• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 List of Tables
 Abstract
 Introduction
 The trace particle technique and...
 A particle following machine
 Initialization and implementat...
 Results of performance tests
 Discussion and conclusions
 Appendix A: Analysis of the data...
 Appendix B: Approximate geometric...
 Appendix C: Tablet digitizer...
 Appendix D: Fortran program...
 Bibliography
 Biographical sketch














Group Title: machine learning approach to following trace particles in turbulent flow /
Title: A machine learning approach to following trace particles in turbulent flow /
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 Material Information
Title: A machine learning approach to following trace particles in turbulent flow /
Physical Description: x, 187 leaves : graphs ; 28 cm.
Language: English
Creator: Crowe, Randel Allan, 1948-
Publication Date: 1977
Copyright Date: 1977
 Subjects
Subject: Turbulence   ( lcsh )
Fluid dynamics   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 183-186.
Additional Physical Form: Also available on World Wide Web
General Note: Defective copy: leaf 96 bound before leaf 95.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Randel Allan Crowe.
 Record Information
Bibliographic ID: UF00097483
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000209981
oclc - 04164208
notis - AAX6800

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Table of Contents
    Title Page
        Page i
        Page i-a
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Figures
        Page vi
        Page vii
    List of Tables
        Page viii
    Abstract
        Page ix
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    The trace particle technique and data acquisition system
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    A particle following machine
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
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        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    Initialization and implementation
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
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        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
    Results of performance tests
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
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        Page 85
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        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
    Discussion and conclusions
        Page 95
        Page 96
        Page 97
    Appendix A: Analysis of the data acquisition system
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
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        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
    Appendix B: Approximate geometric ray trace prism equations
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
        Page 150
        Page 151
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        Page 154
    Appendix C: Tablet digitizer error
        Page 155
        Page 156
        Page 157
        Page 158
        Page 159
        Page 160
        Page 161
        Page 162
    Appendix D: Fortran program listings
        Page 163
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        Page 176
        Page 177
        Page 178
        Page 179
        Page 180
        Page 181
        Page 182
    Bibliography
        Page 183
        Page 184
        Page 185
        Page 186
    Biographical sketch
        Page 187
        Page 188
        Page 189
Full Text

















A MACHINE LEARNING APPROACH TO FOLLOWING
TRACE PARTICLES IN TURBULENT FLOW








By

RANDEL ALLAN CROWE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1977



















































UNIVERSITY OF FLORIDA


3 1262 08552 3156































To Patricia, A Most Special Person










ACKNOWLEDGEMENTS

Deserving much of the credit for the completion of this work is,

of course, my advisor, Professor Gale E. Nevill. Often in dark times,

he gave me needed encouragement and guidance and has my most sincere

thanks. Others on my committee should be noted for their patience as

well as valuable suggestions. Dr. Kurzweg reviewed the fluid mechanics

aspects and Dr. Boykin the dynamics and modeling. Little could have

been accomplished in my graduate work without the very early support

and friendship of Dr. Hemp. Enduring the final push, Dr. Shaffer

has my deepest appreciation. Yet many others should also be included.



Surely, Professor E. Rune Lindgren, and his associates, Drs.

Elkins and Jackman whose project provided this problem, together with

Professor Roland C. Anderson who aided in the optical analysis, should

be recognized for providing unending assistance and counseling. Quietly

in the background are the members of the "MNDC,' a local social group,

too numerous to mention individually, but all very supportive and there

when needed. Unusual was the serendipity and productivity of my con-

versations with Mickey Rogers. All words fail me when I try to express

my appreciation and admiration of my wife, Pat, who gave me incessant

drive and inspiration. To my parents as well as those mentioned and

unmentioned, I would like to express my everlasting gratitude and

indebtedness.










TABLE OF CONTENTS



ACKNOWLEDGEMENTS.... .............................................

LIST OF FIGURES...................................................

LIST OF TABLES....................................................

ABSTRACT..... .................................. ..................

CHAPTER

I INTRODUCTION.............................................

Scope of Work................................ ...... ....

Pertinent Background for the Particle Following
Machine.. ..............................................

II THE TRACE PARTICLE TECHNIQUE AND DATA ACQUISITION SYSTEM..

The Use of Trace Particles for Flow Visualization......

Earlier Particle Following Approaches .................

Qualitative Observations of Particle Motion............

III A PARTICLE FOLLOWING MACHINE .............................

Overview ..............................................

PFM: Input and Output.................................

Image Path Attributes.................................

Following Image Paths: The Decision Process...........

The Image Feature Vector...............................

Modeling the Residual.................................

Control of the PFM ....................................

Measuring Performance.................................

Summary...............................................


Page

iii

vi

viii

ix






CHAPTER Page

IV PFM: INITIALIZATION AND IMPLEMENTATION................... 58

Initialization........................................ 58

The Data ............................................... 66

The Initialization Program ............................. 69

The Particle Following Program......................... 69

Program LEARN: Determination of the Joint
Probabilities......................................... 74

V RESULTS OF PERFORMANCE TESTS.............................. 76

Initialization........................................ 76

Particle Following.................................... 85

Performance of the PFP................................. 90

VI DISCUSSION AND CONCLUSIONS............................... 95

APPENDIX A ANALYSIS OF THE DATA ACQUISITION SYSTEM ............... 98

The Data Acquisition System........................ 98

Approximate Geometric Ray Trace.................... 104

Qualitative Observations of Images................. 107

Geometric Ray Trace................................ 108

Spot Diagrams.................................... 128

Digitization of Film Data.......................... 133

Path Characteristics Using Orthogonal Projections.. 136

APPENDIX B APPROXIMATE GEOMETRIC RAY TRACE PRISM EQUATIONS........ 144

APPENDIX C TABLET DIGITIZER ERROR................................ 155

APPENDIX D FORTRAN PROGRAM LISTINGS.............................. 164

BIBLIOGRAPHY.............................. ......................... 183

BIOGRAPHICAL SKETCH................................... ............ 187










LIST OF FIGURES


FIGURE Page

1-1 DATA ACQUISITION SYSTEM BLOCK DIAGRAM ..................... 3

2-1 THE DATA ACQUISITION SYSTEM ............................... 11

3-1 THE PARTICLE FOLLOWING MACHINE ............................ 22

3-2 DIGITIZED IMAGE PLANE COORDINATES ......................... 22

3-3 IMAGE CANDIDATES AND ERRORS............................... 35

3-4 LINK CLASSES, DECISION STATES AND HYPOTHESES.............. 37

3-5 EXAMPLE OF JOINT TRANSITION PROBABILITY................... 49

3-6 BLOCK DIAGRAM OF THE PARTICLE FOLLOWING MACHINE............ 57

4-1 FRAME-BY-FRAME DATA....................................... 67

4-2 INVERTIBLE PATH DATA...................................... 67

4-3 SIMPLIFIED FLOWCHART OF "INIT"............................ 70

4-4 SIMPLIFIED FLOWCHART OF "PFP"............................. 71

4-5 SIMPLIFIED FLOWCHART OF "LEARN" ........................... 75

5-1 TANGENTIAL AND NORMAL ACCELERATION COSTS.................. 77

5-2 COST FUNCTION CONTOURS................................... 79

5-3 INITIAL IMAGE PATH SEGMENTS: TOP HALF OF IMAGE PLANE..... 80

5-4 INITIAL IMAGE PATH SEGMENTS: BOTTOM HALF OF IMAGE PLANE.. 81

5-5 RESIDUAL JOINT TRANSITION PROBABILITY, s = 1.0............ 86

5-6 RESIDUAL JOINT TRANSITION PROBABILITY, s = 1.2............ 87

5-7 RESIDUAL JOINT TRANSITION PROBABILITY, s = 1.5............ 88

5-8 RESIDUAL JOINT TRANSITION PROBABILITY, s = 2.0............ 89

A-i DEFINITION OF COORDINATE AXES............................. 99

A-2 APPROXIMATE MERIDIONAL RAY TRACE .......................... 106








FIGURE Page

A-3 GRID OF OBJECT POINTS.................................... 109

A-4 MERIDIONAL RAY TRACE: LARGE PRISM ........................ 110

A-5 OFF-AXIS RAY TRACE: LARGE PRISM.......................... 110

A-6 RAY TRACE TOP VIEW....................................... 111

A-7 PRINCIPAL RAYS FROM OBJECT GRID INTERSECTING IMAGE PLANE.. 111

A-8 MERIDIONAL RAY TRACE: SMALL PRISM ........................ 113

A-9 LARGE PRISM RAY LENGTHS FOR FULL LIGHTING.................. 115

A-10 UNNECESSARY CONFUSION FOR LARGE PRISM WITH FULL LIGHTING.. 115

A-11 SMALL PRISM RAY LENGTHS FOR FULL LIGHTING................. 116

A-12 UNNECESSARY CONFUSION FOR SMALL PRISM WITH FULL LIGHTING.. 116

A-13 LIGHTING SCHEMES......................................... 117

A-14 UNNECESSARY CONFUSION FOR HALF AND CIRCLE ILLUMINATION.... 118

A-15 VOLUME SENSITIVITIES (cm3/mm2)........................... 123

A-16 PROBABILITIES FOR CONFUSION AND OVERLAP.................... 124

A-17 HEXAPOLAR ARRAY ........................................... 130

A-18 OBJECT POINT LOCATIONS.... ............................... 130

A-19 SENSITIVITY OF FOCUSING POINT A............................ 131

A-20 SPOT DIAGRAMS FOR POINT A AND A', BEST FOCUS.............. 132

A-21 SPOT DIAGRAMS FOR POINT C AND C', BEST FOCUS SET FOR
POINT A.................................................... 134

A-22 SPOT DIAGRAMS FOR POINTS B AND B', BEST FOCUS ON A........ 135

A-23 ORTHOGONAL STEREOSCOPIC PROJECTION OF A PATH............... 137

B-1 RAY TRACE GEOMETRY.................................... .... 146

C-1 DISTRIBUTION OF PEN LOCATION ABOUT DESIRED IMAGE
LOCATION.................................................. 156

C-2 TWO DIMENSIONAL REGIONS................................... 156










LIST OF TABLES


TABLE Page

5-1 KEY TO FIGURES 5.3 AND 5.4................................ 82

5-2 INITIALIZATION PERFORMANCE ON 207 FRAME-ONE IMAGES........ 84

5-3 PFP PERFORMANCE, s = 2.0.................................. 92

5-4 ISOLATION OF DECISIONS................................... 94

A-la APPROXIMATE PARTICLE MOVEMENT (MM) IN SHUTTER TIME......... 103

A-lb APPROXIMATION USING MAX VELOCITY= UBARO.791............... 103

A-2 PROBABILITY OF TWO PARTICLES OCCURRING IN VI .............. 119

A-3 AVERAGE CONFUSION AND OVERLAP PROBABILITIES FOR SMALL
PRISM..................................................... 126

A-4 AVERAGE CONFUSION AND OVERLAP PROBABILITIES FOR LARGE
PRISM..................................................... 127

C-1 TWO DIMENSIONAL TABLET ERROR.............................. 161

C-2 PROBABILITY OF BIT ERROR FOR GIVEN E....................... 162


viii








Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A MACHINE LEARNING APPROACH TO FOLLOWING
TRACE PARTICLES IN TURBULENT FLOW

By

Randel Allan Crowe

August 1977

Chairman: Gale E. Nevill, Jr.
Major Department: Engineering Sciences

Machine learning is used in a statistical decision theoretic scheme

to follow trace particles in turbulent flow. Data used to demonstrate

performance came from previous experiments studying turbulent flow in a

pipe (Reynolds number less than 6500), and consisted of digitized stereo-

scopic trace particle image locations. Accuracy of the data acquisition

technique is examined with emphasis on optical errors. Particle

location in the pipe is determined from its two stereo images. How-

ever, high trace particle count prevents immediate pairing of particle's

stereo images. Therefore, identification of image paths (sequentially

sampled particle locations) is carried out separately in each view.

The particle following scheme is initialized by short image path segments

found using search procedures based on finding five-image sequences with

minimum velocity variation. Image dynamics are modeled using a constant

acceleration assumption. An image's dynamic state is estimated using

a Kalman filter with finite fading memory. The next image belonging

to the path is sought in the neighborhood of a prediction. A minimum

error rate Bayes decision rule is applied to choose the next image

from the candidates in the neighborhood. The feature vector is








composed of the two most recent filter residuals. Probabilities of

the residual transitions are learned by analyzing previously completed

and verified sequences. It is found that this scheme gives better

performance than earlier attempts, which utilized a constant velocity

assumption, including improved handling of image overlap, confusion,

and measurement noise. It is concluded that a semiautomated system

is necessary where the particle following procedure's output is.

verified by an operator who can change or correct decisions made.














