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4-H. l? ‘f ‘8

  

  

, LIBRARY
Michigan State

   

This is to certify that the

thesis entitled

AUGMENTED RELAXATION LABELING
AND
DYNAMIC RELAXATION LABELING
FOR COMPUTER VISION

presented by

Stephen Andrew Kuschel

has been accepted towards fulfillment
of the requirements for

Ph.D. Computer Science

 

 

degree in

, . ,7
'7 /. fl [’1‘ ,./ ,l/ _// ,
(tick C4 /¢/j’/Je
’/

Major professor

 

Date August 13, 1979

 

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flwi FINES:
25¢ per day per item

RETURNS LIBRARY HAT;
Phat in book return
chm. from dramatic

0308 03
MAR212003

AUGMENTED RELAXATION LABELING
AND
DYNAMIC RELAXATION LABELING
FOR COMPUTER VISION

BY

Stephen Andrew Kuschel

A DISSERTATION

Submitted to

Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Computer Science

1979

ABSTRACT
AUGMENTED RELAXATION LABELING

AND
DYNAMIC RELAXATION LABELING

FOR COMPUTER VISION

BY

STEPHEN ANDREW KUSCHEL

Relaxation labeling is a bottom-up data dependent
method of labeling the objects in a picture through
local consistency criteria over the assigned labels.
The criteria are specified through the use of pairwise
label compatabilities. Relaxation is amenable to
parallel processing. Only homogeneous relaxation has
been studied so far, where information description and
flow is restricted to regular patterns for each pixel
regardless of label.

It is desirable to direct the labeling in a
top-down manner without the restrictions of homogeneous
relaxation processes while retaining the picture level
local consistency criteria. Such direction allows
additional information to be brought to bear on hard to

label areas of the picture.

Augmented Relaxation Labeling is a two stage
relaxation labeling process which contains a separate
iteration stage with a top—down direction capability
for specific pixel labeling. The augmentation is
accomplished by a broadcast process which sends
pertinent information to specifically chosen areas
throughout the picture. The received broadcasts cause
a change in the labelings to reflect the new
information now available. Thus the broadcast process
of the augmented relaxation labeler is a suitable
communication construct between the levels of a
hierarchical description and aids in the labeling of
ambiguous areas of the picture by bringing in top-down
information.

Augmented Relaxation is demonstrated on simple
line drawings. Comparisons show it to be both faster
per iteration and more effective per iteration than
regular relaxation.

Dynamic Relaxation Labeling is a single stage
combination of the feature of broadcasting and the
relaxation paradigm. It is not bound by the
requirements of homogeneity but allows information to
flow within the picture based on the labelings of the
picture and any other knowledge sources available.
Dynamic Relaxation is also demonstrated and compared on

simple line drawings.

General relaxation processes are modeled by the
constructs of Shannon Infonmation Theory. The
information flow of the local consistency criteria and
the broadcast processes are given an information
channel model. The appropriate information measures
are described and quantified for the various relaxation
processes described here. One result is a method for
quantifing the goodness of the compatability
specifications and a second result gives a measure for
the effectiveness of an iteration.

A final section on object position prediction by
relaxation in sequences of pictures containing motion

is included.

TO THE GREATER
HONOR AND GLORY

OF GOD

ii

ACKNOWLEDGEMENTS

It is my great pleasure to acknowledge the gifts that
God has given me. They are not my own, but merely in
my stewardship. By this dissertation it is my hope to
make productive use of them.

I also acknowledge my brothers and sisters in Jesus
Christ in The werk of Christ Community who have
tirelessly encouraged and supported me. I especially
wish to thank Coleen Shireman and Rene McCarty who
freely gave of their time to help with typing and to
Joann Rzepka who helped with the figures.

I further acknowledge Dr. Carl V. Page for his ideas
direction and guidance that made this thesis possible
and to Drs. Richard Reid, John Eulenberg and Erik
Goodman who served on the guidance committee.

Finally, I wish to acknowledge Dr. Harry G. Hedges
who as department chairman always found a way to
provide me with an assistantship or other means of
support throughout my stay at Michigan State.

iii

TABLE OF CONTENTS

LIST OF FIGURES Vi

LIST OF TABLES vii

I. THE LITERATURE ON CONSISTENT LABELING
and RELAXATION LABELING METHODS l
1.1. INTRODUCTION and PROBLEM STATEMENT l
CONSISTENT LABELING 2
5

I.2.
I.3. RELAXATION LABELING
II. BASIC DEFINITIONS AND PROBLEM STATEMENT 10
11.1. THE PURPOSE OF THIS THESIS 10
II.2. DEFINITIONS 11
11.3. EXPERIENCE MATRICES AND BROADCAST l3
PATTERNS
II.3.l. Broadcasts 13
II.3.2. Broadcast Reception 13
II.4. BROADCASTING THRESHOLDS, TOP DOWN
CONTROL 15
II.5. EXAMPLE 15
III. INITIAL TESTING and ALGORITHM DEVELOPMENT 19
111.1. PROBLEM DEFINITION l9
III.2. HOMOGENEOUS UNBIASED RLP 20
III.3. DEFINING THE BROADCAST SYSTEM 25
III.3.l. The Experience Matrices 25
III.3.2. Broadcast Control 29
III.3.3. Broadcast Patterns 30
III.4. TESTING 32
III.4.1. Test Pictures 32
III.4.2. Test Runs 35
III.4.3. Analysis 38
III.5. REDESIGN 42
III.5.l. The New Algorithm 42
III.5.2. Broadcast Strength and
Algorithm Behavior 46
III.5.3. Reveiw S4
III.5.4. TANH SS
III.6. TEST PROGRAM RESULTS 56
III.6.l. The Original Test Picture 56
III.6.2. Picture 2 58
111.7 CONCLUSIONS 60

iv

IV. THEORETICAL ISSUES

IV.1.
IV.

IV.
IV.

IV.2.
IV.

IV.2.2. Multiple Label Experience Matrices

IV.
IV.3.
IV.
IV.
IV.

IV.
IV.
IV.4.

IV.
IV.

ON COMPATABILITY ESTIMATION
1.1. The Relaxation Rule and
Local Consistency
1.2. The Broadcast Rule as a

Relaxation Processor
1.3. Conclusions ‘
DYNAMIC RELAXATION LABELING
2.1. Definitions

2.3. Results

INFORMATION MEASURES

3.1. The Model

3.2. The Measures

3.3. Channel Capacity in the Line
Processors

3.4. Measures for the Line Processors

3.5. Conclusions

CHARACTERIZATION OF FIXED POINTS

IN AN ARLP

4.1. RLP Fixed Points

4.2. Broadcasts and RLP Fixed Points

V. MOTION PREDICTION
V.1. MOTIVATION
v.2. DEFINITIONS
V.2.1. Experience Matrices
V.2.2. Broadcast Patterns
v.3. EXAMPLE
V 4. DISCUSSION
V.4.l. Occlusion
V.4.2. Expanded Label Sets

v.4.3.

V.4.4. Multiple Label Reception

VI. CONCLUSION AND FUTURE PROBLEMS

V1.1.
V1.2.
VI.
VI.

VI.
VI.
VI.

SUMMARY AND CONCLUSION
FUTURE PROBLEMS
2.1. Convergence and Optimality
2.2. On Experience Matrices and
Compatabilities
2.3. Broadcast Parameters
2.4. Hierarchies
2.5. Chess by Broadcasting

BIBLIOGRAPHY

APPENDIX A COMPATABILITY MATRICES

APPENDIX

EXPERIENCE MATRICES

B DRLP PROGRAM

75
75

75

77
83
87
87
89
96
102
102
107

110
113
115
124

124
125

13%
136
130
139
131
132
135
135
135

Motion Prediction and Curve Fittingl36

137

138

138
139
139

140
141
143
143

144

146
149

156

FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

3.10

3.11

3.12

3.13

3.14

3.15

LIST OF FIGURES

RIVER EXAMPLE

LINE TEST GRID WITHOUT NOISE

RELAXATION EFFECTS FROM THE TEST GRID

BROADCASTING PATTERNS
TEST PICTURE #1

FAINT LINES OF PICTURE #1
BROADCAST PROCESS RESULTS
ORIGINAL BEHAVIOR
SPURIOUS LINE EXAMPLE
DESIRED BEHAVIOR

BEHAVIOR OF THE REVISED BROADCAST

ALGORITHM e = .2

\\'
BEHAVIOR OF THE REVISED BROADCAST
ALGORITHM e = .4

\\'
BEHAVIOR OF THE REVISED BROADCAST
ALGORITHM e = .8

\\'
BOUNDED BEHAVIOR CURVES IN THE
NEW ALGORITHM e = .8

\\'

TANH FUNCTION UPDATE BEHAVIOR

RESULTS OF THE REDESIGNED ALGORITHM

RESULTS FROM THE REDESIGNED BROADCAST

ALGORITHM e = .8
\\'

vi

16
23

31
33

33

4O
41

43

50

51

52

53

56

61

62

FIGURE 3.16 RESULTS OF THE RELAXATION ALGORITHM 63

FIGURE 3.17 AUGMENTED RELAXATION RESULTS 65
FIGURE 3.18 ARLP RESULTS WITH EXTRA BROADCASTS 66
FIGURE 3.19 TEST PICTURE #2 ' 68

FIGURE 3.28 RESULTS OF THE RELAXATION ALGORITHM 69
ON PICTURE #2

FIGURE 3.21 ARLP RESULTS ON PICTURE #2 78

FIGURE 3.22 ARLP RESULTS ON PICTURE #2 71
WITH EXTRA BROADCASTS

FIGURE 3.23 BROADCAST BEHAVIOR ONLY ON PICTURE #2 74

FIGURE 4.1 BROADCAST PATTERN FOR NO-LINE LABELS 88

FIGURE 4.2 BROADCAST RESULTS ON PICTURE 81

ALL NODES AND LABELS BROADCASTING 98
FIGURE 4.3 BROADCAST RESULTS ON PICTURE #2

ALL NODES AND LABELS BROADCASTING 92
FIGURE 4.4 DYNAMIC RELAXATION ON PICTURE #2

INTERSECTION THRESHOLD = .75 98
FIGURE 4.5 DYNAMIC RELAXATION ON PICTURE #2

INTERSECTION THRESHOLD = 1.8 181
FIGURE 4.6 BROADCAST CHANNELS 185
FIGURE 4.7 CASCADED CHANNELS 186

FIGURE 4.8 INFORMATION MEASURE NORMS FROM
THE ARLP ON PICTURE #2 116

vii

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

FIGURE

INFORMATION MEASURES FOR EACH NODE
FROM THE ARLP ON PICTURE #2 117

INFORMATION MEASURE NORMS FROM THE
DRLP ON PICTURE #2 128

INFORMATION MEASURES FOR EACH NODE
FROM THE DRLP ON PICTURE #2 121

INFORMATION MEASURE NORMS FROM THE
RLP ON PICTURE #2 123

A SIMPLE SCENE WITH A MOVING OBJECT 133
NODES RECEIVING BROADCASTS 133

EXAMPLE EXPERIENCE MATRICES 134

viii

LIST OF TABLES

TABLE 3.1 EXPERIENCE MATRIX DEFINITIONS 29

TABLE 4.1 BLANK LABEL EXPERIENCE MATRICES 94

TABLE 4.2 INTERSECTION LABEL EXPERIENCE MATRICES 95

ix

I. THE LITERATURE ON CONSISTENT LABELING and
RELAXATION LABELING METHODS

I.1. INTRODUCTION and PROBLEM STATEMENT

The field of computer picture processing and
vision systems is very diverse. A Survey by Kuschel
[1] presents a review of various picture segmentation,
modeling and control techniques, notes on syntax,
semantics and representation, mathematical theories and
psychological models. This survey sould be consulted
for an overview of the field.

Picture processing is mainly concerned with
enhancement, segmentation, compression and storage of
the picture. The problem of Vision differs from picture
processing in that vision requires an analysis of a
scene to be made so that labels can be applied to each
object or region of the scene. Visual models are often
employed as data structures to represent the properties
and relationships of the objects and regions of the
scene.

This thesis focuses on the topic of labeling of
the items in the picture and consequently the
relationships of the items to each other. This work is
especially applicable to the picture level or the low
level of the data structure representing the scene.
Such work is a first approximation to the scene
labeling and often uses world knowledge from higher

levels of the structure. Pertinent papers on labeling

will now be reviewed.
I.2. CONSISTENT LABELING

The work done by Waltz [2] involved the labeling
of line segments and regions of line drawings of
polyhedra. The line junctions are the basis on which
the lines interact and constrain the possible labelings
that each line may take. In other words, two lines
joined at a junction cannot have labels which are
incompatible with the fact that they are joined
together.

The labeling process is straightforward. The
first lines labeled are the outermost lines of the
figure, which form the figure-background boundary.
Figure and background regions are also labeled.
Secondly, lines that are joined to the already labeled
lines are considered and the set of possible labels
they can take on is determined. These new labels are
checked for consistency with existing labels. Any
inconsistent labels are eliminated. The second step is
repeated until all existing lines have been considered
and each line and the regions it bounds, have been
consistently labeled.

Waltz's method is inherently serial and dependent
on prior knowledge of each line segment, its location
and endpoints. The important contribution of the work
was the demonstration that compatability information

about the (labels and hence the) parts of a picture

decreased the set of labelings a picture could take on,
even by several orders of magnitude, and hence
decreased the complexity of picture processing.
Haralick [3,4,5] developed a notation which models
Waltz's constraints and labeling process as well as
serving as a meta-description for several other classic
problems (e.g. graph matching, graph or map coloring
such as the famous four color problem and the
homomorphism and automata equivalence problems). The

notation is as follows:

U = {u ,u , ... u } The set of units or objects to (1)
1 2 M be labeled in a domain
L = {1 ,1 , ... l } The set of labels for unit u (2)
l 2 ki
N
T c U The unit-constraint relation (3)
T is the set of N-tuples of units
which mutually constrain each other.
N may take on several values depend-
ing on the nwmber of units entering
into a constraint relationship.
For example:
N = 3, 4, or 5 lines at one junction.
N
R c (U x L) The unit-label constraint (4)

relation R is a set of 2N-tuples
where (l , ... ,l ) is a permitted
1 N
labeling of units (u , ... ,u ).
l N
The consistent labeling problem is to find all

labelings (l , l , ... l ) for U which are consistent
l 2 M

with the constraints expressed by the relations T and
R. "One general technique used for this problem is a

depth first search. The search procedure fixes labels

to units as long as it can find a label for each new
unit that is compatable according to the constraint
relation R with labels already fixed to previous units"
[3]. As in any tree search, one wishes to minimize
backtracking and thrashing. This is done by the use of
a generalized look—ahead operator § KP where K<N<P<M
with K<P, N and M being parameters of the problem
definition given above.

"To apply the § operator to R we individually
check each 2N-tuple of R. We fix any K of the N units
(in a 2N-tuple of R) to their labels and check every
set of P—K units to determine if there are P-K labels
which make the K fixed labels plus the P-K labels a
consistent labeling of all P units. We repeat this
process for every combination of K out of the N units
in the 2N-tuple. If for any combination of K fixed
units and labels, there is a set of P-K units for which
(a set of) P-K labels to make a consistent labeling
cannot be found, then the 2N-tuple of R is thrown out."
[3] Thus those 2N-tuples which cannot participate in a
global labeling of all units are eliminated.

A second and similar operator, T'KP, will
eliminate only those K-tuples of unit-label pairs in a
2N-tuple of R which do not contribute to a consistent
labeling. This saves the ellimination of the whole

2N-tuple. Examples can be found in [3].

Haralick presents results on the power of Q KP and
Y'KP, their interrelationships and their effects on R.
Indications of how One might apply this system to
example problem domains is also given. For scene
labeling, Haralick uses N=2 and predicates such as
'next-to', 'above', 'behind' etc. When two units
(objects) such as a lamp and a table are related by
these predicates, then that relation becomes part of
the sets T and R. Such a system is overly simplistic,
however, as it presupposes a well segmented picture and
T and R relations of great complexity and irregularity.
Hence it could not be a 'stand alone' labeling system,
but rather it is more suitable as one level of a
hierarchical system at a time, such as the object

level.

