grammar.
In Figure 2.3, one can augment the two parses by probabilities ρ
and 1 − ρ, respectively. We write this as a stochastic production rule:
A → a · b | c · c;
ρ|(1 − ρ).
(2.14)
Here “|” means an alternative choice and is represented by an “Ornode.” “·” means composition and is represented by an “And-node”
with an arc underneath. One may guess that the interpretation B has
a higher probability than C, i.e., ρ > 1 − ρ in natural images.
2.5
Stochastic Grammar with Context
In the rest of this paper, we shall use an And–Or tree defined by a
stochastic grammar but we will augment it to an And–Or graph by
adding relations and contexts as horizontal links. The resulting probabilistic models are defined on the And–Or graph to represent a stochastic context sensitive grammar for images.
A simple example of this in language, due to Mark, Miller and
Grenander augments the stochastic grammar models with word cooccurrence probabilities. Let ω = (ω1 , ω2 , . . . , ωn ) be a sentence with n
words, then bi-gram statistics counts the frequency h(ωi , ωi+1 ) and all
290 Background
word pairs, and therefore leads to a simple Markov chain model for the
string ω:
p(ω) = h(ω1 )
n−1
h(ωi+1 |ωi ).
(2.15)
i=1
In [48], a probabilistic model was proposed to integrate parse tree model
in (2.9) and the bi-gram model in (2.15) for the terminal string, by
adding factors h∗ (ωi , ωi+1 ) and re-normalizing the probability:
n(ω)
n−1
1 ∗
∗
h (ωi+1 , ωi ) ·
p(γj ).
p(pt(ω)) = h (ω1 )
Z
i=1
(2.16)
j=1
The factors are chosen so that the marginal probability on word pairs
matches the given bi-gram model. Note that one can always rewrite the
probability in a Gibbs form for the whole parse tree and strings,


n−1
 n(ω)

1
λ(γj ) −
λ(ωi+1 , ωi ) ,
(2.17)
p(pt(ω); Θ) = exp −


Z
j=1
i=1
where λ(γj ) = − log p(γj ) and λ(ωi+1 |ωi ) = − log h∗ (ωi+1 |ωi ) are parameters included in Θ. Thus the existence of the h∗ is a consequence of the existence of exponential models matching given
expectations.
However, the left-to-right sequence of words may not express the
strongest contextual effects. There are non-local relations as the arrows
in Figure 2.8 show. First interjections mess up phrases in language. The
italicized words in the sentence split the text flow. Thus the “next”
relation in the bi-gram is not deterministically decided by the word
order but has to be inferred. Second the word “what” is both the object
of the verb “said” and the subject of the verb “is.” It connects the
What I just said, though I cannot be completely sure, is perhaps real.
Fig. 2.8 An English sentence with non-local “next” relations shown by the arrows and the
word “what” is a joint to link two clauses.
2.6 Three New Issues in Image Grammars in Contrast to Language
291
two clauses together. Quite generally, all pronouns indicate long range
dependencies, link two reusable parts and carry context from one part
of an utterance or text to another. In images one shall see many different
types of joints that combine parts of objects, such as butting, hinge,
and various alignments that similarly link two reusable parts. As we
shall discuss in a later section, each node may have many types of
relations in the way it interacts with other nodes. These relations are
often hidden or cannot be deterministically decided and thus we shall
represent these potential connections through some “address variables”
associated with each node. The value of an address variable in a node ωi
is an index toward another node ωj , and the node pair (ωi , ωj ) observes
a certain relation. These address variables have to be computed along
with the parse tree in inference.
In vision, these non-local relations occur much more frequently.
These relationships represent the spatial context at all levels of vision
from pixels and primitives to parts, objects and scenes, and lead to various graphical models, such as Markov random fields. Gestalt organizations are popular examples in the middle level and low-level vision. For
example, whenever a foreground object occludes part of a background
object, with this background object being visible on both sides of the
foreground one, these two visible parts of the background object constrain each other. Other non-local connections may reflect functional
relations, such as object X is “supporting” object Y.
2.6
Three New Issues in Image Grammars in Contrast
to Language
As we have seen already, an image grammar should include two aspects:
(i) The hierarchic structures (the grammar G) which generate a large
set of valid image configurations (i.e., the language L(G)). This is especially important for modeling object categories with large intra-class
structural variabilities. (ii) And the context information which makes
sure that the components in a configuration observe good spatial relationships between object parts, for example, relative positions, ratio of
sizes, and consistency of colors. Both aspects encode important parts
of our visual knowledge.
292 Background
Going from 1D language grammars to 2D image grammars is nontrivial and requires a major leap in technology. Perhaps more important
than anything else, one faces enormous complexity, although the principles are still simple. The following section summarizes three major
differences (and difficulties) between the language grammars and image
grammars.
The first huge problem is the loss of the left-to-right ordering in
language. In language, every production rule A → β is assumed to generate a linearly ordered sequence of nodes β and following this down
to the leaves, we get a linearly ordered sequence of terminal words.
In vision, we have to replace the implicit links of words to their left
and right neighbors by the edges of a more complex “region adjacency
graph” or RAG. To make this precise, let the domain D of an image I
have a decomposition D = ∪k∈S Rk into disjoint regions. Then we make
an RAG with nodes Ri and edges Rk — Rl whenever Rk and Rl
are adjacent. This means we must explicitly add horizontal edges to our
parse tree to represent adjacency. In a production rule A → β, we no
longer assume the nodes of β are linearly ordered. Instead, we should
make β into a configuration, that is, a set of nodes from VN ∪ VT plus
horizontal edges representing adjacency. We shall make this precise
below.
Ideas to deal with the loss of left-to-right ordering have been proposed by the K. S. Fu school of “syntactic pattern recognition” under
the names “web grammars” and “plex grammars” [22], by Grenander
in his pattern theory [28], and more recently by graph grammars for
diagram interpretation in computer science [60]. These ideas have not
received enough attention in vision. We need to study the much richer
spatial relations for how object and parts are connected. Making matters more complex, due to occlusions and other non-local groupings,
non-adjacent spatial relations often have to be added in the course of
parsing.
One immediate consequence of the lack of natural ordering is that
a region has very ambiguous production rules. Let A be a region and a
an atomic region, and let the production rules be A → aA | a. A linear
region ω = (a, a, a, . . . , a) has a unique parse graph in left-to-right ordering. With the order removed, it has a combinatorial number of parse
293
2.6 Three New Issues in Image Grammars in Contrast to Language
a
(a)
a
a
a
a
a
(b)
Fig. 2.9 A cheetah and the background after local segmentation: both can be described by
an RAG. Without the left-to-right order, if the regions are to be merged one at a time, they
have a combinatorially explosive number of parse trees.
trees. Figure 2.9 shows an example of parsing an image with a cheetah.
It becomes infeasible to estimate the probability p(ω) by summing over
all these parse trees in (2.10).
Therefore we must avoid these recursively defined grammar rules
A → aA, and treat the grouping of atomic regions into one large region
A as a single computational step, such as the grouping and partitioning
in a graph space [3]. Thus the probability p(ω) is assigned to each object
as a whole instead of the production rules. In the literature, there are a
number of hierarchic representations by an adaptive image pyramid, for
example, the work by Rosenfeld and Hong in the early 80s [34], and the
multi-scale segmentation by Galun et al. [23]. Though generic elements
are grouped in these works, there are no explicit grammar rules. We
shall distinguish such multi-scale pyramid representation from parse
trees.
The second issue, unseen in language grammar, is the issue of image
scaling [45, 80, 82]. It is a unique property of vision that objects appear
at arbitrary scales in an image when the 3D object lies nearer or farther from the camera. You cannot hear or read an English sentence
at multiple scales, but the image grammar must be a multi-resolution
294 Background
images
sketches
primitives
Fig. 2.10 A face appears at three resolutions is represented by graph configurations in three
scales. The right column shows the primitives used at the three levels.
representation. This implies that the parse tree can terminate immediately at any node because no more detail is visible.
Figure 2.10 shows a human face in three levels from [85]. The left column shows face images at three resolutions, the middle column shows
three configurations (graphs) of increasing detail, and the right column
shows the dictionaries (terminals) used at each resolution, respectively.
At a low resolution, a face is represented by patches as a whole (for
example, by principle component analysis), at a middle resolution, it is
represented by a number of parts, and at a higher resolution, the face
is represented by a sketch graph using smaller image primitives. The
sketch graphs shown in the middle of Figure 2.10 expands with increasing resolution. One can account for this by adding some termination
rules to each non-terminal node, e.g., each non-terminal node may exit
the production for a low resolution case.
∀A ∈ VN ,
A → β1 | · · · |βn(A) | t1 | t2 |,
(2.18)
2.7 Previous Work in Image Grammars
295
where t1 , t2 , ∈ VT are image primitives or image templates for A at certain scales. For example, if A is a car, then t1 , t2 are typical views (small
patches) of the car at low resolution. As they are in low resolution, the
parts of the cars are not very distinguishable and thus are not represented separately. The decompositions βi , i = 1, 2, . . . , n(A) represent
the production rules for higher resolutions, so this new issue does not
complicate the grammar design, except that one must learn the image
primitives at multiple scales in developing the visual vocabulary.
The third issue with image grammars is that natural images contain a much wider spectrum of quite irregular local patterns than in
speech signals. Images not only have very regular and highly structured objects which could be composed by production rules, they also
contain very stochastic patterns, such as clutter and texture which are
better represented by Markov random field models. In fact, the spectrum is continuous. The structured and textured patterns can transfer
from one to the other through continuous scaling [80, 84]. The two categories of models ought to be integrated more intimately and melded
into a common model. This raises numerous challenges in modeling and
learning at all levels of vision. For example, how do we decide when we
should develop a image primitive (texton) for a specific element or use
a texture description (for example, a Markov Random Field)? How do
we decide when we should group objects in a scene by a production
rule or by a Markov random field for context?
2.7
Previous Work in Image Grammars
There are four streams of research on image grammars in the vision
literature.
The first stream is syntactic pattern recognition by K. S. Fu and his
school in the late 1970s to early 1980s [22]. Fu depicted an ambitious
program for scene understanding using grammars. A block world example is illustrated in Figure 2.11. Similar image understanding systems
were also studied in the 1970–1980s [33, 54] The hierarchical representation on the right is exactly the sort of parse graph that we are pursuing today. The vertical arrows show the decomposition of the scene
and objects, and the horizontal arrows display some relations, such as
296 Background
Scene A
scene A
wall N
objects B
background C
1
L
1
T
object D
floor M
X
D
E
2
2
M
N
Y
2
Z
object E
L
T
X
Y
2
Z
2
relation 1: support = {(M,D), (M,E)}
relation 2: adjacency = {(L,T), (X,Y), (Y,Z), (Z,X), (M,N)}
Fig. 2.11 A parser tree for a block world from [22]. The ellipses represents non-terminal
nodes and the squares are for terminal nodes. The parse tree is augmented into a parse
graph with horizontal connections for relations, such as one object supporting the other, or
two adjacent objects sharing a boundary.
support and adjacency. Fu and collaborators applied stochastic grammars to simple objects (such as diagrams) and shape contours (such as
outline of a chromosome). Most of the work remained in 1D structures,
although the ideas of web grammars and plex grammars were also studied. This stream was disrupted in the 1980s and suffered from the lack
of an image vocabulary that is realistic enough to express real-world
objects and scenes, and reliably detectable from images. This remains
a challenge today, though much progress has been made recently in
appearance based methods, such as PCAs, image primitives, [31], code
books [17], fragments and patches [38, 77]. It is worth mentioning that
many of these works on patches and fragments do not provide a formalism for composition and that they lack the bond structures studied
in this paper.
The second stream are the medial axis techniques for analyzing
2D shapes. For animate objects represented by simple closed contours,
Blum argued in 1973 [8] that medial axes are an intuitive and effective
representation of a shape, in contrast to boundary fragments. Leyton
proposed a process grammar approach to these in 1988 [43]. He argued
that any shape is a record of motion history, and developed a grammar for the procedure for how a shape grows from a simple object, say
a small circle. A shape grammar for shape matching and recognition
via medial axes was then developed by Zhu and Yuille in 1996 [91].
297
2.7 Previous Work in Image Grammars
S
1
2
8
9
17
3
6
9
7
8
4 5
14
14
S
13 10
15
10
7
15
16
17
3
6
16
13
11 12
2 14
5
11 12
(a)
(b)
(c)
Fig. 2.12 (a) A dog and its decomposition into parts using the medial axis algorithm of
[91]. (b) The shock graph of a goat with its shock tree in (c) adopted from [68]. The root
of the tree is the node at the “hip” of the goat marked by a square.
An example is shown on the left in Figure 2.12. The dog should be
read as a node A in the parse tree and the fragments below it as the
child nodes for a production rule that expands the dog into its limbs,
trunk, head, and tail. The circles are the maximal circles on which the
medial axis is based and allow one to create horizontal arrows between
the parts, so that the production yields not merely a set of parts but a
configuration.
A formal shock graph was studied by Zucker’s school including Dickinson [40], Kimia [67], Siddiqi et al. [41, 64, 68]. They reverse Leyton’s
growth process by collapsing the shape using the distance transform.
The singularities in the process create “shocks,” for example, when
two sides of the leg of a dog collapse into an axis. Thus different sections of their skeleton are characterized by the types of singularity
and record the temporal record of the shape’s collapse. Figure 2.12
shows on the right the shock graph of a goat from [68]. The vertical
arrows in their shock tree are very different from those in the parse
tree. In the shock tree the child nodes are a younger generation that
grow from the parent nodes, thus the two graphs have quite different
interpretations.