CHAPTER I

INTRODUCTION


Scope of Work

Turbulence has evaded detailed and complete understanding by scores

of scientists over many generations. Recent attempts at its description

suffer from the difficulty of obtaining good experimental verification

which is needed in any modeling effort. In particular, two separate

experiments that have been reported utilized trace particles to facilitate

the detailed quantitative description of the structure of turbulence in

pipes. Johnson (1974) and Jackman (1976) used neutrally buoyant particles

in water and a novel prism-cinematographic data recording system to

study transition and low Reynolds number flows (3500 Re 6500).

Breton (1975) used small glass spheres in trichloroethelyne (TCE)

and a unique "distributed camera" system* to study a flow with Reynolds

number of the order of 100,000. These workers were interested in a

Lagrangian description of the turbulence. That is, a turbulent model

written in terms of the individual traces of fluid elements. These

experiments generated stereoscopic film records of temporally sampled

particle paths, and both suffered from the inability to automatically

reduce these data to the individual path sequences.






*This consisted of two sets of twenty lenses placed along a test section
of the pipe.






2

Quantitative flow visualization requires taking data by an optical

system that records two different views (stereo views) of the particles

in the flow. Each view is a projection of the three-dimensional par-

ticle paths onto a two-dimensional film plane. By measuring the image

locations in the film plane, the two views of a particle can be used to

determine its three-dimensional position. This is what is termed photo-

grammetry. Breton (1975) used a sufficiently small number of particles

to allow straightforward determination of stereo pairs. Johnson (1975)

and Jackman (1976), however, had high particle counts making the immedi-

ate determination of stereo image pairs impossible. Instead, they deter-

mined the two-dimensional image paths ( a process called particle following)

of all images and then compared the paths to determine which were stereo

pairs (conjugate paths). A high particle count yields a better and more

reliable visualization. But the use of more particles increases the

burden on the data reduction procedures. Breton (1975) and Jackman (1976)

used automated procedures as much as possible but were not successful at

attaining full automation. Automatic data analysis is difficult and will

not become a reality without a thorough understanding of the data

acquisition system, its errors, the particle path characteristics and

the concepts of particle following. The work presented here considers

Jackman's experimental arrangement and the errors in his data acquisition

system. Considered in detail is the particle following problem.

Data reduction consists of several identifiable subproblems which,

together, form a linear data acquisition and reduction system (See

Figure 1-1). Once the experimental system has been defined, the first

problem is to acquire quality stereo images of the trace particles.














































DATA ACQUISITION SYSTEM BLOCK DIAGRAM


FIGURE 1-1









This is an acute optics problem in the case of Johnson's (1975) prism

system. The present work includes an in depth analysis of the optical

properties of the pipe and prism system. Appendix A presents details of

the analysis. Second, the images must be recorded on film and their lo-

cations digitized for use by the analysis programs. Jackman's (1976)

use of a graphic tablet system for data entry is treated in Appendix C.

The primary concern of this work is the third step; converting the

digitized film data into image paths. The approach taken here involves

three procedures: prediction, decision making, and learning. The

combination of these processes to perform the image following task is

termed the Particle Following Machine (PFM). The PFM takes the digitized

image locations and determines which images represent the same particle

from frame to frame (sample to sample). It outputs image paths which can

be compared to find the conjugate paths. Then, a transformation is used

to determine the three-dimensional particle paths. These last two aspects

of the data analysis are not considered in the current work but they are

discussed along with the actual evaluation of the results in terms of

fluid turbulence theory by Johnson (1974), Breton (1975), and Jackman (1976).

The principal contribution of this work is the development of a

theoretical basis for the PFM. This approach to following images of

trace particles makes use of a combination of estimation, decision and

pattern recognition theory. The PFM is presented in a general format

and could be applied to other problems where a group of identical objects

are being tracked and the motion of an object is fairly consistent over

time. Performance tests of a specific implementation of the PFM were

made using a computer code written in FORTRAN. Some modifications would

be necessary to make this implementation an economic production program.









A secondary contribution, the optical analysis of the prism system

and digitization error of a data tablet, is important to the continued

vitality of the experimental procedure. The results of the analysis

performed gives insight into the data acquisition system and the charac-

ter of the input to the PFM.

Pertinent Background for the Particle Following Machine

The particle following task is simply stated as determining which

images in the digitized film data belong to a given particle. An image

path is found by determining which image in each frame of data belongs

to the specified particle. Therefore, for each frame, a decision must

be made as to which image belongs to the particle being followed. This

leads to the task being considered a pattern recognition problem. The

image paths are the desired patterns to be identified. The images of

a frame of data must be classified as belonging to the given particle

or not. To make these classifications, a feature vector is necessary.

Here, the entire feature vector is used in the decision process. The

class probabilities for a given feature vector are learned by using

previously identified image paths. The use of the entire feature vector

and a set of given classifications is referred to as machine learning,

i.e. parallel pattern recognition with supervised learning (Hunt, 1975).

Synonymously, Hunt uses the term adaptive pattern recognition while

Duda and Hart (1972) refer to it as simply adaptive learning. Further,

the particle following task is a compound decision problem since more

than one decision is being made at each step.

The PFM uses residuals of an adaptive tracking filter as components

of the feature vector. Finite memory filters can be used as tracking

filters as shown by Jazwinski (1970). An alternate technique, used in








the present work is the finite fading memory filter of Tarn and

Zaborszky (1970). These filters are successful at tracking because

they eliminate possible divergence of the Kalman filter. The divergence

of the Kalman filter is considered by Price (1968). Adaptive radar

tracking systems following manned maneuvering targets use similar track-

ing filters. Singer (1970) uses a Kalman filter to track targets with

known maneuvering capabilities. Singer and Behnke (1970) present a

comparison of five tracking filters concluding that the Kalman filter

requires the most computation but provides measures of tracking error

statistics. McAulay and Denlinger (1973) present a tracking filter

that tracks by changing its dynamic model of the target when a maneuver

is detected. This system requires knowing a target's allowed maneuvers

beforehand. Adaptive Kalman filters are treated in Gelb (1974). Back-

ground on learning and adaptive systems can be found in Tsypkin (1971

and 1973). The general area of adaptive systems is scanned by Lainiotis

(1976).

Since the feature vector is constructed from filter residuals, it

is a time series. It is assumed here to be a Markov process with the

next state only dependent on the immediately previous state. This

use of the history of the feature vector makes the PFM an example of

making decisions in the context of the current situation (i.e. the way

the system got to its current state is important). Modeling the fea-

ture vector is then partially involved with time series analysis (Box

and Jenkins, 1970, and Robinson, 1967) and partially estimation theory

(Jazwinski, 1970).








The most interesting aspect of the PFM is its need to first

decide which image in a frame belongs to the particle being followed.

Once this decision has been made, normal estimation theory applies, i.e.

the last estimate can be updated with information from the next measure-

ment. The particle following task is therefore a combination of appli-

cations of decision theory and estimation theory. No directly applicable

literature was found that discussed the underlying theory of such a

system. This unique problem, then, requires as basic a development

as possible to provide a foundation on which to build a viable production

system. This is evident from the lack of success of other attempts

at automatic particle following. Breton (1975) avoided the problem

entirely by resorting to manual entry of conjugate images and having a

very low particle/image density. Jackman (1976) tried to use a deter-

ministic approach that assumed constant velocity particles and no

measurement noise. This effectively reduced the amount of decision

making, but when a choice had to be made, it was arbitrary. Its per-

formance was fairly poor and the images were eventually followed by

manual effort. Since the desired image paths are known to be subsets

of the data (collections of images), one approach might be to test all

image combinations with a heuristic cost function. It would then

be assumed that paths with minimum cost would be the desired image

paths. With the image density that Jackman used, the number of

possible combinations is very large; at ten times the density, the

problem would be enormous. Nilsson (1971), and Newell and Simon (1972)

discuss heuristic problem solving. For this approach, the search of

possible combinations is heuristically guided by rules that "seem"









appropriate. These rules are ad-hoc but possibly simpler than the

decision theoretic approach taken here. It is difficult to base such

rules on theory and, typically, they provide neither insight to their

accuracy nor guidance for their improvement. The heuristic approach

is useful in solving the initialization problem. (See Chapter IV.)

Initially, there is no information available as to the dynamic state

of an image. By severely limiting the search and accepting some'error,

a heuristic approach is taken to identify short image paths. These

initial path segments are then sufficient to start the particle following

procedure. Application of this initialization procedure to the particle

following task would ignore much information that is readily available

such as image state and measurement noise.

After a brief description of the data acquisition system and a

discussion of some aspects of the analysis of the optical system in

Chapter II, the PFM is described in Chapter III. Chapter IV presents

the details of initialization and programming a specific implementation

of the particle following machine while results and conclusions are

presented in Chapters V and VI respectively.














CHAPTER II

THE TRACE PARTICLE TECHNIQUE AND DATA ACQUISITION SYSTEM


Presented in this chapter is the background of the trace particle

technique and data acquisition system used by Johnson (1974) and

Jackman (1976). In addition, results of qualitative and quantitative

analyses of the system are discussed. These results provide a foundation

for the development of the particle following machine in Chapter III.

Details of the analysis are given in Appendix A.

The Use of Trace Particles for Flow Visualization

There exist numerous techniques for flow visualization. Their

purpose is to reveal the motion of a fluid element over time in com-

plicated flows. Additives such as dye, wax and rosin spheres, dust,

hydrogen bubbles, aluminum powder and mica flakes have all been tried.

Ideally, the trace material used follows the fluid motion precisely

without altering the flow structure by its presence. Examples of flow

visualization may be found in Reynolds (1883), Prandtl (1904), Fage and

Townend (1932), Prandtl and Tietjens (1934), Lindgren (1954-1969), Coles

(1965), Kline et al (1967), Corino and Brodkey (1969), Kim et al.(1971),

Johnson (1974), Breton (1975), Jackman (1976), Johnson et al.(1976), and

Elkins et al.(1977).

The use of pliolite trace particles is due to Nychas, Hershey, and

Brodkey (1973). These particles are desirable because they are opaque

(reflect light well), of selectable size, and almost neutrally buoyant.

Johnson (1974) gives a lengthy argument as to their abilities to follow

the fluid motion faithfully.









Johnson (1974) introduced the use of a prism to generate the stereo

images required for quantitative measurements. This technique requires

only one camera, simplifying somewhat the required equipment compared

to the special distributed camera of Breton (1975).

Quantitative descriptions are typically not obtainable from flow

visualization techniques due to the complicated equipment and analysis

required. Breton (1975) traced only a few particles compared to Johnson

(1974). Jackman (1976) was able to attain one or two particles per cubic

centimeter which approaches a reasonable particle count (henceforth re-

ferred to as density). A goal of the current work of Lindgren and his

associates is to increase the density by an order of magnitude (Lindgren,

1977). As the density is increased, the work required in data reduction

increases tremendously and fully or semiautomatic procedures are vital.

Johnson performed all particle following by hand. Jackman was able to

develop semiautomated data entry, and path matching (finding conjugate

paths) procedures but resorted to following the images manually. Breton

reported failure of his automatic system to follow image:path pairs and

essentially used a machine-assisted manual entry procedure.

The present work is a study of the particle following problem.

In order to understand the basis for the development of the particle

follower, the experimental system and data acquisition equipment need

to be understood. The remainder of Chapter II describes the apparatus

and discusses the results from analysis of the optical system.

Earlier Particle Following Approaches

Of primary concern to the current work is the data acquisition

system used by Jackman. Mounted at the observation station of his

experimental apparatus is a 90 prism as shown in Figure 2-1. Light


















































LLi







CJ LLJ
\U <
LJL LL


0


LU


o)
00
C-



-AJ
I
Lii






I-
LU





















Lid
1.


Lii Lii


-a-


LUJ

0)
Ct-







0





cr -







L--
LL





3:









projected down the pipe reflects off the trace particles making them

highly visible. An observer looking into the pipe through the prism

sees two views of the particles in the pipe, one view in each of the

two prism faces. A particle is invertible if it has an image in both

views. Its location inside the pipe can then be calculated. The

region for which a particle's images are invertible depends on the

prism parameters, the pipe parameters, and the observer's position.

To enhance the capability of the system, two prisms were used. The

smaller is appropriate for studies of motion close to the wall; the

larger to make observations deep into the pipe. A Bell and Howell

35mm movie camera was used to record the particle images. Its film

rate was adjusted to suit the flow rate (faster flows--more particle

motion--faster film rate required). Two small light sources mounted

on the prism were used as reference points for frame alignment during

subsequent data conditioning and analysis. The display of a Hewlett

Packard digital counter was also recorded on each frame to allow

accurate determination of the sample rate.