1.3. RELAXATION LABELING

Relaxation is a well known method for finding
solutions to linear systems by iterative approximation.
The residual vector R is used as a measure of how well

K
the solution vector x satisfies the system. The

general form of a relaxation system is given below:

K
R = A x - c

Rosenfeld et al [6] also developed a notation
following Waltz [2] for assigning a label to an object

from a set of possible labels for that object. The

notation follows a relaxation system with the choice of
the labels at each pixel being a solution vector x.
However the concept of constraints was buried by simply
postulating a subset of consistent labels as opposed to
all possible labels. The special power of relaxation
labeling is that it can be computed in parallel for
each pixel. Rosenfeld et al extended their work to
include fuzzy labels and probabilistic labels. It was
shown that all three produced a set of consistent
labelings after a suitable number of iterations.
General relaxation was later shown by Haralick to be a
subcase of his T’KP operator [3].

Following Rosenfeld et a1, Zucker and Mohammed [7]
added compatability functions (constraints) for the
units and further analyzed and developed the
probabilistic parallel algorithm, now called a
Relaxation Labeling Process or RLP. Their notation:

L = {X ,k , ... X } The label set (5)
1 2 k

p (A) The probability that node i (6)
1 (unit i) has label X.
29.0.) = 1
I

P (AIA') The compatability of label A
ij for node i, with label X'
of neighboring node j. (7)
0 < P .(XIX') < 1
13

The idea of a specific set of unit N-tuples, or
2N-tuples of unit-label pairs that constrain each

other, is not used here to define the compatability

values p ()lk'). Instead, the units are presumed to
13'

be elements of a regular rectangular grid, rather than
nodes in a general graph. Constraining relationships
of the units are based on grid position instead of on
individual node interrelationships. This is the basic
idea for what is called a 'homogeneous' RLP. A
homogeneous RLP is defined as having the following
properties:
1. Each node (unit) uses the same label set (8)
2. Each node has the same relationship (constraining
ability) with its neighbor nodes. This set of

neighbors is termed a relaxation neighborhood.

3. Each node uses the same compatability functions,
P (XIX'), to specify this constraining relationship.

in

4. The compatability functions are unbiased e.g. they
favor no single label in and of themselves.

These properties allow for an easy implementation
of a label probability updating rule for each node that
can be computed in parallel. This is given as the

following product updating rule:

K J k K
p (A) '{T 2 p (AIA') :3 ()0)
n+1 i j x' ij j
p (A) = ----------------------------------- (9)
i K J k K
Ep (x) TI 2 p (me) p (>0)
1 j x ij j

Labeling is accomplished by giving each label of

each node an initial probability based on the

underlying data. Label probabilities are updated by an
iteration of the product updating rule. (The K in the
rule indicates the iteration index.) Each iteration
modifies the label probabilities based on whatever
”world knowledge" resides in the compatability

measures: p ()l)‘). After several iterations or when
13

the probabilities converge to a fixed point, the label
with the greatest probability is chosen as the label
for that node. Thus the RLP operates on two sources of
information, the initial probabilities and the
compatability estimates.

If the fixed point gives a unique interpretation

for each label (unique interpretation matrix of p (A)
i

or a UIM ) where one label of node i has value 1.8 and
all other labels of node 1 having value 8; the label
choice is certain. However, if the fixed point is
ambiguous: where several labels have non-zero
probabilities, the neighborhood local consistency
contribution has become negligible. In other words
”the picture contains no local information suitable for
resolving the ambiguities. In such cases control
should be passed to another process..." [9].

The product rule converges to the fixed point
faster than its earlier predecessor, the arithmetic

update rule. The speed increase is accomplished

through a series approximation to the arithmetic
updating rule, raised to a power, thus yielding a
geometric or product rule. Consequently small changes
specified by the local neighborhoods are magnified.
The derivation details are given in [9,8]. The topic
is discussed further in section IV.l.

Eklundh [18] developed a measure of convergence in
a picture from the relaxation definition. The
residual, R, is computed between the solutions of
successive iterations using the Euclidean norms of the
solutions. Computations may be stopped when the
residual is small enough to indicate that the labelings
are reaching a fixed point.

Zucker and Mohammed [8] have applied their RLP to
the problem of detecting lines in a noisy picture. It
has functioned well, separating the lines frOm the
noise. Their success is due in part to an extension to
a two level RLP where relaxation updates both levels as
well as serving as the communication vehicle between
levels. The higher level is designed to provide
direction to the lower level through inter-level

compatability functions.

II. BASIC DEFINITIONS AND PROBLEM STATEMENT
II.l. THE PURPOSE OF THIS THESIS

Relaxation is basically a bottom-up, data
dependent operation amenable to parallel processing.
The homogeneous RPL updates every label of every node
at every iteration using the same compatability
functions based on each node's immediate neighbors.
Thus each node can be processed identically and in
parallel on a lockstep parallel machine (where the same
operation is performed by each CPU at each time step).

It is desired to update specific node label pairs
based on top-down world knowledge direction, apart from
and in addition to the homogeneity requirements while
preserving the parallel nature of the RLP algorithm.
The work of this thesis is to define and implement a
basic Augmented Relaxation Labeling ProcessOr (ARLP)
which contains the ability for top down direction.

This augmentation guides overall picture labeling and
aids in hard to label areas of the picture where a
unique labeling cannot be arrived at directly. Picture
results are given to illustrate ARLP and RLP behaviors
and properties.

Secondly, attention is focused on the inter-label
compatability functions which define the world
knowledge and its use. The result is the definition of
a Dynamic Relaxation Labeling Processor (DRLP), based
on the features of the augmentation. The DRLP is a

18

 

11

general, non-homogeneous, relaxation labeler with

greater speed and discrimination than the RLP or ARLP.

II.2. DEFINITIONS

Augmented Relaxation Labeling was first described
by Carl V. Page in 1978. This thesis presents the first
implementation and testing. The following definition is
adapted from Page [11] and Page and Kuschel [12].

Augmented relaxation labeling is an RLP with an
added information broadcasting and receiving process.
The augmentation allows a pixel or node in the data
structure to broadcast information about its label set
beyond its immediate neighbors. The received broadcast
is used to update the label probability distribution of
the nodes receiving the broadcast. The top down
control mechanism decides which nodes may brOadcast,
and when, as well as the direction and pattern of the
broadcasts. The decision is made based on structural
and statistical information in the picture and through
the knowledge sources comprising the controller.

The basic structure of the ARLP is an iterative
two stage process. In the first stage, information is
broadcast and received. Labels are appopriately
updated. In the second stage, the local consistency
compatabilities are applied Via the RLP. The first
stage is then re-initiated using the refined

information as a starting point. The iterations

12

continue until either convergence on a consistent
picture label set (a fixed point) occurs or a resource
bound stops computations.

Picture information in both stages is represented
by the same construct, namely a probability
distribution over the labels. World knowledge is still
represented numerically as the interaction of a label
pair as was done in the RLP. A formal definition

follows.

DEFINITION 2.1.
An Augmented Relaxation Labeling System (ARLP) is
a system defined as:

A = < R, B, T > (18)
where R is a relaxation labeling process; T is the
control process which determines the cycling between
relaxation and broadcasting and B is a broadcast:

B = < Bu, G, E > (11)
where Bu is a set of broadcasting units: picture lines,
regions, nodes or other units, or a separate knowledge
source; G is the set of grammars which determine the
broadcast pattern and E is the set of experience
matrices containing the knowledge about the label
relationships by which broadcasts update the

probabilities of nodes receiving the broadcast.

13

II.3. EXPERIENCE MATRICES and BROADCAST PATTERNS
II.3.1. Broadcasts

The broadcast from a node contains the identity or
present labeling of that node. The most likely label
at the node conveys that information and so forms a
suitable message for a broadcast. The broadcast
reaches those nodes defined in the "broadcasting
pattern". The choice of a broadcasting pattern for a
node and label is semantically dependent on the label.
Other factors such as the node position in the picture
description, its relation to other nodes and the
certainty of the label for that node may also play a
part. In general, these patterns can be specified by
the two dimensional grammar G, or any other convenient
representation for the elements of the rectangular
pixel grid or the nodes in a general graph.i
II.3.2. Broadcast Reception

The effect of a broadcast on a node is determined

by an experience matrix E[) .A ,...,X ,...,X ] where
1 2 i b

the A are the broadcast labels received by the node.
i

E is a square kxk column stochastic matrix where k is
the number of labels in the label set. Each row of the
matrix gives the "experience” for the updating of one
label at the receiving node. Each element of a row

describes the ”experience” the system has about the

14

row-column label pair (the A,X' pair in (13) below).
The label probabilities of the receiving node i

are given as a k x 1 column vector p , and are updated
i

according to the following equation:

p =E[>\ ....pA] P (12)
b

i l 1

Using e as a general element of E with,X as a row

X,>\'
index and A' as a column index we can write (12) in the
node label form similar to (9) from Zucker [7, 8].
k
P (M =2 e P (X) (13)
i . kw i 3
3:1 3

We can further define a "broadcast neighborhood"
of the receiving node i as being those nodes which have
affected node i through the broadcast process. The
intention of the definition is to allow an analogous

term for the relaxation neighborhood defined in section

1.2 at (8).

II.4. BROADCASTING THRESHOLDS, TOP-DOWN CONTROL

The decision as to whether a node should broadcast
its labeling is made in general in a top-down fashion
by the knowledge source or control process, T, in
authority. The decision may be made in specific
situations where obtaining convergence on a label set
for a region is difficult due to noise or picture
complexity. In simpler cases the decision is more
uniformly applied . An example of the latter would be
to allow every node to broadcast its most probable label
if the probability exceeds a given threshold. The
threshold may be picture wide or dependent on the node,
label and/or location. It is possible that behavior
similar to a neural net can be obtained by allowing
different thresholds at different points, depending on
factors such as time, previous broadcast and reception

patterns.

II.5. EXAMPLE

Suppose we have an image containing a river which
is crossed by a highway, shown schematically (without a
grid) in Figure 2.1. Suppose further that, after
several passes of an augmented relaxation algorithm, we
arrive at the situation where the region indicated by
the circle bordering an island, bridge, riverbank and
water is not definitively labeled (with high

probability) as "river". The control process, T, will,

15

16

 

Figure 2.1
RIVER EXAMPLE

17

in effect, broadcast the existence of the river segments
to nodes nearby. The reception of the broadcast labels
will cause transformation of corresponding probabilities.
The broadcast pattern can be modeled by a two
dimensional stochastic grammar as in the example below.

R
rt

R -> R R
rt rt rt

R (14)

Notation: The leftmost nonterminal location is occupied
by the underlined symbol. Overlapping symbols at the
same pixel of the same type are merged into one.

Thus a node i in the range of both broadcasts, h
horizontal units from the left and k horizontal units
from the right receives the broadcast

((R .ph ). (R .qk )) (15)
1f rt

where ph and qk describe the grammar rule and the new
picture position it is applied at.

If ph>t, a threshold, and qk>t, we use this information

by transforming the label probability distribution

18
(P (X ),...,p (X )) by an experience matrix E(R ,R ).
i 1 i b 1f rt

The purpose of this experience matrix E(R ,R )
1f rt

is to move probability from labels which are unlikely
to lie along the path of a difficult to perceive river
to labels which tend to explain the difficulty. So for

our example let us define:

a *
* R B I D A U F O *
* *
* l 8 8 1/4 1/4 1/4 1/4 1/4 *
* 8 1 8 1/4 1/4 1/4 1/4 1/4 *
* 8 8 1 1/4 1/4 1/4 1/4 1/4 *
E(R ,R )= * 8 8 8 1/4 1/4 1/4 1/4 1/8 *
1f rt * 8 8 8 8 8 8 8 8 *
* 8 8 8 8 8 8 8 8 *
* 8 8 8 8 8 8 8 8 *
* 8 8 8 8 8 8 8 1/8 *

H'
I,
A
H
0‘
V

Thus if the distribution at a point i is

R B I D A U F O t

(8.2, 8.1, 8, 8, 8, 8.2, 8.2, 8.3) p (17)

After receiving the broadcast labels R ,R ,
1f rt

p is updated according to (12) using (16) giving:
i

R B I D AUF 0t (18)
(8.375, 8.275, 8.175, 8.1375, n, 8, 8, 8.8375) = P
Thus, the probabilities of labels for "river",
"bridge”, "island”, and "dam" are increased at the

expense of "urban", "factory” and ”other".

 

III. INITIAL TESTING and ALGORITHM DEVELOPMENT
III.1. PROBLEM DEFINITION

A problem space is desired that will provide a
small and workable class of pictures to test an ARLP
program. A set of pictures containing line (or edge)
information is chosen following Zucker [8]. Lines are
well defined and the relationships of pixels forming a
line are easily quantified into compatabilities that
encourage line formation and continuation. This domain
provides an extension to previously recognized work and
a set of non-trivial problems to work on. (see also
[8]. [7]. [2])

The purpose of the augmentation is to aid in hard
to label areas of the picture, or to enhance a
particular label or feature of the picture. Noise,
fading, gaps, and spurious information are classic
problems in line and edge images. Such problems are an
appropriate testing ground to compare the performance
of the relaxation labeling algorithm with its augmented
counterpart. It will be shown that the augmentation
does indeed provide additional resolving power over the
RLP.

The following sections detail the definition and
construction of the RLP and broadcast test programs for
pictures containing lines. The results on some simple
16x16 pixel pictures are given. A revision of Page's
initial broadcast definitions is undertaken based on the

19

28

test results. Revised results are then presented for

each system separately and for the combined ARLP.

III.2. HOMOGENEOUS UNBIASED RLP

The first task of an initial test is to establish
a running program that implements an unbiased
homogeneous relaxation labeling algorithm, R, as
defined in (8), (9) and (18) from Zucker [7,8] for the
picture domain described above. The label set for each
node (pixel) is the possible orientation the line may
have (as a directional vector). For edge data the
orientation gives the relative positions of the light
and dark components of the edge. For example, an edge
along the X axis with the lighter component of the edge
above the axis and the darker component below would be
orientation 1. If the darker component is above and
the lighter component below, an orientation of 5 is
assigned.

0

 

9 No line information in the node

I X Orientation l, 8 to the horizontal
| l
I o
I A OR 2 , 45 to the horizontal
| 2
I O O
A = I . .
l . . (19)
I C O
I o
l A OR 8 , 278 to the horizontal
| 8
I
| X
l
|

21

The relaxation neighborhood for the compatability
functions (7) was defined as the 8 adjacent neighbor
nodes in a regular rectangular grid. The compatabil-
ities themselves were chosen as in [8] for node i with
label X in relation to neighboring node j with label
X'. They are defined by the cosine function in (28)
below. This design encourages straight line formation
while neighboring lines become less compatable as the

angle between them increases.

l cos(OR(X)-B) cos(OR(X')-B) (28a)
|
l

r (X,X') = I cos(2[OR(X)-B]) if X' = NO LINE (28b)
ij |
l

| l X,X' both NO LINE labels (28c)

|_._
-l < r < 1 (28d)
ij ‘

where the r are the compatabilities; OR(X) is the
13'

orientation (angle) of label A; and B is the angle of

the imaginary vector between nodes i and j. These

compatabilities are then converted into conditional

probabilities through the normalization below:

(r._()\,).') + l)
13
p .(XIX') = w --------------------- (21a)
13 A g (r._()..).') + l)
1)
8< p < 1 (21b)
ij

22

; P (XIX') = l for all A' eg the p are (21c)
ij 13‘
column stochastic

where p are the conditional probability estimates of
ii

the compatibilities which sum to one over the A labels

for all A' at each i j node pair. The p values for
ii

each of the 8 neighbors are given in the appendix.

The w values are adjusted by iteration for each

A

label A in (22) subject to (21c) so that the
compatabilities will have no bias towards any
particular label. The w values are giVen in the
appendix for sums that are constant to 7 decimal
places. These values were recomputed at Zucker's
direction since the original values were not available.
In some applications, a bias may be desirable in the
compatabilities. In the line domain, all lines are
equally probable, a priori, regardless of orientation.
Hence, no bias is desired.