The third stream can be seen as a number of works branching out
from the school of pattern theory. Grenander [28] defined a regular pattern on a set of graphs which are made from some primitives which he
298 Background
called “generators.” Each generator is like a terminal element and has
a number of attributes and “bonds” to connect with other generators.
Geman and collaborators [6, 27, 36] proposed a more ambitious formulation for compositionality which is quite similar to that developed
in this paper. Moreover, they seek to create not only computer vision
systems but models of cortical vision mechanisms in animals. In sharp
contrast to our approach, they make the overlapping of their reusable
parts into a central element of their formalism. This overlapping is used
to allow parts to compute their “binding strength” depending on any
and all features of this overlap. It is also the key, in their system, to
synchronizing the activity of the neurons expressing the higher order
parts. As a proof of concept, they applied the compositional system to
handwritten upper case letter recognition and to licence plate reading
[36]. The work in this paper belongs to this approach, cf. an attribute
grammar to parse images of the man-made world [32], and a context
sensitive grammar for representing and recognizing human clothes [9].
These will be reviewed in later sections.
Finally, the sparse image coding model can be viewed as an attribute
SCFG. In sparse coding [56, 69], an image is made of a number of n
independent image bases, and there are a few types of image bases,
such as Gabor cosine, Gabor sine, and Laplacian of Gaussian etc. These
bases have attributes θ = (x, y, τ, σ, α) for locations, orientations, scales
and contrasts, respectively. This can be expressed as an SCFG. Let S
denote a scene, A an image base, and a, b, c the different bases.
S → An ,
n ∼ p(n) ∝ e−λo n ,
A → a(θ) | b(θ) | c(θ),
θ ∼ p(θ) ∝ e−λ|α| ,
where p(θ) is uniform for location, orientation and scale. Crouse et al.
[13] introduce a Markov tree hierarchy for the image bases and this
produces an SCFG.
3
Visual Vocabulary
3.1
The Hierarchic Visual Vocabulary — The “Lego Land”
In English dictionaries, a word not only has a few attributes, such as
meanings, number, tense, and part of speech, but also a number of
ways to connect with other words in a context. Sometimes the connections are so strong that compound words are created, for example,
the word “apple” can be bound with “pine” or “Fuji” to the left, or
“pie” and “cart” to the right. For slightly weaker connections, phrases
are used, for instance, the work “make” can be connected with “something” using the prepositions “of” or “from,” or connected with “somebody” through the prepositions “at” or “against.” Figure 3.1 illustrates
a word with attributes and a number of “bonds” to connect with other
words. Thus a word is very much like a piece of Legos for building toy
objects.
The bonds exist more explicitly and are much more necessary
in the 2D image domain. We define the visual vocabulary in the
following.
299
300 Visual Vocabulary
Make
Attributes
meaning
plural
tense
part of speech
noun
verb
adverb
...
apple
. from sth
pine
pie
Fuji
cart
. of sth
. at sb
. against sb
(a)
(b)
Fig. 3.1 In an English dictionary, each word has a number of attributes and some conventional ways to connect to other words. In the first example, the word “make” can be
connected to “something” or “somebody.” The word “apple” has strong bonds with other
words to make compound words “pine-apple,” “Fuji-apple,” “apple-pie,” “apple-cart.”
Definition 3.1 Visual vocabulary. The visual vocabulary is a set of
pairs, each consisting of an image function Φi (x, y; αi ) and a set of d(i)
bonds (i.e., its degree), to be eventually connected with other elements,
which are denoted by a vector βi = (βi,1 , . . . , βi,d(i) ). We think of βi,k
as an address variable or pointer. αi is a vector of attributes for (a) a
geometric transformation, e.g., the central position, scale, orientation
and plastic deformation, and (b) appearance, such as intensity contrast, profile or surface albedo. In particular, αi determines a domain
Λi (αi ) and Φi is then defined for (x, y) ∈ Λi with values in R (a grayvalued template) or R3 (a color template). Often each βi,k is associated
with a subset of the boundary of Λi (αi ). The whole vocabulary is thus
a set:
∆ = {(Φi (x, y; αi ), βi ) : (x, y) ∈ Λi (αi ) ⊂ Λ},
(3.1)
where i indexes the type of the primitives.
The conventional wavelets, Gabor image bases, image patches, and
image fragments are possible examples of this visual vocabulary except
that they do not have bonds. As an image grammar must adopt a
multi-resolution representation, the elements in its vocabulary represent visual concepts at all levels of abstraction and complexity. In the
3.2 Image Primitives
301
following, we introduce some examples of the visual vocabulary at the
low, middle, and high levels, respectively.
3.2
Image Primitives
In the 1960s–1970s, Julesz conjectured that textons (blobs, bars, terminators, crosses) are the atomic elements in the early stage of visual
perception for local structures [37]. He found in texture discrimination
experiments that the human visual system seem to detect these elements with a parallel computing mechanism. Marr extended Julesz’s
texton concept to image primitives which he called “symbolic tokens” in
his primal sketch representation [49]. An essential criterion in selecting
a dictionary in low level vision is to ensure that they are parsimonious
and sufficient in representing real-world images, and more importantly
they should have the necessary structures to allow composition into
higher level parts. In this subsection, we review a dictionary of image
primitives proposed in Guo et al. [31] as a formal mathematical model
of the primal sketch. Many other studies have come up with similar
lists, including studies which are based on the statistical analysis of
small image patches from large databases [35, 42, 66].
Illustrated in Figure 3.2(a), an image primitive is a small image
patch with a degree d connections or bonds which are illustrated by
the half circles. The primitives are called blobs, terminators, edges
or ridges, “L”-junctions, “T”-junctions, and cross junctions for d =
0, 1, 2, 3, 4, respectively. Each primitive has a number of attributes for its
geometry and appearance. The geometric attributes include position,
orientation, scale, and relative positions of the bonds with respect to
the center. The appearance is described by the intensity profiles around
the center and along the directions perpendicular to the line-segment
connecting the center and the bonds. For instance, a d = 2 primitive
could be called a step edge, a ridge/bar, or double edge depending
on its intensity profile. Each bond of the primitive is like an arm or
hand. When the bonds of two primitives are joined by matching the
two half circles, we say they are connected. Figure 3.2(b) illustrates
how a “T”-shape is composed through 3 terminators, 3 bars, and 1
“T”-junction.
302 Visual Vocabulary
(b)
(a)
Fig. 3.2 Low level visual vocabulary — image primitives. (a) Some examples of image
primitives: blobs, terminators, edges, ridges, “L”-junctions, “T”-junction, and cross junction
etc. These primitives are the elements for composing a bigger graph structure at the upper
level of the hierarchy. (b) is an example of composing a big “T”-shape image using 7
primitives. From [30].
In the following, we show how these primitives can be used to represent images. We start with a toy image in Figure 3.3 to illustrate the
model and a real image in Figure 3.4.
In Figure 3.3, the boundaries of the two rectangles are covered by
4 “T”-junctions, 8 “L”-junctions, and 20 step edges. We denote the
domain covered by an image primitive Φsk
i by Λsk,i , and the pixels
covered by these primitives, which are called the “sketchable part” in
[31], are denoted by
Λsk =
n
sk
Λsk,i .
(3.2)
i=1
The image I on Λsk is denoted by Isk and is modeled by the image
primitives through their intensity profiles. Let be the residual
noise.
Isk (x, y) = Φsk
i (x, y; αi , βi ) + (x, y),
(x, y) ∈ Λsk,i , i = 1, 2, . . . , nsk .
(3.3)
3.2 Image Primitives
303
A
A
B
B
(a)
(b)
Fig. 3.3 An illustrative example for composing primitives into a graph configuration. (a)
is a simple image, and (b) is a number of primitives represented by rectangles which cover
the structured parts of the image. The remaining part of the image can be reconstructed
through simple heat diffusion.
(a) input image
(d) remaining texture pixels
(b) sketch graph
configuration
(e) texture pixels clustered
(c) pixels covered by primitives
(f) reconstructed image
Fig. 3.4 An example of the primal sketch model. (a) An input image I. (b) The sketch
graph – configuration computed from the image I. (c) The pixels in the sketchable part
Λsk . (d) The remaining non-sketchable portion are textures, which are segmented into a
small number of homogeneous regions in (e). (f) The final synthesized image integrating
seamlessly the structures and textures. From [31].
304 Visual Vocabulary
The remaining pixels are flat or stochastic texture areas, called nonsketchable, and are clustered into a few homogeneous texture areas
Λnsk = Λ\Λsk =
n
nsk
Λnsk,j .
(3.4)
j=1
They can be reconstructed through Markov random field models conditional on Isk ,
Insk,j | Isk ∼ p(Insk | Isk ; Θj ).
(3.5)
Θj is a vector-valued parameter for the Gibbs model, for example, the
FRAME model [90].
Figure 3.4 shows a real example of the primal sketch model using
primitives. The input image has 300 × 240 pixels, of which 18, 185 pixels (around 25%) are considered sketchable. The sketch graph has 275
edges/ridges (primitives with degree d = 2) and 152 other primitives for
“vertices” of the graph. Their attributes are coded by 1, 421 bytes. The
non-sketchable pixels are represented by 455 parameters or less. The
parameters are 5 filters for 7 texture regions and each pools a 1D histogram of filter responses into 13 bins. Together with the codes for the
region boundaries, total coding length for the textures is 1, 628 bytes.
The total coding length for the synthesized image in Figure 3.4(f) is
3, 049 bytes or 0.04 byte per pixel. It should be noted that the coding
length is roughly computed here by treating the primitives as being
independent. If one accounts for the dependence in the graph and
applies some arithmetic compression schemes, a higher compression
rate can be achieved.
To summarize, we have demonstrated that image primitives can
compose a planar attribute graph configuration to generate the structured part of the image. These primitives are transformed, warped,
and aligned to each other to have a tight fit. Adjacent primitives are
connected through their bonds. The explicit use of bonds distinguishes
the image primitives from other basic image representations, such as
wavelets and sparse image coding [47, 56] mentioned before, and other
image patches and fragments in the recent vision literature [77]. The
bonds encode the topological information, in addition to the geometry
3.3 Basic Geometric Groupings
305
and appearance, and enable the composition of bigger and bigger structures in the hierarchy.
3.3
Basic Geometric Groupings
If by analogy, image primitives are like English letters or phonemes,
then one wonders what are the visual words and visual phrases. This is
the central question addressed by the gestalt school of psychophysicists
[39, 88]. One may summarize their work by saying that the geometric
relations of alignment, parallelism and symmetry, especially as created
by occlusions, are the driving forces behind the grouping of lower level
parts into larger parts. A set of these composite parts is shown in
Figure 3.5 and briefly described in the caption.
It is important to realize that these groupings occur at every scale.
Many of them occur in local groupings containing as few as 2–8 image
primitives as in the previous section. We will call these “graphlets” [83].
But extended curves, parallels and symmetric structures may span the
whole image. Notably, symmetry is always a larger scale feature but
one occurring very often in nature (e.g., in faces) and which is highly
detectable by people even in cluttered scenes. Parallel lines also occur
(a)
(e)
(i)
(b)
(f)
(j)
(c)
(g)
(k)
(d)
(h)
Fig. 3.5 Middle level visual vocabulary: common groupings found in images. (a) extended
curves, (b) curves with breaks and imperfect alignment, (c) parallel curves, (d) parallels
continuing past corners, (e) ends of bars formed by parallels and corners, (f) curves continuing across paired T-junctions (the most frequent indication of occlusion), (g) a bar occluded
by some edge, (h) a square, (i) a curve created by repetition of discrete similar elements,
(j) symmetric curves, and (k) parallel lines ending at terminators forming a curve.
306 Visual Vocabulary
(a)
(b)
Fig. 3.6 An example of graphlets in natural image. The graphlets are highlighted in the
primal sketch. These graphlets can be viewed as larger pieces of lego. From [24].
frequently in nature, e.g., in tree trunks. The occlusion clue shown in
Figure 3.5 is especially important because it is not only common but
is the strongest clue in a static 2D image to the 3D structure of the
scene. Moreover, it implies the existence of an “amodal” or occluded
contour representing the continuation of the left and right edges behind
the central bar. This necessitates a special purpose algorithm to be
discussed below. Figure 3.6 shows an image with its primal sketch on
the right side with its graphlets shown in dark line segments.
These graphlets are learned through clustering and binding the
image primitives in a way discussed in Equation (2.1). Each cluster in
this space is an equivalence class subject to an affine transform, some
deformation, as well as minor topological editing. These graphlets are
generic 2D patterns, and some of them could be interpreted as object
parts.
3.4
Parts and Objects
If one is only interested in certain object categories segmented from the
background, such as bicycles, cars, ipods, chairs, clothes, the dictionary
will be object parts. Although these object parts are significant within
each category or reusable by a few categories, their overall frequency
307
3.4 Parts and Objects
β 25
β11
β21
β24
g2
β23
β31
g1
g3
β22
β13
β12
β32
β33
Fig. 3.7 High level visual vocabulary — the objects and parts. We show an example of
upper body clothes made of three parts: a collar, a left and a right short sleeves. Each
part is again represented by a graph with bonds. A vocabulary of part for human clothes
is shown in Figure 3.8. From [9].
is low and they are often rare events in a big database of real-world
images. Thus the object parts are less significant as contributors to
lowering image entropy than the graphlets presented above, and the
latter are, in turn, less entropically significant than the image primitives
at the low level.
We take one complex object category — clothes as an example.