Jackman used a graphic tablet digitizer to encode the image lo-

cations. The graphic tablet system consisted of a Scriptographics

Tablet Digitizer connected on-line to a Tektronix graphic terminal

interfaced to a central IBM 370/165 computer facility. Using a photo-

graphic enlarger, the film of a turbulent flow (Re=3500) was projected

onto the eleven-inch square tablet surface. The particle images were

then entered one-by-one using the tablet's locator pen. The images,

once encoded, were translated and rotated to a reference position by

use of the frame's reference points. His data consisted of 55 frames









and roughly 13,000 points whose entry into the computer through the

tablet was straightforward but tedious and highly susceptible to

operator error. Graphic output was used to verify data and quickly

showed most needed corrections. Once the image paths were identified

(manually), FORTRAN programs were used to find conjugate paths (particle

paths seen through both prism faces). More processing yielded the three-

dimensional particle paths. (Jackman found approximately 150 pairs of

invertible image paths. Breton's system made all images invertible,

but he used only 10-20 particles.) Use of the computer greatly reduced

the time used for data analysis; from the many months that Johnson*

used to a matter of hours. The procedures established by Jackman were

faster, but required a great deal of human-computer interaction for

data entry and manual particle following.

Jackman's attempted but abandoned automatic particle following

procedure estimated the location of the next image from the last two

by assuming the particle's velocity constant and the measurement per-

fectly accurate. A window was constructed around the estimate and

the next image of a path was sought in this region. This was a straight-

forward, deterministic approach, yet it failed to follow particle paths

for a significant distance and made many errors in assigning an image






*Johnson mounted the film as slides and using a standard projector,
enlarged film records of a Re=6500 turbulent flow onto graph paper.
He then recorded by hand the particle images' locations, and manually
determined the motion in time of each image path.









to its correct path. As is evident from this (and has been known to

researchers in computer vision for a long time) programming a machine

to do even as simple a human task as merging points into lines is not

easy. If frames are superimposed, the particle paths are immediately

evident to a human observer, yet to program a machine to find these

correct paths is not elementary.

Qualitative Observations of Particle Motion

There are a number of useful qualitative observations pertinent

to the discussion of the data acquisition system and image path at-

tributes. Direct observation of turbulence for Reynolds numbers from

3500 to 6500 was performed to obtain the following information. The

system was designed to study particle motion using a Lagrangian or

referential description of fluid flow. That is, motion of the fluid

elements is being observed which is strikingly different from more

common fluid flow descriptions using Eulerian or spatial descriptions.

For flows in the pipe, a strong axial velocity exists so that particles

neither back up nor traverse circles. The observer is given a feeling

of circular fluid motion, but if any specific particle is followed, none

is seen. Furthermore, particles are not observed (by the eye) to have

any random "jumpy" motion as is the case in Brownian motion and their

paths are seen to be smooth trajectories, typically of small curvature.

(Meant here is the curvature in the projection; the three-dimensional

curvature is nearly impossible to observe since the particle's stereo

image and hence location in the pipe is not known.)

Particles are observed throughout the field of view to have widely

different speeds with faster particles giving the effect of being deeper

in the pipe. Bright, slowly-moving particles can sometimes be paired










to their stereo image. Observation of these shows that the motions

of the two stereo images are not particularly similar.

At low Reynolds numbers, large periods of laminar flow are seen.

Here, as expected, particles travel in paths parallel to the pipe

axis; slow particles near the wall, faster particles in the center.

As the Reynolds number is increased through transition, regions of

disturbed motion become more frequent. This turbulence occurs for

isolated periods and is referred to as a slug flow. As a turbulent

slug approaches the viewing station, observed particles begin to deviate

from their laminar trajectories with increasing violence of motion

until the slug has passed, when a sudden calming occurs. Observation

of particle paths at a higher Reynolds number (Re=4500) shows that the

projected paths vary in direction over time, smoothly, not erratically,

with at most 10 to 15 degrees of slope. This can be expected to in-

crease for higher Reynolds numbers but it does indicate how dominant

the mean velocity is.

These observations imply that following images should be straight-

forward since the expected paths are smooth and slowly changing. This

is not the case however. With Jackman's data, one major problem was the

large measurement noise incurred with the use of the tablet for data

entry and too few references for frame alignment. As the optical analy-

sis shows, there are other significant errors (See Appendix A).

There are errors due to the pipe-prism refractive properties,

perspective, camera imperfections, and particle movement. It is shown

in Appendix A, that during the exposure time, a particle may move on

the order of one millimeter in the pipe. This causes elongation of the








16
particle images and a loss of accuracy in the position of their center.

This error has been eliminated from future experiments by the use of

a strobed lighting system synchronized to the camera (Lindgren, 1977).

In addition, coma, caused by use of too low an f number on the camera

lens can be corrected by increasing the f number. This requires more

illumination which has been increased to some extent. An increased

f number also reduces the effect of astigmatism due to the pipe and

prism. The spot diagrams in Appendix A demonstrates the focusing

problems typical of astigmatism. A larger f number has a greater depth

of field bringing more of the pipe into focus and keeping the image size

small. Other errors are more significant.

Determining the location of a particle from the location of its

two stereo images requires transforming the image locations into inter-

secting rays in the pipe. Both Johnson and Jackman wrote analytic

approximations that were adequate in a region near the meridional

plane. A full three-dimensional analysis requires an iterative tech-

nique to determine a ray's direction and is therefore much more compli-

cated than the straightforward "prism equations" given in Appendix B.

It reveals that considerable error can occur in a particle's position

when using these approximations. This error increases as the camera

is moved closer to the pipe. The camera cannot be too far from the

pipe because the illumination is not adequate to sufficiently expose

the film at the film rate required and the lenses available. Thus,

full three-dimensional analysis is necessary. The close proximity

of the camera and the pipe causes this perspective problem. It also

causes another error. When projecting a line parallel to the pipe










axis into the image plane, the two stereo images are not scaled the

same, i.e. the foreshortening of the two images is different. This

causes sampled image locations of the two image paths not to be above

one another, increasing the difficulty of finding stereo image pairs.

Perspective errors are obscured in Jackman's data because of the large

measurement noise. As the system is improved and measurement noise

reduced, they will become more important and if not considered, will

reduce the capabilities of the data acquisition system.

There are other problems intrinsic to systems using projections.

The process of projecting the three-dimensional images onto an image

plane causes a loss of information. Two such projections are required

just to recover the three-dimensional position of the particle. When

many particles are involved, images overlap and particles shadow

other particles. Stereo pairs of images cannot be readily identified.

A particle following machine must handle these phenomena to achieve a

reasonable performance level. The statistical approach taken in

Chapter III requires knowledge of the probability of occurrence of

these problems. In particular, two separate situations need to be

considered: overlap and confusion.

Overlap occurs when a particle comes between the camera and anoth-

er particle deeper in the pipe. Since the optical system has a finite

resolution, there is a region, something like a shadow, that exists for

each particle in which a second particle will be overlapped. This region

varies in size and shape depending on the particle's location. Any

particle in the shadow of another particle cannot be resolved by the

data acquisition system. If the two particles' images overlap only










partially, the center of the resultant image does not represent the

true location of either image and an error will be incurred when the

location of this image is digitized. This problem will always occur

for a system with finite resolution and must be considered in any error

analysis. In qualitative observations, the overlap of particles was

observed infrequently. As the particle density is increased, overlaps

can be expected to increase in occurrence. It can be concluded that

this "instrumentation" noise will always exist, perturbing the true

locations of the images.

Confusion is the term applied to the occurrence of images existing

near each other in the film plane even though their respective particles

may not be close to each other in the pipe. This is a result of the

projection of three-dimensional locations into two dimensions. There

are two types of confusion: necessary and unnecessary. Necessary

confusion occurs when invertible paths (particle paths with two image

paths) have images at a given time located in the same region. This

cannot be avoided. Unnecessary confusion occurs when paths that are

not invertible (paths with only one image path and hence not resolvable

into their three-dimensional positions) have images which occur in

the neighborhood of an invertible image. This problem can be corrected

to some extent by limiting the illumination in the pipe to a volume

that can be resolved by the prism (only invertible images are produced).

Of course, images occurring in the same region but at different times

cause no problems. Confusion cannot be observed directly since the

individual projections give no information as to particle location;

any image could possibly belong to an invertible particle.










In summary, the possibility of overlap means that image locations

cannot be considered as perfectly accurate, some measurement noise will

always exist. The fact that confusion will always occur to some extent

means that image paths may not necessarily be considered isolated. There-

fore, it cannot be assumed that a small window drawn around an image

that is larger than the system resolution will contain only one image.

This has implications about any image following scheme. If an estimate

is made of an image location, no matter how small a variance this esti-

mate may have, it is possible that other than the correct image is

arbitrarily close to the estimate. Hence, no estimator can "isolate"

a path, i.e. a decision as to which image is correct will always have

to be made. Finally, to make matters worse, both of these problems

increase in severity as particle density increases. The analysis in

Appendix A shows that data collected by Jackman can be considered

relatively sparse. He had a particle density on the order of one

particle per cubic centimeter. The probabilities of overlap and con-

fusion for this case are very small and allow the data to be reduced

even with the large measurement noise. Higher densities would have

less chance of being reduced correctly and hence would require a smaller

particle density or a smaller illuminated region. The sparse nature

of the data explains why it can be reduced at all.

The data acquisition system described has been shown to have

a number of errors present; some correctable, some not. Optical error

sources were identified which can typically be reduced in severity or

eliminated by better design. Basic phenomena of stereo projections were

discussed and quantified for use in the particle follower theory.








20
A simplified projection system is considered in Appendix A.

This shows that the characteristics of the two projections, while

directly related to the particle path, are not particularly similar.

This means that there is very little information available to identify

a particle's stereo images. This has impact on particle following in

the sense that it limits the possible techniques available. The pro-

cedures developed in this work are limited to identification of image

paths as opposed to a more global approach that incorporates some

three-dimensional information.















CHAPTER III

A PARTICLE FOLLOWING MACHINE


Overview

Presentation of the theory of a Particle Following Machine (PFM)

begins with the formal definition of its input and output. Then image

path attributes are presented and models developed. Next, the decision

process used by the PFM to follow a path is given. This process requires

use of a Kalman filter and, from its residual, an image feature vector

is defined. After the residuals have been modeled, the details of the

decision process are stated. The use of learning to find some of the

probabilities needed for the decision process is then discussed. Finally,

control of the PFM and measurement of its performance are presented.

Figure 3-1 is an abstract description of the PFM. The image data

array Sik. (described in the next section) forms the input to the PFM.
1j k
Using its "concept of paths," the PFM sorts images into groups which are

composed of all images of one particle. These image sequences are denoted

as paths P p = 1, 2, ...N, where N is the total number of paths found.

The PFM is then a decision maker; identifying patterns (paths) described

only by their "concept." This "concept" is formulated in terms of pro-

vided rules (models) and learned probabilities. The remainder of this

chapter formally defines the PFM and develops procedures to accomplish its

task. A summary of the process is given in the last section. Figure 3-6

shows the details of the PFM described in the summary but may be used to














INPUT


THE PARTICLE FOLLOWING MACHINE

FIGURE 3-1


REGION FOR IMAGES
THROUGH TOP OF PRISM


imox


-~ I --


im ax `*.

THROUGH BOTTOM OF PRISM



IMAGE PLANE COORDINATES

FIGURE 3-2


OPTICAL AXIS
(Uaa)

Jmin


j Sijk


DATA


FRAME
BOUNDARY-


flow


DIGITIZED








23

aid understanding throughout the chapter as the specific formulation is

developed.


PFM: Input and Output

The input data to the PFM is assumed to be a digitized and "reduced"

version of cinematographic records. These records consist of pictures

(copies of the front image plane) taken every T seconds. They could be

produced by an analog or digital video source as well as a film camera.

Reduction of the picture data consists of estimating the center of the

finite sized images. (Even with known errors in shape and location of

the image, this is currently the only reasonable way of determining the

image location.) Unavoidable errors exist in these locations (measure-

ments) which are due to the phenomena discussed in Chapter II (overlap,

optical abberation and particle motion). In the case of Jackman's data,

the error of manual entry of the image centers and reference points was

also present.