2 w p (XIX') = constant (22)
X' A 11

The RLP thus described was first tested on a
simulated test grid of lines of orientation 1 and of
varying probabilities. No noise was present. The
picture and label probabilities are given in Figure
3.8. The behavior of the algorithm is given in Figure

3.1. The curve shows the change in p (A) for l
i

23

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This curve shows the behavior of the relaxation
processor as measured by the test grid of figure 3.0. The
curve shows the change in label probability at one node, i,
from one iteration. Any portion of the curve above the
reference diagonal represents an increase in the probabi-
lity of the label, and any portion below, a decrease. Such
above-below behavior is desireable in the RLP since it
must both encourage useful labels and discourage noise (low
probability labels) at the same time without any direction
as to which label it should encourage.

This behavior varies with line length and noise, and
at line ends, generally by raising the value of the diagonal
intercept.

Figure 3.1
RELAXATION EFFECTS FROM THE TEST GRID

25

iteration of the RLP. The starting value is labeled OLD
p (X) on the X axis and the resulting value NEW p (A)
o; the Y axis. A diagonal is also shown for refeience
as this represents no change from the iteration. Any
portion of the curve above the diagonal represents an
increase in label probability, and any portion below -
a decrease.

Hence for the RLP to encourage line formation, the

initial probabilites must be in the range

3

” 8.75 < P (A) < 1.8.
i

The presence of noise, gaps in the lines or line end
points, or even the presence of lines of differing
orientation nearby, all effect the range, usually by
increasing the lower boundary.

The developement of the test RLP processor

following Zucker[7,8] is now complete.

III.3. DEFINING THE BROADCAST SYSTEM
III.3.l. The Experience Matrices

The effects on the label probabilities of a node
receiving a broadcast is determined by the experience
matrix, E, for the set of received labels as given in
section 11.2 and (12), (13). Ideally for every
combination of received broadcasts there would be a
specific experience matrix. This quickly becomes
impractical as the number of labels grow and the

broadcast neighborhood around a node increases (or

26

conversely as broadcast patterns extend further).
Instead of many complex matrices, one experience
matrix was developed for each of the labels, e.g.

E[x ],E[X 1,...,E[x ]. These matrices are given in the
1 2 9

appendix. They are chosen to encourage a straight line
of the particular label they represent. Slow curves
are less strongly encouraged and sharp corners are
least encouraged.

These criteria are the same as in the RLP
compatabilities given in (28). The behavior of the
experience matrix update should be such that lines are
not lengthened, shortened or spread. In addition,
since each application of an E matrix is due to the
reception of a broadcast label, the matrix need only
describe the growth in probability of the received
label. A behavior curve for the broadcast system like
Figure 3.1 will be entirely above the diagonal. In
other words, the behavior of this system depends on
the reception of a broadcast and not on the label
probability at the receiving node. (see also section
III.3.2 and III.5.1 where label probability is
discussed.)

To implement these criteria, each matrix row will
denote the percent of label probability to take from
other labels and give to the label represented by that

row. Hence an entry of .758 in row 1, column 3 for

27

label 1 indicates that label 1 will receive 75% of the
label probability from label 3. Label 9 is the no-line
or blank pixel label. The diagonal element in row 3,
column 3 of .25 indicates the retention of 25% of its
original probability by label 3. The actual numbers
chosen here are simple combinations of the negative
powers of 2. Thus they are easily computable. The
ordering of the numbers reflects the criteria for curve
encouragement given above.

It should be noted that the experience matrices
are not unbiased. In fact they are intentionally very
biased, for the particular label they represent. Their
application is directed by the broadcast controller to
specific nodes where that bias is to be applied. Hence
they are fundamentally different in design from the RLP
compatabilities which are applied uniformly everywhere.

A node receiving the broadcast of a single line
label simply accesses that label's matrix and updates
using equation (12). A node receiving broadcasts from
several different nodes and line labels computes an
experience matrix by averaging the single line label
matrices as in (23). Blanks are discussed below.

E1) I) I°°°l)\] =

l b
- X BIA] (23)
1 2 b b j=l °

3

An average was chosen as it is well known that as

a statistic the average is the best linear unbiased

28

estimator. However, such a choice is not without
flaws. If a corner or intersection of lines exists in
the picture, the cOrner node will receive broadcasts of
both labels. The resulting experience matrix will
encourage each label equally, resulting in the
retention of an ambiguous labeling at that node. There
are two immediate solutions here. One would be to
simply choose one of the labels and encourage it. The
other would be to set up particular labels denoting a
corner or intersection and encourage those labels.

This problem will be discussed further in section 111.6
and in section IV.2.

Blanks or no—line labels, A , are not allowed to
broadcast in this initial test sistem. This choice is
made to test the enhancing abilities of the experience
matrices and hence of the broadcast definitions. The
processing of blanks is left to the RLP cycle of the
ARLP. The functioning of the experience matrix
definitions is summed up in Table 3.1 given below.

The definitions are to be applied for any pair of

incoming broadcasts at one node.

29

 

Table 3.1
EXPERIENCE MATRIX DEFINITIONS

1. Single Label Broadcasts

BIA , A 1 for j=k --> E[x.]
j k 3

2. Ignore Blank Labels

E[x , X ] for A a blank --> E[X ]
j k k j

3. Averaging Of Single Line Label Broadcasts

El) . A 1 for j=k --> BIA ] + EIX 1
j k j k

 

III.3.2. Broadcast control

The decision as to whether a broadcast is made or
not, is generally decided in a top down fashion by the
knowledge source or control process T in authority, as
given in (18). The broadcasting units, Bu in (11). are
defined to be the line pixels themselves. (Throughout
the paper the more general term "node" will be
retained.) The broadcast is then the most likely line
label at that node. The decision to broadcast is made
by comparing the most likely line label at each node to
a threshold: t. If the threshold is exceeded by a line
label, the label is allowed to broadcast. Blank labels

are not allowed to broadcast as above. There is more

38

on this topic in section IV.2. The threshold value was
arbitrarily chosen at .24 but with an eye to being
small enough to allow weak lines to form and yet large
enough to exclude most noisy points from broadcasting.

It should be noted that the presence and value of
a broadcasting threshold is of theoretical importance.
A homogeneous relaxation system should have no bias
towards any label as discussed above. Thus a picture
containing no information (all labels equally likely)
should remain unchanged by the ARLP. If the same
threshold t is used for each label and t is greater
than l/k , where k is the number of labels, then the
broadcast control will also be unbiased. Since the
test system has 9 labels we need t=.24 > .lll=l/9 .
III.3.3. Broadcast Patterns

The choice of a broadcasting pattern is
semantically dependent on the label and other factors.
Line labels are defined as broadcasting directionally
along the orientation of the label. Figure 3.2 shows
the patterns for two labels. Broadcasts were
arbitrarily allowed to extend four pixels from the
broadcasting node in each direction as specified by the
label. Thus in a 16x16 picture, a broadcast can extend
halfway across the picture. This decision is
implemented through node positions in the rectangular
grid rather than as a more general grammar G over the

nodes as given in (11).

31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The broadcasting pattern of a label is semantically
dependent on the label. This figure shows the patterns
for two line labels, 3 and 8. The broadcast is allowed
to extend four pixels from the broadcasting pixel in the
direction of the label as indicated by the asterisks (*).

Figure 3.2
BROADCASTING PATTERNS

32

An alternate choice to a discrete, set, distance
would be to allow broadcasts to extend to the picture
boundary. The purpose of the broadcast is to put the
information where it is needed most, based on the
semantic information available. The choice of a local
or global broadcast pattern is chosen on this basis. A
more practical consideration is that the computational
overhead rises as the broadcast patterns increase. A
choice of a distance of one offers no advantage over
the RLP. In the small pictures used here, four is a
compromise choice.

As in radio and TV broadcasts, the signal strength
decreases with distance traveled. The strength may be
described by a continuous measure such as
e**(-distance) or a discrete measure based on the
number of pixels or nodes traveled. This topic is not
part of the original broadcast algorithm. It is

further discussed in section III.5.2.

III.4. TESTING
III.4.1. Test Pictures

The suggested broadcast augmentation to the RLP
from Page [11] has now been described for a set of
pictures containing line information. This broadcast
system will now be tested separately as was the RLP in

section 111.1 to establish its behavior characteristics.

33

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35

Several pictures have been generated and tested.
Lines are represented in printed output by a single
digit which denotes the label direction. The picture
output presents the most probable label at each node,
be it a line (digit) or a no-line (blank) label. Low
intensity lines will not appear in the original picture
as the no-line label is often more probable.

The first picture used here is given in Figure
3.3. It is designed to contain all of the test cases
given in section III.l. The line of orientation 8
starts strong and slowly fades out. The curve slowly
changes direction but has gaps in it and also fades at
the top. There is also a very weak line of orientation
2. A copy of the test picture showing all the low
intensity lines is given in Figure 3.4.

Noise was added randomly to 18% of the labels in
the picture. A random number was compared to the 18%
threshold for each label of each node. If the
threshold was met, then a second random number was
generated, multiplied by .25 and included in the
picture as the probability for that label. Hence the
maximum noise probability is .25 (which is greater than
the broadcasting threshold t=.24). The sum of the
probabilities for each node is then set to 1.8.
III.4.2. Test Runs

The results of a test run are given below in

Figure 3.5. As the broadcast algorithm iterated, the

36

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38

probability of the line labels increased for lines
actually in the picture. However, the labels
introduced by noise increased, broadcast and
propagated, becoming Visible in later outputs (Figures
3.5c, 3.5d). The real lines also extended and
propagated past their actual positions. As a result,
by the 4th iteration the noise and propagated lines
overwhelmed the picture.

Combining the relaxation algorithm with the
broadcast algorithm into the ARLP did not offset the
noise effects. The noise grew much faster (becoming
more probable) than the real line label probabilities
(much like the behavior of cancer).

Thus the broadcast definitions from Page [11] do
not meet the criteria set for them in section III.l
and section III.3. A simple mathematical analysis of
of (12) and (13) pinpoints the difficulty.

III.4.3 Analysis

The updating behavior of (12) and (13) was plotted
for the simple case of a two label system, e.g., one
line label x, and a no-line or blank label, X'. The line
label is broadcast and the broadcast received at node i.
Equation (24) gives the update based on the received line
label as would (13). This is a representative analysis
since the most common neighbor labels of high probability

are blanks and not other line labels.

39

p (A) e p (A) + e p (A') (24)
i 111 121

p (A) + e (l-P (A)) since the original
i 12 i label always retains
188% of its probability
ell=l.8, and in a two
label system
po') = <1—p.(x))

l l
The family of lines for varying e is given in
12

Figure 3.6. The graph depicts the problem well.
Labels of low initial probability are greatly increased
while labels that were highly probable initially
received a small increase. Worst of all, labels with
zero initial probability can instantly become very
probable. This is the source of the noise increase and
line extension-propagation problem.

To illustrate this difficulty consider node i of

Figure 3.7 with p (X3) = 8. Allow it to receive a
i

broadcast of label 3 from the nodes above it and let

e = .4. After one iteration we would have:
12

p (3) p (3) + .4(l-p (3)) (25a)
1 1 1

8.8 + .4 * 1.8 = .4

NEW pi(A)

0.0

1.0
q

.3.

 

40

 

‘015
e17,
ex}
[If]. 09
ON} dfib
8°
00
9
,lb
8‘}
n I J g n l j n 1 _L
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.00 .1 02 O3 .4 05 O6 07 .8 09 100
OLD pi (A)

proposed broadcast algorithm.
pi(1)) grew much faster than high probability ones.
caused the line extension and noise problems depicted in
Figure 3.5 and lead to the redesign of the algorithm.

pi (A) = pick) + e12(1-pi(x>>

Figure 3.6

ORIGINAL BEHAVIOR

Figure 3.6 presents the original behavior of Page's
Low probability pixels (OLD

This

41
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X ‘ 8 - X
X 8 2 X.
X 8 X
X 8 X
X 8 3 X
X £3 3 X
X 3 X
X :3 8 X
X- _§ 8 X
X 2 [_] 8 X
x 2 I . x
X 2 pi(3)=0.0 X
X 1. 1 l 12 X
X X
X X

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Figure 3.7 shows the position of a chosen node i with
probability p1(3) ='0.0. There is no initial probability of a
line of orientation 3 at this node. The node will however,
receive the broadcasts of the line segment of orientation 3
that lies directly above it in the picture. In the originally
proposed algorithm, the reception of these broadcasts could
establish a positive probability of label 3 at node i. The
result is a spurious line segment.

Figure 3.7
SPURIOUS LINE EXAMPLE

42

after 2 iterations:

P.‘3> p (3) + .4<1-p (3)) (25b)
1 1 , i

= .4 + .4 * .6 . = .64
And so on until p (3) 3 l where there was no initial
i

evidence of a label 3. As Figure 3.6 indicates, the

larger e is, the worse the behavior is.
12

The solution is to alter the linear
additive-summation nature of the broadcast update
algorithm so as to forbid the increase of initially
zero probability labels, to allow small increases for
small initial values and to quickly map large initial

probabilities to 1.8.

III.5. REDESIGN
III.5.1. The New Algorithm

The basic algorithm behavior that is desired is
shown in Figure 3.8 as a family of convex curves for

varying values of e . One family of curves meeting
12

the design criteria was developed by introducing a

p (X) factor to the computation of the Experience
1

values as shown in (26) and (27).

It is also desired to have each broadcast

accurately represent the amount of knowledge or

43

1.0.

NEW pi(A)

 

 

 

OLD pi(A)

The desired behavior of the broadcast algorithm is
shown here in contrast to the original behavior shown in
Figure 3.6. Here labels with low initial probability receive
a small increase from one broadcast reception regardless of
the value of eAA that is used. Initially moderately probable
labels receive the largest increase while highly probable labels
move quickly to probability one.

Figure 3.8
DESIRED BEHAVIOR

44

certainty at the broadcasting node. A broadcast

strength factor, str , is introduced at this point to

A
represent the level of knowledge at the broadcasting

node in the calculation of the e. values. The

XX'
equations are given below for the two label case (26)

and the general case (27).

P (A) = e P (X) + str 6 P (X)(l-P (X)) (26)
i 11 i X 12 i i
_—_ for X = X'

1.8 and a broadcast of (27a)

label X received

I
l
I
I
|
l p (X)*str *e when a broadcast of (27b)
l i X XX' label X received
I and X f X'

e' =I

X,X' :

I l-p (X)*str *e A = X' (on the diagonal)
l i X XX' unless case 1 applies (27c)
l
l
l
| 8.8 elsewhere. (27d)
|.__

with 8< e' < l

XX'
and e' =l.8 for all X' as before.
X XX'

Equations (12) and (13) then become

p = B'IA ....X 1p (28)
i 1 b i

45

k
P_(X) =.Z e' p_(x )
1 3:1 A.Aj 1 J (29)
J
The simplest case of a strength factor, str , is
A
b p (A)
m (30)
str = Z ----------------------
X j=l dist(node m to node i)

where p (A) is the probability of the broadcasting
m

label at the broadcasting node m and dist is some
measure of the distance from node m to node i. The
distance calculation can be appropriately chosen to fit
the application.

Although (27) looks confusing it is simply chosen
so as to allow E' to remain a kxk column stochastic
matrix. As an aid to understanding, consider the
following example where a broadcast of label 3 from
node m reaches node i:

Let STR = 2.8 and p (3) = .75
3 i

and the e values in the third row are:

XX'

.6 .5 1.0 .5 .6 .65 .6 .5 .5

E' is then formed as follows: since a broadcast of
label 3 was received the third row of E is

I9
str * p (3) * e I from (27b)
3 i 3,X' X'll

46

which is:
e e e e e e e e e
31 32 33 34 35 36 37 38 39
.9 .75 1.8 .75 .9 .925 .9 .75 .75
with a third element of 1.8 from (27a). The diagonal

is then l-row3 by (27c) to retain column stochasticity

and zero elsewhere by (27d). Thus we have

.25
.25

l l

I .1 8.8 8.8 I

I8.8 .25 I

I .9 .75 1.8 .75 .9 .925 .925 .9 .75 .75 .75 I

| .25 l

'[X l = l .1 |
3 I .875 8.8 l
I .875 I

| . l I

I 8.8 .25 I

I l

I I

| l

I

This revised experience matrix formula is used in
the broadcst algorithm as described in (28) and (29).
All other details of the broadcast operation remain as
previously defined. The properties and behavior of the
new definitions are now investigated.
III.5.2. Broadcast Strength and Algorithm Behavior.