Figure 3.7 shows how a shirt is composed of three parts: a collar, a
left, and a right short sleeves. In this figure, each part is represented by
an attribute graph with open bonds, like the graphlets. For example,
the collar part has 5 bonds, and the two short sleaves have 3 bonds to
be connected with the arms and collar. By decomposing a number of
instances in the clothes category together with upper body and shoes,
one can obtain a dictionary of parts. Figure 3.8 shows some examples
for each category.
Thus we denote the dictionary by
(x, y; αi ), βi ) : ∀i, αi , βi .}
∆cloth = {(Φcloth
i
(3.6)
308 Visual Vocabulary
a
e
b
f
c
d
g
h
g
Fig. 3.8 The dictionary of object parts for cloth and body components. Each element is a
small graph composed of primitives and graphlets and has open-bonds for collecting with
other parts. Modified from [9].
As before, Φcloth
is an image patch defined in a domain Λcloth
which
i
i
does not have to be compact or connected. αi controls the geometric
and photometric attributes, and βi = (βi1 , βi2 , . . . , βid(i) ) is a set of open
bonds. These bonds shall be represented as address variables that point
to other bonds. Some upper-cloth examples that are synthesized by
these parts are shown in Figure 9.7.
In fact, the object parts defined above are not so much different from
the dictionaries of image primitives or graphlets, except that they are
bigger and more structured. Indeed they form a continuous spectrum
for the vision vocabulary from low to high levels of vision.
By analogy, each part is like a class in object oriented programming, such as C++. The inner structures of the class are encapsulated,
only the bonds are visible to other classes. These bonds are used for
communication between different object instances.
In the literature, Biederman [5] proposes a set of “geons” as 3D
object elements, which are generalized cylinders for representing 3D
man-made objects. In practice, it is very difficult to compute these
generalized cylinders from images. In comparison, we adopt a view
based representation for the primitives, graphlets, and parts which can
be inferred relatively reliably.
4
Relations and Configurations
While the hierarchical visual vocabulary represents the vertical compositional structures, the relations in this section represent the horizontal
links for contextual information between nodes in the hierarchy at all
levels. The vocabulary and relations are the ingredients for constructing
a large number of image configurations at various level of abstractions.
The set of valid configurations constitutes the language of an image
grammar.
4.1
Relations
We start with a set of nodes V = {Ai : i = 1, 2, . . . , n} where Ai =
(Φi (x, y; αi ), βi ) ∈ ∆ is an entity representing an image primitive,
a grouping, or an object part as defined in the previous section. A number of spatial and functional relations must be defined between the
nodes in V to form a graph with colored edges where the color indexes
the type of relation.
Definition 4.1 Attributed Relation. A binary relation defined on
an arbitrary set S is a subset of the product set S × S
{(s, t)} ⊂ S × S.
309
(4.1)
310 Relations and Configurations
An attributed binary relation is augmented with a vector of attributes
γ and ρ,
E = {(s, t; γ, ρ) : s, t ∈ S},
(4.2)
where γ = γ(s, t) represents the structure that binds s and t, and
ρ = ρ(s, t) is a real number measuring the compatibility between s
and t. Then S, E is a graph expressing the relation E on S. A k-way
attributed relation is defined in a similar way as a subset of S k .
There are three types of relations of increasing abstraction for the
horizontal links and context. The first type is the bond type that connects image primitives into bigger and bigger graphs. The second type
includes various joints and grouping rules for organizing the parts and
objects in a planar layout. The third type is the functional and semantic
relation between objects in a scene.
Relation type 1: Bonds and connections. For a set of nodes V =
{Ai : i = 1, 2, . . . , n} defined above, each node Ai ∈ V has a number
of open bonds {βij : j = 1, 2, . . . , n(i)} shown by the half disks in the
previous section. We collect all these bonds as a set,
Sbond = {βij : i = 1, 2, . . . , n, j = 1, 2, . . . , n(i)}.
(4.3)
Two bonds βij and βkl are said to be connected if they are aligned
in position and orientation. Therefore the bonding relation is a set of
pairs of bonds with attributes:
Ebond (S) = {(βij , βkl ; γ, ρ)},
(4.4)
where γ = (x, y, θ) denote the position and orientation of the bond.
The latter is the tangent direction at the bond for the two connected
primitives. ρ is a function to check the consistency of intensity profile
or color between two connected primitives.
The trivial example is the image lattice. The primitives Ai ,
i = 1, . . . , |Λ| are the pixels. Each pixel has 4 bonds βij , j = 1, 2, 3, 4.
Then Ebond (S) is the set of 4-nearest neighbor connections. In this
case, γ = nil is empty, and ρ is a pair-clique function for the intensities
at pixels i and j. Figures 3.5 and 3.7 show more examples of bonds
for composing graphlets from primitives, and composing clothes from
4.1 Relations
311
parts. Very often people use graphical models, such as templates, with
fixed structures where the bonds are decided deterministically and thus
become transparent. In the next subsection, we shall define the bonds
as random variables to reconfigure the graph structures.
Relation type 2: Joints and junctions. When image primitives are
connected into larger parts, some spatial and functional relations must
be found. Besides its open bonds to connect with others, usually its
immediate neighbors, a part may be bound with other parts in various ways. The gestalt groupings discussed in the previous section are
the best examples: parts can be linked over possibly large distances
by being collinear, parallel, or symmetric. To identify this groupings,
connections must be created flagging this non-accidental relationship.
Figure 4.1 displays some typical relations of this type between object
parts.
Some of these relations also contribute to 3D interpretations. For
example, an ellipse is a part that has multiple possible compositions.
If it is recognized as a bike wheel, its center can function as an axis
and thus can be connected to the tip of a bar (see the rightmost of
Figure 4.1). It could also be the rim of a tea cup, and then the two
ends of its long axis will be joined to a pair of parallel lines to form
a cylinder. In Figure 2.8, we discussed a phenomenon occurred in language where the word “what” is shared by two clauses. Similarly we
have many such joints in images, such as hinge joints, and butting
joints.
Hinged
Butting
Concentric
Attached
Colinear
Parallel
Radial
Bar-circle
Fig. 4.1 Examples of spatial relations for binding object parts. The red dots or lines are the
attributes γ(s, t) of joint relation (s, t) which form the “glue” in this relation. From [59].
312 Relations and Configurations
As Figure 4.1 shows, two parts can be hinged at a point. For example, two hands of a clock have a common axis. For a set of parts in an
image S = V , the hinge relation is a set
Ehinge (S) = {(Ai , Aj ; γ(Ai , Aj ), ρ(Ai , Aj ))}.
(4.5)
Here γ is the hinge point and ρ = nil. In a butting relation, γ(Ai , Aj )
represents the line segment(s) shared by the two parts. The line segment
is shown in red in Figure 4.1. Sometimes, two parts may share two line
segments. For example the handle of a teapot or cup share two line
segments with the body.
Relation type 3: Object interactions and semantics. When letters
are grouped into words, semantic meanings emerge. When parts are
grouped into objects, semantic relations are created for their interactions. Very often these relations are directed. For example, the occluding relation is a viewpoint dependent binary relation between object
or surfaces, and it is important for figure-ground segregation. A view
point independent relation is a supporting relation. A simple example
is shown in Figure 2.11. Let S = V be a set of objects,
Esupp = {M, D, M, E},
Eoccld = {D, M , E, M , D, N , E, N }.
(4.6)
The represents directed relation and the attributes γ, ρ are omitted. There are other functional relations among objects in a scene. For
example, a person A is eating an apple B Eedible (S) = {A, B}, and
a person is riding a bike Eride (S) = {A, C}. These directed relations
usually are partially ordered.
It is worth mentioning that the relations are dense at low level, such
as the bonds, in the sense that the size |E(S)| is in the order of |S|,
and that they become very sparse (or rare) and diverse at high level. At
the high level, we may find many interesting relations but each relation
may only have a few occurrences in the image.
4.2
Configurations
So far, we have introduced the visual dictionaries and relations at various levels of abstractions. The two components are integrated into what
we call the visual configuration in the following.
4.2 Configurations
313
Definition 4.2 Configuration. A configuration C is a spatial layout
of entities in a scene at certain level of abstraction. It is a one layer
graph, often flattened from hierarchic representation,
C = V, E.
(4.7)
V = {Ai , i = 1, 2, . . . , n} is a set of attributed image structures at the
same semantic level, such as primitives, parts, or objects and E is a
relation. If V is a set of sketches and E = Ebonds , then C is a primal
sketch configuration. If E is a union of several relations E = Er1 ∪ · · · ∪
Erk , which often occurs at the object level, then C is called a “mixed
configuration.” For a generative model, the image on a lattice is the
ultimate “terminal configuration,” and its primal sketch is called the
“pre-terminal configuration.” Note that E will close some of the bonds
in V and leave others open; thus we may speak of the open bonds in a
configuration.
We briefly present examples of configurations at three levels.
First, for early vision, the scene configuration C is a primal sketch
graph where V is a set of image primitives with bonds and E = Ebonds
is the bond relation. For example, Figure 3.3(b) illustrates a configuration for a simple image in Figures 3.3(a), and 3.4(b) is a configuration for the image in Figure 3.4(a). These configurations are attributed
graphs because each primitive vi is associated with variables αi for its
geometric properties and photometric appearance. The primal sketch
graph is a parsimonious “token” representation in Marr’s words [49],
and thus it is a crucial stage connecting the raw image signal and the
symbolic representation above it. It can reconstruct the original image
with perceptually equivalent texture appearance.
Second, for the parts to object level, Figure 9.7 displays three possible upper body configurations composed of a number of clothes’ parts
shown in Figure 3.8. In these examples, each configuration C is a graph
with vertices being 6–7 parts and E = Ebond is a set of bonds connecting the parts, as it was shown in Figure 3.7.
Third, Figures 4.2(a) and 4.2(b) illustrate a scene configuration at
the highest level of abstraction. V is a set of objects, and E included
314 Relations and Configurations
sky
occluded
sky
sky
lower
head
head
occluded
field 1
upper
head
occluded
field 2
body
body
field
(a) image
(b) layer 1 configuration
field
(c) layer 2 configuration
Fig. 4.2 An illustration of scene configuration. (a) is a scene of a man in a field.
(b) is the graph for the highest level configuration C = V, E, V is the set of 4 objects
{sky, f ield, head, body} and E = Eadj ∪ Eocclude includes two relations: “adjacency” (solid
lines) and“occlusion” (dotted arrows). (c) is the configuration at an intermediate level in
which the occlusion relation is unpacked: now the dotted arrows indicate two identical sets
of pixels but on separate layers.
two relations an “adjacency” relation in solid lines
Eadj = {(sky, f ield), (head, body)},
(4.8)
and a directed “occlusion” relation in dotted arrows,
Econtain = {head, sky, head, f ield, body, f ield}.
(4.9)
In summary, the image grammar which shall be presented in the
next section is also called a “layered grammar.” That is, it can generate
configurations as its “language” at different levels of detail.
4.3
The Reconfigurable Graphs
In vision, the configurations are inferred from images. For example, in
a Bayesian framework, the graph C = V, E will not be pre-determined
but reconfigurable on-the-fly. That is, the set of vertices may change,
so does the set of edges (relations). Therefore, the configurations must
be made flexible to meet the demand of various visual tasks. Figure 4.3
shows such an example.
On the left of the figure is a primal sketch configuration Csk for
the simple image shown in Figure 3.3. This is a planar graph with 4
“T”-junctions. In this configuration two adjacent primitives are connected by the bond relation Ebond . The four “T”-junctions are then
4.3 The Reconfigurable Graphs
A
a1
b1
t1
a1
315
a2
a2
b2
t2 b 3
b1
b2
b3
b4
b5
b6
B
b4
t3
b5
t4
b6
a4
a3
a3
(a)
a4
(b)
Fig. 4.3 (a) A primal sketch configuration for a simple image. It has four primitives for
“T”-junctions — t1 , t2 , t3 , t4 . It is a planar graph formed by bonding the adjacent primitives. (b) A layered (2.1D sketch) representation with two occluding surfaces. The four
“T”-junctions are broken. The bonds are reorganized. a1 is bonded with a3 , and a2 is
bonded with a4 .
broken in the right configuration, which is called the 2.1D sketch [53]
and denoted by C2.1sk . The bonds are reorganized with a1 being connected with a3 and a2 with a4 . C2.1sk includes two disjoint subgraphs
for the two rectangles in two layers. From this example, we can see
that both the vertices and the bonds must be treated as random variables. Figure 4.4 shows a real application of this sort of reconfiguration
in computing a 2.1D sketch from a 2D primal sketch. This example
is from [25]. It decomposes an input image in Figure 4.4(a) into three
layers in Figures 4.4(d)–(f), found after reconfiguring the bonds by completing the contours (red line segments in Figures 4.4(b) and 4.4(c))
behind and filling-in the occluded areas using the Markov random field
region descriptor in the primal sketch model. From the point of view of
parse structures, we need to add new nodes to represent the extra layers present behind the observed surfaces together with “occluded by”
relations. This is illustrated in Figure 4.2(c). This is a configuration
which has duplicated three regions to represent missing parts of the
background layer.
A mathematical model for the reconfigurable graph is called the
mixed Markov model in [20]. In a mixed Markov model, the bonds
are treated as nodes. Therefore, the vertex set V of a configuration
316 Relations and Configurations
(a) input image
(b) curve completion at layer 2
(c) curve completion at layer 3
(d) layer 1
(e) layer 2 after fill-in
(f) layer 3 after fill-in
Fig. 4.4 From a 2D sketch to a 2.1D layered representation by reconfiguring the bond
relations. (a) is an input image from which a 2D sketch is computed. This is transferred to
a 2.1D sketch representations with three layers shown in (d), (e), and (f), respectively. The
inference process reconfigures the bonds of the image primitives shown in red in (b) and
(c). From [25].
has two type of nodes — V = Vx ∪ Va . Vx include the usual nodes for
image entities, and Va is a set of address nodes, for example, the bonds.