Continuous coordinate axes, (u,v), are defined for the image plane

as shown in Figure 3.2. An image center (ul, v1) is quantitized with a

resolution of (Au, Av). If i and j are the discrete indices of the u and

v axes respectively, the discrete axes can be defined as u = iAu and

v = jAv. Therefore, an image center (ul, 1) in the frame would have

integer indices i = l, j = 1
S-' j = (round-off to integer is implied). The
Au Av
digitized location, then, has an error of +Au and +Av for the u and v

axes respectively. The discrete axes have the same origin as their con-

tinuous counterparts and have ranges corresponding to the size of a

picture frame. Without loss of generality, the coordinates were chosen










such that the u axis is in the same direction as the pipe's x axis and

the v axis is in the z direction if the camera and prism are aligned

properly (see Appendix A). Then, images typically move in the increasing

u direction (due to mean flow rate) and the top section of the prism

generates images in the top half of the front image plane; the bottom

section generates images in the bottom half. This forces top images to

have positive v (and j) values and bottom images to have negative values

while the u (and i) value will always be positive. The frame boundary

is defined by limits on (i, j):

u range: 0 i i m i 0 u r i Au
max max

v range: j mi j < j : j i Av v j max Av
min max mm max

where j max is positive and j min negative. Then the optical axis inter-
max mnu
i Au
sects the front image place at (u v ) = ( max 0).
2
As mentioned earlier, the sample period is T seconds so frame k

contains images at time t = kT. Time is assumed to begin at zero for

frame one (t = 0) and increase to a final time t = k T where k
o f max max
is one minus the number of frames of data available. Therefore, the

range of k is: 0 k s k The data from an experiment and hence the
max.

input of the PFM may then be represented by the binary array

1 if image present
S =
ik 0 otherwise.

where (i, j, k) have the limits previously discussed. Array S can be

considered as the output of a fictitious sensor that performs all the

preprocessing; each (i, j) plane representing one film frame at time k.









A particle in the field of view (therefore having at least one

image) undergoes continuous motion that is sampled by the camera. Thus

it is recorded as a sequence of image locations which are defined as an

image path. In general, an image path P for particle p that is n samples

long may be written as the sequence:

Image Path P = {S.. S. ..."'' Sijn (1)
p 1j1 132 ijn

where any Sik = 1, k = 0, 1, 2, ..., n. Path P is then a subset of all
ijk p
image locations.

When a particle generates two image paths (stereo views), ideally the

two paths can be used to determine the particle's location in the pipe.

The two paths are defined as invertible image paths. One path will neces-

sarily be in the top region of the front image plane (j > 0); the other

in the lower region (j < 0). When a particle generates only one image

path, its position cannot be determined and the image path is termed non-

invertible. Several problems occur when the errors in the data acquisition

system are accounted for. Image locations have errors that make invertible

image paths as defined above not necessarily invertible. The inversion

requires tracing rays from the image locations into the pipe. Ideally

the two rays from a stereo image will intersect at the particle's posi-

tion but the measurement error prevents this. In addition, images may

be inadvertently generated or lost. (See Chapter IV for examples.) This

can occur for a number of reasons, but it means that some images may not

represent particles and some particle's image may be missing thereby

adding spurious images that need to be eliminated and leaving gaps in

paths that need to be filled. Finally, there is the possibility of overlapping









images. This means that an image may belong to more than one path

prohibiting any possible one-to-one relationship between images and

particles.

The PFM does not determine invertible image paths. It finds all

image paths in the data leaving the determination of conjugate paths to

a later procedure. Therefore, (1) represents the output of the PFM.

The PFM's task has now been set. Its input and output have been

stated precisely in this section. The following sections build the

internal procedures that the PFM uses to accomplish its task.


Image Path Attributes

Since the input of the PFM is strictly array S, it is important to

know the characteristics of image paths that it is expected to contain.

One must be careful not to give image paths characteristics found in the

three-dimensional particle paths. The PFM can only operate on what it

can "see," i.e. the two-dimensional projections. It is interesting to

note that very little is known about the input data. This is discussed

further in Chapter IV when the initialization problem is considered. What

is of interest here are the characteristics of an image path: What inform-

ation is available; what is important; what can be determined:

From the definition of an image path given in the last section,

fixed attributes can be identified as follows.

1. An image path consists of a subset of S.
2. An image path will be in either the top or bottom part
of the film plane.
3. The time index of an image path increases monotonically.










Variable attributes include:

1. Image dynamic state (motion, velocity, acceleration) changing
with time.
2. Image location perturbed by measurement noise.
3. Confusion (local image density) varying along a path.
4. A conjugate image path may or may not exist.
5. Sample rate uncertainty.

Fixed attributes are hard and fast rules that define what constitutes an

image path. In any array S, there are many possible image paths, each

with its own qualities expressed by the variable attributes. To determine

the true image paths, these quantities must be measured and compared to

the internal path concept. The PFM, then, must filter the true paths

from the set of all possible paths (candidates). To develop a procedure

to accomplish this, the quality of the variable attributes must be

considered.

The dynamics of the image are, of course, closely tied to the

dynamics of the particle (see Appendix A). If a perfect dynamic model

of the particle existed, there would be no need for the experiment; the

structure of turbulence could be easily simulated. Since this is not

the case, it is necessary to model the particle (and image) motion. The

approach taken here uses a model that is known to be simpler than the

actual process. This is done to allow the PFM to be flexible and follow

images for different Reynolds number flows without having to modify the

model. The way that the PFM uses the model error to aid in tracking the

images will be discussed shortly. First, the image path model is iden-

tified. Using the results of qualitative observations made of the

particle motion presented in Chapter II, an appropriate model for a

particle path is a constant acceleration process. This results in paths









with smooth trajectories and if the acceleration is small, they will

have small curvature. Image paths are assumed to have similar char-

acteristics but only be two-dimensional. If r(t) is the position of an

image, it may be written as


r(t) = u(t)i + v(t)j


where i and j are unit vectors in the +u and +v directions. The con-

tinous path is assumed to be twice differentiable so velocity and

acceleration of the image may be written as


i(t) = v(t) = u(t) i + #(t)j

_(t) = a(t) = u(t) i + v(t)j


The components of these vectors describe the dynamic state of an image

and are used to make the image state vector,


u(t)
v(t)
t u(t)
x(t) = u(t)
u(t)
v(t)


At some initial time, t = 0, the initial state vector is


u(O)
v(O)
x(0)

(O(O))









Assuming a(t) is constant, we have


i(t) = v(t)

_(t) = a(t)

i(t) = 0


i+Fx


where


The state transition matrix 4(t), for this stationary system, is then


found (see e.g. Gelb, 1974). By definition,


P(t) = eF(t)


which is expanded to yield


4(t) = I + Ft +


22 3 3
+2 2.3
2 2.3


t2/2 0
0 t2/2
0 0
+ 0
0 0
0 0
0 0


fl
0
0
0
0
0


0
0

0
0
0
0










1 0 t 0 t2/2 0 1
0 1 0 t 0 t2/2
0 0 1 0 t 0
0 0 0 1 0 t
00 0 0 1 0
000 0 0 1


For the sampled system with sample period T, we can then write


{xk+ = (T) {x}


where


1 0 T 0 T2/2 0
0 1 0 T 0 T2/2
(0 0 1 0 T 0
0 0 0 1 0 T
0 000 1 0
0 0 0 0 1


This is a recursive state transition relation for an image based on an

assumption of constant acceleration. There is no process noise so this

is a deterministic process. Actually, the acceleration will be a random

process due to the measurement noise so the state becomes stochastic.

The expected range of the states are estimated from the qualitative

observations in Chapter II. The image position, of course, must remain

within the frame boundaries. The image's velocity, u, will be limited

by the fluid's maximum axial velocity as discussed in Appendix A. The

velocity, 7, will be smaller than u. Recall from Chapter II, that the

image paths were found to form at most a 10-150 angle with the x axis of

the pipe. Hence v can be expected not to be more than 25% of u. The

accelerations, u and v are expected to be small and slowly changing. Note

that a zero acceleration reduces the image motion to straight lines.










Hence this model is expected to be adequate for short sections of an

image path. With a dynamic model of an image path and expected state

values, the PFM can begin to sort the proper image paths from all the

candidates.

In addition to the dynamic model, something is known of the measure-

ment noise. It is assumed that the measurement can be represented as a

linear combination of image state and additive white gaussian noise, i.e.


z = H 4+ w
z-k :I

where zk is the measurement vector, H is the output matrix and wk is a

white Gaussian process with mean zero and covariance R.* Since the

measurement consists of the (u, v) location of an image,


1 0 0 0 000
0 1 0 0 0 0


The combined errors due to the optics and the digitizer (manual or

automatic) are assumed to be gaussian and uncorrelated. The covariance

matrix is then written as



uR 0
R = R u
R= R




This is recognized to be only approximate since there are other identifi-

able sources of error that could be individually modeled. This is not

done because, for the present, the measurements have large gaussian noise



*The use of w as the measurement noise should not be confused with some
conventions where w is process noise.








32
components (see Appendix C) and, in addition, a simple model is adequate

for the tracking problem where suboptimal estimation is accepted. It

should be noted that one source of error that was ignored and is not

particularly easy to model, was the frame alignment in the digitization

procedure. This error does reduce the performance demonstrated in

Chapter V, but the alignment problem will be significantly reduced in

future work (Lindgren, 1977) making it allowable to ignore this error for

the present.

The last three variable image attributes are more difficult to model.

Confusion comes from high local image densities causing less confidence

in the resulting decision. It is uncertain whether or not special pro-

cedures could be applied to regions with high confusion. One possible

approach would be to use the conjugate image paths and perform three-

dimensional tracking. In the pipe, the particles might be more separated

making the correct decision more obvious. This approach would require

knowledge of the conjugate paths and a trial-and-error search of all

combinations of the candidate images would be necessary to find them.

(Recall that conjugate images are not yet known to the PFM.) This was not

incorporated in the current work and would require more consideration

before being implemented. Any benefit could easily be outweighed by the

added cost since the paths traced cannot be checked for exact accuracy,

only compared to what a human would choose. Furthermore, finding

conjugate paths is not a simple matter. Due to the measurement errors

and the characteristics of the optics, conjugate images do not have the

i location. Finding paths that have a maximum similarity of horizontal








33

(i axis) positions works for random measurement noise but does not con-

sider the optical shifting and scaling that may take place (see Appendix

A). Hence for high densities where confusion is large, conjugate paths

would also be difficult to find, leaving the PFM with more work and

possibly no more information.

The path attributes, fixed and variable, allow a candidate path to

be rated as to its possibility of being a true image path. The next

section shows how the attribute metrics are applied.



Following Image Paths: The Decision Process

To begin the process, it is necessary that a partial image path

(a segment) be identified. This is referred to as the initialization

problem and is treated in Chapter IV. Then there are three possible sub-

problems that can be defined: the forward, backward, and central problem.

The first two are labels given respectively to the extention of a segment

forward and backward in time. The backward problem is of interest

because it might be used to resolve some cases where abnormally large

measurement noise occurred or high confusion exists. The central problem

is a combination of the forward and backward cases. This involves con-

necting two segments to make one long segment. Only the forward problem

is considered in the present work.

To consider the forward problem in detail, it is assumed that a

segment of path P for particle p has been followed up to time k-l. This

is denoted as P k-. Using this segment, a prediction is made for the
location of the next image in the path:
location of the next image in the path:









-k= H(-) = H-k-1

where xk (+) is the estimate of the state of the last image of segment
-k-i

P k- and xk(-) is the predicted state of the next image in the segment.

This relation is obtained by taking the expectation of the measurement

equation and using the state transition equation. It is assumed that

the desired images) that extend segment P [k-l will be in a window, W,

constructed around the estimate zk. The images in the window for the PFM
-k
are defined as candidate images,


{g : (S = 1) and (i, j) is in W}
ijk
q = 1, 2, ..., qk


where q is the (i, j) location of image q in W at time k. Connecting
k
segment Ppk-1 to a candidate image forms a forward link (simply called

a link) from time k-1 to time k. Since there is uncertainty as to which

link or links are true links (those that accurately represent the real

particle paths), a link probability is defined as:


Sk = Pr{P connects to candidate q}
pq p k-1 -

Figure 3-3 summarizes these definitions. At the center of the window is

the prediction zk. Neighboring images (the i's) are shown to link to the

k
prediction by the candidate link vectors (v's) with probabilities
pq
The result of the decision process is described by the decision state

vector apk = {a(1), a(2), ..., a(q )} where a(q) is the decision of true

or false for the link to image q. The i subscript represents different

possible decision states. The number of different decision states depends






















LINK PROBABILITY k
pl


1
7 CANDIDATE



Akl CANDIDATE ERROR


- PREDICTION




2kqk


II
I
k p
pq


j


il --


Sk
4^


IMAGE CANDIDATES AND ERRORS


FIGURE 3-3










on the number of candidate images, qk.* For qk = 1, (only one candidate

link), there are only two possible decision states: the link is true

(ai = {1}) or not (a2 = {0}). For q = 2, there are four possible deci-

sion states: either both are false (a = {0,0}), one link or the other

is true (a = {1,0}, a = {0,1}), both links are true (a = {1,1}).
2 3 -4
In general, the decision states have three categories (called Hypotheses):

HO: No true links - path stops

H1: One true link - path continues normally

H2: More than one true link - path branches

Figure 3-4 summarizes the link classes and decision state hypotheses,

where co is defined as the false link class, and cl, the true link class.