III.5.2.1. Bounds for Column Stochasticity
in the E' Matrix.

In order to retain column stochastic E' matrices

it is necessary to bound the values of str and e

X XX'

for all p (R) values in (27). We can begin with str .
1 Al

47

The test program uses simple Euclidean distance in
(38). Hence with a broadcast pattern as in Figure 3.8
we have:

8 < str < 2*[1/1 + 1/2 + 1/3 + 1/4] 3 4.84 (31)
X

Returning to the two label example of (26), we can

differentiate to find a maximum to guide the analysis.

d
“" P (X) + str *e *P(A)(l-P(X)) = 8 (32)
d p i X XX' 1 i
i
d 2
--—- p (X) + 4.84 e PIA) - 4.84 e p (A) = 8
d p. i M' i )w i
l
1 + 4.84 e -2*4.84 e p (A) = 8
AN xx i
1.8
max e = -----------------
XX' (4.84*(1-2 P (X))
i
lim max e = -1/[4.84*(l-2)] 3 .287
pom —> 1 )x'
1
Hence the maximum value of e 3 .287 insures that a
XX'

column stochastic E' matrix results from the definition
in (27) for the line test program. This result may
also be expressed in general in terms of str as in (33).

A

max e = ---------
XX' max str (33)

48

The result in (33) is of some value in practice even
though it is derived from the two label case. Since
the experience matrices are column stochastic and this
result applies to one column, we can simply apply it to
all columns. In practice it is also found that l<str<3
is a common range. A slightly more flexible bound on
max e results from the lower str value.

XX'
III.5.2.2. Strength and Behavior.

The behavior of the revised broadcast update

algorithm is pictured in Figure 3.9 for e =.2. Using

XX'
the two label case as a simplifying tool, the figure
presents a family of curves for varying strength values

and a single e' value. The graph presents the new

XX'

p (X) values after one broadcast cycle has been
i

received and processed. To obtain an estimate of the

new p value, simply find the initial or old p (A)
i i

value of interest, read up to the strength of broadcast

received, str , and over to the y axis for the new

A

value of p (X).
i

For example an initial p (X) = .6 receiving a
i

broadcast of strength 3 would attain a new value of

P (X) 3.75 by one iteration of (28).
i

49

III.5.2.3. Unbounded e' Values
XX'

A simpler computational form of (27) is now

presented which allows a greater change in p (X) value
1

in one iteration than the maximum a computed matrix

from (27) can provide. Values of e' result that are

XX'
greater than one and hence the e' matrix must be

re-normalized. The formlua is given in (34) below.

E' = STR E (34a)
XX' X XX'

where STR is a kxl column vector containing the

A

row coefficients for the E matrix containing the
strength of the broadcast label X and the p (X).
i
T
STR = [ l ... l, p (X) * str , l ... 1] (34b)
X i X

The effect of (34) is illustrated in Figure 3.18

and 3.11 where the new p take on Values greater than
i

one. This difficulty is resolved in the renormalizing
process, enforcing the upper bound of 1.8 on all values

of e' .

XX'

Since the renormalizing must take place, (34) is a
simpler computational form for multiple label
broadcasts where the experience matrices must be
averaged (or re-normalized). The resulting algorithm

behavior is shown in Figure 3.12. Small initial p (X)
i

50

 

 

é; NORMHLIZED
E MRTRIX

g O<STR<S
EX.X’ :.2

513.00 o'.20 1111f? IRL (3.150 ciao 11.00

A family of curves is presented for varying values of the
strength of a received broadcast at a constant experience matrix
value. The two label case is used here for simplicity. To read
the graph, find the initial value of a node receiving a broadcast
of label on the X axis. Read up the graph to the strength of the
received broadcast and over to the corresponding Y axis value.
This gives the effect of one iteration of the algorithm.

Figure 3.9
BEHAVIOR OF THE REVISED BROADCAST ALGORITHM

51

  

NORMHLIZED
E MRTRIX.
O<8TR<5
EX.X’ =.4

 
  

 

 

D

D

c13.00 0120 0140 0180 0180 1100
INITIHL PI

Due to the strength factors, values in the experience matrix
greater than .207 can cause probabilities that are greater than
one to result in the broadcast processes. Figures 3.10 and 3.11
give this behavior. The graphs are read as given in Figure 3.9.

Figure 3.10
BEHAVIOR OF THE REVISED BROADCAST ALGORITHM

52

 

 

 

B3 NORMRLIZED
E MRTRIX
i 0<8TR<5
E/\ r /\ / Z - 8
20 .00 0'.20 0140 0160 01.80 7.00
INITIRL PI

BEHAVIOR OF THE REVISED BROADCAST ALGORITHM

S3

 

  

 

 

H3 BOUNDED
E MFI T R I X .
g U<STR<5
EX . X ’ : - 8
C13100 0'.20 0'.40 0'.50 01.80 F .00

INITIFIL PI

The results of Figure 3.11 are re-presented only with an
enforced bound on the power of the broadcast updating process.
All probabilities are forced to the correct limit of 1.0.

Figure 3.12
BOUNDED BEHAVIOR CURVES IN THE NEW ALGORITHM

54

are especially boosted by this change.

III.5.3. REVIEW

A review of the first stage of broadcasting is

appropriate here to put each element in order and in

perspective.

1.

Each node is tested to decide if it should
broadcast. If it broadcasts, it does so according
to the broadcast pattern for that label. For an
example see Figure 3.2. The test may be a simple
threshold such as p()) > t. An example of this is
given in section III.2.2

As each node receives a broadcast label, it stores
the received label and adds the strength of that
broadcast to the accumulated strength for that
label at that node according to (30).

When all broadcasts have been received, the
experience matrix for each broadcast label at each
node is computed according to (27) or (34) and as
in the above example.

If more than one label is received by a node the
experience matrix for that node is computed by
averaging as in (23) or through the use of
specially defined multiple label experience
matrices.

Each node's label probabilities are updated by the
experience it receives from the broadcasts using

(28) or (29).

55

6. The second stage of homogeneous unbiased relaxation
occurs to establish local consistency and to
complete the augmented relaxation labeling
process.

III.5.4. TANH
An alternate method for broadcast update does not

involve an experience matrix at all but is based on a

simple function update. The function TANH is a convex

monotonically increasing function asymtotic to 1.0 and

so is well suited to our needs. The update equation is

 

I
| TANH (s*p (X)) if a broadcast of A (35)

 

P (k) = | 1 was received
1 l

I

l TANH(p (X)/s) if no broadcast of X

I i was received
where s = str + 1.6 str defined as in (30)
and X p (X) = 1.0 as always.

X i

The definition involves a small change in the strength
factor usage since TANH ( 1.9 ) = 7/4 = .7853... The s
strength factor here then shifts the TANH values into
the desired range. The behavior of this method is
shown in Figure 3.13 again using the two label
simplification and enforcing the conditions in (35)
(normalizing).

The application of the method is straightforward.
once all broadcasts are received at a node, (35) is

applied to each label x of p .
i

56

 

NORMQLIZED
TRNH[PIJ

1 < S < 6
828TR+1

 

 

 

C’0'.00 0'.20 0140 o'.so o'.80 11.00
IN I T IRL PI

An alternate method to the experience matrices for
broadcast update is to use the TANH function as given in
equation (35). The power of the method is given here as
a family of curves for various strengths of the received
broadcast. The curves are read as given in Figure 3.9.
The behavior here should be compared to the behavior of
the experience matrix update method given in Figure 3.12.

Figure 3.13
TANH FUNCTION UPDATE BEHAVIOR

57

III.6. TEST PROGRAM RESULTS.
III.6.1. The Original Test Picture

Figures 3.14-3.18 present five results of the
redesigned algorithm on the original picture of Figure
3.3. Using exactly the same experience matrix values

e that produced Figure 3.5d, the new algorithm of (27)
MI

and (28) gave the results of Figure 3.14. It is clearly
seen by examining the highlighted probabilities that
even the faintest line is now clear and strong. There
are no noise propagation or line extension problems
after four iterations of the broadcast cycle of the
algorithm only. The label probabilities converge to a
unique interpretation for the labels shown after
several more iterations. The picture background
remains unchanged, spotted with noise, since only line
labels are broadcast at this time by the definitions
section III.3.2. Nodes where blank labels are most
likely, are affected in the relaxation phase.

Figure 3.15 shows the results of a stronger
experience matrix, namely that of Figure 3.12. The
advantage here is that fewer iterations of the
broadcast system alone are required to gain the large
change in label probabilities needed for weak lines.

The relaxation system alone does not give

satisfactory results on picture 1 due to the low

58

initial probabilities of the lines. The results of
such an attempt are given in Figure 3.16. Sufficient
iterations of the broadcast system are required to
build up the label probabilities to the .75 level as
detailed in Figure 3.1. Figure 3.17 shows the poor
results from the ARLP with only one broadcast cycle.
Figure 3.18 gives the more acceptable results from
three broadcast cycles before entering the relaxation
system. Some effects from noise can be seen in Figure
3.18 from the relaxation phase.

The node containing a line intersection (where two
lines of orientation 3 and 8 are present) represents
the problem of an ambiguous labeling predicted in
section 3.3.1, equation (23). The problem is due to
the averaging of the separate experience matrices, and
thus encouraging both labels. The result is a low
maximum label probability (.575 in Figure 3.15) at the
node. A solution to the problem is given in chapter IV.
III.6.2. Picture 2

Picture 2 shows two rectangles and a triangle.
Each of the sides of the figures has a different
initial probability. Intersecting lines and
overlapping corners provide additional problems.
Random noise with a maximum label probability of .25
exist over 5% of the picture. Figure 3.19 gives an

output of picture 2 with the associated initial label

59

probabilities (again highlighted). Figure 3.2% shows
the effect of the relaxation algorithm only. Only the
stongest and longest lines survive. With the augmented
system, only a small weak portion of one rectangle and
part of the triangle are lost as shown in Figure 3.21.
The loss is due in fact to the relaxation cycle and not
the broadcast cycle. Hence a doubled broadcast cycle
initially can again remedy the problem completely as
shown in Figure 3.22. A result from the broadcast
cycles only is given in Figure 3.23 for further

comparisions.

60

III.7. CONCLUSIONS

Let us review the goals of the augmented relaxation
labeling process and the design revisions that allowed
their achievement. The inclusion of a broadcast process
augments the RLP with a separate top-down controllable,
label and node specific enhancing vechicle. The RLP
local consistency and homogeneity criteria are retained.
The ARLP is still amenable to parallel processing.

Selectable levels of enhancing power are available
for the lines through the use of varying experience
matrix values as given in section III.5. Broadcasts
reflect the information at the broadcasting node through
the strength factor, str, which decreases with the
distance the broadcast travels. A full review of the
broadcast process is given is section III.5.3.

Test picture refinement is better due to the
augmented system versus the RLP alone. About the same
number of iterations are required in the ARLP versus
the RLP. In the current line picture implementations,
the broadcast cycle runs in .55-.6O seconds while the
RLP cycle consumes 10.7 seconds per cycle. Result:
enhancement can be specifically directed to detail low
probability or fading lines that the RLP alone misses

at very low cost.

61

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IV. THEORETICAL ISSUES

IV.l. ON COMPATABILITY ESTIMATION

IV.1.1. The Relaxation Rule and Local Consistency
Local label consistency is achieved through

estimation of node-label compatabilities. It is

effected through the summation approximation of p (A)
i

in the relaxation rule (9). This approximation is
based on the labels of the single neighbor node j and

is given in brackets in (9) reproduced here and again

in (36).
K I_ K _|
p (x) TTI X p (MN) p (x) I
K+1 1 j I_>c ij j __
p (M = ---------------------------------------- (9)
1 K K
Zp (x) TT 2 p (Mx') p (w)
x 1 J A' ij j
" K I— K "I
p m = p (ME ) = I 2 p mm p (m I (36)
i i j I_>\' ij j _|
K
The estimate p (A) is also denoted as p (XIE ) as the
i i j

estimate of the label A|at node i based on the evidence
E at node j and iteration K. It is risky to estimate

p (X) on a single neighbor's labels. A more precise
i

estimate involves more neighbors in some neighborhood
pattern or group around node i. Zucker et a1 [8],
chose the neighborhood as the eight adjacent nodes in

the rectangular grid as explained in section 1.3,

75

76

equation 8. The simple arithmetic average of the of

the estimates on each neighbor is used as in (37), (38):

A K K K
p (A) = p (ME . .2 ) =2 c 2 p lex'w (m (37)
i i 1 j j ij x ij j
K
p (M Z c 2 p (Mm p (m
K+l i j ij x ij j
p m = ------------------------------------------- <38)
i K
2 p m 2 c I p (MN) p on
x i j ij x' ij j

The estimate in (37) forms the original arithmetic
updating rule given in (38). Each neighbor

contribution is individually weighted by a factor C .
13'

This estimate is later revised based on the closed form
for the series of (37) to form the present product
updating rule (9), (39). For a discussion of this
derivation see [7].

K K

. .B.) =TT Z p”(>\|>.')p.(>\') (39)
l J 3 >8 13 3

One of the assumptions behind the approximations

" K
P. (k) = 19. (ME
1 l

in (37) and (38) is that of independence over the
neighbors. If it is true that one node's labels are
independent of those nodes and labels around itself, it
is a strange situation to have in the estimation of
local consistency. Each label of each node i is
approximated by those nodes and labels independent of

the label at node i. Obviously it is a statistical

77

simplification which has succeeded by some measure, but
it seems more sensible to attempt to measure local
consistency using as much statistical dependency amongst
the neighbors as is available. This is further detailed
the next section through multiple label compatabilities.
IV.l.2. The Broadcast Rule as a Relaxation Processor
It is instructive to consider the broadcast
process as a relaxation process for comparison purposes.
The following Theorem establishes the equivalence.
THEOREM 4.1 The revised broadcast process of (28),
(29), and (34) can be modeled as a relaxation
process with variable compatabilities and
variable relaxation neighborhoods.

PROOF: We need to work (28), (29), and (34) into

the same mathematical form as (9) and (37) - (39).

p=E'[>\.....>\]p (28)
b 1

i l '
Defining E' as in (34) requires a renormalization so we
write (28) in single label form with the added

factor for restochasticizing as given below.

p (k) = -------------------------- (40a)

x* A}* l b 1
p (k) = ----------------------------- (40b)
1
Z Z 6' [>\ I ~00 I >\ ] P (A*)
>\ M >03 1 b 1
Here (40a) uses the row vector e' , the kth row of E'

>\

78

in a matrix multiplication form with the column vector

p . Equation (40b) is the scalar version of (40a). The
i

normalization is shown as being computed on the final

values of p rather than by forming a column stochastic
i

E' as is done in practice.