The latter are like the pointers in the C language. These address nodes
reconfigure the graphical structure and realize non-local relations. It
was shown that a probability model defined on such reconfigurable
graphs still observes a suitable form of he Hammersley-Clifford theorem
and can be simulated by Gibbs sampler.
By analogy to language, the bonds in this example correspond to
the arrows in the English sentence discussed in Figure 2.8 for nonlocal context. As there are many possible (bond, joint, functional, and
semantic) relations, each image entity (primitives, parts, objects) may
have many random variables as the “pointers.” Many of them could be
empty, and will be instantiated in the inference process. This is similar
to the words “apple” and “make” in Figure 3.1.
5
Parse Graph for Objects and Scenes
In this chapter, we define parse graphs as image interpretations. Then
we will show in the next chapter that these parse graphs are generated
as instances by an And–Or graph. The latter is a general representation
that embeds the image grammar.
Recall that in Section 2.2 a language grammar is a 4-tuple G =
(VN , VT , R, S), and that a sentence ω is derived (or generated) by a
sequence of production rules from a starting symbol S,
S
γ1 ,γ2 ,...,γn(ω)
=⇒
ω.
(5.1)
These production rules form a parse tree for ω,
pt(ω) = (γ1 , γ2 , . . . , γn(ω) ).
(5.2)
For example, Figure 2.6 shows two possible parse trees for a sentence
“I saw the man with the telescope.”
This grammar is a generative model, and the inference is an inverse
process that computes a parse tree for a given sentence as its interpretation or one of its best interpretations. Back to image grammars, a
configuration C is a flat attributed graph corresponding to a sentence ω,
and a parse tree pt is augmented to a parse graph pg by adding horizontal links for various relations. In previous chapter, Figure 2.11(b)
317
318 Parse Graph for Objects and Scenes
has shown a parse graph for a block work scene, and Figure 1.1 has
shown a parse graph for a football match scene.
In the following, we define a parse graph as an interpretation of
image.
Definition 5.1 Parse graph. A parse graph pg consists of a hierarchic parse tree (defining “vertical” edges) and a number of relations E
(defining “horizontal edges”):
pg = (pt, E).
(5.3)
The parse tree pt is also an And-tree whose non-terminal nodes are
all And-nodes. The decomposition of each And-node A into its parts
is given by a production rule which now produces not a string but a
configuration:
γ : A → C = V, E.
(5.4)
A production should also associate the open bonds of A with open
bonds in C. The whole parse tree is a sequence of production rules
pt(ω) = (γ1 , γ2 , . . . , γn ).
(5.5)
The horizontal links E consists of a number of directed or undirected
relations among the terminal or non-terminal nodes, such as bonds,
junctions, functional and semantic relations,
E = E r 1 ∪ Er 2 ∪ · · · ∪ Er k .
(5.6)
A parse graph pg, when collapsed, produces a series of flat configurations at each level of abstraction/detail,
pg =⇒ C.
(5.7)
Depending on the type of relation, there may be special rules for producing relations at a lower level from higher level relations in the collapsing process. The finest configuration is the image itself in which
every pixel is explained by the parse graph. The next finest configuration is the primal sketch graph.
319
(a)
(b)
...
...
Fig. 5.1 Two parse graph examples for clocks which are generated from the And–Or-graph
in Figure 6.1. From [86].
The parse graph, augmented with spatial context and possible
functional relations, is a comprehensive interpretation of the observed
image I. The task of image parsing is to compute the parse graph from
input image(s). In the Bayesian framework, this is to either maximize
the posterior probability for an optimal solution,
pg∗ = arg max p(pg|I),
(5.8)
or sampling the posterior probability for a set of distinct solutions,
{pgi : i = 1, 2, . . . , K} ∼ p(pg|I).
(5.9)
Object instances in the same category may have very different configurations and thus distinct parse graphs. Figure 5.1 displays two parse
graphs for two clock instances. It has three levels and the components
are connected through three types of relations: the hinge joint to connect clock hands, a co-centric relation to align the frames, and a radial
relation to align the numbers.
As it was mentioned in Section 2.6, objects appear at arbitrary
scales in images. As shown in Figure 2.10, a face can be decomposed
into facial elements at higher resolution, and it may terminate as a
whole face for low resolution. Therefore, one remarkable property that
distinguishes an image parse graph is that a parse graph may stop
at any level of abstraction, while the the parse tree in language must
stop at the word level. This is the reason for defining visual vocabulary
at multiple levels of resolution, and defining the image grammar as a
layered grammar.
6
Knowledge Representation with And–Or Graph
This chapter addresses the central theme of the paper — developing
a consistent representation framework for the vast amount of visual
knowledge at all levels of abstraction. The proposed representation is
the And–Or graph embedding image grammars. The And–Or graph
representation was first explicitly used in [9] for representing and recognizing a complex object category of clothes.
6.1
And–Or Graph
While a parse graph is an interpretation of a specific image, an And–Or
graph embeds the whole image grammar and contains all the valid parse
graphs. Before introducing the And–Or graph, we revisit the origin of
grammar and its Chomsky formulation in Sections 2.1 and 2.2.
First, we know each production rule in the SCFG can be written as
A → β1 | β2 · · · | βn(A) ,
with A ∈ VN , β ∈ (VN ∪ VT )+ .
(6.1)
Therefore each non-terminal node A can be represented by an Ornode with n(A) alternative structures, each of which is an And-node
composed of a number of substructures. For example, the following rule
320
6.1 And–Or Graph
321
is represented by a two level And–Or tree in Figure 2.3.
A → a · b | c · c;
ρ|(1 − ρ).
(6.2)
The two alternatives branches at the Or-node are assigned probabilities
(ρ, 1 − ρ). Thus an SCFG can be understood as an And–Or tree.
Second, we have shown in Figure 2.4 that a small And–Or tree can
produce a combinatorial number of configurations — called its language. To represent contextual information in the following, we augment the And–Or tree into an And–Or graph producing a context sensitive image grammar.
In a previous survey paper [89], the first author showed that any
visual pattern can be conceptualized as a statistical ensemble that
observes a certain statistical description. For a complex object pattern, its statistical ensemble must include a large number of distinct
configurations. Thus our objective is to define an And–Or graph, thus
its image grammar, such that its language, i.e., the set of valid configurations that it produces, reproduces the ensemble of instances for the
visual pattern.
An And–Or graph augments an And–Or tree with two new features.
1. Horizontal lines are indicate to show relations, bonds, junctions, and semantic relations.
2. Relations at all levels are augmented on the And–Or
graph to represent hard (compatibility) or soft (statistical)
constraints.
3. The children of an Or-node may share Or-node children. It
represents a reusable part shared by several production rules.
The sharing of nodes reduces the complexity of the representation and thus the size of dictionary. Other possible sharings
may be useful: see, for example, Section 2.3.
In Chapter 1, Figure 1.3(a) has shown a simple example of an And–
Or graph. An And–Or graph includes three types of nodes: And-nodes
(solid circles), Or-nodes (dashed circles), and terminal nodes (squares).
The Or-nodes have labels for classification at various levels, such as
322 Knowledge Representation with And–Or Graph
scene category, object classes, and parts etc. Due to this recursive
definition, one may merge the And–Or graphs for many objects or
scene categories into a larger graph. In theory, the whole natural image
ensemble can be represented by a huge And–Or graph, as it is for
language.
By assigning values to these labels on the Or-node, one obtains an
And-graph — i.e., a parse graph. The bold arrows and shaded nodes
in Figure 1.3(a) constitute a parse graph pg embedded in the And–
Or graph. This parse graph is shown in Figure 1.3(b) and produces a
configuration shown in Figure 1.3(d). It has four terminal nodes (for
primitives, parts, or objects): 1, 6, 8, 10 and the edges are inherited from
their parent relations. Both nodes 8 and 10 have a common ancestor
node C. Therefore the relation B, C is propagated to 1, 6 and 1, 8.
For example, if B, C includes three bonds, two bonds may be inherited
by 1, 8 and one by 1, 6. Similarly the links 6, 10 and 8, 10 are
inherited from C, D.
Figure 1.3(c) is a second parse graph and it produces a configuration
in Figure 1.3(e). It has 4 terminal nodes 2, 4, 9, 9 . The node 9 is a
reusable part shared by nodes C and D. It is worth mentioning that a
shared node may appear as multiple instances.
Definition 6.1 And–Or Graph. An And–Or graph is a 6-tuple for
representing an image grammar G.
Gand−or = S, VN , VT , R, Σ, P.
(6.3)
S is a root node for a scene or object category, VN = V and ∪ V or is
a set of non-terminal nodes including an And-node set V and and an
Or-node set V or . The And-nodes plus the graph formed by their children are the productions and the Or-nodes are the vocabulary items.
VT is a set of terminal nodes for primitives, parts, and objects (note
that an object at low resolution may terminate without decomposition directly), R is a number of relations between the nodes, Σ is
the set of all valid configurations derivable from the grammar, i.e.,
its language, and P is the probability model defined on the And–Or
graph.
6.1 And–Or Graph
323
The following is more detailed explanation of the components in the
And–Or graph.
1. The Non-terminal nodes include both And-nodes and Ornodes VN = V and ∪ V or ,
V and = {u1 , . . . , um(u) },
V or = {v1 , . . . , vm(v) }.
(6.4)
An Or-node v ∈ V or is a switch pointing to a number of possible And-nodes, the productions whose head is v.
v → u1 | u2 · · · | un(v) ,
u1 , . . . , un ∈ V and .
(6.5)
We define a switch variable ω(v) for v ∈ V , that takes an
integer value to index the child node.
ω(v) ∈ {∅, 1, 2, . . . , n(v)}.
(6.6)
By choosing the switch variables in the Or-nodes, one obtains
a parse graph from the And–Or graph. The switch variable is set to empty ω(v) = ∅ if v is not part of the parse
graph. In fact the assignments of Or-nodes at various levels of the And–Or graph corresponds to scene classification
and object recognition. In practice, when an Or-node has a
large n(v), i.e., too fat, one may replace it by a small Or-tree
that has n(v) leaves. We omit the discussion of such cases for
clarity.
An And-node u ∈ V and either terminates as a template
t ∈ VT or it can be decomposed into a set of Or-nodes. In the
latter case, the relations between these child nodes are specified by some relations r1 , . . . , rk(u) ∈ R shown by the dashed
horizontal lines in Figure 1.3. We adopt the symbol :: for representing the relations associated with the production rule or
the And-node.
u → t ∈ VT ; or
u → C = (v1 , . . . , vn(v) ) :: (r1 , . . . , rk(v) ),
vi ∈ V, rj ∈ R.
The termination rule reflects the multi-scale representation.
That is, the node u may be instantiated by a template at a
relatively lower image resolution.
324 Knowledge Representation with And–Or Graph
2. The Terminal node set VT = {t1 , . . . , tm(T ) } is a set of
instances from the image dictionary ∆. Usually it is a
graphical template (Φ(x, y; α), β) with attributes α and open
bonds β. Usually, each t ∈ VT is a sketch graph, such as the
image primitives.
3. The Configurations which are produced from the root node
S are the language of the grammar: Gand−or ,
Gand−or
L(Gand−or ) = Σ = Ck : S =⇒ Ck k = 1, 2, . . . , N . (6.7)
Each configuration C ∈ Σ is a composite template, for example, the cloth shown in Figure 3.7. The And–Or graph in
Figure 1.3(a) contains a combinatorial number of valid configurations, e.g.,
Σ = {(1, 6, 8, 10), (2, 4, 9, 9), (1, 5, 11), (2, 4, 6, 7, 9), . . .}. (6.8)
The first two configurations are shown on the right side of
Figure 1.3.
4. The relation set R pools over all the relations between nodes
at all levels.
(6.9)
R = Em = {est = (vs , vt ; γst , ρst )}.
m
These relations become the pair-cliques in the composite
graphical template. When a node vs is split later, the link
est may be split as well or may descend to specific pairs of
children. For example, in Figure 1.3 node C is split into two
leaf nodes 6 and 8, then the relation (B, C) is split into two
subsets between (1, 6) and (1, 8).
5. P is a probability model defined on the And–Or graph. It
includes many local probabilities - one at each Or-node to
account for the relative frequency of each alternative, and
local energies associated with each link e ∈ R. The former
is like the SCFG and the latter is like the Markov random
fields or graphical models. We will discuss the probability
component in the next subsection.
6.1 And–Or Graph
325
clock
and-node
or-node
leaf-node
hands
3 hands
numbers
Arabic
2 hands
no
hand
hour
hand
frames
minute
hand
second
hand
...
...
outer
ring
...
inner
ring
central
ring
Roman
no
no
frame number
a1
... a12
r1
1
... 12
I
... r12
no ring
...
... XII
...
Fig. 6.1 An And–Or graph example for the object category — clock. It has two parse graphs
shown in Figure 5.1, one of which is illustrated in dark arrows. Some leaf nodes are omitted
from the graph for clarity. From [86].
Before concluding this section, we show an And–Or graph for a clock
category [86] in Figure 6.1. Figure 6.1 has shown two parse graphs as
instances of this And–Or graph. The dark bold arrows in Figure 6.1
are the parse tree shown in Figure 5.1(a).
Another And–Or example is shown in Figure 9.6. It is a subgraph
extracted, for reason of clarity, from a big And–Or graph for the upper
body of human figure [9]. Figure 9.7 displays three cloth configurations
produced by this And–Or graph.