Also given is the number of states in each hypothesis. For example, let

qk = 3. Then there are three candidate links and eight different decision

states. Of these states, one is H0, three are H1, and four are H2. If

pk
all decision states, a were of equal probability, H2 would have the most
i
chance of occurring. Of course this is not the case as will be indicated

shortly. In general, for qk candidate images, there are 2 k possible
pk Aqk
decision states, pk i = 1, 2, ..., 2 of which one type is HO and q

are H1. The remaining are of H2. Each state is mutually exclusive making

the three hypotheses mutually exclusive. Therefore

3 2 k
SPr{H.} = 1 and Pr{a.} = 1
1 -1
i=l i=l

As qk increases, the PFM must discriminate between an increasing number

of decision states making the work increasingly difficult.



*Decision states are the different vectors, a1, whose components are
link states a(q), q = 1, ..., qk"
K'











Link Classes


Decision State
Hypotheses


qk H H1 2

1 1 1 0

2 1 2 1

3 1 3 4

4 1 4 11

5 1 5 26


Number of
in the



Example for qk = 3


Link
Class


C0:
c :
H :

H1:
H2:


False link
True link

no links true

one link true

more than one link true


Decision States Resulting
Hypotheses H0, H1, H2




Decision
State


'1':
'O':


true link: c1

false link: c0


LINK CLASSES,


DECISION STATES

FIGURE 3-4


AND HYPOTHESES


a1 S2 a3 4 26 6 a-7 48

0 1 0 0 1 0 1 1

0 0 1 0 1 1 0 1

0 0 0 1 0 1 1 1










To formalize the decision process, we consider a simpler example

as a guide. Assume that an object has feature vector C. Two possible

classifications exist for the object: co and cl. If there is no cost

for a correct classification and unit cost for an error, it can be

shown that the minimum error rate Bayes decision rule is:


Decide cl if Pr{cI } > Pr{col[}.


This says that cl is chosen if the probability of cl given the feature

vector C is greater than the probability of cO given C. (For a derivation

of this result, see Duda and Hart, 1973.) For the PFM, there are several

possible decision states to choose from. The feature vector becomes a

matrix of feature vectors and the problem is termed a compound decision

problem.

The possible decision states for the forward problem may be written

as (the subscripts p and k are dropped for simplification):


a., i = 1, 2, ..., n
-i


where n = 2 Representing the feature matrix as E, the minimum error

rate Bayes decision rule becomes:


Decide a. if Pr{a. I} > Pr{a. i} V. (2)
-i -J J # i

where ac. represents one state (a vector of qk link decisions). Bayes rule

is used to calculate the required probabilities. This rule relates the

probabilities required in the decision rule (a posteriori probabilities)

to the a priori probabilities. This is written as








Pr{|ia.} Pr{a,}
Pri 15E} = --
Pr{S}

where ai is one particular decision state. Formulating the decision

process in such a manner is of little direct use. Some assumptions must

be made to reduce the calculation of these probabilities to a more

tenable form. The term Pr{5ia.} is the probability of the features, E,

given a decision state, a.. Here, it is reasonable to assume that the

features are independent of one another. We can write

qk
Pr{ ia.} = Pr{(_ a(q)}
q=1

where the symbol n is used to express the product, and Pr{q la(q)} is
-q
the probability of the feature vector q occurring for a classification

a(q) of the link to image q. Note that a(q) represents a choice of

either class co or cl. The term Pr{E} is the probability of the features

for all classifications. This term acts as a normalization constant and

can be ignored for the present situation. Finally, Pr{a.} is the
-1
a priori probability of the decision state a.. As discussed previously,
--1

the decision states have categories H0, HI, and H2. It will be assumed

that all a 's classified as H1 have equal probability. The same assumption
--i
*
is made for H2. Of course, only one ai can be classified as HO. It

is therefore possible to compute the probabilities required for the deci-

sion rule. In order to calculate these probabilities, the feature

vector must be specified and modeled.




This is the weakest assumption. The hypothesis H2 contains cases of
of single, double and higher branches which could be given different prob-
abilities. However, a properly chosen window size will keep qk small
and the possibilities of higher order branches low.










Note on alternate decision rules. The Bayes decision rule is very

general and complicated. It is reasonable to ask; is this necessary?

Considering other alternatives, this decision rule does appear very

reasonable. An alternate rule might be to arbitrarily choose one

image in the window and assume the error generated would not be

significant to the results. A second possibility would be to choose the

image in the window that is closest to the predicted location.
The first rule might be acceptable if the window were very small.

This would require small measurement error and a better dynamic image

path model. The same would also be necessary for the second rule to

be viable. Improving the resolution and reducing the measurement error

of the system are certainly desirable. However, improving the dynamic

model can only be done after more is known of the structure of the

turbulence. This is what is being sought in flow visualization research.

In addition, the closest image to the prediction, because of confusion,

may not be the correct image. No matter how good the estimator, there

will be some variance. This manifests itself as a region of uncertainty

around the prediction and results in a finite probability of an image

intervening between the correct image and the prediction. Therefore,

the minimum error rate rule (2) based on the estimator error statistics

is used and its performance is expected to be statistically better

than these alternate procedures.





*
This rule would result from assuming the feature vector to be gaussian
and have zero mean. This would not necessarily be valid.










The Image Feature Vector

The feature vector, required by the decision process, should

be as simple as possible, but contain sufficient information for

decisions to be made at an acceptable error rate. With this as the

goal, the feature vector chosen is derived from the relative location

vector (candidate error vector), kq Recall that this was defined as


S = qq Hx (-)
-kq --k k


(see Figure 3.3). This looks very similar to the residual in the

Kalman filter and the connection will be made shortly. First, the

Kalman filter used for estimation and prediction, will be presented.

Then the feature vector will be modeled and used in the final forms

of the decision rule.

Estimation and Prediction. The Kalman filter has been extensively

covered in the literature (see Kalman, 1960, Bryson and Ho, 1969,

Jazwinski, 1970, and Gelb, 1974) so only the results will be presented.

Refer to Figure 3.6 at the end of this chapter for a block diagram of

the decision and estimation system.

As described previously, the state x has a recursive transition

equation

-k -k-I

where D is the stationary transition matrix. The measurement equation

was given as

z = H w+ wk

where wk is a zero mean, uncorrelated gaussian noise process with
-K










covariance R. If x is the state estimate, its error is written as

k = k xk*
It can be shown that minimization of the cost function
-T
J = E{ x Qx }

where Q is any positive semidefinite matrix and E{ } is the expectation

operator, for a linear, unbiased, estimator yields the following Kalman

filter relations (as in Gelb, 1974):



Prediction
Equations Pk = P (+) Tk


(+) = k(- + Kk -k xk

Update Pk(+) = {I KH P(-)
Equations
Kk= Pk(-) H HPk(-)H + R}-1


Here, the symbols (-) and (+) are used to indicate the estimate before

and after a measurement is taken. The matrix P is the covariance of

the error, x, and K is the Kalman gain. The initial values for the system

are x (-) and P (-). The first measurement updates these estimates to
--o o
x (+) and P (+). A prediction can then be made and the process repeats.
--o o
This is an optimum filter in that it gives an estimate with minimum

error covariance.

For the ideal case where the process is perfectly modeled by

the state transition equation, the covariance of the error becomes very

small making the gains, K, very small which essentially makes the output

independent of the measurements. The residual









u
= fk] =
k
:k v H
Vk


which is the difference between the measurement and the predicted

state estimate becomes a white noise (uncorrelated, gaussian) process

and hence contains no information. For the PFM, it is known that the

dynamic model, as for most real systems, is not precisely accurate.

This leads to divergence of the estimate from the actual state.

The problem of divergence is treated by Price (1968). To eliminate

divergence, solutions typically involve improving the system models

(using more states) or using a filter with only a finite memory. The

latter technique forces the system to ignore input in the distant past

and keeps the covariance matrix from becoming very small. This causes

the filter to "track" the input. Another alternative is to add arbi-

trary process noise which also keeps the covariance from going to zero.

All of these techniques have good and bad points and their usefulness

is dependent on the application. The estimator that is used in the

present work is the finite fading memory filter. (See Tarn and

Zaborszky, 1970, Sachs and Sorenson, 1971, and Miller, 1971.)

The idea behind the finite fading memory filter is that new

measurements are weighted more heavily to make the filter "forget"

earlier measurements. This is appropriate for the PFM since it is

the tracking function that is desired as opposed to a truly optimal

estimator. The error covariance matrix for measurements that occurred

at time j is increased by
k-j
R. = s R kzj.
J










The normal covariance, R, is a constant. The present time is k and

s 2 1. The resulting filter equations are the same except for the

covariance which is


P'(-) = s 0 P' T
k k-i

where the prime indicates that this is a different covariance matrix

than normal. The value of s is empirically determined. The residual

vk has different characteristics as s is changed. This is demonstrated

in Chapter V. Essentially, s selects how much memory the filter has.

With s = 1.0, the filter reduces to the normal Kalman filter and all

past points affect the prediction and state estimate. This is not

desirable for the PFM because the residual may diverge. As s increases,

the past has less affect on the estimate causing the residual to always

remain finite. By doing this, the residual becomes more consistent

and therefore modelable as is shown in the next section. Intuitively

it seems apparent that if the residual remains finite, never crossing

zero (i.e. has a non-zero mean), a current value would be correlated to

the distant past. By using values of s other than one, the autocorre-

lation of the residual becomes small between a present value and past

value, eventually becoming dependent only on its next to last value.

This is the assumption used in order to model the residual.


Modeling the Residual

The residual, presented in the last section, is a discrete sto-

chastic process that "contains information" as to the difference between

the image dynamic model and the real process. Use of the residual to










modify parameters is a type of adaptive filtering which is a form of

machine learning. An example of a system that uses adaptive techniques

to modify the process noise is found in Jazwinski (1970) and to change

the system dynamic model in McAulay and Denlinger (1973). Other examples

are also found in Gelb (1974). However, the PFM does not use these

techniques. It operates at a more basic level because it must first

identify its next measurement. That is, it must choose which image in

the next frame i:ost likely belongs to the particle being followed.

To model the residual, several assumptions are necessary. First,

it is assumed that the residual can be characterized as a Markov chain.

Therefore, we can write


Pr{l k-1, k-2..'" } = Pr{Ikk-1


That is, the joint conditional probability of k given all past residuals

is dependent only on k_-1, not the entire history of the chain. In

addition, it is assumed that the statistics for each path':s residual

are stationary and applicable to all transitions on all paths (i.e.
u v
ergodic). Finally, it is assumed that the elements vk and vk of the

residual vector v are independent.

Using the assumptions made for the residual, the feature vector

for candidate q is specified as

= }T
-kq 1-kq' -k-


In words, the feature vector of a candidate image is a vector composed
*
of candidate residual, vkq, and the last residual yk-. The matrix of



The last residual, v ,was the candidate residual selected at time
k-1 and hence does not Aave a q subscript.









features, E, is then

= ^l k2' "'' kq k


This represents all the features used to make the forward following

decision for path P k-l'

To actually make a decision, the decision rule is written out

explicitly. Recall that the probability of a decision state given the

feature is

Pr{'*a.} Pr{a.}
Pr{a E} = -
Pr{E}



Considering Pr{2a} = Hk Pr{e.la(j)}
q=l



where Pr{ qla(j)} = Pr{vkj, uk la(j)}Pr{rv v, lla(


because the u and v residuals are assumed independent. The probabilities

in this last relation are joint conditional probabilities of the last two

u and v residuals given the candidate's link state a(j).