There are two tracks of the derivation from this
point. The first uses the definition of averaged
experience matrices from (23) for multiple label
broadcasts. The second retains the original
definitions of special multiple broadcast matrices. We
shall pursue the first and begin by substituting in the
definitions of E' from (23) and (34) into (40a) and

(46b) for those labels receiving a broadcast:

I b l
IZpIM str ethI p
|'=1 1 j x x j' I i
|_ j ____|
p (X) = --------------------------------------- (416)
1
z e' [>\ , ... y). ] p
X X l b 1
IT _|
I X p (x ) str e [k ] | p (A*)
Z l'=l i j x WI 3' I i
w |_ j __|
p m = ---------------------------------------- (41b)
2 2 e' [k , ... ,X ] P (X*)
>\ M M" l 1

Again (41a) presents the vector form and (41b) the

scaler form. Bringing in (36) and regrouping we have:

___ p (A I .___
| b ' j I
l X P (A ) --------- e [A ] I P (A*)
Z Ij=1 1 3 dist(i.jI xx* 3 I 1
I* I___ I
P_()) = ----------------------------------------- (42a)
1
Z X 9' [>\ I 00- IX 1 P (A*)
A x* ;x* 1 b 1
I“B‘ 1 ___I
. I,Z -------:- p (A ) e [K ] p (I II p <x*I
2 |J=l dist(i.3I i j xx* j j j I i
M I_ I
p_(>\) = ----------------------------------------- (42b)
1
Z 2 e' [A . ... .x 1 p Ix*I
A A* AA* 1 b 1
l‘B‘ —_—I
p (x*I I.) c _ p Ix*Ix ) p,(x I I
i |3=1 1) in J J j I
* I___ I
p (A) = ----------------------------------------- (42C)
1
IT _I
pIx*II Z c . p (A*IA I p (A I I
Z Z i I=l 11 in j i j I
A x* I___ ___I
with p. (x*lx ) = p (x ) e [k.] a new compatability

13 j i j AA* 3 estimate for labels
receiving a broadcast

l/ a neighbor weighting
ij / dist(ipj) factor

and c

Labels not receiving a broadcast simply substitute

the e factor from (34) into (40). Since this factor
AA'

is almost always zero, the derivation is not presented.
Form (42c) is a similar form to the arithmetic

updating rule as in (38). The neighborhood estimate of

80

p. is again a sum of pair-wise label compatabilities as

1
shown in the brackets above. The major difference is
in the computation of the neighborhood contribution.
The summation over X' (over all the labels of the
neighbor nodes) in (9) and (37) is gone. Instead only
the single broadcast label is used. In the line
processor of chapter 3 the broadcast label is the most
likely label although it can be any label the control,
T, chooses.

The derivation of the second track for multiple
label experience matrices begins from (40b). Again the
subtitution of the definitions of E' and the further
reorganizations form an equation similar to (41b). The
summation over the broadcast labels (over the index j)

from averaging is now absent. Substituting (34) we have:

X* X XX* 1 b 1
p.(X) = ---------------------------------- (43a)
1

Z Z s e' [X . .X] p (m

X X* X XX* 1 b 1

Where 8 is the combined or assigned strength, and S

X
is the Xth element of the row vector 8. Since these
experience matrices are free to encourage any or all of
the received labels, or even to encourage different
labels, it is difficult to characterize the S values
for labels in general. To continue this analysis, we

will treat those labels which are encouraged and simply

81

retain the broadcast strengths as defined in (30), (34).
Other labels are updated by the e' factor of (34) alone.
As above, e' is often zero and the derivation is omitted.

Expanding the definitions of S and e' we have

1.9
2 --------- PIX)e pIX)p(X*)
X* dist(iyj) i j XX* j j i
pOIXI= ----------------------------------------- (43b)
1
Z 2 e' [X . .X] p (X*)
X X* XX* 1 b 1
1.0
Again let c = ---------
i dist(i,j)

and let p (X*|X ) = p (X ) e so we have
ij j i j XX*

U

I
I

i I113 33
I

I
p_<x*I I c. pogwa p.(X.)
I 1 13 J J

U

Equation (43c) is the analog to (42d) above and the

theorem is proven. **********

When multiple label experience matrices are used,

the estimate of p is not based on sums of pair-wise
i

compatability measures as in the RLP as seen in (36),

(37a) and (38). Instead, direct N-ary compatabilities

82

result through the appropriate choice of experience
matrices and strength values as shown in the following
corollary.
COROLLARY 4.1.1. Multiple Label Experience Matrices
which combine the received broadcast strengths
to encourage a single label make use of direct
N-ary compatability estimation.

PROOF: The key is in the definition of the strength

factor S. Let S be defined as was (34b) as follows:

b
s = [1,...,1, _TI p 0.) str ,1,...,1] (44)
j=1 i 1' II

where the product of the received strengths is in the
kth position in the row vector S when X is the label to
be encouraged at node i.

Substituting (44) into (43a) and expanding the

definition of str we have:

X 1 1 __ * l b j=l j
p (X) = ----------------------------------------- (45a)
1
Z 2 e' [X . .X] p (m
X X* XX* 1 b 1
b 1.0
where c = TT' --------- from str in (44)

i j=l dist(i,j)

Now let p (X*|X ...X ) = e [X ...X ] and we have:
ij 1 b XX! 1 b

83

I b I
XCP(X*)|P(X*IX...X)TIP(X)I
>3 1 i I ij 1 b j ' '__I

ll
...
LJ

The neighborhood estimate of p (X) is the
i

expression in brackets above and given again in (46).

A

P (X) = P (X*|X ...X) 19'— P (X) (46)
i ij 1 b j=l j j

This estimate is the desired N-ary compatability
measure desired. **********

The estimate obtained in (46) should be compared
to the RLP estimates from (36) and (39). The RLP
estimates are sums of products of pairwise compatabil-
ities. As such they are estimates of the correct form
of the equation as used in (46). The DRLP estimate of
(46) is free to exploit the label interdependence
information that is available in the multiple label
conditionals.

IV.l.3. Conclusions

The following conclusions can be drawn about the
broadcast process as described:

1. The broadcast rule can be viewed as an RLP with
dynamic relaxation neighborhood and dynamic
compatability measures. This variability is under

program control and dependent on the incoming picture

84

and the semantic world knowledge about the labels. It
is carried out through the broadcast process and seen
mathematically in (42c) and (43). Thus the information
in the picture is more efficiently exploited. (see
also section IV.3) '

It is also important to note that the dynamic case
is closer to the original definition of Waltz [2] and
the meta-description of Haralick [3,4,5]. Waltz used
particular information about lines and junctions to
resolve ambiguity in labeling. The experience matrices
are set up to hold information about particular labels
as opposed to the RLP compatability matrix which
contains information about all the lines and labels.
Both Waltz and Haralick use semantically defined
N-tuples of nodes in a general graph to desribe these
labeling relationships. The RLP has only pairwise
label compatabilities. True N-ary relations are
possible through the reception of multiple label
broadcasts for which special experience matrices exist
as shown in corollary 4.1.1. and in (46).

Results from a dynamic relaxation system with
multiple label experience matrices is given in section
IV.2.

2. Dynamic relaxation is computationally quicker to
perform due to the absence of the innermost summation.
This is born out in practice where the relaxation rule

computed straightforwardly as in (9) requires 6.5 to

85

6.6 seconds per iteration while the broadcast rule
requires .35 to .6 seconds per iteration. Part of the
savings rests on the fact that not all nodes broadcast
or receive a broadcast. If every node broadcasts and
receives then approximately 1.8 seconds per iteration
are required. (These figures are taken from the timing
data in the pictures of chapters III.6 and IV.2.)

3. Dynamic relaxation can be seen as a first term
approximation to the neighborhood contribution series
(36) of the full RLP when the broadcast label is the
most likely label. The overhead of finding the most
likely label at each node is akin to the ordering of
each branch of a tree sort prior to the sorting of the
whole list. A faster sort results overall. The same
is true here: broadcasting is an improvement over the
product rule as the product rule improves the
arithmetic rule. The gain is not only in execution
time but also in resolving power as shown in the
results of chapter 3.

It is interesting to note that previous attempts
to improve the speed and convergence power of the RLP
have centered on homogeneity. The solution is in
irregularity instead (just as with the complexity
solutions Waltz found).

Haralick has shown that general relaxation is an
NP complete problem. Dynamic relaxation is also NP

complete but faster by a factor of k, the number of

86

labels in the label set.
4. A method for defining the variable compatabilities
is given by (42) based on a constant world knowledge

factor e and the broadcast strength. The world

XX'
knowledge here reflects any information available about
structure and interrelationships of objects. There is a
certain flexibility in the numerical specification of
the world knowledge due to the broadcast strength. One
need not be exact as suitable scaling is done by the
strengths for each broadcast.
5. Averaging single label experience matrices for
multiple label broadcasts also requires the assumption
of statistical independence over the neighboring nodes,
as seen in the resulting similarity of (42) and (37).
Hence averaging of single label experience matrices is
no worse statistically than the RLP. The importance of
the developement of multiple label matrices is
highlighted since such matrices embody the multiple
label conditional probability information and as such
are free to exploit statistical dependencies for
consistency approximations.
6. Convergence time and fixed point behavior of the
broadcast process is dependent on the definitions of
the broadcast patterns. This is further discussed in

section IV.4.

IV.2. DYNAMIC RELAXATION LABELING
IV.2.1. DEFINITIONS

The mathematics behind the concept of dynamic
relaxation was presented in the preceeding section by
the development of equations (42c) (43d) and (45). In
dynamic relaxation the broadcast algorithm is used as a
stand alone system. Since every node must be processed
in such a system, every node must receive a broadcast.
To accomplish this, most every node will also send a
broadcast. Every label will particpate, including the
blank label, and not just the line labels as in the
ARLP.

The broadcasting pattern for no-line or blank
labels is given in Figure 4.1. A broadcast of a blank
is sent to the 8 adjacent nodes of the broadcasting
node. The experience matrix for the reception of a
single blank label is given in the appendix.

Experience with blanks shows them to often be either
picture background and or boundary nodes for lines. As
background, the single label experience matrix for a
blank uses a constant value to describe its relation to
the different line labels. This allows the experience
matrix to control background noise of any label
equally.

The experience of blanks as line neighbors should
encourage the formation of lines. This follows from
the character of a line and is easily seen in the

87

88

 

 

 

 

 

 

* * *
* .45 *
* * *

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The blank or no-line 1abel’6 has a broadcasting
neighborhood as shown in Figure 4.1. The pixels

represented by an asterisk (*) can be reached in one
broadcast.

Figure 4.1
BROADCASTING PATTERN FOR NO-LINE (BLANK) LABELS

89

relationship of lines to blanks in a 3x3 pixel
neighborhood. Simple averaging of the blank experience
matrix given above with the line label experience
martices used previously does not give satisfactory
results. One matrix encourages the line while the
other treats it as noise to be discouraged. The
expectation of such an average is that weak lines would
be lost and line ends would be shortened.

The results of such a system are given in Figure
4.2 and 4.3 for the test pictures. In picture 1 the
low probability lines and line ends are lost to blanks.
In both pictures noise is adequately controlled
although the convergence rate for lines is slower.
Simply adjusting the values of the experience matrix is
insufficient to resolve the problem. Too low of a
value will not affect the noise. Too large a value
removes the lines. The solution is the same solution
that was suggested for the line intersection problem as
indicated in section III.3.l.
IV.2.2. Multiple Label Experience Matrices

Multiple label experience matrices are introduced
to solve ambiguity problems when multiple labels are
received. Matrices for blank and line label
combinations and for line intersections are developed.

When blanks are present in a received broadcast

the experience matrix used is chosen based on two factors:

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94

the number of blank broadcasts received and the
received strength of the line label broadcasts. There

are three possible outcomes as given by the table

below.

Table 4.1
BLANK LABEL EXPERIENCE MATRICES

1. Line encouragement

E[b, X] => Ele for str *p (X) > 0.0
X i
and 6 or fewer blank
labels (b) received

2. Noise control
E[b, X] => E[b] for str *p (X) = 0.0
X i

or when 8 blank labels
(b) are received

3. Noise control, default
E[b]/3.fl + EIX] for str *p (X) >a.0

E[bl X] => ----------------- x 1
2.0 and 7 or 8 blank (b)

labels received

The choice of the count of blank label broadcasts
received is based on the relationships of the line in a
3x3 pixel neighborhood. The strength factor str*p (X),
is simply an existence criteria of label X at the 1
receiving node. Case 3 is in some sense an indeterminant
case where iteration must resolve the ambiguity. The
strength of the blank broadcast is reduced so as not to

completely dominate the line label broadcast even

though the label existence may be due to noise.

95

When several different line labels are received,
the possibility of.a line intersection exists. To
label an intersection, an additional label, I, is added
to the system. The label I is encouraged based on the
received strength of the labels as given below.

Table 4.2
INTERSECTION LABEL EXPERIENCE MATRICES

4. Intersection
E[X , X ] => E[I] when str * p (X) > t
l 2 Xj i

for 2 or more labels j.
t a suitable threshold

5. Indeterminant

E[X ] + E[X ] for str *p (X) > t
E[>\I>\]=> 1 2 xi
1 2 -------------- for less than 2
2.0 labels

The choice of a threshold in case 1 is dependent
on noise present in the picture. The application of
the I label is not appropriate simply when a broadcast
for 2 labels is received. Noise may exist at the node
to make the received strength non zero. In such a case
the intersection label could be inappropriately chosen.
If there is a strong presence of several lines at the
node then an intersection is likely. In the
indeterminant case, iteration must resolve the

difficulty.

96

In both tables the definitions of the multiple
label matrices is given for a label pair. The
definitions should be considered as operators on the
received label set where necessary. For example, a

received label set of [X,, X , X , blank] can apply
1 2 3

both rule 1 and rule 4 to yield E[I] or perhaps rules 3
and 5 to yield the default case of an average over the
single label matrices.
IV.2.3. Results

The dynamic relaxation labeling process (DRLP) was
applied to the second test picture. The results are
given in Figures 4.4 and 4.5. The results of Figure
4.4 use a threshold of .75 for cases 4 and 5. After
one iteration, most of the noise is overcome, many of
the lines are uniquely labeled, and the intersection
label has appeared for the line intersections at the
sides of the rectangles. In the second iteration, the
intersection label is applied to the triangle rectangle
intersection. Since so many labels shared this
intersection an iteration was needed to build up the
broadcast line strength to meet the threshold. By the
5th iteration almost every point is uniquely labeled.

These results should be compared to the results of
Figures 3.14-3.20. Correct convergence is much more
quickly obtained. The rectangle corners are not line
intersections as drawn. One line ends and the other

begins. In this case the discrimination is correct.

97

Corner labels could likewise be added to reflect the
different types of corners that Waltz describes [2].

The results of Figure 4.5 use a threshold of l.@.
After 4 iterations, the intersection label was not
applied at the triangle rectangle intersection. Almost
all other nodes are uniquely and correctly labeled.

Hence the choice of threshold value is significant.

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IV.3. INFORMATION MEASURES

The effect of an iteration of a relaxation or of
an broadcast process can be modeled using information
theory. This model is presented below. The following
sections present interpretations of the common
information measures and actual values for the RLP and
broadcast systems studied.

IV.3.l. The Model

The Shannon information communication model
[l4],[lS], centers around a channel through which a set
of messages pass. A set of source symbols, 8', makes
up these messages. Each symbol has an attached
probability which describes the probability or
frequency with which the symbol is transmitted through
the channel. The output message is composed of the
symbols in the output symbol set, 8. Again there is a
probability distribution over the output symbols
describing their probability or frequency of being
received. The channel is described by a column
stochastic conditional probability matrix which yields
the probability of a particular output symbol given the
input symbol.

In a relaxation labeling process or broadcast
process the information used to update the probability
distribution of the label set,,X, at a node, i, comes
from the probability distribution over the label sets,
X', of the neighboring nodes, j, according to the

102

103

relaxation or broadcast neighborhoods. The relaxation
systems fit the Shannon model as follows. The input
symbol set is the set of labels at a single neighboring
node with the associated probabilities. The channel
output is the neighbor contribution to the updating of
the label set at the receiving node as a set of
probabilities over the label set,X. The channel matrix
is simply the compatability matrix, p.. or the

13
experience matrix E'. A message is a single label.

The existence of a channel between any two nodes
is dependent on the relaxation and broadcast
neighborhoods. For the RLP, the homogeniety property
requires a channel between a node and each of its 8
adjacent neighbors. In the spirit of homogeniety, this
channel should have identical properties for each
neighbor. In the line processor, the compatabilities
depend on the angle between the node and neighbor (see
(20)). Hence there are eight separate and distinct
channels, one for each neighbor position.

In the broadcast system, a channel exists between
any two groups of nodes that is desired as specified by
the broadcasting patterns. In the line processor, all
broadcasts of one label received at a node are pooled
together and the strengths summed (see (30)). Hence, a
channel exists from a group of nodes with the same

broadcast label to the single receiving node. This

104

arrangement is depicted in Figure 4.6.

In the process of an iteration, information flows
through the channels from the neighbor nodes j to the
receiving node i. This information influences the
probabilities over the labels at node i as described in
IV.1. If on a later iteration, information from i
reaches node k, there is the effect of a cascade of
channels bringing information about j's labels to k
through i. Consider the case where node j is allowed
to broadcast directly to node k. A direct channel
exists and the information about j reaches k in one
iteration of the broadcast system. In the RLP however,
two iterations are required and there is the additional
loss to noise in cascaded channels as is well known.
The situation is illustrated in Figure 4.7.