In summary, an And–Or graph Gand−or defines a context sensitive graph grammar with VT being its vocabulary, VN the production
rules, Σ its language, R the contexts. Gand−or contains all the possible parse graphs which in turn produce a combinatorial number of
configurations. Again, the number of configurations is far larger than
326 Knowledge Representation with And–Or Graph
the vocabulary, i.e.,
|VN ∪ VT | |Σ|.
(6.10)
This is a crucial aspect for representing the large intra-category structural variations.
Our next task is to define a probability model on Gand−or to make
it a stochastic grammar.
6.2
Stochastic Models on the And–Or Graph
The probability model for the And–Or graph Gand−or must integrate the Markov tree model (SCFG) for the Or-nodes and the
graphical (Markov) models for the And-nodes. Together a probability model is defined on the parse graphs. The objective of this
probability model is to match the frequency of parse graphs in an
observed training set (supervised learning will be discussed in the next
section).
Just as the language model in Equation (2.17) defined probabilities
on each parse tree pt(ω) of each sentence ω, the new model should
define probabilities on each parse graphs pg. As pg produces a final
configuration C deterministically when it is collapsed, thus p(pg; Θ)
produces a marginal probability on the final configurations with Θ
being its parameters. A configuration C is assumed to be directly
observable, i.e., the input, and parse graph pg are hidden variables
and have to be inferred.
By definition IV, a parse graph pg is a parse tree pt augmented
with relations E,
pg = (pt, E).
(6.11)
For notational convenience, we denote the following components in pg.
• T (pg) = {t1 , . . . , tn(pg )} is the set of leaf nodes in pg. For
example, T (pg) = {1, 6, 8, 10} for the parse graph shown by
the dark arrows in Figure 1.3. In applications, T (pg) is often
the pre-terminal nodes with each t ∈ T (pg) being an image
primitive in the primal sketch.
6.2 Stochastic Models on the And–Or Graph
327
• V or (pg) is the set of non-empty Or-nodes (switches) that
are used pg. For instance, V or (pg) = {B, C, D, N, O}. These
switch variables selected the path to decide the parse tree
pt = (γ1 , γ2 , . . . , γn ).
• E(pg) is the set of links in pg.
The probability for pg is of the following Gibbs form, similar to Equation (2.17),
p(pg; Θ, R, ∆) =
1
exp{−E(pg)},
Z(Θ)
where E(pg) is the total energy,
λv (ω(v)) +
E(pg) =
v∈V
+
or (pg)
t∈T (pg)∪V
(6.12)
λt (α(t))
and (pg)
λij (vi , vj , γij , ρij ).
(6.13)
(i,j)∈E(pg)
The model is specified by a number of parameters Θ, the relations set
R, and the vocabulary ∆. The first term in the energy is the same
as the SCFG. It assigns different weights λv () to the switch variables
ω(v) at the or-nodes v. The weight should account for how frequently
a child node appears. Removing the 2nd and 3rd terms, this reduces to
an SCFG in Equation (2.9). The second and third terms are typical singleton and pair-clique energy for graphical models. The second term is
defined on the geometric and appearance attributes of the image primitives. The third term models the compatibility constraint, such as the
spatial and appearance constraint between the primitives, graphlets,
parts, and objects.
This model can be derived from a maximum entropy principle under
two types of constraints on the statistics of training image ensembles.
One is to match the frequency at each Or-node, just like the SCFG, and
the other is to match the statistics, such as histograms or co-occurrence
frequency as in standard graphical models. Θ is the set of parameters
in the energy,
Θ = {λv (), λt (), λij (); ∀v ∈ V or , ∀t ∈ VT , ∀(i, j) ∈ R}.
(6.14)
328 Knowledge Representation with And–Or Graph
Each λ() above is a potential function, not a scalar, and is represented
by a vector through discretizing the function in a non-parametric way,
as it was done in the FRAME model for texture [90]. ∆ is the vocabulary for the generative model. The partition function is summed over
all parse graph in the And–Or graph Gand−or or the grammar G.
exp{−E(pg)}.
(6.15)
Z = Z(Θ) =
pg
7
Learning and Estimation with And–Or Graph
Suppose we have a training set sampled from an underlying distribution
f governing the objects.
obs
Dobs = {(Iobs
i , pgi ) : i = 1, 2, . . . , N } ∼ f (I, pg).
(7.1)
The parse graphs pgobs
are from the groundtruth database [87] or coni
sidered missing in unsupervised case. The objective is to learn a model
p which approaches f by minimizing a Kullback–Leibler divergence,
p∗ = arg min KL(f ||p)
f (I, pg) log
= arg min
pg∈Ωpg ΩI
f (I, pg)
dI.
p(I, pg; Θ, R, ∆)
(7.2)
This is equivalent to the ML estimate for the optimal vocabulary ∆,
relation R, and parameter Θ, as it was formulated in [59]
(∆, R, Θ)∗ = arg max
N
obs
log p(Iobs
i , pgi ; Θ, R, ∆) − (VT , VN , N ),
i=1
(7.3)
where (VT , VN , N ) is a term that shall balance the model complexity
w.r.t. sample size N but also account for the semantic significance of
329
330 Learning and Estimation with And–Or Graph
each elements for the vision purpose (human guided here). The latter is
often reflected by utility or cost functions in Bayesian decision theory.
Learning the probability model includes three phases and all three
phases follow the same principle above [59].
1. Estimating the parameters Θ from training date Dobs for
given R and ∆,
2. Learning and pursuing the relation set R for nodes in G
given ∆,
3. Discovering and binding the vocabulary ∆ and hierarchic
And–Or tree automatically.
In the following we briefly discuss the first two phases. There is no
significant work done for the third phase yet.
7.1
Maximum Likelihood Learning of Θ
For a given And–Or graph hierarchy and relations, the estimation of Θ follows the MLE learning process. Let L(Θ) =
N
∂L(Θ)
obs
obs
i=1 log p(Ii , pgi ; Θ, R, ∆) be the log-likelihood, by setting ∂Θ =
0, we have the following three learning steps.
1. Learning the λv at each Or-node v ∈ V or accounts for the
frequency of each alternative choice. The switch variable at
v has n(v) choices ω(v) ∈ {∅, 1, 2, . . . , n(v)} and it is ∅ when
v is not included in the pg. We compute the histogram,
#(ω(v) = i)
,
hobs
v (ω(v) = i) = n(v)
j=1 #(ω(v) = j)
i = 1, 2, . . . , n(v).
(7.4)
#(ω(v) = i) is the number of times that node v appears with
ω(v) = i in all the parse graphs in Ωobs
pg . Thus,
λv (ω(v) = i) = − log hobs
v (ω(v) = i),
∀v ∈ V or .
(7.5)
2. Learning the potential function λt () at the terminal node
t ∈ VT . ∂(Θ)
∂λt = 0 leads to the statistical constraints,
Ep(pg;Θ,R,∆) [h(α(t)] = hobs
t ,
∀t ∈ VT .
(7.6)
7.2 Learning and Pursuing the Relation Set
331
In the above equation, α(t) are the attributes of t and h(α(t))
is a statistical measure of the attributes, such as the hisis the observed histogram pooled over all the
togram. hobs
t
occurrences of t in Ωobs
pg .
1 .
(7.7)
hobs
1 z − < α(t) ≤ z +
t (z) =
#t t
2
2
#t is the total number of times, a terminal node t appears
in the data Ωobs
pg . z indexes the bins in the histogram and is the length of a bin.
3. Learning the potential function λij () for each pair relation
(i, j) ∈ R. ∂(Θ)
∂λij = 0 leads to the following implicit function,
Ep(pg;Θ,R,∆) [h(vi , vj )] = hobs
ij ,
∀(i, j) ∈ R.
(7.8)
Again, h(vi , vj ) is a statistic on vi , vj , for example, a histogram on the relative size, position, and orientation, appearance etc. hobs
ij is the histogram summed over all the occurrence of (vi , vj ) in Dobs .
The equations (7.5), (7.6), and (7.8) are the constraints for deriving
the Gibbs model p(pg; Θ, R, ∆) in Equation (6.12) through the maximum entropy principle.
Due to the coupling of the energy terms, both Equations (7.6) and
(7.8) are solved iteratively through a gradient method. In a general
case, we follow the stochastic gradient method adopted in learning the
FRAME model [90], which approximates the expectations Ep [h(α(t))]
in Equation (7.6) and Ep [h(vi , vj )] in (7.8) by sample means from a set
of synthesized examples. This is the method of analysis-by-synthesis
adopted in our texture modeling paper [90]. At the end of this chapter, we show the sampling and synthesis experiments on two object
categories — clock and bike in Figures 7.1 and 7.2.
7.2
Learning and Pursuing the Relation Set
Besides the learning of parameters Θ, we can also augment the relation sets R in an And–Or Graph, and thus pursue the energy terms
332 Learning and Estimation with And–Or Graph
(a)
(b)
(c)
(d)
(e)
Fig. 7.1 Learning the And–Or graph parameters for the clock category. (a) Sampled clock
examples (synthesis) based on SCFG (Markov tree) that accounts for the frequency of
occurrence. (b–e) Synthesis examples at four incremental stages of the minimax entropy
pursuit process. (b) Matching the relation positions between parts, (c) further matching the
relative scales, (d) further pursuing the hinge relation, (e) further matching the containing
relation. From [59].
in
(i,j)∈E(pg)λij (vi , vj ) in the same way as pursuing the filters and
statistics in texture modeling by the minimax entropy principle [90].
Suppose we start with an empty relation set R = ∅ and thus p =
p(pg; λ, ∅, ∆) is an SCFG model. The learning procedure is a greedy
pursuit. In each step, we add a relation e+ to R and thus augment
model p(pg; Θ, R, ∆) to p+ (pg; Θ, R+ , ∆), where R+ = R ∪ {e+ }.
e+ is selected from a large pool ∆R so as to maximally reduce KLdivergence,
e+ = arg max KL(f ||p) − KL(f ||p+ ) = arg max KL(p+ ||p),
(7.9)
7.2 Learning and Pursuing the Relation Set
333
Fig. 7.2 Random sampling and synthesis of the bike category. From [59].
Thus we denote the information gain of e+ by
def
δ(e+ ) = KL(p+ ||p) ≈ f obs (e+ )dmanh (hobs (e+ ), hsyn
p (e+ )).
(7.10)
In the above formula, f obs (e+ ) is the frequency that relation e+ is
observed in the training data, hobs (e+ ) is the histogram for relation e+
over training data Dobs , and hsyn
p (e+ ) is the histogram for relation e+
over the synthesized parse graphs according to the current model p.
dmanh () is the Manhanonabis distance between the two histograms.
Intuitively, δ(e+ ) is large if e+ occurs frequently and tells a
large difference between the histograms of the observed and the synthesized parse graphs. Large information gain means a significant
relation e+ .
334 Learning and Estimation with And–Or Graph
Algorithm 7.1. Learning Θ by Stochastic Gradients
Input: Dobs = {pgobs
i ; i = 1, 2, . . . , M }.
obs obs from D obs for all fea1. Compute histograms hobs
v , ht , hij
ture/relations.
2. Learn the parameters λv at the Or-nodes by Equation (7.5).
3. Repeat (outer loop)
4. Sample a set of parse graphs from the current model
p(pg; Θ, R, ∆) Dsyn = {pgsyn
i ; i = 1, 2, ..., M }
syn
from Dsyn for all feature/
5. Compute histograms hsyn
t , hij
relations
6. Select a feature/relation that maximizes the difference between
obs. vs syn. histograms.
7. Set λ = 0 for the newly selected feature/relation.
8. Repeat (inner loop)
9. Update the parameters with stepsize η
− hobs
δλt = ηt (hsyn
t ),
t
δλij =
ηij (hsyn
ij
−
hobs
ij ),
∀t ∈ VT ,
∀(i, j) ∈ R.
Sample a set of parse graphs and update the histograms.
syn
obs
− hobs
10. Until |hsyn
t | ≤ and |hij − hij | ≤ for the selected feat
ture/relations.
syn
obs
− hobs
11. Until |hsyn
t | ≤ and |hij − hij | ≤ for all features and
t
relations.
Equations (7.6) and (7.8) are then satisfied to certain precision.
7.3
Summary of the Learning Algorithm
In summary, the learning algorithm starts with an SCFG (Markov tree)
and a number of observed parse graphs for training Dobs . It first learns
the SCFG model by counting the occurrence frequency at the Or-nodes.
Then by sampling this SCFG, it synthesizes a set of instances Dsyn . The
sampled instances in Dsyn will have the proper components but often
have wrong spatial relations among the parts as there are no relations
7.4 Experiments on Learning and Sampling
335
specified in SCFG. Then the algorithm chooses a relation that has the
most different statistics (histogram) over some measurement between
the sets Dobs and Dsyn . The model is then learned to reproduce the
observed statistics over the chosen relation. A new set of synthesized
instances is sampled. This iterative process continues until no more
significant differences are observed between the observed and synthesized sets.
Remark 1. At the initial step, the synthesized parse graphs will
match the frequency counts on all Or-nodes first, but the synthesized
parse graphs and their configurations will not look realistic. Parts of
the objects will be in wrong positions and have wrong relations. The
iterative steps will make improvements. Ideally, if the features and statistical constraints selected in Equations (7.6) and (7.8) are sufficient,
then the synthesized configurations
syn
: pgsyn
−→ Ciobs , i = 1, 2, . . . , M }.
Ωsyn
i
C = {Ci
(7.11)
should resemble the observed configurations. This is what people did
in texture synthesis.
Remark 2. Note that in the above learning process, a parse graph
obs
pgi contributes to some parameters only when the corresponding
nodes and relations are present in pgobs
i .