At this point, it is useful to present an explicit example for the

situation qk = 2. There are four (2 ) link states given as:


Link Class

a(l) a(2)
a 0 0
-1

2 1 0
Decision '1' = true
a 0 1
State '0' = false
a 1 1
--4 _____






47

We can then write (while reverting to vector notation for the residuals)


Pr{~ljE} Pr{EI-I} Pr{~l} = Pr{kl,.l C o} Pr{vk2' 1'-1 } Pr

Pr{a2E} = Pr{Ea2} Pr{a2} = Pr{vk, Ikl c} Pr{k2'-k-l c} Pr{2}
(3)
Pr{aB E} = Pr{(a3} Pr{ 3} = Pr{vkl'kI co} Pr{4k2,'k-J1co} Pr{(a

Pr{aIE} = Pr{Ea4} Pr{(4} = Pr{ykl,4 k cl} Pr{vk2' k-lci) Pr{a4}


To interpret this, the quantities on the left are referred to as the a

posteriori probabilities of the decision state, i.e. the probability of

the decision state given the features. These are only proportional to the

right hand side because Pr (E) has been eliminated from the denominator.

(Recall that this is just a scale factor.) The Pr(a ) term is the a

priori decision state probability and is conditioned by the likelihood

of the feature given the state, Pr(E Ia). Recall that the decision state,

aR, is the vector composed of all link class decisions. Then, for example,
-i
with a, = {co, co}, we have


Pr{Bla Pr{v ,v klco} Pr{v v C}
Pr{ = -kl'-k-l P-k2'-k-1l}

since features are assumed independent. The quantity Pr{.kl,_k-_lco} is

the joint probability of candidate residual Yk, and the last residual k-1

given a class co (false) link. As another example, consider


Pr{Ea2} = Pr{yl,kllcI} Pr{k2',k_- co}.


Here, Pr{vkl,kl-cl} is the joint probability of the candidate residual
1
and the last residual given a true link to candidate image k. After

making a few observations, a useable relation will result. Terms like









Pr(v kqv -lc} = Prt{ Ico} refer to the probability of the feature

vector E = {(vkq ,vk} given a false link condition. It can be reasoned
-q
that an image with a confusing false link can occur anywhere in the

window with an equally likely chance. Therefore, any one particular

feature vector has a probability of occurrence that is equal to one

divided by the total number of possible feature vectors. (See Chapter V

for a specific example.) Terms such as Pr{. Jc1} are the probability of
-J
a true link. to the image q with feature vector E Figure 3-5 shows
-K -q
1 2
an example for the case qk = 2. Two candidate images, 1k and k are

found in the window around the prediction zk. Candidate residuals are

v and v. A past residual of is assumed to have occurred at
-kl -k2 -k-1
time k-l. The sketch shows contours of a possible residual joint prob-

ability Pr{vq ,U k cI} = constants L1, L2, or L3. If, for example,

k-1 = d, then the joint transition probabilities Al andA2 are found, i.e.


A1 = Pr{v, dlcl}


A2 = Pr{vk2 dc}


as shown. The joint probability shown is unimodal but, in general, no

constraints are placed on its form so it could be multi-modal. Since the

residual statistics are assumed stationary, it can be expected that this

quantity could be learned from examples of correct image path sequences.

Before discussing the learning aspect further, the a priori probabilities,

Pr{a) need to be examined in more detail.

The decision state, a., was previously classified according to three

hypotheses: the path stops, continues normally, or branches. It can be










S: prediction 1
1 2 2k
k, : candidate images

~V' k2: candidate residuals
u
u k1j
u


k2 2
Ik2

2

IMAGE PLANE CONFIGURATION
IMAGE PLANE CONFIGURATION


v


u_
v
"k2


U
v Candidate
Vq Residual


A
A2-----
I


L3 k Pr{vq uk-l11 c1
= constant


u
vk-1
Last
Residual


I k2


EXAMPLE OF JOINT TRANSITION PROBABILITY

FIGURE 3-5









expected that the chances are much greater that the path will continue

normally rather than stop or branch if there are no missing images and

the image density is sufficiently small to limit:the number of overlaps.

As shown in Appendix A, the overlap expectation is a function of particle

density and system resolution. The chance that a path stops is negligible

(if not near the edge of a frame) except in the case of data that were

input using the graphics tablet. For tablet data entry images are

easily omitted and will cause the performance to suffer. In essence,

the a priori probabilities define expectations of what the decision

results should be. They weight the calculations of the a posterior prob-

abilities and, hence, directly affect the selected decision state.

The a priori probabilities vary as the number of candidates changes.

When there are no candidate images, qk = 0, it is assumed that the path

stops. Therefore, Pr{H0} = 1.0. For the case of only one candidate,

it is assumed that the path will most likely continue, linking to the

candidate (Pr{H1} = .99, Pr{Ho) = .01). For qk = 0 or 1, branching can-

not occur so Pr{H2) = 0. When qk = 2, branching can occur so Pr{H2}

is just the probability of image overlap calculated in Appendix A. These

probabilities were found to be negligible for the case of two or more

images. Therefore, when qk = 2, decision state 4 = {cl,c1} is highly

unlikely. (Thus branching is assumed negligible.) If qk = 3, one state

is {clC]cl} which has a very small likelihood. Other states such as

{clcico} or {clcocl} would have the same probability as aq = {cici}. It

is clear from the analysis presented in Appendix A that the current data

available is a rather sparce situation since the chance of overlap and









confusion are so small. This is good for the PFM because the expected

number of decisions to be made is small and reduce to picking the one

most likely link. It is important to remember that as the density or

lighting situation changes, the probabilities of the hypotheses, HO, HI,

and H2, will change. Returning to the example for q = 2. Define

PO = Pr{ lIco}, q = 1, 2, so (3) can be written


Pr{allE} (P0)2Pr{H0}

Pr{c2IE} AIP0 Pr{H1I
2

Pr{ 3|} POA2 Pr{H}1
2

Pr{aI} AIA2 Pr{H2}


where Al and A2 are the probabilities Pr{l ci} and Pr{ 2lc2}. Now,
it is assumed that Pr{H0} and Pr{H2} are very small compared to Pr{H1}.

Therefore, the decision amounts to a choice between states a2 and a .

Bayes minimum error rate decision rule (2) reduces to:

Choose a2 if A > A2 and a3 otherwise.

When A and A2 are very small, the probabilities for a a 3 and a are

small allowing the possibility of a Therefore, the decision rule becomes:

Choose al if POPr{Ho} > APr{Hi} and POPr{Ho} > A2Pr{Hi}
2 2


This just says that when neither link has a probability greater than the

chance of a confusing image being present, the path should stop decisionn

state a ). Finally, when A1 and A2 are both large, state a becomes

theoretically possible. The decision rule would be









Pr{H1}
choose a. if A Pr{H2} > POr{H
2

and

Pr{H1)
A2Pr{H2} > POr{H
2


This completes the example. A decision can be made as to which is the

most probable state. It is evident that the a priori assumptions are

very important in the stop-path and branch decisions but do not enter

in the normal condition where one link of the candidate links is chosen.

Similar results can be obtained for other values of qk.

Learning. An important aspect of the PFM is the determination of the

residual transition probabilities. It has been shown that this reduces

to determining the joint probability of the feature vector given a true

link, Pr{ Jcl}. Used here is a nonparametric procedure of identifying

the feature probabilities. (See Duda and Hart, 1973.) To determine these

feature likelihood, the PFM is given path examples that have been pre-

viously identified by manual effort, or alternatively, by previous output

(which has been verified by an operator) of the PFM. An identified image

path is a sequence of images that have true links. Therefore, no deci-

sion has to be made, the filter can be given the sequence as input. As

the filter follows this path a residual will be generated


V1' I 2' 3""'Vk-l' yk'

This sequence contains examples of residual transitions for true links.

By using several paths (the training sample), many example transitions

are generated. The residuals are quantified and the transitions -v to









k are counted to form a histogram of transition probability. There

are two histograms that result:


Pr{v v cl} and Pr{q v, v Cl
kq' k- kq' k-i

These are used by the PFM to make decisions on new data as discussed pre-

viously in this section. The decisions made by the PFM are then dependent

on the training sample. As the characteristics of the training sample

change, the PFM's decisions will change and thereby adapt to the char-

acteristics of the flow being studied.



Control of the PFM

With the capability of making a decision as to the link classes, the

PFM can follow image paths. At each step, the decision state probabilities

can be calculated and the decision state with the largest probability is

chosen. The three possible results are simply the state hypotheses as

presented earlier. It is expected that the normal response will most

often be taken (i.e. only one link found; Hi). When H0 is decided, the

path can simply be stopped. When H2 is decided, branches have been identi-

fied. They can be saved for following after the primary path (the path

with largest link probability) has been found. Dealing with branches is

very difficult; many possible combinations of links occur. For example,

if a path branches and then comes back together or, in general, any case

where a path is split, there is a question as to whether this represents

the true case or is a result of confusion. The only real answer to such

a question is to do everything possible to avoid branches. It is worth-

while to introduce the term isolated path. This is a path whose link









state decisions were all for the case of qk = 1. That is, no branch

decisions were necessary; the window around the estimate contained only

one image. This is what would be considered a "nice" path. If the

measurement noise is small enough and the estimator fairly accurate,

these paths are a reality. Data consisting of isolated paths would have

a high confidence level. Any other situations of confusion and overlap

would have a lower confidence level. As particle density is increased,

the possible confidence will decrease and more operator interventions

may be necessary to resolve confusing cases. The limit to the particle

density, then, is where the cases become so obscure that the operator

cannot be confident in his decision. The limit to the abilities of the

PFM to follow paths is clear. When the link state probabilities are

very close to each other (the decision boundaries), the decision is not

obvious; the PFM is indecisive. However, for reasonable measurement

noise and density, the PFM control decisions are expected to be correct

more often than not. By monitoring the probabilities likelihoodd) of

the decisions made, an operator can be warned of possibly bad decisions

and poor performance.



Measuring Performance

Perhaps the most frustrating aspect of the particle following

problem is the inability to know precisely what images belong to the

same particle. This means that any measure of error or performance of

the PFM is subject to interpretation. Obviously isolated paths can be

quickly identified by a human observer as well as the PFM. But when

confusion occurs, the true image path is only conjecture. Both the







55

human and the PFM must make decisions, and both will make mistakes. The

determination of performance of the PFM becomes a relative comparison,

i.e. does the path found by the PFM agree or disagree with the human's

belief? The human, of course, has the ability of taking more factors

into account than the PFM. However, the PFM is expected to be more con-

sistent in applying its concept of what an image path is. If the PFM is

made more sophisticated, it would be expected that its decisions would

agree to more of an extent with the human. However, there is a trade-off.

If a semi-automatic procedure is used that has a good performance and

only a few links are wrong and need human intervention, it might be best

to avoid making a more costly and complicated PFM. This can only be

judged with a great deal of experience using the program. For the

present, the performance of the PFM is judged against results given by

Jackman. No procedures were developed to handle special cases so the

error rate is not artificially decreased. Nevertheless, the performance

is fairly good as will be shown in Chapter V. When automatic data

acquisition techniques are used and the variance of an image's location

is reduced (probably by an order of magnitude), it can be expected that

the present PFM will provide satisfactory results. It is conceivable

that when errors are detected, the path in question can be discarded.

Alternatively, it can be kept if the choice is between two images that

would not alter the description of the flow field significantly. The

choice of how to deal with these "differences of opinion" allows the

experimenter to select his experimental accuracy and confidence. It

may be discomforting to know that the performance cannot be stated










quantitatively, but at least an acceptable level of confidence is

realizable.



Summary

The PFM has the task of identifying image paths (patterns) in the

data array Sijk. The approach taken here assumes that an initial path

segment has been identified which the PFM is to extend forward in time.

Referring to Figure 3-5, candidate images, J, q = 1, 2, ..., qk, are

found in a region (the window) around a prediction, zk, of the location

of the next image of the particle being followed. A decision must be

made as to which candidate images) is (are) most likely the correct

imagess. To do this, the probability of the transition vk-1 to -kq

is assumed known. The images with the most likely transition prob-

abilities are selected as true links. If more than one image is selected,

branching occurs. The PFM continues following only one branch, saving

the remaining branches to be followed at a later time. If no images

pass the requirements, or no images are in the window, the path is halted.

The normal situation is for only one image to be selected with index q .

This defines the measurement, zkq*. A Kalman filter is used to generate

the prediction, and its residuals are used as features in the decision

process. The sequence z k = k k + 1, ..., k + n 1 is the
-k o o o p
identified path, where k is the initial frame of the path and n is the

number of frames spanned by the path.







































































LU
I.-
V)

V)


C)
0




























Z r--4
Z *-
i- z


0
LL


I-

LL<


I-






LL
I-















-J
Q














CHAPTER IV

PFM: INITIALIZATION AND IMPLEMENTATION


Presented in this chapter is a discussion of the procedures used

to implement the Particle Following Machine (PFM). The material pre-

sented is not vital to the understanding of results given in Chapter V,

but will aid in the comprehension of where the results came from.