Such cascades are the way local consistency is
developed into a more global consistency in an RLP. The
cascades in each case are combinations of the eight
known channels. The broadcast system, in contrast has
as many direct global channels as world knowledge
permits. In fact, each channel is custom tuned by the
parameters of (29) to carry its particular broadcast
directly. Cascades of broadcast channels also exist
spanning even greater distances. It is difficult to
compare such complex systems in a general mathematical

way and will not be attempted here.

105

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NODE
CHANNEL 1

 

 

 

The single channel used by the four nodes broadcasting
label X5 to the receiving node i is depicted.) In the broad-
cast process, identical labels arriving at a node all use
the same channel. Compare this figure to Figure 3.2.

Figure 4.6
BROADCAST CHANNELS

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ITERATION I
ZODE
X l
J'
NODE
:9
CHANNEL 1
NODE
k
ITERATION I + 1
NODE
A
"-~a-—- j
I
\ NODE
‘~:::::::::=[> 1
CHANNEL
‘ NODE
h __i) k
CHANNEL

 

 

 

Information about label A at node j is transmitted to
node i at iteration I. This information is further transmitted
to node k at the next iteration. The resulting cascaded channel
from j to k with the attendant losses due to noise models the
RLP local consistency process.

Figure 4.7
CASCApED CHANNELS

107

IV.3.2. The Measures

The standard entropy and channel measures are
applied and interpreted below. The source entropy,
H(X'), gives the average amount of information in each
label of the source alphabet.

(47)
k
0.0 < H(X') = -2 p (X') 109 p (X') < 109 k

X' j 2 j 2
A value of zero for H(X') indicates a unique labeling,
where one label has probability 1.0. A value of log k
indicates that each label is equally likely or that the
picture labeling at that node is completely ambiguous.
In the case of a high entropy, it is expected that
there would be little information available to the
local consistency estimation. The broadcast threshold
system of the line processor excludes this case as
described in section III.3.l.

Output entropy is similarly defined for the
neighbor contributions (eg (37)) over the channel
output label set‘X. This is different than the final
label probabilities from one iteration of the RLP (eg
(36)) because of the normalizing process. A second
output entropy can be defined over the final label
probabilities if desired. To distinguish between the
two we will refer to the first as channel output
entropy and the second as final output entropy.

One very basic measure of the effectiveness of the

algorithms is the decrease in entropy from the initial

108

label probabilities, the source entropy, to the final
output entropy, over the final label probabilities. To
compute the entropy drop, subtract the final output
entropy from the source entropy as computed by (47).
Two measures of the channel are the dispersion and
equivocation. The Dispersion or channel entropy,
H(X|X') is a measure of the uncertainty of the received

label X when it is known which X' was sent.

H(X|X' = - E E P(X'.X) 109 P(X|X') (48)
X' X 2

0.0 < H(X|X') < log k
2

The dispersion is a measure of information lost in
transmission, dispersed from the input symbol over the
output symbols. The higher the value, the more the
input is spread out or lost. In a deterministic
channel, the dispersion is identically zero as each
input is mapped to one output (eg each column of the
channel matrix has one value of 1.0). Dispersion is
usually desired to be as low as possible.

The Equivocation or noise entropy, H(X'|X) is a
measure of information lost in transmission due to
noise. It is the uncertainty about the received
symbol, X', averaged over all received symbols. Again
a small value here is prefered. In a lossless channel
the equivocation is zero (eg each row of the channel

matrix has one value of 1.0).

109

k

k
H(X'IX)= - g ; pLX'.X) log p(X'|X) (49)
' 2

0.0 < H(X'|X) < log k
2
The equivocation is the measure of loss often used in
channel cascades. An ideal channel is both lossless
and deterministic.

The Mutual Information, I(X',X), is the average
amount of information received per symbol transmitted.
As such it is a measure of the information transmitted
through the channel. In the relaxation model this
corresponds to the amount of information received for
label updating from the neighbor(s) on the other end of
the channel. Obviously the larger this value is, the

more effective the iteration can be in the updating

process.
0.0 < I(X'.X) = H(X') - H(X'IX) < 109 k (50)
I(X'.X) = H(X) - H(XIX') 2
I(X'.X) = I(X.X')

The Channel Capacity, C, is the maximum
information that can be received per symbol
transmitted, using an average symbol. It is a measure
of the channel's ability to transfer information
clearly, without loss.

0.0 < C = max I(X,X') < log k (51)
over p(X') 2

110

The channel capacity is then a measure of the maximum
average information received per symbol transmitted.
As such, it is a measure of the amount of knowledge

in the compatabilities on how to use the neighbor
information. As the channel capacity rises, so does
the potential information flow from the neighbor nodes
to the node being updated. The measure of actual
information flow is given by the mutual information

as stated above.

The difference between the mutual information and
the channel capacity reflects how well the capacity is
used for a given input. As is seen in (50) above, the
capacity is greater when loss due to equivocation and
dispersion is low. In a deterministic channel
therefore, C is the maximum: log k. The channel
capacity is in general very difficult to compute. It
can be done for the special channels mentioned above
and by a Gauss-Jordan (relaxation) technique if the
channel matrix is non—singular.

IV.3.3. Channel Capacity in the Line Processors

The channel capacity of the RLP compatability
matrices cannot be directly computed since they are
singular. The experience matrices are also singular in
general. An exception where capacity can be computed
is the case of the reception of strong label
broadcasts. The single label experience matrices and

the intersection label matrix become deterministic

111

channels for strong enough broadcasts. This allows at
least a partial characterization of DRLP capacity as a
function of the broadcast strength parameter: str .

X

This is seen in the following theorem.

THEOREM 4.2. The channel capacity of the single label
experience matrices is dependent on the broadcast
strength with a maximum of log k.

PROOF: We will show that the single label experience

matrices approach a deterministic channel for large

enough values of str . The channel capacity for the

X

deterministic channel is then computed.

The single label matrix is described in (34) as a
row vector for the broadcast label and a diagonal. The
elements in the row are monotonically increasing,

dependent on str , to a maximum of 1.0. The diagonal

A.

elements decrease correspondingly to maintain column
stochasticity. This behavior is illustrated in Figure

3.12 and in (52).

lim e' = 1.0 for the Xth row (52)
str -> 00 XX'

X

lim e' = 0.0 elsewhere
str -> 00 XX'

X

(str need really only approach a small integer as shown
in Figure 3.12 rather than infinity.)
For sufficiently strong broadcasts, the resultant

experience matrix has only a single entry, a 1.0, in

112

each column and is thus a deterministic channel. The

particular strength value needed can be deduced from

the initial experience matrix values in (34) and as

shown in the behavior curves as described in section

III.5.

In the deterministic channel, the dispersion (48)

is zero. Substituting in the experience matrix scalers

from (34) and (52) the desired relationship between

channel capacity and broadcast strength is established.

k k
H(XIX')= -Z Z p(X.X') log p<XIX')
X' X 2
k k
9 - Z 2 p(X') p(X|X') log p(X|X')
X' X 2
k k
= ’ 2 ; P(X') 8' log e'
X' XX' 2 XX'
lim H(X|X') = 0.0
Str -> oo
lim I(X,X') = H(X)-H(XIX') = H(X)
str -> oo
lim C = lim max I(X,X) = H(X)
str -> oo str->00 X'

(48)

(53a)

(53b)

(54)

(55)

(56)

In the limit of (54), as e' goes to 1.0 for the

Xth row the log e' factor goes to 0.0. In all other

terms the factor e' goes to zero and so every term in

the series goes to zero and the dispersion becomes

113

zero. Hence the mutual information depends only on the
source entropy as given in (55). Since the channel
capacity is the maximum mutual information, and the
mutual information depends on broadcast strength, for
strong enough broadcasts the channel capacity depends
only on the source entropy H(X). Hence the channel
capacity is log k for strong broadcasts. **********

It is important to note the dependence of the
mutual information on the broadcast strength. It is
the mutual information that is the measure of the
channel effectiveness. Hence the effectiveness of the
broadcast system increases as the strength of the
broadcasts increase. This is intuitively pleasing
since the strength is designed to represent the
certainty of the label being broadcast.

IV.3.4. Measures for the Line Processors-

The information measures defined in section IV.2.2
are computed for each node in the second test picture
for the RLP, ARLP and DRLP. A picture wide norm for the
entropy drop, mutual information and dispersion is
given for each iteration. For the broadcast systems,
there is only one channel per node. So computation is
straightforward. In the relaxation systems, the mutual
information and dispersion are computed for each of the
8 channels and then averaged to get single measures for
the node. The norms are averages over the nodes

processed by that particular system.

114

The measures are presented in Figures 4.8-4.11 for
the ARLP and DRLP systems. Figure 4.8 is a repeat of
the ARLP output of Figure 3.21 but including the
information norms. Figure 4.9 presents 16x16 entry
tables of node measures. For brevity, only the first
two iterations are presented. Figures 4.10 and 4.11
present similar results for the DRLP output of Figure
4.4. Figure 4.12 presents the results of Figure 3.20
from the RLP for comparison.

Several observations are in order. The dispersion
values for the relaxation system is close to the
maximum of logzk = 3.1699 while the dispersion for the
broadcast systems is close to the minimum as predicted
in Theorem 4.2. What is happening is that the RLP is
spreading the input information out over the entire
label set, then choosing particular portions of the
information based on the receiving node probabilities
and renormalizing it. The effect is similiar to
mashing a potatoe, finding the largest remaining lump
and using only the lump. The reason the RLP has any
success at all is because the compatabilities are set
to make lumps representing compatable labels a little
larger than the others.

The broadcast systems by contrast, concentrate the
input information around the received label as directed

by the broadcast reception. This is a far more

115

appealing way to utilize world knowledge.

The node entropy drop figures are also intriguing.
The broadcast systems have a greater ability to
decrease the labeling entropy. As Figure 4.9 shows,
the relaxation system actually increases label entropy,
increases ambiguity, for some nodes.

The mutual information figures, channel
effectiveness measures, also are uniformly better for
the broadcast systems. In fact for the RLP the mutual
information for a particular channel can even be zero.
This indicates that on the average label, no
information was received from the neighbors at all.
IV.3.5. Conclusions

The information model provides a well recognized
and understood model for the relaxation process.

Simple numerical measures result by which the
compatability or experience matrices can be evaluated
as channels. The channel capacity gives a measure of
potential, while the dispersion and equivocation show
the nature of the information losses. Mutual informa-
tion is the performance measure for an iteration. It
can be a useful measure for the stopping of computation
which is not worth the cost for the information gained.
Such measures give insight into design criteria as
well. Matrices can be trained or chosen to meet

specific operational or selection criteria.

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121

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IV.4. CHARACTERIZATION OF FIXED POINTS IN AN ARLP
IV.4.1. RLP Fixed Points

Zucker et a1 [7] characterize the fixed point
(where all label probabilities remain unchanged by
succesive iterations) of an RLP. There are three cases
which result in a fixed point and only these three.

The cases are based on the label probabilities p (A)
i

and the neighborhood contribution q (A) to that label
i

probability in the RLP local consistency estimation.

Case 1: p (A) = Q for all i, A. The corresponding
1

q(A)=B for all A where B is a scaler including 0.
i

Case 2: p (A) = 0 for some A with the corresponding
1

q (A) = B as above. The remaining q (A) are
i 'i

arbitrary (eg for those A such that p (A) = 0)
1

Case 3: p (A) = l for one A at each i and p (A) = 0 for
i i

all other A. This case provides a unique labeling
at each node i.
To understand these cases consider the nature of a fixed

point. At a fixed point we necessarily have

p - p = g (or < epsilon) (57a)

124

125

By expanding the definition of an iteraton we have:

K
I q (A) I
K I i l
p (A) I --------------- - 1 I = (5 (57b)
1 I K K |
I X p (AM; (A) I
I A i i I

Because the factor in brackets must be zero we have:
K K K

q (A) = X p (A) q (A) (57c)
i A i i

where q (A) is defined as p (A) in (37). Since the right
i i

side of (57c) is a constant for each i we have case 1.

If p (A) = 0 it is clear that q (A) may be arbitrary
i i

which is case 2. Case 3 is the simple identity q = q .
i i

Whenever the RLP fixed point is characterized by
case 1 or 2 it is clear that the local consistency
neighborhoods have no further information to disambiguate
the labelings. The information must come from the
broadcasts.

IV.4.2. Broadcasts and RLP Fixed Points

When a broadcast arrives at a node, the change in
probability at the node is described by the behavior
curves of Figures 3.9 to 3.13. This section describes
the effect of a broadcast on a node i when the broadcast

is received by a neighbor of node i.

126

Assume a node i is at a case 1 fixed point. (case
2 being a subcase of case 1 it is sufficient for now to
consider case 1 only). Allow a neighbor node ”a" to
receive a broadcast of label A'.' The effect of the
broadcast on the fixed point behavior at node i is
characterized by the following derivation.

The neighborhood contribution of the RLP given
in (57c) is as in (37) which is repeated here as (58).

K K
q.(A) = Z P. W; P._(A|A') p.(A') = B (58)

1 A 1 3 ' 13 3

Since only one node has received a broadcast the
neighborhood contribution from the other nodes remain
constant. Thus we can expand (58) to highlight the

change at node "a":

(59a)

l—J —II_ _I

q (A) = 2 p (AII TT E p(AIA')P (A'HI; p(AIA')P(A)I
i A i :j=a ' ij j H ' ia a I

I‘ "I I“ ‘I
I p (A)lCon(A)l I § p (AIA')p (A')| (59b)
A ' I_ ' 1a a _I

1 |_ _|

by definition of the fixed point having constant
neighborhood contributions, Con(A) is a set of

constants, l for each A.

127

We need now only be concerned with the second
quantity in brackets, the linear combination of the

p and p , and its behavior for each A.
ia a

Consider n(A) as this neighborhood contribution from
neighbor a for label A of node i
k
n(A) = 2: p (AIA') p (A') (60)
A' ia a

K
Now for node i to remain at the fixed point the n (A)

K+l
must equal n (A) where the K+l indicates the

broadcast iteration. Hence we need to study (61)

k K K ? k K+l K+l
2 p (AIA') p (A') = E p (AIA') p (A') (61a)
A' ia a A' ia a
K ? K+l
or simply: n (A) = n (A) . (61b)

By definition of homogeneity we have:
K K+l (62a)
P (AIA') = P (AIA')
ia ia
And by the definition of a broadcast to the neighbor

node only we have:

(62b)

K K+l
p (A') # p (A') for one A* and for

a a at least one other

A' say A+

we can further state

K+l K

P (A*) > p (A*) (62¢)

a a

128

K+1 K
p (A+) < P (A+) (62d)
a a

now if p (AIA*) = P (AIA+) then we have
ia ia ‘

K K+1
n (A) = n (A)

and the RLP remains at a fixed point at node i in Spite
of the received broadcast at neighbor node a. This is
an unlikely event. In the more general case of (61)
where there are several A+ in (62) it becomes much more
difficult to analyze the linear combinations of (61),

yet equality is even more unlikely.

In conclusion, to the extent which we can define
the behavior of (61) is the extent to which we can say
that a broadcast reception by a node moves the RLP away
from a case 1 fixed point for the nodes in the
relaxation neighborhood. Page [16] has further shown
that the change must be sufficient to change the
Eigenvectors of the governing matrix to obtain a change
in the fixed point behavior. The governing matrix here
is the matrix Q of the Q(A) for all labels‘A (contribution
of the relaxation neighborhood). Hopefully the movement
will be sufficient and towards an unambiguous labeling.

If we consider a broadcast received at node i only
and not at any neighbors we can write (58) following (59):

K+1

q (A) = P (A) Con(A) (63)
i A i

129

since there has been no change in any of the neighbors
contributions. We are left in an identical position in
terms of analyzing the linear combination of (63) as we
were for (59). There exists the remote possibility
that the broadcast will not change the fixed point.

Let us assume there is a change away from the
case 1 fixed point. If the change is sufficient, the
iterations will move the RLP towards a new, and
hopefully unambigious, fixed point. If not, the
iterations will simply return the RLP to the same fixed
point.