7.4
Experiments on Learning and Sampling
In [89], the first author showed a range of image synthesis experiments
by sampling the image model (ensembles) for various visual patterns,
such as textures, texton processes, shape contours, face etc. to verify the
learned model in the spirit of analysis-by-synthesis. In this subsection,
we show synthesis results in sampling the probabilistic ensemble (or
the language) defined by the grammar, i.e., sampling the typical configurations from the probabilistic model defined on the And–Or graph.
Gand−or
(7.12)
C ∼ L(Gand−or ) = (Ck , p(Ck )) : S =⇒ Ck .
This is equivalent to first sampling the parse graphs,
pg; ∼ p(pg; Θ, ∆),
(7.13)
336 Learning and Estimation with And–Or Graph
and then producing the configurations,
pg → C.
(7.14)
Figure 7.1 illustrates the synthesis process for a clock category
whose And–Or graph is shown previously in Figure 6.1. The experiment
is from (Porway, Yao and Zhu) [59]. Each row in Figure 7.1 shows five
typical examples from the synthesis set Ωsyn
pg in different iterations. In
the first row, the clocks are sampled from the SCFG (Markov tree) in a
window. These examples have valid parts for clocks shown in different
colors, but there are no spatial relations or features to constrain the
attributes of the component or layouts. Thus the instances look quite
wrong. In the second row, the relative positions of the components (in
terms of their centers) are considered. After matching the statistics of
the synthesized and observed sets, the sampled instances look more
reasonable. In the third, fourth, and fifth rows, the statistics on the
relative scale, the hinge relation between clock hands, and a containing
relation are added one by one. The synthesized instances become more
realistic configurations.
Figure 7.2 shows the same random sampling and synthesis experiment on another object category — bike. With more spatial relations
included and statistics matched, the sampled bikes from the learning
models become more realistic from (a) to (d).
The synthesis process produces novel configurations not seen in the
observed set and also demonstrates that the spatial relations captured
by the And–Or graph will provide information for top-down prediction
of object components. Figure 9.9 shall show an example of top-down
prediction and hallucination of occluded parts using the learned bike
model above.
In a recent experiment on a recognition task with 33 object categories [44], Lin et al. used the synthesized samples to augment the
training set and showed that the generalized examples can improve the
recognition performance by 15% in comparison to the expertiments
without synthesized examples.
8
Recursive Top-Down/Bottom-Up Algorithm
for Image Parsing
This chapter briefly reviews an inference algorithm with three case
studies of image parsing using grammars by the author and collaborators. The first case is a generic grammar for man-made world scenes.
The compositional objects include buildings (indoor or outdoor) and
furniture [32]. The second is a more restrictive grammar for human
clothes and upper body [9]. The third case [86] applies the grammar
for recognizing five object categories — clock, bike, computer (screen
and keyboard), cup/bowl, teapot. In both cases, the inference is performed under the Bayesian framework. Given an input image I as the
terminal configuration, we compute a parse graph pg that maximizes
a posterior probability
pg∗ = arg max p(I|pg; ∆sk )p(pg; Θ, ∆).
pg
(8.1)
The likelihood model is based on the primal sketch in Section 3.2, and
the prior is defined by the grammar model in Equation (6.12).
In the following, we briefly review the computing procedures, and
refer to the original papers [32] and [9] for more details.
The And–Or graph is defined recursively, as is the inference
algorithm. This recursive property largely simplifies the algorithm
337
338 Recursive Top-Down/Bottom-Up Algorithm for Image Parsing
design and makes it easily scalable to arbitrarily large number of object
categories.
Consider an arbitrary And-node A in an And–Or graph. A may
correspond to an object or a part. Without loss of generality, we assume
it can be either terminated into one of n leaves at low resolution or
decomposed into n(A) = 3 parts,
A → A1 · A2 · A3 | t1 | · · · | tn .
(8.2)
This recursive unit is shown in Figure 8.1.
In this figure, each such unit is associated with data structures which
are widely used in heuristic searches in artificial intelligence [58].
• An Open List stores a number of weighted particles (or
hypotheses) which are computed in bottom-up process for
the instances of A in the input image.
• A Closed List stores a number of instances for A which are
accepted in the top-down process. These instances are nodes
in the current parse graph pg.
Thus the inference algorithm consists of two basic processes that
compute and maintain the Open and Closed lists for each unit A.
The bottom-up process creates the particles in the Open lists in two
methods.
(i) Generating hypotheses for A directly from images. Such
bottom-up processes include detection algorithms such as
t1 t2
tn
A
A
t1
t2
A
tn
A1.A2 . A3
open list (weighted particles for hypotheses)
closed list (accepted instances)
A1
A2
A3
Fig. 8.1 Data structure for the recursive inference algorithm on the And–Or graph. See text
for interpretation.
339
Adaboosting [21, 78], Hough transform etc. for detecting the
various terminals t1 , . . . , tn without identifying the parts. The
detection process tests some image features. These particles
are shown in Figure 8.1 by single circles with bottom-up
arrows. The weight of a detected hypothesis (indexed by i)
is the logarithm of some local marginal posterior probability
ratio given a small image patch Λi ,
i
ωA
= log
p(Ai |F (Iλi ))
p(Ai |Iλi )
i
≈
log
= ω̂A
.
p(Āi |Iλi )
p(Āi |F (Iλi ))
Ā means competitive hypothesis. For computational effectiveness, the posterior probability ratio is approximated by
posterior probabilities using local features F (Iλi ) rather than
the image Iλi . For example, in face detection by Adaboosting
[78], the strong classifier can be reformulated as a posterior
probability ratio of face vs. non-face [21, 63].
(ii) Generating hypotheses for A by binding a number of k
(1 ≤ K ≤ n(A)) parts from the existing Open and Closed
lists of its children A1 , A2 , . . . , An(A) . The binding process will
test the relationships between these child nodes for compatibility and quickly rule out the obviously incompatible compositions. In Figure 8.1, these hypotheses are illustrated by a
big ellipse containing n(A) = 3 small circles for its children.
The upward arrows show existing parts in the Open or Closed
lists of the child nodes, and the downward arrows show the
missing parts that need to be validated in the top-down process. The weight of a bound hypothesis (indexed by i) is
the logarithm of some local conditional posterior probability
ratio. Suppose a particle Ai is bound from two existing parts
Ai1 and Ai2 with Ai3 missing, and Λi is the domain containing
the hypothesized A. Then the weight will be
i
= log
ωA
≈ log
p(Ai |Ai1 , Ai2 , IΛi )
p(Ai1 , Ai2 , IΛi |Ai )p(Ai )
=
log
p(Āi |Ai1 , Ai2 , IΛi )
p(Ai1 , Ai2 , IΛi |Āi )p(Āi )
p(Ai1 , Ai2 |Ai )p(Ai )
i
,
= ω̂A
p(Ai1 , Ai2 |Āi )p(Āi )
340 Recursive Top-Down/Bottom-Up Algorithm for Image Parsing
where Ā means competitive hypothesis. p(Ai1 , Ai2 |Ai ) is
reduced to tests of compatibility between Ai1 and Ai2 for computational efficiency. It leaves the computation of searching
for Ai3 as well as fitting the image area IΛA to the top-down
process.
The top-down process validates the bottom-up hypotheses in all the
Open lists, following the Bayesian posterior probability. It also needs
to maintain the weights of the Open lists.
i , the top-down process
(i) Given a hypothesis Ai with weight ω̂A
validates it by computing the true posterior probability ratio
i stated above. If Ai is accepted into the Closed list of A.
ωA
This corresponds to a move from the current parse graph
pg to a new parse graph pg+ . The latter includes a new
node Ai – either as a leaf node or as a non-terminal node
with children Ai1 , . . . , Ain(A) . The criterion of the acceptance
is discussed below. In a reverse process, the top-down process
may also select a node A in the Closed list, and then either
deletes it (putting it back to the Open list) or disassembles
it into independent parts.
(ii) Maintaining the weights of the particles in the OPEN Lists
after adding (or removing) a node Ai from the parse graph.
It is clear that the weight of each particle depends on the
competing hypothesis. Thus for two competing hypotheses A
and A which overlap in a domain Λo , accepting one hypothesis will lower the weight of the other. Therefore, whenever
we add or delete a node A in the parse graph, all the other
hypotheses whose domains overlap with that of A will have
to update their weights.
The acceptance of a node can be done by a greedy algorithm that
maximizes the posterior probability. Each time it selects the particle
whose weight is the largest among all Open lists and then accepts it
until the largest weight is below a threshold.
Otherwise, one may use a stochastic algorithm with reversible
jumps. According to the terminology of data driven Markov chain
341
Monte Carlo (DDMCMC) [73, 74], one may view the approximative
i as a logarithm of the proposal probability ratio. The accepweight ω̂A
tance probability, in the Metropolis–Hastings method [46], is thus
q(pg+ → pg) p(pg+ |I)
·
a(pg → pg+ ) = min 1,
q(pg → pg+ ) p(pg|I)
q+ (Ai )
i
i
exp{ωA − ω̂A } ,
= min 1,
q(Ai )
where q+ (Ai ) (or q(Ai )) is the proposal probability for selecting Ai to
be disassembled from pg+ (to be added to pg).
For the stochastic algorithm, its initial stage is often deterministic
when the particle weights are very large and the acceptance probability
is always 1.
We summarize the inference algorithm in the following:
Algorithm 8.1. Image Parsing by Top-down/Bottom-up
Inference
Input: an image I and an And–Or graph.
Output: a parse graph pg with initial pg = ∅.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Repeat
Schedule the next visit note A
Call the Bottom — Up(A) process to update A’s Open lists
(i) Detecting terminal instances for A from images
(ii) Binding non-terminal instances for A from its children’s
Open or Closed lists.
Call the Top — Down(A) process to update A’s Closed and Open
lists
(i) Accept hypotheses from A’s Open list to its Closed list.
(ii) Remove (or disassemble) hypotheses from A’s closed lists.
(iii) Update the Open lists for particles that overlap with current
node.
Until a certain number of iteration or the largest particle weight
is below a threshold.
342 Recursive Top-Down/Bottom-Up Algorithm for Image Parsing
The key issue of the inference algorithm is to order the particles in
the Open and Closed lists. In other words, the algorithm must schedule
the bottom-up and top-down processes to achieve computational efficiency. For some visual patterns, like human faces in Figure 2.10, it
is perhaps more effective to detect the whole face and then locate the
facial components. For other visual patterns, like the cheetah image in
Figure 2.9, it is more effective to work in a bottom-up fashion. More
objects, like the two examples in the following two subsections, need to
alternate between the bottom-up and top-down processes.
The optimal schedule between bottom-up and top-down is a long
standing problem in vision. A greedy way for scheduling is to measure
the information gain of each step, either a bottom-up testing/binding or
a top-down validation, divided by its computational complexity (CPU
cycles). Then one may order these steps by the gain/cost ratio. A special
case is studied in [7] for coarse-to-fine testing. Many popular algorithms
in AI heuristic search [58] or the matching pursuit [47] can be considered
deterministic versions of the above algorithm. In DDMCMC [73, 92],
the algorithm always performs all the necessary bottom-up tests before
running the top-down process. As does the feedforward neural networks
[61]. This may not be the optimal schedule.
9
Three Case Studies of Image Grammar
9.1
Case Study I: Parsing the Perspective Man-Made
World by Han and Zhu
In this case, the grammar has one class of primitives as the terminal
nodes (i.e., VT ), which are 3D planar rectangles projected on images.
Obviously rectangles are the most common elements in man-made
scenes, such as buildings, hallways, kitchens, living rooms, etc. Each
rectangle a ∈ VT is made of two pairs of parallel line segments in 3D
space, which may intersect at two vanishing points through projection.
The grammar has only two types of non-terminal nodes (i.e., VN ) —
the root node S for the scene and a node A for any composite objects.
The grammar has six production rules as shown in Figure 9.1. The
scene node S generates m independent objects (rule r1 ). An object
node A can be instantiated (assigned) to a rectangle (rule r5 ), or be
used recursively by the other four production rules: r2 — the line production rule that aligns a number of rectangles in one row, r3 — the
mesh production rule that arranges a number of rectangles in a matrix,
r4 — the nesting production rule that has one rectangle containing the
other, and r6 — the cube production rule that aligns three rectangle
343
344 Three Case Studies of Image Grammar
r1
r3
scene
S
::=
S
r5
mesh
::=
A
instance
A
A
::=
A2m
A2
A1
Am
A11
mxn
r4
line
A
::=
a
rectangle
A1m
m
r2
A
A
r6
nesting
A
::=
cube
A
A
::=
A
A1
A1
A2
Am
m
A1
A2
A3
A2
A3
A1
A2
A3
line production rule
A1 A2
nesting production rule
A1
A2
cube production rule
Fig. 9.1 Six attribute grammar rules for generic man-made world scenes. This grammar features a single class of primitives — rectangle and four generic organizations — line, mesh, cube, and nesting. Attributes will be passed between a node
to its children and the horizontal lines show constraints on attributes. See text for
explanation.
into a solid shape. The unknown numbers m and n can be represented
by the Or-nodes for different combinations.
Each production rule is associated with a number of equations that
constrain the attributes of a parent node and those of its children.
These rules can be used recursively to generate a large set of complex
configurations. Figure 9.2 shows two typical parsing configurations —
(b) a floor pattern and (d) a toolbox pattern, and their corresponding
parse graphs in (a) and (c), respectively.
The parsing algorithm adopts a greedy method following the general description of Algorithm 8.1. For each of the 5 rules r2 , . . . , r6 , it
maintains an Open list and a Closed list. In an initial phase, it detects
an excessive number of rectangles in by a bottom-up rectangle detection process and thus fill the Open list for rule r5 . Each particle consists
of two pairs of parallel line segments.