Three logically separate routines are outlined: the initialization

program (INIT), the Particle Following Program (PFP), and the learn-

ing program (LEARN). Listings of the source codes are given in

Appendix D. Before presenting the details of the programs, it is

necessary to present the background reasoning to the initialization

problem. The PFM, presented in Chapter III, assumes that an initial

segment has been identified and precedes to follow the image path

using its path concept. Finding initial segments must be done with

much less information and is therefore rather difficult. Once the

initialization problem has been presented, the discussion of the

programs is preceded with a short description of the data structures

used by the programs.

Initialization

The Initialization Problem. If the initialization of the particle

paths were simple, the whole problem could be reduced to a series of

initialization steps. However, the problem of finding initial path

sequences is not simple. Required here is a scheme that can take an

arbitrary image location and find its closest successors or predecessors

58









representing the path. This must be done without any knowledge of the

path's attributes. Especially important and lacking is any information

concerning the particle's velocity (speed and direction). The data

contains particles traveling at different speeds and many different

directions. The fact that there is a strong general velocity in the

x direction is useful, but as the turbulence increases, it becomes

less dominant. Difficult cases to consider here are the possible

backward motion of a slow particle due to measurement error and the

possible close proximity of a slow and fast particle. In the latter

case, the slow particle has a much higher spatial sampling frequency

that may possibly obscure the faster path whose spatial frequency is

much lower.

Considering this problem in terms of spatial sampling gives some

insight to the difficulties involved. The particles have different

velocities so when sampled at a constant time rate will travel different

directions and distances. Since, in the initialization problem, there is

no information available concerning the particle velocity, the problem

becomes one of identifying the spatial sampling rate of paths of arbi-

trary shape. To make matters much worse, large measurement noise

exists in data currently available making the spatial sampling rate

highly variable. If this were not the case, Fourier transform tech-

niques could be utilized to advantage. Here, spatial transforms of

the data would indicate strong sampling frequencies and directions which

could be identified through filtering. The fact that paths are not

straight causes some problems, but as discussed earlier, the paths

are roughly straight over five time samples allowing one to look for










straight line segments. With such a small number of points in the

assumed line, however, measurement error obscures any trend in

the spatial frequency.

The beauty of using spatial transforms is that they utilize all

possible information available in the initialization step. This in-

formation consists of knowing that in any region of the data array S,

there are straight lines that have different directions and different

spatial sampling rates. Therefore, the result of the initialization

step should be groupings of the images by path segments. The number of

groups are unknown, but each group contains up to a certain number of

images (the number of time samples considered in the region) whose

time indices are monotonically increasing. The reason for the unknown

number of groups is that the spatial dimension of the region under

consideration is arbitrary and hence may sever image paths leaving

a segment not starting at the time boundary of the region. This view

of the problem is that of a constrained clustering problem which could

also apply to particle following. The difference is that the particle

follower makes use of path history. Therefore, initialization is a

special case of the particle follower. That is, the probability of

an image in the next time sample connecting to one in the current

time is uniform; all images have equal chance. This is in essence

Jackman's initialization assumption. He then used a constant velocity

criterion to locate the next image. If only one image was located,

he assumed he had found the start of a path, otherwise he arbitrarily

chose one. This approach also returns to the problem of considering

all image path possibilities. As the image density increases, the









number of possible paths increases and the possibility of confusion

increases, thus making this approach far less desirable.

An Initialization Procedure. The final form of the initialization

program is an implementation of heuristic search procedures. A

particle in the first frame to be analyzed is chosen as the base or

reference. A window is constructed around this location with a size

sufficient to contain the next four images of the particle represented

by the reference image. All possible three-image paths are then found

and a cost calculated for each. The path with the smallest cost is

possibly a valid path. If other paths have costs within 10% of the

minimum cost, they are retained for further processing. Three-image

paths are not typically adequate for initial path segment identification;

there is too high a chance that three arbitrary images will have an

accidentally low cost. Therefore, a fourth and fifth frame of data

is considered. A new candidate path is formed by connecting each of

the low-cost paths identified in frames one through three to each image

in the window at frame four. The last three images of each candidate

are used to calculate a three-image path cost as previously described.

The cost from the first three and the last three images are summed

to make a cost for the four-image paths. Those paths with costs within

5% of the minimum are considered likely initial segments. Now, frame

five images are linked to the candidates. Again, the last three

images are used to calculate a cost which is added to the total cost

of the path calculated up to frame four. Finally, those paths having

costs within 3.3% of the minimum are selected as initial path segments

to be used as input to the particle following program. The criteria









for closeness changes from 10% to 3.3% because the costs are being

added. This causes the total cost to increase, so, by decreasing the

percentage, the criteria remains roughly the same. Note that this

technique does not guarantee that the paths found are the overall min-

imum cost paths because only three images are used at a time. This

was done in lieu of using all possible five-image paths which would

be very time consuming and expensive.

To be more specific, we define the reference image position

(in frame k ) as r = (i j ). A region surrounding this point in

space and time (called a window) is defined as


i i < i
w w
min max

Window R = j m w
w w w
min max

k k k + 4
o o




The size of the window is selected to include five samples of the

fastest particle. The qualitative observations discussed in Chapter II

indicate that images typically travel in the +u direction with small

inclinations to the v = 0 axis. Allowing for the possibility of a

noisy, slow particle backing up, i is selected slightly less than
w.
min
i Then i is set at roughly six times the maximum expected
max
velocity. Finally, j and j are set to allow the fastest image
max min
to be inclined approximately twenty-five degrees up or down. With

the window boundaries fixed, the image data set can be searched

for all images within the window. The resulting list of images is










defined as the candidate image list. Assuming that there is a true

path segment among all possible paths that can be constructed from

the candidates, one approach would be to calculate the cost of each

path as a function of its five images. Then the minimum cost path

would be the path most likely to be the true path. In order to save

computation time, a slightly modified procedure is used.

A cost function is defined for any three-image path. This cost

function is a heuristic whose specification was guided by qualitative

observations as well as trial and error. Basically, the cost is designed

to be minimal for a straight, evenly sampled path with a spatial sampling

rate (absolute speed) that is reasonable.

Consider a window containing n1 images in frame k + 1, n

images in frame k + 2, etc. Then, nI links connect the reference
o 1
image at O- to r (images p in frame k + l,p = 1,2,...,nl).
q
Each of these links could connect to any of the n2 images E2

in the third frame (time k + 2) making the total number of possible

three-image paths, n1 x n2. For each two-image link, a first dif-

ference is defined as

k to k + 1 : v r r V
o o -0 -1 -O p


k + 1 to k + 2: v r r V
o o 1 -2 -1 pq


The first difference is a vector quantity that is a first approximation

to velocity. The cost function, J is then defined for each two-link
pq
(three-image) path segment using the reference image, image p and image q.

The form for the cost is









V v

J 1 --
J = +
pq C C




The first term represents a change in magnitude of velocity, i.e. a

tangential acceleration. The second term finds the change in direction

(cosine of the angle) between the first and second link with the

result shifted to be zero when the directions are identical. This

term is an approximation to the direction of normal acceleration.

The reasoning behind the choice of this cost function comes from the

qualitative results that indicated that a short path segment was a

straight line, i.e. zero normal and tangential acceleration. Mini-

mization of the cost yeilds a path with minimum tangential and

normal acceleration. By selecting a value of the scale factors C1

and C and a maximum allowed cost, J ,a the performance of the
2' max
initialization procedure can be adjusted to be specific for most of

the correct initial segments.

Three-image segments are the shortest segments for which the path

accelerations can be calculated. It was found that there is a good

chance that incorrect candidate paths have a smaller cost than the

true path, especially where image density is high. Therefore, a limit,

C3, is placed on the maximum speed, I I, l'vql etc. Limiting the

allowed angle is not desirable since slow particles can have large

fluctuations in direction due to noisy measurements; the observation

of small inclinations being valid only for particles with nominal

speed.








65

Extending the search to five images reduces the number of errors

even further. To accomplish this, the search procedure looks for low-

cost paths in the first three frames. Then, using frame four (k0 + 3),

each of the path segments in the reduced candidate list, is extended

r
to all images r~, r = 1,2,...,n and the first difference is found

from

qr = r r V
-2 -3 -2 qr


Then, the cost of the last three-image segments is calculated by
pq qr
1 q
+ r + 2

pqr
1 2

where p and q are the image indices for the candidate paths and r is

the index for images in frame four. The cost, J is added to the
pqr
candidate costs, J and the low cost paths are determined by finding

those paths whose costs are within 5% of the minimum. This procedure

s
is then repeated using the images r4, s = 1,2,...,n in frame five.

Here, the first difference is


rs s r
v = r -r V
-.3 -4 -3 r,s


and the cost becomes
qr rs
v v
-2 -3
1 -
rs, qq qr I rs
3 -2 2 3
J =+
qrs C
2 3

where q and r are indices of images in the candidate paths and s is

the index of images in frame five. This cost is added to the candidate

costs to give a total five-image path cost for a given candidate path:










J =J +J + J
tot pq pqr qrs



Those paths whose overall cost is within 3.3% of the minimun cost are

the final initial path segments and are used as input to the particle

following program. Chapter V shows the results of using this initial-

ization procedure and discusses some of the special problems encountered.

The Data

Since this work focused on the theory of particle following,

new experimental data was not taken from the flow apparatus. Instead,

the data analyzed by Jackman (1976) was utilized. This consisted of

55 frames of raw image locations as generated by manual entry via

the data tablet as well as the manually identified invertible paths.

This data was collected using the small prism for a flow with Re = 3500

at 200C. A film rate of 25.7 frames per second was used to expose

Tri-X film through a 100mm lens fitted with a close-up attachment.

Images by Frame. Data, as entered via the tablet, consists of the

image locations in the film plane (i, j, k). At these locations,

array S, (see Chapter II), has the value '1'. The image locations

were entered systematically, but not necessarily in any sorted order.

Therefore, this data was sorted by i value which allows somewhat faster

windowing. Each frame of data contains different numbers of images

which, for the present purposes, are best stored as a vector of all

images and addressed via an index to the first image of each frame.

Three vectors are then used to represent the frame-by-frame data:

NPART, KINDEX and DATA. (See Figure 4-1.) NPART and KINDEX must have

dimension at least as large as the number of frames needed. (Not all








DATA

'jI1 J2 _2

INDEX

Frame 1
Pointer
N DATA
Frame 2 # of images # of images # of images
Pointer in Frame 1 in Frame 2 in Frame 3

Frame 3
Pointer



FRAME-BY-FRAME DATA

FIGURE 4-1


ID START FRAME LENGTH

il J '2 J2 '3 J3

IMAGE LIST

ID I START FRAME LENGTH
I l i 2 J2 i3 J3


IMAGE LIST


INVERTIBLE PATH DATA


FIGURE 4-2









frames are necessarily used.) DATA has dimension at least as large

as two times the number of images since each image location has an i

and j value. The numbers obtained from the tablet were integers but

when used by these programs, they were stored as real numbers. This

allows addressing of the data as a complex array equivalenced to the

array DATA. Then, when convenient, both i and j values of an image

may be retrieved by only one index. The frame-by-frame data were

rotated and translated to common axes and, without loss of generality,

the origin was set at the lower right hand corner of the tablet as

opposed to the image plane coordinates used in Chapter III. This

merely shifts the j values making them always positive. The portion

of this data set used by any program is currently stored in core which

simplifies the programming and speeds up execution.

Invertible Path Data. The second major data set consists of the in-

vertible paths identified by Jackman. These are the path sequences

P Each sequence has a length, a start frame number and the image

locations. This data set is accessed sequentially, path by path and

is not saved in core. Its structure is shown in Figure 4-2. The

paths are not in any particular order, but each path has an identifier,

ID, that was assigned by Jackman. Conjugate paths were the only

paths put in the data set, but they were not necessarily located

next to each other. When comparing the output of the PFM to the

paths in this data set, it was necessary to build an index that located

a path by its first point. However, because this data was manually

identified and punched, many errors occurred (they were typically small

since gross errors were corrected) making it impossible to compare









the results automatically. In addition, the paths in this data set

include generated points that were used to fill in missing data points.

The PFP was not built to handle this case since new automatic data

entry procedures will not have this problem (Lindgren, 1977).

The Initialization Program

The FORTRAN program written to perform the initialization procedure

is outlined in Figure 4-3. The main program, INIT, primarily handles

the selection of a reference image, various counters, and outputting

of the results. Subroutine REDDAT establishes the data arrays contain-

ing the image locations, frame start pointers and frame lengths as

discussed previously. Options allow the selection of printing inter-

mediate steps, plotting the results and storing or recalling the data

in the directly usable form generated by REDDAT. Input data to INIT

also includes selection of the time and space window boundaries. Sub-

routine WINDOW searches the data for images in a specified window.