The situation of the reception of multiple label
broadcasts is even harder to analyze because of the
uncertain nature of the experience matrices for these
broadcasts. We cannot state (62c) quite so cleanly as
there may be several A* s which behave so. This in
turn further complicates the analysis of (61).

With a no-information fixed point the unbiased RLP
should remain at the case 1 fixed point. This is the
situation where there is no information in the picture
at all, each label being equally probable. In this
picture there is no unique labeling in existence since
there is nothing to label. It should be noted that the
broadcast system will not broadcast any label here, as
described in section III.2.l. Thus the "Correct

Labeling” will be maintained.

V. MOTION PREDICTION
v.1. MOTIVATION
The behavior of an ARLP noted in section III.3.3

and Figure 3.6 has an unexpected redeeming application.

The property that allows an ARLP to create an non-zero
probability for a label that had zero probability
caused line extensions and noise propagation. When
motion is present in a sequence of pictures the ARLP
can utilize this property to predict the position of
the moving object in the next picture in sequence. In
other words, the result of an iteration on picture ”i"
can be used to predict the position of objects in, and
hence the labeling of, picture 'i+1”. The inverse of
the prediction mechanism models the removal of the
moving object from its present position.
v.2. DEFINITIONS
V.2.1. Experience Matrices

Two types of experience matrices are required to
transform a picture with moving objects into a model
for the next picture in sequence. These are the
"create" and the ”remove” matrices. The create matrix
is used when a node receives a broadcast of a label
that describes an object in motion. The matrix
restructures the label set at that node so the object
will appear at that position in the next iteration.

The matrix is a kxk column stochastic matrix as
described in section II.3. The row of the matrix

130

131

corresponding to the received label will contain ones
in the columns of labels that describe background
objects which the moving object will hide from view,
and zeroes for foreground objects which hide the moving
object.

The remove matrix restores the label probability
of background objects previously occluded. It is used
when a broadcast of a label for a departing object is
received at the node presently containing the object.

The row of the matrix corresponding to the
broadcasting label A is all zeroes, effectively setting

P (A) = 0.0. The corresponding column details which
1

background labels will receive the probability mass
formerly belonging to the moving object. The diagonal
elements are all 1's in the other matrix columns.
Again the remove matrix is kxk column stochastic.

The restoration of background labels by the remove
matrix is by no means a clear cut issue. It is here
that the local consistency properties of the RLP are
desired. Once an initial labeling of the region is
accomplished, the RLP will reconcile small
difficulties.

V.2.2. Broadcast Patterns

Two distinct broadcasts must be made as opposed to

the single label broadcast of the previous definition.

The first is a broadcast of the label in the direction

132

of the moving object. This broadcast, when received,
will use an appropriate create matrix in the update
calculations (12) and (13) so as to establish the
moving object in its correct position. The second
broadcast of the label is to the node containing the
object. The remove matrix will be used in this case.

The actual broadcasting unit will be a knowledge
source concerned with trajectories and curve
fitting rather than the node itself. This is necessary
due to problems of occlusion which may cause the moving
object to temporarily vanish. Under the old definitions
such an occluded object could never reappear. (This
effect parallels an infant who equates sight with
existence.)
v.3. EXAMPLE

Consider the example of a simple scene involving
four labels: an airplane, sky, ground and clouds as
pictured in Figure 5.1. For the sake of simplicity, let
us assume that the plane is traveling in a horizontal
trajectory. The broadcast pattern is given in Figure
5.2. The circles show nodes which receive a type 1
broadcast and using a create matrix. Squares show a
type 2 broadcast and remove matrix. Example experience
matrices are given in Figure 5.3. The broadcast update
procedes as in (12) and (13). The relaxation cycle
then restores the local consistency of returning

background labels.

133

 

 

 

W

A SIMPLE SCENE WITH A MOVING OBJECT

 

 

 

 

 

A IPLANE
1

AZ-SKY

A3'GROUND

AA'CLOUD

0 Those nodes receiving a type 1 broadcast

D Those nodes receiving a type 2 broadcast

Figure 5.2
NODES RECEIVING BROADCASTS

134

P S G C
I__ __
l l l l .8 | FIG. 5.3a A create matrix
E[p1=laaaa|s
l l I 0 0 0 0 IS for a cloud as a background
I 0 0 0 .2 I
I__ .__| label.
P S G C
I__ __I
I l l l .2 IP FIG. 5.3b A create matrix
B[p]=loeoa|s
l 2 I 0 0 0 0 IG for a cloud as a foreground
I 0 fl 0 .8 IC
I__ __I object.
P S G C
I__ __l _
l 0 0 0 0 IP FIG. 5.3c A remove matrix
E[P ] = l.75 l 0 0 IS for sky and ground as the
2 I.25 9 l D IG background objects that
I 6 0 a 1 IC regain visibility.
I__ __I
P = Plane S = Sky G =Ground C = Cloud

Figure 5.3
EXAMPLE EXPERIENCE MATRICES

135

v.4. DISCUSSION
V.4.l. Occlusion

The presence of an object like a cloud raises
several interesting issues about the level of world
knowledge needed by a motion detection system. Some
depth or distance information is needed for each object
in order to determine whether the plane will pass in
front of or behind (or in) the clouds. Once this is
known the correct experience matrix can be chosen.

The matrix in Figure 5.3a is an example of the
cloud as a background object while allowing for a small
possibility that the cloud may occlude the plane (the
.2 in row 4 column 4). Figure 5.3b is the example for
the cloud as a foreground object, this time with a
small possibility of the plane peeking through the
clouds. The cloud may completely vanish behind the
plane just as the sky does or vice versa.

V.4.2. Expanded Label Sets

Additional labels with special semantic properties
can be of use here. For example: two varieties of
cloud labels, one designating an occluding cloud and a
second a background cloud. The choice is then in which
label to support rather than in how much to support a
single label.

The same situation exists with the ground label.

We can differentiate level ground from occluding

136

mountains and from background mountains. The plane can
then fly in front of or behind the mountains and land
on the level ground. Landing disables the broadcast
function as the plane is no longer moving. In each
case the particular experience matrix needed is
requested in the broadcast mechanism.

V.4.3. Motion Prediction and Curve Fitting

Broadcasts are controlled by the knowledge source
in charge of the trajectory of the moving object. This
trajectory is continuously updated by a curve fitting
routine. Actual positions are compared to predicted
positions for fit refinements or to reflect a change in
course. The knowledge source must have a good
understanding of kinematics.

Three dimensional trajectories involve more than
simple translation. Translational information only
positions the broadcast pattern. The broadcasting unit
must include rotation and scale information in chosing
the size and shape of the broadcast pattern. A plane
that curves and flies off into the distance is a
perfect example.

If there is uncertainty in the prediction
mechanism as to the next position, several possible
positions may be selected. Multiple type one

broadcasts are sent, one to each of the various

137

possible positions. A strength factor is employed to
reflect the certainty of the predicted positions. The
resulting multiple images are resolved with the
labeling of the next picture in the sequence. The new
trajectory is then entered into the curve fitting
routine.

V.4.4. Multiple Label Reception

A large, slow moving or partially stationary
object may generate type one and type two broadcast
patterns which overlap. Hence a node receives
contradictory experience which requires both the
re-creation and removal of the object. It is
intuitively clear that the type one broadcast must take
precedence. The object does not vanish but rather just
moves along. The type two broadcast may be ignored, or
in special cases, modify the type one matrix.

Reception of two type one broadcasts of different
labels presents a slightly different situation. The
solution is semantically dependent on the labels
involved. A moving cloud can cause a broadcast as well
as a plane. The foreground and background information
in this case will be the deciding factor as to which
object will remain visible. Both examples are cases
where a special multiple label experience matrix is a
necessity. Averaging the single case matrices can not

be expected to give the desired result.

VI. CONCLUSION AND FUTURE PROBLEMS
VI.1. SUMMARY AND CONCLUSION

This thesis has examined relaxation labeling
systems for computer vision. The idea of augmenting
the homogeneous relaxation system of Zucker [7,8] with
the broadcast processes of Page [11] has been studied.

The definitions of the brodcast process have been
further revised and tested on a series of simple line
drawings. This Augmented Relaxation Labeling
Processor, (ARLP), is shown to give a more correct
labeling in hard to label or ambiguous areas of the
picture. The added power of the ARLP is due to the
world knowledge available about the structure and
interrelationships of the objects in the picture. This
knowledge is embodied in the experience matrices (a
form of pairwise label compatability matrices.)

The broadcast process is also modeled as a
relaxation process over the entire picture. To
accomplish this, an experience matrix must be defined
for every label in the picture and every node must
receive a broadcast. The relaxation process is no

longer a homogeneous one since the broadcasts use the

label dependent experience matrices and label dependent
broadcasting patterns in every neighborhood
contribution computation. The resulting processor is
termed a Dynamic Relaxation Labeling Processor, (DRLP).
The DRLP is likewise tested on simple line drawings and

138

139

shown to be superior to both the ARLP and RLP, as well
as significantly faster per iteration.

The neighborhood contribution process is modeled
using Shannon Information Theory. The neighbor
labelings serve as the channel input, the compatability
and experience matrices model the channel, and the
channel output is the neighborhood contribution. The
relaxation methods are the examined in the light of the
measures describing the channel model. The RLP is
found to have an unacceptably high dispersion while the
DRLP becomes a deterministic channel with zero
dispersion for strong broadcasts. The other standard
measures of mutual information, entropy drop and
equivocation also favor the DRLP.

A final section on motion prediction by
broadcasting was presented. The developement of two
types of experience matrices, the create and remove

matrices, facilitate this task.

VI.2. FUTURE PROBLEMS
VI.2.l. Convergence and Optimality

One important topic in the study of relaxation
systems is the convergence properties of the system.
This thesis has only scratched the surface of
convergence properties for augmented relaxation
systems. The initial direction in section IV.4 can be
followed. The convergence to a fixed point depends on

the broadcast patterns and strengths and the number of

140

received broadcasts. This further complicates any
analysis however.

For dynamic relaxation systems, much of the work
already done can apply when suitable modifications are
made for the variability in the broadcast neighborhoods
and compatabilities versus the homogeneous constraints
in the relaxation processor. This is not easy as one
major reason for the introduction of the homogeneity
requirements was to simplify the mathematics of the
descriptions.

Even if a mathematical model is difficult, several
convergence measures are potentially available. A
measure of convergence can be directly based on the
information measures described in section IV.3,
especially the mutual information and entropy drop
measures. All that is needed is a method of evaluating
the numbers that result across a picture set. Another,
more predictive measure can be developed from the
broadcast strengths. As the strengths rise, the
picture converges more quickly.

VI.2.2. On Experience Matrices and Compatabilities
VI.2.2.1. Optimality

Several avenues exist for the definition of an
optimal relaxation system. A useful area to optimize
is the compatability and experience matrices. The
channel measures of dispersion and equivocation are

obvious choices here. An optimal channel has zero

141

dispersion and equivocation. So should an optimal
experience or compatability matrix. Whenever the
channel capacity can be computed, it also is a useful
statistic.

Another approach would be to look at labeling
error statistics or to look at the convergence
statistics above. A experience or compatability matrix
producing lowest error or fastest convergence, or both,
is describable as optimum.

IV.2.2.2. Defining and Training

Building experience matricies is only a simple task
for simple pictures. One key is to have the experience
as concentrated on a single label as possible. The field
of artificial intelligence has a wealth of avenues for
the specification of knowledge and experience from
training samples to rote, statistical or analytical
abstractions for learning, to procedure based knowledge
sources. Any and all of these can be appropriate.
VI.2.3. BroadCast Parameters
VI.2.3.l. Language Aspects

One area of Page's original definitions that has
not been devloped is the use of two dimensional
grammars for the specification of broadcasting
patterns. The various languages describable and their
attendant resolving power is a wide open area for
research. For example, is a context sensitive language

any more (or less) powerful or optimal or sensitive

142

than a context free language.

A second research topic is the description of a
label algebra incorporating the broadcast grammars to
describe the updating process. The operations of the
algebra on the labels describe the interactions from
the broadcasts and the resulting labelings. Such a
model allows the discussion of algorithm behavior in
the abstract. The resulting properties describe the
system and may even hold a clue to the convergence

question. Such algebras can describe both the regular
and motion prediction systems.
VI.2.3.2. Thresholds

Much work can be done in the area of thresholds
for broadcasting and for broadcast reception. The
major work is to specify various thresholds and the
picture effects that result. Where exactly should the
receiving threshold be set is an example.
VI.2.3.3. Labels

The depth of picture description depends on the
breadth of the label set available. For line
processors, the label set from Waltz [2] should be
included rather than a simple intersection label as was
done for the DRLP. Additional labels require additional
experience, however and the resulting combinational

explosion must be dealt with. Hopefully the
simplification that resulted from the label constraints

in Waltz's work will carry over here in the number of

143

special purpose experience matrices needed.

A different idea is that of having broadcast
permeable and broadcast reflective labels. In a sense
a high receiving threshold is similar to a permeable
label. The broadcast has no effect, or just passes by.
A reflective label is interesting but would play havoc
with most descriptive models.

IV.2.4. Hierarchies

A major area for future work is to expand the ARLP
and DRLP to a multi-level hierarchy with descriptive
abstraction such as Minsky's frames. Such a
hierarchical description allows for a greater variety
of top down controllable broadcasts. A test of the
broadcast algorithm as a inter-level communication
device is also testable then. Such a system allows the
algorithms to be tested on more complex pictures.
VI.2.5. Chess by Broadcasting

An interesting application of the broadcast
paradigm is the determination of the squares on a chess
board which can be attacked by a pawn or piece. To see
this, simply consider Figure 3.2 to be the partial
broadcast definition of a rook or bishop. The marked
(*) squares show those squares to which the piece can
move in one turn. These marked squares can the

broadcast for position prediction at the next lookahead

ply-

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

BIBLIOGRAPHY

"Computer Vision: A Literature Survey”, Kuschel,
Stephen A. unpublished paper, May 1978

"Generating Semantic Descriptions from Drawings
of Scenes with Shadows", Waltz, D. MIT Thesis,
1972, also available in The Psychology of Computer
Vision, P. Winston, Ed. pp.l9—9l McGraw Hill, 1976

"The Consistent Labeling Problem", Haralick, R. M.
and Shapiro, L. G. unpublished paper, Jan 1978

"Scene Analysis, Homomorphisms and Arrangements"
Haralick, R. M. in Machine Vision, Academic Press

1978

"Scene Matching Problems", Haralick, R. M.
unpublished paper, May 1978

"Scene Labeling by Relaxation Operation", Rosenfeld,
A., Hummel, R. and Zucker, 8. IEEE Transactions

on Systems, Man and Cybernetics Vol. SMC-6, No. 6

p. 420-433 (June 1977)

"Analysis of Probabilistic Relaxation Processes”,
Zucker, 8., Computer Vision and Graphics Laboratory,
Department of Electrical Engineereing, McGill Univ.
Montreal, Quebec TR78-3R. Also available in Proc.
IEEE Conf. on Pattern Recognition and Image
Processing, 1978. ‘

"A Hierarchical Relaxation System for Line Labeling
and Grouping", Zucker, S. and Mohammed, J., Computer
Vision and Graphics Laboratory, Department of
Electrical Engineering, McGill University, Montreal,
Quebec TR78—llR 1978. Also available in Proc. of the
IEEE Conf. on Pattern Recognition and Image
Processing, 1978.

"Relaxation Processes for Scene Labeling: Convergence
Speed and Stability”, Zucker, 5., Krishnamurthy, E.V.
and Haar, R., IEEE Trans. on Sys. Man and Cyber.

Vol. SMC-B, No. 1. (Jan. 1978)

"Convergence Properties of Relaxation", Eklundh, J.
and Rosenfeld, A. Computer Science Center, University
of Maryland, College Park, MD. TR-70l MSC-76-23763
October 1978.

144

[ll]

[12]

[l3]

[14]

[15]

[16]

145

”A Study of Augmented Relaxation Labeling for Image
Understanding”, Page, Carl V., Department of Computer
Science, Michigan State University, unpublished paper
October 1978

"Augmented Relaxation Labeling", Page, Carl V., and
Kuschel, Stephen A., Department of Computer Science,

"Symbols for the Manipulation of Natural Language",

C. D. Paren; Ph.D thesis, Department of Computer
Science, Michigan State University, 1852.

”The Mathematical Theory of Computation", Shannon,
Claude, E, and Weaver, Warren. University of Illinois
press. Urbana. 1949.

"Information and Coding Theory", Ingels, Franklin, M.
Intext, Scranton PA. 1971.

"Strong Stability Problems for Probabalistic Machines"
Carl V. Page, Information and Control Vol.15 No.6 1969

 

APPENDIX A

COMPATABILITY MATRICES

EXPERIENCE MATRICES

The pi‘1 label compatabilities are given below as defined in equation (20)

M
.1
<1
p.
O
p.
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a. 61-!

146

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COL TOTALS

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ALUES FOR

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147

TOTALS

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COLOTOTALS

fidchDLOxOlN-flmo
IshmtOMMmmmC’
ODDQdeHv-{O
0 0 g 0 . . . . .5

mmmoaaamao
”noncomnoo
aoocNNNHOD
0 . 0 . O . . . . O

H

JJ'LDJiOQLfiv-INO
\D‘DJ QJ‘J'NJ’I‘OO
OQQQHT'HflV‘O
0 0 o 0 . 0 g Q 0 0
v4

JJmJQ‘DLfifiNO
\OKD-J'uaJ-J'Nd‘mu
omomafladdo
O . . . 0 . . . . 0

.4

.ID

HflNMNNNONu
3 ”MMfiCO‘OO'DND
fifiv-(HDOOOHD
of . . 0 . 0 . 0 0 .v-‘O

co
I
UflHNy-{v-«Iv-OMKOHO
HH“M@U‘[B"‘OHU
wNNNv-Ioooouo
Z . 0 . . . . . . 0 0
.4
(Z
O
ILU‘ONOQJ'JMJmD
OOQMOOMNQO
mfififlaogWODv-CD
m . o . . . . . . 0 .
D
...!
d
>O‘O‘IDxOJ-1'M—fmo
\OOQMOQMNQO
fiHHv-h-cctnaodc
H . 0 . . 0 0 0 . g .

a_ H

COL TOTALS

NNO‘O‘LOtoN4foD
hrs-:Nxotorsv-AOI:
«Acorn-(cacao
0 0 0 O 0 0 0 0 0 0

O‘O‘JOGQO‘JO‘D
NNNfiHv-{d4fv46
fidflOHv—(NNUD
0 0 0 . O 0 . o 0 .

ouomm.+4tm«—L:o
fifi¢moczoomo
dfiocdadmco
o . 0 . 0 0 0 O . 0

.4

M-‘ONNMMOQNQ
OOHniuoc‘ome
fifiv-h-h-h-{oofio
0 0 0 0 0 0 0 0 0 0

fl

NnNNMMOQNQ
DD"! HUHO'HI’MD
HHfiHHHODv-GO
0 . . 0 0 0 . 0 0 g

d

7

QQM'Oflw-ILDLBCDLJ
T‘lH-‘DCJMVJMv-imo
fiHNNfiHODoo
a: 0 0 . 0 0 0 O O . .

H

6
CD
I

UOQNONNTJNMO

mHadcu—tdfiafio
w 0 0 0 0 0 0 g 0 0 0

.4

ALU

>Mnomfldv4v-lao

ooomoaoooo
'Jdaflodde-Ific
H O . 0 O 0 0 0 0 0 0

a H

148

COL TOTALS

NNmaxomw-komo
ocmmmmmramo
NNDDCC‘HNQQ
0 O 0 0 0 O 0 0 0 0

vi

nonmrulnmNI-nao
OOmO‘O‘O‘ooo‘o
HHQDODde-JO
0 0 0 0 0 0 0 . o 0

mmmdmmmmmo
:meaoompo
OOdHHfifiooo
0 O . 0 O . O. 0 . 0

ommdmnmnoc
:J'Nowon-Dmho
auHflHHHooo
. . 0 . . . . 0 . .

aommmmo‘ahu
J'JMfiGQQNNo
OOHNNNHDDO
0 0 0 . O 0 0 0 . 0

«I

HBCR ” 5

ushmm-oaoonwm:
HmwaHHw-‘Io'o‘oc:
UCOflv-{Hfificafio
Z . . 0 . . g g . 0 .
v4
0:
D
ammococdhoo
NNO‘MmmNcooo
MHfl'DQOuv-‘Hv-‘D
w . 0 . . 0 o . g . .
D (a
.J
4
>mm®¢066rfl~mo
KNO‘MmmNQoo
fidfiaoocfidfio
H . . 0 . . 0 0 0 . 0
a a

09839637939 1.0044923346

09938537609

.9509470375

1o009h923346 1.0066812174 1.0086629013

.7.9999996996

7.9999997970

.7.9999999343

8.0000009690 8.0000001920 7.9999995060

709999996996

149

EXPERIENCE MATRICES FOR ARLP

IAEMQI

1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

IAEHLZ

.500

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000

IAEHL3

.250
0.000
.750
0.000
0.000
.0.000
0.000
0.000
0.000

.500'

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
1.000
0.000
0.000
0:000
0.000
0.000
0.000
0.000

0.900
.500
.500

0.000

0.000

0.000

0.000

0.000

0.000

.750
0.000

.290
0.000
0.000
0.000
0.000
0.000
0.000

0.000
.500
.500

0.000

0.000

0.000

0.000

0.000

0.000

0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000

.875
0.000
0.000

.125

.0.000

0.000
0.000
0.000
0.000

0.000

.750
0.000

.250
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000

.1.000
0.000
0.000
0.000

'0.000

0.000
0.000
0.000
0.000

0.000

.875
0.000
0.000

.125
0.000
0.000
0.000
0.000

0.000
0.000

.750
0.000

.250
0.000
0.000
0.000
0.090

.875
0.000
0.000
0.000
0.000

.125
0.000
0.000
0.000

0.000

1.000-

0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.875
0.000
0.000

.125
0.000
0 0000
0.000

.750
0.000
0.000
0.000
0.000
0.000

.250
0.000
0.000

0.000

.875
0.000
0.000
0.000
0.000

.125
0.000
0.000

0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000

0.000

.750
0.000
0.000
0.000
0.000
0.000

.250
0.000

0.000
0.000

.875
0.000
0.000
0.000
0.000

.125
0.000

.400
0.000
0.000
0.000
0.000
0.000
0.060
0.000

.600

0.000

0400
0.000
0.000
0.000
0.000
0.000
0.000

.600

0.000
0.000

.400
0.000
0.000
0.000
0.000
0.000

.600

IAKHL4
.125
0.000
0.000
.875
'0.000
0.000
0.000
0.000

0.000.

LABEL5

0.000
0.000
-0.000
0.000
1.000
0.000
0.000
0.000
0.000

IAEHL6

.125
9.000
0.000
0.000
0.000

.875
0.000
0.000
0.000

0.000

.250
0.000

.750
0.000
0.000
0.000
0.000
0.000

0.000

.125
0.000
0.000

.875
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000

0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.250
0.000

.750
0.000
0.000
0.000
0.000

0.000
0.000

.125
0.000
0.000

.8075
0.000
0.000
0.000

150

0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.250
0.000

.750.

0.000
0.000
0.000

0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

.500

.500
0.000
0.000
0.000

0.000
0.000
0.000

.750
0.000

.250
0.000
0.000
0.000

0.000
0.000
0.000
0.000
.500
.500

0.000‘

0.000
0.000

0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000

0.000
0.000
0.000

.875
0.000
0.000

.125
0.000
0.000

0.000
0.000
0.000
0.000

.750
0.000

.250
0.000
0.000

0.000
0.000
0.000
0.000
0.000
..500

.500
0.000
0.000

0.000
0.000
0.000
1.000
0.000
~0.ooo
0.000
0.000

0.000'

0.000
0.000
0.000
0.000

.975
0.000
0.000

.125

0.000.

0.000
0.000
0.000
0.000
0.000

.750
0.000

.250
0.000

0.000
0.000
0.000

.400
0.000
0.000
0.000

.600

0.000
0.000
0.000
0.000

.400
0.000
0.000
0.000

O 600

0.000
0.000
0.000
0.000
0.000

.400
0.000
0.000

.600

IAKBL7

.250
0.000
0.000
0.000
0.000
0.000

.750
0.000
0.000

IAKHLB

.500
9 .000
0.000
0.000
0.000
0.000
0.000

.500
0.000

IAEML9
.990
0.000
0.000
0.000
0.000
0.000
0.000
0.000
.010

0.000

.125
0.000
0.000
0.000
0.000

.875
0.000
0.000

0.000

.250
0.000
0.000
0.000
0.000
0.000

.750
0.000

0.000

.990
0.000
0.000
0.000
0.000
0.000
0.000

.010

0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000

0.000
0.000

.125
0.000
0.000
0.000
0.000

.875
0.000

0.000
0.000

.990
0.000
0.000
0.000
0.000
0.000

.010

151

0.000
0.000
0.000

.125
0.000
0.000

.875
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000
10000
0.000

0.000
0.000
0.000

.990
0.000
0.000
0.000
0.000
5.010

0.000
0.000
0.000
0.000

.250
0.000

.750
o .000
0.000

0.000
0.000
0.000
0.000

.125
0.000
0.000

.875
0.000

0.000
0.000
0.000
0.000

.990
0.000
0.000
0.000

.010

-0.000

0.000
0.000
0.000
0.000

.500

.500
o .000
0.000

0.000
0.000
0.000
0.000
0.000

.250
0.000

.750
0.000

0.000
0.000
0.000
0.000
0.000

.990
0.000
0.000

.010

0.000
0.000
0.000

‘0.000

0.000
0.000
1.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000

.500

0.000

0.000
0.000
0.000
0.000
0.000
0.000

.990
0.000

.010~

0.000
0.000
0.000
0.000
0.000
0.000

.500

.500
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.990

.010

0.000
0.000
0.0000
0.000
0.000
0.000

.400
0.000

.600

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.400

.600

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000

IABHLI

1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

IAEH02

.500

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

IABEL3

.250
0.000

.750
.0.000
0.000
0.000
0.000
0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0-000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
.500
o 500

0.000

0.000

0.000

0.000

0.000

0.000

0.000

152

EXPERIENCE MATRICES FROM THE

.750
0.000

.250
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
.500
.500

0.000'

0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.87

0.000

0.000'

.125
0.000
0.000
0.000
0.000
0.000
0.000

0.000

.750
0.000

.250
0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000
0.000

1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000

.875
0.000
0.000

.125
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.750
0.000

.250
0.000
0.000
0.000
0.000
0.000

.875
0.000
0.000
0.000
0.000

.125
0.000
0.000
0.000
0.000

0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

'o.ooo
0.000
.875
0.000
0.000
.125
0.000
0.000
0.000
0.000

IMHI

.750
0.000
0.000
0.000
0.000
0.000

.250
0.000
0.000
0.000

0.000

.875
0.000
0.000
0.000
0.000

.125
0.000

0.000

0.000

0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000
0.000

0.000

.750
0.000
0.000
0.000
0.000
0.000

.250
0.000
0.000

0.000
0.000

.875
0.000
0.000
0.000
0.000

.125

'0.000

0.000

.600
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.400
0.000

0.000

.600
0.000
0.000
0.000
0.000
0.000
0.000

.400
0.000

0.000
0.000

.600
0.000
0.000
0.000
0.000
0.000

.400
0.000

.800
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.200

0.000

.800
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.200

0.000
0.000

.800
0.000
0.000
0.000
0.000
0.000
0.000

.200

1AKHQ4

.125
0.000
0.000

.875
0.000
0.000
0.000
0.000
0.000
0.000

0.000

.250
0.000

.750
0.000
0.000
0.000
0.000
0.000
0.000

IABHLS

0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000

LMEL 6

.125
0.000
0.000
0.000
0.000

.875
0.000
0.000
0.000
0.000

0.000

.125
0.000
0.000

.875
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000

0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.250
0.000

.750
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.125
0.000
0.000

.875
0.000
0.000
0.000
0.000

0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.250
0.000

.750
0.000
0.000
0.000
0.000

153

0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.750
0.000

.250
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

.500

.500
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000

.875
0.000
0.000

.125
0.000
0.000
0.000

O .000
0.000
0.000
0.000
..750
0.000

.250
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

.500

.500
0.000
0.000
0.000

0.000
0.000
0.000
1.000
0.000
0.000
0.000

0.000'

0.000
0,000

0.000
0.000
0.000
0.000

.875
0.000
0.000

.125
0.000
0.000

0.000
0.000
0.000
0.000
0.000

.750
0.000

.250
0.000
0.000

0.000
0.000
0.000

.600
0.000
0.000
0.000
0.000

.400
0.000

0.000
0.000
0.000
0.000

.600
0.000
0.000
0.000

.400
0.000

0.000
0.000
0.000
0.000
0.000

.600
0.000
0.000

.400
0.000

0.000
0.000
0.000

.300
0.000
0.000
0.000
0.000
0.000

.200

0.000
0.000
0.000
0.000

.800
0.000
0.000
0.000
0.000

.200

0.000
0.000
0.000
0.000
0.000

.800
0.000
0.000
0.000

.200

IAEEL

.250
0.000
0.000
0.000
0.000
0.000

.750
0.000
0.000
0.000

LMEL
.500
0.000
0.000
0.000
0.000
0.000
0.000
.500
0.000
0.000

7

.125
0.000
0.000
0.000
0.000

;875
0.000
0.000
0.000

8
0.000
.250
0.000
0.000
0.000
0.000
0.000
.750
0.000
0.000

0.000 0.000 0.000

0.000.0.000

0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000

0.000
0.000

.125
0.000
0.000
0.000
0.000

.875
0.000
0.000

0.000

.‘25
0.000
0.000

.975
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000

LABEL 9 FOR INTERSECTIONS

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000

0.000

.500
0.000
0.000
0.000
0.000
0.000
0.000
0.000

0.000
0.000

.500
0.000
0.000
0.000
0.000
0.000

.500
0.000

0.000
0.000
0.000

.500
0.000
0.000
0.000
0.000

.500
0.000

154

0.000
0.000

00.000

0.000
.250

.750
0.000
0.000
0.000

0.000
0.000
0.000
0.000

.125
0.000
0.000

.875
0.000
0.000

0.000
0.000
0.000
0.000

.500
0.000
0.000
0.000

.500
0.000

0.000
0.000
0.000
0.000
0.000

.500

0:00
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

.250
0.000

.750
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000
0.000

0.000
0.000
0.000
0.060
0.000
0.000

.500

.500
0.000
0.000

0.000
0.000
0.000
0.000
0.000

.500
0.000
0.000

.500
0.000

0.000
0.000
0.000
0.000
0.000
0.000

.500
0.000

.500
0.000

0.000
0.000
0.000
0.000
0.000
0.000

.500

.500
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.500

.500
0.000

0.000
0.000
0.000
0.000
0.000
0.000

.600
0.000

.400
0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.600

.400
0.000

0.000
0.000“
0.000
0.000
0.000
0.000

.390
0.000
0.000

.200

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.900
0.000

.200

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000

0.000

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.500

.500

LABEL 10 FOR BLANKS

.800
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.200

0.000
.800
0.000.
0.000
0.000
0.000
0.000
0.000
0.000
.200

0.000
0.000

.390
0.000
0.000
0.000
0.000
0.000
0.000

.200

0.000
0.000
0.000

.900
0.000
0.000
0.000
0.000
0.000

.700

155

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.200

0.000
0.000
0.000
0.000
0.000

.300
0.000
0.000
0.000

.200'

0.000
0.000
0.000
0.000
0.000
0.000

.800
0.000
0.000

.200

0.000
0.000
0.000
0.000
0.000
0.000
0.000

.800
0.000

.200 .

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000

.800

.700

0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000

APPENDIX B

DRLP PROGRAM

u7l1317°

11V £.50£!!

156
0r1-1

73/73

‘k'ICl-l' THESIS

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07/13/79

11N ‘.60533

157

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73/73

PROGRkH 1NESIS

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4.6‘AS3

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