345
9.1 Case Study I: Parsing the Perspective Man-Made World by Han and Zhu
r6
r4
r2
r4
d
r4
a
b
c
r4
e
r4
b
d
e
r2
c
f
g
a
b
c
d
c
e
e
(b)
(a)
f
g b
(c)
a
d a
(d)
Fig. 9.2 Two examples of rectangle object configurations (b) and (d) and their corresponding parse graphs (a) and (c). The production rules are shown as non-terminal nodes.
The top-down and bottom-up computation has been illustrated in
Figure 1.2 for a kitchen scene. Figure 1.2 shows a parse graph under
construction at a time step, the four rectangles (in red) are the accepted
rectangles in the Closed list for r5 . They activated a number of candidates for larger groups using the production rules r3 , r4 , r6 , respectively,
and three of these candidates are then accepted as non-terminal nodes
A, B, and C, respectively. The solid upward arrows show the bottom-up
binding, while the downward arrows show the top-down prediction.
Figure 9.3 shows the five Open lists for the candidate sets of the
five rules. At each step the parsing algorithm will choose the candidate
with the largest weight from the five particle sets and add a new nonterminal node to the parse graph. If the particle is in the r5 Open list,
it means accepting a new rectangle. Otherwise the algorithm creates a
non-terminal node and inserts the missing children in this particle into
their respective Open lists for future tests.
r2
r4
r3
r6
r5
Fig. 9.3 Illustration for the open lists of the five rules.
346 Three Case Studies of Image Grammar
Fig. 9.4 Some experimental results. The first row shows the input images. The second row
shows the computed rectangle configurations. From [32].
Figure 9.4 shows three examples of the inference algorithm. The
computed configuration C for each image consists of a number of rectangles arranged in generic structures. More discussions and experiments
are referred to [32].
Figure 9.5 shows two ROC curves for performance comparison
in detecting the rectangles in 25 images against human annotated
groundtruth. One curve shows the detection rate (vertical axis) over
the number of false alarms per image (horizontal axis) for pure bottomup method. The other curve is for the methods integrating bottomup and top-down. From these ROC curves, we can clearly see the
dramatic improvement by using top-down mechanism over the traditionally bottom-up mechanism only. Intuitively, some rectangles are
nearly impossible to detect using the bottom-up methods and can
only be recovered through the context information using the grammar
rules.
9.2
Case Study II: Human Cloth Modeling and Inference
by Chen, Xu, and Zhu
The second example, taken from [9], represents and computes clothes by
And–Or graph. Unlike the rigid rectangle objects in the first example,
9.2 Case Study II: Human Cloth Modeling and Inference by Chen, Xu, and Zhu
347
Fig. 9.5 ROC curves for the rectangle detection results by using bottom-up only and, using
both bottom-up and top-down. From [32].
human clothes are very flexible objects with large intra-category structural variations.
The authors in [9] took 50 training images of college students sitting
in a high chair with good light conditions and uniform background to
reduce occlusion and control illumination. An artist was asked to draw
sketches as consistent as possible on these images. From the sketches,
they manually separate a layer of sketches corresponding to shading
folds and textures (e.g., shoe lace, text printed on T-shirt), and then
decompose the remaining structures into a number of parts: hair, face,
collar, shoulder, upper and lower arms, cuff, hands, pants, shoes, and
pockets. Some of the examples are shown in Figure 3.8. The largest two
categories are hands and shoes. The hands have many possible configurations — separate or held/crossed. The 50 pairs of hands collected
are not necessarily exhaustive. However, an interesting observation in
the experiment is that human vision is not very sensitive to the precise
348 Three Case Studies of Image Grammar
Arms
B
B
D
C
...
...
C
...
F
E
...
...
E
...
...
Fig. 9.6 The And–Or graph for arms as a part of the overall And–Or graph.
hand gesture/poses. If a test image has a hand configuration outside of
our training category, the algorithm will find a closest match and simply paste the part at the hand position without noticeable difference.
Therefore complex parts, such as hands and shoes, can be treated less
precisely.
With these categories, an And–Or graph is constructed manually
to account for the variability of configurations. A portion of the And–
Or graph for arms and hands is shown in Figure 9.6. Intuitively, this
And–Or graph is like a “mother template” and it can produce a large
set of configurations including configurations not seen in the training
set. Figure 9.7 displays three configurations produced by this And–Or
graph.
This And–Or graph is then used for drawing clothes from images
using a version of algorithm II. The algorithm makes use of the bottomup process for detecting parts that are most discriminable, such as face,
skin color, shoulder. Then it activates top-down searches for predicted
parts based on the context information encoded in the And–Or graph.
Figure 9.8 shows three results of the computed configurations. These
graphical sketches are quite nice for they are generated by rearranging
the artist’s parts. Such results have potential applications in digital
arts and cartoon animations.
9.3 Case Study III: Recognition on Object Categories by Xu, Lin, and Zhu
g1
g1
g2
g3
g2
349
g1
g3
g2
g3
g7
g4
g4
g6
g5
g5
g7
g6
g7
g4
g5
g6
Fig. 9.7 Three novel configurations composed of 6,5,7 sub-templates in the categories,
respectively. The bonds are shown by the red dots.
Fig. 9.8 Experiment on inferring upper body with clothes from images. From [9].
9.3
Case Study III: Recognition on Object Categories
by Xu, Lin, and Zhu
The third example, taken from [86], applies the top-down/bottom-up
inference to five object categories — clock, bike, computer (screen and
350 Three Case Studies of Image Grammar
keyboard), cup/bowl, and teapot. The five categories are selected from
a large scale ground truth database from the Lotus Hill Institute. The
database includes more than 500,000 objects over 200 categories parsed
in And–Or graphs [87]. The probabilistic models are learned for these
And–Or graphs using the MLE learning presented in the previous section. The clock and bike sampling results were shown in Figures 7.1
and 7.2.
The And–Or graphs together with their probabilistic models represent the prior knowledge above the five categories for top-down inference. Figure 9.9 shows an example of inferring a partially occluded
bicycle from clutter.
In Figure 9.9, the first row shows the input image, an edge map,
and bottom-up detection of the two wheels using Hough transform. The
input image
edge map
bottom up detection
top-down predict 1
top-down predict 2
top-down predict 3
match 1
imagine 1
match 3
match 2
imagine 2
imagine 3
Fig. 9.9 The top-down influence in inferring a partially occluded bike from clutter.
From [86].
9.3 Case Study III: Recognition on Object Categories by Xu, Lin, and Zhu
351
Hough transform method is adopted to detect parts like circles, ellipses,
and triangles. The second row shows some top-down predictions of bike
frame based on the two wheels. The transform parameters of the bike
frame are sampled from the learned MRF model. As we cannot tell the
front wheel from the rear at this moment, the frames are sampled for
both directions. We only show three samples for clarification. The third
row shows the template matching process that matching the predicted
frames (in red) to the edges (in blue) in the image. The one with minimum matching cost is selected. The fourth row shows the top-down
hallucinations (imaginations) for the seat and handlebar (in green). As
these two parts are occluded. The three sets of hallucinated parts are
randomly sampled from the And–Or graph model, in the same way as
random sampling of the whole bike.
Finally, we show a few recognition examples in Figure 10.1 for the
five categories. For each input image, the image on its right-side shows
the recognized parts from the image in different colors. It should be
mentioned that the recognition algorithm is distinct from most of the
classification algorithms in the literature. It interprets the image by a
parse graph which includes the classification of categories and parts on
the Or-nodes, and matches the leaf templates to images, and hallucinates occluded parts.
10
Summary and Discussion
This exploratory paper is concerned with representing large scale visual
knowledge in a consistent modeling, learning, and computing framework. Specifically two huge problems must be solved before a robust
vision system is feasible: (i) large number (hundreds) of object and
scene categories; and (ii) large intra-category structural variation. The
framework proposed to tame these two problems is a stochastic graph
grammar embedded in an And–Or graph, which can be learned from a
large annotated dataset.
First, to represent intra-category variation, the grammar can create a large number of configurations from a relatively much smaller
vocabulary. The And–Or graph acts like a reconfigurable mother template, and assembles novel configurations on-the-fly to interpret novel
instances unseen before.
Second, to scale up to hundreds of categories, the And–Or graph is
recursively designed. Thus one can integrate, without much overhead,
all categories into one big And–Or graph. The learning and inference
algorithms are designed recursively as well. This permits large scale
parallel computing.
352
353
Fig. 10.1 Recognition experiments on five object categories. From [86].
There are two open issues for further study.
(i) Learning and discovering the And–Or graph. As it was proposed in a series of recent works [17, 52, 59, 81, 86], the
objective is to map the visual vocabulary including dictionaries at all levels of abstraction and all visual aspects. This
task can be formulated in theory under a common learning principle, that is to put the dictionary ∆ into the maximum likelihood learning process. The various information
criteria, such as the binding strength, mutual information,
354 Summary and Discussion
minimax entropy, will come naturally out of this learning
process.
However, the ultimate visual vocabulary is unlikely to be
learned fully automatically from statistical principles, as the
determination of the vocabulary must take the purposes of
vision into account. This argues for a semi-automatic method
which is being carried out at the Lotus Hill Institute. Human
users, guided by real life experience, psychology and vision
tasks, define most of the structures, and leaving the estimation of parameters and adaptation to computers. The computers, at a more sophisticated stage, should be able to find
and pursue the addition of novel elements in their dictionaries. So far, And–Or graphs have been constructed for over
200 object and scene categories, including aerial images, at
the Lotus Hill Institute [87].
(ii) Scheduling and ordering of top-down and bottom-up processes. When we have a big And–Or graph with thousands of
nodes organized hierarchically, we can imagine that the computing process is like a many-story factory with thousands
of assembly lines. Intuitively, each assembly line corresponds
to the Open and Closed lists of a node in the And–Or graph.
With all these assembly lines sharing only one CPU (or even
multiple CPUs), it is crucial to optimize the schedule to maximize the total throughput of the factory. Traditionally, vision
algorithms always start with bottom-up processes to feed
the assembly lines with raw materials (proposing weighted
hypothesis), for example, the DDMCMC [73, 92], and feedforward neural networks [61]. Due to the multi-resolution
property, each node in the And–Or graph can be terminated
immediately and thus the raw material can be sent to the
assembly lines at all stories of the factory directly, instead
of going up story-by-story. This strategy is supported by
human vision experiments [18, 70] that show humans can
detect scene and object categories as fast as we detect the
low level textons and primitives.
355
There has been a long standing debate over the roles of top-down
and bottom-up processes [76]. We believe that this debate can only be
answered numerically not verbally. That is to say, we need to compute,
numerically, the information gain of each operator, either top-down or
bottom-up, over the ensemble of real-world images.
Acknowledgments
The authors thank Drs. Stuart Geman, Yingnian Wu, Harry Shum,
Alan Yuille, and Joachim Buhmann for their extensive discussions and
helpful comments. The first author also thanks many students at UCLA
(Hong Chen, Jake Porway, Kent Shi, Zijian Xu) and the Lotus Hill
Institute (Liang Lin, Zhenyu Yao, Tianfu Wu, Xiong Yang, et al.) for
their assistance. The work is supported by a NSF grant IIS-0413214 and
an ONR grant N00014-05-01-0543. The work at the Lotus Hill Institute
is supported by a Chinese National 863 grant 2006AA01Z121.
356
References
[1] S. P. Abney, “Stochastic attribute-value grammars,” Computational Linguistics, vol. 23, no. 4, pp. 597–618, 1997.
[2] K. Athreya and A.Vidyashankar, Branching Processes. Springer-Verlag, 1972.
[3] A. Barbu and S. C. Zhu, “Generalizing Swendsen-Wang to sampling arbitrary
posterior probabilities,” IEEE Transactions on PAMI, vol. 27, no. 8, pp. 1239–
1253, 2005.
[4] K. Barnard et al., “Evaluation of localized semantics: Data methodology, and
experiments,” Tech. Report, CS, U. Arizona, 2005.
[5] I. Biederman, “Recognition-by-components: A theory of human image understanding,” Psychological Review, vol. 94, pp. 115–147, 1987.
[6] E. Bienenstock, S. Geman, and D. Potter, “Compositionality, MDL priors, and
object Recognition,” in Advances in Neural Information Processing Systems 9,
(M. Mozer, M. Jordan, and T. Petsche, eds.), MIT Press, 1998.
[7] G. Blanchard and D. Geman, “Sequential testing designs for pattern recognition,” Annals of Statistics, vol. 33, pp. 1155–1202, June 2005.
[8] H. Blum, “Biological shape and visual science,” Journal of Theoretical Biology,
vol. 38, pp. 207–285, 1973.
[9] H. Chen, Z. J. Xu, Z. Q. Liu, and S. C. Zhu, “Composite templates for cloth
modeling and sketching,” in Proceedings of IEEE Conference on Pattern Recognition and Computer Vision, New York, June 2006.
[10] Z. Y. Chi and S. Geman, “Estimation of probabilistic context free grammar,”
Computational Linguistics, vol. 24, no. 2, pp. 299–305, 1998.
[11] N. Chomsky, Syntactic Structures. Mouton: The Hague, 1957.
357
358 References
[12] T. F. Cootes, C. J. Taylor, D. Cooper, and J. Graham, “Active appearance
models–their training and applications,” Computer Vision and Image Understanding, vol. 61, no. 1, pp. 38–59, 1995.
[13] M. Crouse, R. Nowak, and R. Baraniuk, “Wavelet based statistical signal processing using hidden Markov models,” IEEE Transactions on Signal Processing,
vol. 46, pp. 886–902, 1998.