Data is required to have been sorted by i values which speeds up the

search significantly. Subroutine TRAK determines the minimum cost

paths. The FORTRAN source code is listed in Appendix D and contains

additional documentation of these routines.

The Particle Following Program

The implementation of the particle following machine is the

Particle Following Program (PFP). This routine takes the initial path

segments identified by routine INIT and attempts to follow them for

a specified number of frames. A simplified flow chart for the PFP

is given in Figure 4-4.









READ OPTIONS, INPUT DATA

READ IMAGE DATA
FOR FRAMES ONE TO NTS

PLOT IMAGE DATA FOR
FRAMES NTSO NTS

SELECT A REFERENCE IMAGE
I S- ,

FIND ALL IMAGES IN WINDOW
BASED ON REFERENCE

FIND MINIMUM PATH(S)
FOR REFERENCE

NO
DONE?

YES
WRITE RESULTS



I
PLOT PATH SEGMENTS

END


SUBROUTINES
(REDDAT)








(WINDOW)


(TRAK)


SIMPLIFIED FLOWCHART OF "INIT"

FIGURE 4-3









SUBROUTINES

(PFPRD)


(LRNMAT)




(FILTER)


(UPDTE)


(WEST)


(DECIDE)



UPDATE )


END


SIMPLIFIED FLOWCHART OF"PFP"


FIGURE 4-4










This program begins by reading the options which consist of

writing or recalling the image data in a directly useable form (as

in INIT), and writing intermediate results. Subroutine PFPRD is

functionally identical to Subroutine REDDAT used by INIT. However,

larger data arrays are specified since PFP needs at least as many

frames of data as images in the desired path. LRNMAT reads the joint

probability matrices used in the decision process. These matrices

are the output of the program LEARN which is discussed in the next

section.

After the preliminaries, an initial segment is read. This con-

sists of five sequential image locations assumed to start in frame

one. (Only paths starting in frame one were considered for the demon-

stration of the particle follower.) The path state is then initialized.

The location of the first image is assumed to be the path's estimated

location. The initial velocity is calculated from the first difference

of the first two images in the initial segment, and the acceleration is

assumed to be zero. These values specify the initial state x (-).

The initial covariance matrix is specified as


10
10
5
P (-) =5
1
1


which reflects fairly large uncertainty in the initial position, less

for the initial velocity and even less for the initial acceleration.

Initialization of the filter in this manner forces the first two images









to be used and their resulting error to be zero. Other possibilities

exist such as forcing all five images in the initial segment to be

used and calculating an approximate initial state from them. By

utilizing only the first two images, some errors in the initial paths

can be eliminated, and some incorrect initial paths can be eliminated

entirely. (Recall that the initial paths are not guaranteed to be

correct.)

Once the initial estimates are fixed, the Kalman filter sub-

routine FILTER is initialized. This is a multiple entry routine which,

on initial call, sets the $, H, and R matrices used in the Kalman

filter equations. Entry UPDTE, calculates a new estimate using the

latest measurement, and entry EST, calculates the predicted state

(the estimated state prior to a measurement). Once the updated initial

state has been determined, PFP enters a loop that attempts to follow

the path as far as possible.

The following loop consists of making a prediction by calling EST,

windowing the estimate with WEST, choosing the next image with DECIDE,

and finally, updating the state estimate using the new image (measurement)

with UPDTE. Windowing the PFP is two-dimensional since only the images

surrounding the prediction, at the same time, are required. The actual

search for candidate images is the same as for subroutine WINDOW used by

INIT. Subroutine DECIDE, calculates the candidate residuals, and using

the joint probability matrices, selects the candidate image that would

produce the most probable error. A check is made to determine whether

the decision is isolated and a message is printed if it is not. In

addition, a simple check is performed to determine whether the









choice made was different from the minimum error choice (choosing the

image closest to the predicted location). If it was different, another

message is printed. Finally, if the probability of the chosen image

is less than the minimum allowed, DECIDE sets a flag to stop the PFP

from following the path further. After a candidate image has been

chosen for use as the next measurement, the UPDTE entry to the filter

routine is called to update the state estimate. The following-loop

continues until the path is stopped by subroutine DECIDE, or the speci-

fied number of images have been identified for the path. The results

are then printed and the next initial segment is used. After all

initial segments have been followed, PFP is finished. The PFP source

is given in Appendix D with additional documentation.

Program LEARN: Determination of the Joint Probabilities

Routine LEARN is very similar to PFP. it takes a given path

(identified by earlier manual effort), and applies the Kalman filter.

The transitions of the residuals are quantified and counted, thereby

developing histograms of the joint probability to be used in PFP. A

simplified flow chart is given in Figure 4-5.

A path is read by the program and initial state estimates are

made in precisely the same manner as in PFP. The same routine FILTER

is used to propagate the state estimate over time. After the path has

been "followed" (the measurements in this case are known beforehand),

the residuals are added into the probability histograms by routine SAVE.

After all the paths to be analyzed have been studied, the resulting

joint probability matrices are output. The source and additional

documentation for LEARN is given in Appendix D.









SUBROUTINE


(FILTER)


UPDATE )



(EST)



(UPDTE)







(SAVE)


SIMPLIFIED FLOWCHART OF "LEARN"


FIGURE 4-5













CHAPTER V

RESULTS OF PERFORMANCE TESTS


Initialization

Program INIT was used to find initial path segments composed of

five images with the first image in frame one. The data used was col-

lected by Jackman (1976). Identification of initial path segments

is dependent on the values of parameters C1, C2 and C3 used to calculate

the cost of three-image segments. To observe how these parameters af-

fect the cost, consider a three-image segment (il, j1), (i2, j2), and

(i3, j3). Define vector A as a vector from image one to two, and

vector B as a vector from image two to three. Then the cost function

may be written as


Total = T + N



where JT = IIA
CI


A B
1 1
IA
JN
C 2


The cost is then a sum of two costs, JT and JN which can be interpreted

as instantaneous tangential and normal acceleration costs respectively.

Figure 5-1 shows the costs JT and J as functions of IBI IAI and


76


























JT: COST OF TANGENTIAL ACCELERATION


JN: COST OF NORMAL ACCELERATION



TANGENTIAL AND NORMAL ACCELERATION COSTS


FIGURE 5-1









A B
II I, (= cosine of the angle between A and B) for various values of

C1 and C2. As the parameters C1 and C2 increase, the cost of a specific

difference in speed or angle decreases.

This, in effect, decreases the cost of the vector velocity varia-

tion between A and B. For the results given here, the parameter values

found to give best performance were as follows:


C = 40.


C = 1.0


C = 30.0

Recall that the last parameter, C3, is a hard upper limit on A and B.

The upper limit on cost, J was arbitrarily set to one. Therefore,
max
many combinations of J and J exist that produce costs less than Jmax
T N max
Figure 5-2 shows the contours of Jtotal. The contour, Jtotal = 1.0,

is the limit, so any point within this contour would represent an

allowed three-image segment.

Figures 5-3 and 5-4 show the initial path segments as found by

INIT. The locations of the symbols '1' through '5' represent locations

of images in frame one to five. (This figure is just the superposition

of the first five frames of data.) A proper path segment consists of

five sequentially numbered images. Figure 5-3 shows data from the

"top" section of the frames, i.e. one stereo view. Table 5-1 describes

labeled segments in these two figures. Important to note here, is that

the output of INIT contains several different kinds of errors. Most

errors are a result of data entry from the tablet. When using the

tablet for data entry, numerous images were accidentally omitted or














79













































L-j
C) LL









F-j
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'LI) (O ~ I



LL


I-



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tL










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I





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T












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II.

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01 1
IT' r of









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r., n1 Id


U5,









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42 7





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riv3
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TABLE 5-1

KEY TO FIGURES 5-3 AND 5-4


NOTES (SEE FIGURE 5.3)


NOTES (SEE FIGURE 5.4)


LABEL


LABEL


A NORMAL FIVE-IMAGE SEGMENTS

B TABLET MISPLACED IMAGE ONE

C MISSING IMAGE IN FRAME TWO

D PATH SEGMENT BEGAN OUT OF FIELD-OF-VIEW

E IMAGE INCORRECTLY STARTED, "JUMPS" TO
ALTERNATE PATH

F IMAGE MOVES OUT OF FIELD-OF-VIEW

G TABLET MISPLACED IMAGE FOUR


I,K INITIAL PATH COST TOO HIGH

J ISOLATED IMAGE ONE

L DARK PATH TOTALLY WRONG "CUTS-ACROSS"
PATHS

M FIRST THREE IMAGES CORRECT

N DOUBLE ENTRY OF IMAGE ONE

P UPPER IMAGE ONE LINK INCORRECT

Q IMAGE THREE OF LOWER PATH INCORRECT BUT
CORRECTABLE










entered twice. In addition, a failure of the tablet to correctly

locate an image could occur because of operator error. This caused

some image locations to be displaced to the left as seen in Figures

5-3 and 5-4. These errors are not considered in the overall perform-

ance because they are expected to have been eliminated by technique

modifications as discussed previously. The significant errors in

these results occur when the image density is high and incorrect initial

paths are selected. Some of the errors can be corrected by the particle

following program (see last section, Performance of the PFP). Another

error is not starting a path for an image in frame one. This usually

results from the cost of the path being too high and can be corrected

by increasing J However, this must be done carefully because many
max
erroneous paths may also be allowed. The parameter values chosen re-

present a trade-off in this regard. Very few "good" images in frame

one were lost.

Performance of INIT. Table 5-2 contains the results of initialization

of frame-one images. The different types of errors that occur are

categorized by whether or not an image in frame one was "started,"

i.e. an initial segment was found starting with a frame-one image.

An initial segment can have one or more incorrect members, be in

error because of a double entry of the image's location, or be the

result of an inadvertently added frame-one image. If an initial

path is not found for an image, there are a number of possible reasons.

The cost of a correct path may be too large, images may be missing or

inadvertently added, the tablet could have caused a displacement of an

image, or the correct path does not have five images in the field of view.









Except for the first of these errors, too high a cost, these errors

are a result of using the data tablet and are unavoidable. Of the

254 images in frame one, 47 were classified as images with unavoidable

data entry errors and were disregarded. Of the remaining 207 images,

148 were started correctly, 31 were started with at least the first

two images correct, 11 were started incorrectly (totally wrong) and

15 were not started because their cost was too high. These results

are expressed as percentages in Table 5-2. Note that initial segments

with the first two images correct are sufficient to start the PFP and

may possibly be correctable. Initial segments that are totally wrong

may be followed by the PFP but would be expected to be aborted as

short (up to five-image) paths since the chance of these inadvertent

paths being longer is very small.


TABLE 5-2

INITIALIZATION PERFORMANCE ON 207 FRAME-ONE IMAGES


IMAGES STARTED

CORRECT FIVE-IMAGE 72%
SEQUENCES CORRECT OR
87% CORRECTABLE
INCORRECT BUT WITH 15%
THE FIRST TWO IMAGES
CORRECT

TOTALLY WRONG 5%


IMAGES NOT STARTED

COST TOO HIGH 8%










Particle Following

Learning Results. Program LEARN generated the joint probabilities

as described in Chapter III from a training sample of fifty arbi-

trarily selected image paths that had been identified by Jackman. The

resulting probability distributions are dependent on the finite fading

memory parameters. Figures 5-5 through 5-8 show the distributions for

s = 1.0, 1.2, 1.5, and 2.0. Using other arbitrary groups of fifty

paths gives reasonably similar results. These graphs are interpreted

in the same manner as the example discussed in Chapter III. In general,

these results are fairly similar for the different values of s. As s

is increased, there is a slight change in Pr {v vql _} to change from

positively correlated to negatively correlated. The Pr{v qIv 1} also

becomes negatively correlated. Some distributions are seen to be

multi-modal, but only slightly, All of the distributions have non-

zero means and show some dependence (correlation). Checks were made

of the cross-correlations between u and v residuals and no significant

dependence was found. Furthermore, correlations were not found to be

significant between kq and v v etc. Therefore, the assumptions
--kq k-2 k-3
made in Chapter III about the residual appear reasonable.

The probabilities for the u and v transitions are seen to be dif-

ferent in character. One possible explanation for this comes from

considering the odd image shapes. Analysis of the tablet error in

Appendix C assumed images to be circular. As shown in Appendix A,

particle movement causes the images to be lengthened in the u direction.

Therefore, the variance of the u location of an image should be larger

than for the v location. This effect shows itself in the u residual























































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