[14] S. J. Dickinson, A. P. Pentland, and A. Rosenfeld, “From volumes to views:
An approach to 3D object recognition,” CVGIP: Image Understanding, vol. 55,
no. 2, pp. 130–154, 1992.
[15] D. L. Donoho, M. Vetterli, R. A. DeVore, and I. Daubechie, “Data compression
and harmonic analysis,” IEEE Transactions on Information Theory, vol. 6,
pp. 2435–2476, 1998.
[16] L. Fei-Fei, R. Fergus, and P. Perona, “Learning generative visual models from
few training examples: An incremental Bayesian approach tested on 100 object
categories,” Workshop on Generative Model Based Vision, 2004.
[17] L. Fei-Fei, R. Fergus, and P. Perona, “One-Shot learning of object categories,”
IEEE Transactions on PAMI, vol. 28, no. 4, pp. 594–611, 2006.
[18] L. Fei-Fei, A. Iyer, C. Koch, and P. Perona, “What do we perceive in a glance
of a real-world scene?,” Journal of Vision, vol. 7, no. 1, pp. 1–29, 2007.
[19] M. Fischler and R. Elschlager, “The representation and matching of pictorial
structures,” IEEE Transactions on Computer, vol. C-22, pp. 67–92, 1973.
[20] A. Fridman, “Mixed markov models,” Proceedings of Natural Academy of Science USA, vol. 100, pp. 8092–8096, 2003.
[21] J. Friedman, T. Hastie, and R. Tibshirani, “Additive logistic regression: A statistical view of boosting,” Annals of Statistics, vol. 38, no. 2, pp. 337–374,
2000.
[22] K. S. Fu, Syntactic Pattern Recognition and Applications. Prentice-Hall, 1982.
[23] M. Galun, E. Sharon, R. Basri, and A. Brandt, “Texture segmentation by
multiscale aggregation of filter responses and shape elements,” Proceedings of
ICCV, Nice, pp. 716–723, 2003.
[24] R. X. Gao, T. F. Wu, N. Sang, and S. C. Zhu, “Bayesian inference for layered
representation with mixed Markov random field,” in Proceedings of the 6th
International Conference on EMMCVPR, Ezhou, China, August 2007.
[25] R. X. Gao and S. C. Zhu, “From primal sketch to 2.1D sketch,” Technical
Report, Lotus Hill Institute, 2006.
[26] S. Geman and M. Johnson, “Probability and statistics in computational linguistics, a brief review,” in Int’l Encyc. of the Social and Behavioral Sciences,
(N. J. Smelser and P. B. Baltes, eds.), pp. 12075–12082, Pergamon: Oxford,
2002.
[27] S. Geman, D. Potter, and Z. Chi, “Composition systems,” Quarterly of Applied
Mathematics, vol. 60, pp. 707–736, 2002.
[28] U. Grenander, General Pattern Theory. Oxford University Press, 1993.
[29] G. Griffin, A. Holub, and P. Perona, “The Caltech 256,” Technical Report,
2006.
[30] C. E. Guo, S. C. Zhu, and Y. N. Wu, “Modeling visual patterns by integrating
descriptive and generative models,” IJCV, vol. 53, no. 1, pp. 5–29, 2003.
References
359
[31] C. E. Guo, S. C. Zhu, and Y. N. Wu, “Primal sketch: Integrating texture and
structure,” in Proceedings of International Conference on Computer Vision,
2003.
[32] F. Han and S. C. Zhu, “Bottom-up/top-down image parsing by attribute graph
grammar”. Proceedings of International Conference on Computer Vision, Beijing, China, 2005. (A long version is under review by PAMI).
[33] A. Hanson and E. Riseman, “Visions: A computer system for interpreting
scenes,” in Computer Vision Systems, 1978.
[34] T. Hong and A. Rosenfeld, “Compact region extraction using weighted
pixel linking in a pyramid,” IEEE Transactions on PAMI, vol. 6, pp. 222–229,
1984.
[35] J. Huang, PhD Thesis, Division of Applied Math, Brown University.
[36] Y. Jin and S. Geman, “Context and hierarchy in a probabilistic image model,”
in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, New York, June 2006.
[37] B. Julesz, “Textons, the elements of eexture perception, and their interactions,”
Nature, vol. 290, pp. 91–97, 1981.
[38] T. Kadir and M. Brady, “Saliency, scale and image description,” International
Journal of Computer Vision, 2001.
[39] G. Kanisza, Organization in Vision. New York: Praeger, 1979.
[40] Y. Keselman and S. Dickinson, “Generic model abstraction from examples,”
CVPR, 2001.
[41] B. Kimia, A. Tannenbaum, and S. Zucker, “Shapes, shocks and deformations I,”
Interantional Journal of Computer Vision, vol. 15, pp. 189–224, 1995.
[42] A. B. Lee, K. S. Pedersen, and D. Mumford, “The nonlinear statistics of
high-contrast patches in natural images,” IJCV, vol. 54, no. 1/2, pp. 83–103,
2003.
[43] M. Leyton, “A process grammar for shape,” Artificial Intelligence, vol. 34,
pp. 213–247, 1988.
[44] L. Lin, S. W. Peng, and S. C. Zhu, “An empirical study of object category
recognition: Sequential testing with generalized samples,” in Proceedings of
International Conference on Computer Vision, Rio de Janeiro, Brazil, October 2007.
[45] T. Lindeberg, Scale-Space Theory in Computer Vision. Netherlands: Kluwer
Academic Publishers, 1994.
[46] J. S. Liu, Monte Carlo Strategies in Scientific Computing. NY: Springer-Verlag,
p. 134, 2001.
[47] S. Mallat and Z. Zhang, “Matching pursuit in a time-frequency dictionary,”
IEEE Transactions on Signal Processing, vol. 41, pp. 3397–3415, 1993.
[48] K. Mark, M. Miller, and U. Grenander, “Constrained stochastic language models,” in Image Models (and Their Speech Model cousins), (S. Levinson and
L. Shepp, eds.), IMA Volumes in Mathematics and its Applications, 1994.
[49] D. Marr, Vision. Freeman Publisher, 1983.
[50] D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human segmented
natural images and its application to evaluating segmentation algorithms,”
ICCV, 2001.
360 References
[51] H. Murase and S. K. Nayar, “Visual learning and recognition of 3-D objects
from appearance,” International Journal of Computer Vision, vol. 14, pp. 5–24,
1995.
[52] K. Murphy, A. Torralba, and W. T. Freeman, “Graphical model for recognizing
scenes and objects,” Proceedings of NIPS, 2003.
[53] M. Nitzberg, D. Mumford, and T. Shiota, “Filtering, segmentation and depth,”
Springer Lecture Notes in Computer Science, vol. 662, 1993.
[54] Y. Ohta, Knowledge-Based Interpretation of Outdoor Natural Color Scenes.
Pitman, 1985.
[55] Y. Ohta, T. Kanade, and T. Sakai, “An analysis system for scenes containing
objects with substructures,” in Proceedings of 4th International Joint Conference on Pattern Recognition, (Kyoto), pp. 752–754, 1978.
[56] B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field
properties by learning a sparse code for natural images,” Nature, vol. 381,
pp. 607–609, 1996.
[57] B. Ommer and J. M. Buhmann, “Learning compositional categorization
method,” in Proceedings of European Conference on Computer Vision, 2006.
[58] J. Pearl, Heuristics: Intelligent Search Strategies for Computer Problem Solving.
Addison-Wesley, 1984.
[59] J. Porway, Z. Y. Yao, and S. C. Zhu, “Learning an And–Or graph for modeling
and recognizing object categories,” Technical Report, Department of Statistics,
2007.
[60] J. Rekers and A. Schürr, “A parsing algorithm for context sensitive graph
grammars,” TR-95–05, Leiden University, 1995.
[61] M. Riesenhuber and T. Poggio, “Neural mechanisms of object recognition,”
Current Opinion in Neurobiology, vol. 12, pp. 162–168, 2002.
[62] B. Russel, A. Torralba, K. Murphy, and W. Freeman, “LabelMe: A database
and web-based tool for image annotation,” MIT AI Lab Memo AIM-2005-025,
September 2005.
[63] R. E. Schapire, “The boosting approach to machine learning: An overview,”
MSRI Workshop on nonlinear Estimation and Classification, 2002.
[64] T. B. Sebastian, P. N. Klein, and B. B. Kimia, “Recognition of shapes by editing
their shock graphs,” IEEE Transactions on PAMI, vol. 26, no. 5, pp. 550–571,
2004.
[65] S. M. Sherman and R. W. Guillery, “The role of thalamus in the flow of information to cortex,” Philosophical Transactions of Royal Society London (Biology),
vol. 357, pp. 1695–1708, 2002.
[66] K. Shi and S. C. Zhu, “Visual learning with implicit and explicit manifolds,”
IEEE Conference on CVPR, June 2007.
[67] K. Siddiqi and B. B. Kimia, “Parts of visual form: Computational aspects,”
IEEE Transactions on PAMI, vol. 17, no. 3, pp. 239–251, 1995.
[68] K. Siddiqi, A. Shokoufandeh, S. J. Dickinson, and S. W. Zucker, “Shock graphs
and shape matching,” IJCV, vol. 35, no. 1, pp. 13–32, 1999.
[69] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, “Shiftable
multi-scale transforms,” IEEE Transactions on Information Theory, vol. 38,
no. 2, pp. 587–607, 1992.
References
361
[70] S. Thorpe, D. Fize, and C. Marlot, “Speed of processing in the human visual
system,” Nature, vol. 381, pp. 520–522, 1996.
[71] S. Todorovic and N. Ahuja, “Extracting subimages of an unknown category
from a set of images,” CVPR, 2006.
[72] Z. W. Tu, X. R. Chen, A. L. Yuille, and S. C. Zhu, “Image parsing: Unifying
segmentation, detection, and recognition,” International Journal of Computer
Vision, vol. 63, no. 2, pp. 113–140, 2005.
[73] Z. W. Tu and S. C. Zhu, “Image segmentation by data-driven Markov chain
Monte Carlo,” IEEE Transactions on PAMI, May 2002.
[74] Z. W. Tu and S. C. Zhu, “Parsing images into regions, curves and curve groups,”
International Journal of Computer Vision, vol. 69, no. 2, pp. 223–249, 2006.
[75] M. Turk and A. Pentland, “Eigenfaces for recognition,” Journal of Cognitive
Neuroscience, vol. 3, p. 1, 1991.
[76] S. Ullman, “Visual routine,” Cognition, vol. 18, pp. 97–157, 1984.
[77] S. Ullman, E. Sali, and M. Vidal-Naquet, “A fragment-based approach to object
representation and classification,” in Proceedings of 4th International Workshop
on Visual Form, Capri, Italy, 2001.
[78] P. Viola and M. Jones, “Rapid object detection using a boosted cascade of
simple features,” CVPR, pp. 511–518, 2001.
[79] W. Wang, I. Pollak, T.-S. Wong, C. A. Bouman, M. P. Harper, and J. M.
Siskind, “Hierarchical stochastic image grammars for classification and segmentation,” IEEE Transactions on Image Processing, vol. 15, no. 10, pp. 3033–3052,
2006.
[80] Y. Z. Wang, S. Bahrami, and S. C. Zhu, “Perceptual scale space and it applications,” in International Conference on Computer Vision, Beijing, China, 2005.
[81] M. Weber, M. Welling, and P. Perona, “Towards automatic discovery of object
categories,” IEEE Conference on CVPR, 2000.
[82] A. P. Witkin, “Scale space filtering,” International Joint Conference on AI.
Palo Alto: Kaufman, 1983.
[83] T. F. Wu, G. S. Xia, and S. C. Zhu, “Compositional boosting for computing
hierarchical image structures,” IEEE Conference on CVPR, June 2007.
[84] Y. N. Wu, S. C. Zhu, and C. E. Guo, “From information scaling laws of natural
images to regimes of statistical models,” Quarterly of Applied Mathematics,
2007 (To appear).
[85] Z. J. Xu, H. Chen, and S. C. Zhu, “A high resolution grammatical model for face
representation and sketching,” in Proceedings of IEEE Conference on CVPR,
San Diego, June 2005.
[86] Z. J. Xu, L. Lin, T. F. Wu, and S. C. Zhu, “Recursive top-down/bottomup algorithm for object recognition,” Technical Report, Lotus Hill Research
Institute, 2007.
[87] Z. Y. Yao, X. Yang, and S. C. Zhu, “Introduction to a large scale general purpose groundtruth database: Methodology, annotation tools, and benchmarks,”
in 6th International Conference on EMMCVPR, Ezhou, China, 2007.
[88] S. C. Zhu, “Embedding Gestalt laws in Markov random fields,” IEEE Transactions on PAMI, vol. 21, no. 11, 1999.
362 References
[89] S. C. Zhu, “Statistical modeling and conceptualization of visual patterns,”
IEEE Transactions on PAMI, vol. 25, no. 6, pp. 691–712, 2003.
[90] S. C. Zhu, Y. N. Wu, and D. B. Mumford, “Minimax entropy principle and its
applications to texture modeling,” Neural Computation, vol. 9, no. 8, pp. 1627–
1660, November 1997.
[91] S. C. Zhu and A. L. Yuille, “Forms: A flexible object recognition and modeling system,” Interantional Journal of Computer Vision, vol. 20, pp. 187–212,
1996.
[92] S. C. Zhu, R. Zhang, and Z. W. Tu, “Integrating top-down/bottom-up for
object recognition by data-driven Markov chain Monte Carlo,” CVPR, 2000.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement