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Lyapunov-Based Range and Motion Identification for Affine and Non-Affine 3D Vision Systems

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Lyapunov-Based Range and Motion Identification for Affine and Non-Affine 3D Vision Systems
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2008

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Coordinate systems ( jstor )
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Homography ( jstor )
Image processing ( jstor )
Image rotation ( jstor )
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Paraboloids ( jstor )
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LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION FOR AFFINE
AND NON-AFFINE 3D VISION SYSTEMS
















By

SUMIT GUPTA


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Sumit Gupta






























To my parents, my sister, and Amrita.















ACKNOWLEDGMENTS

I express my most sincere appreciation to my supervisory committee chair,

mentor, and friend, Dr. Warren E. Dixon. His contribution to my current and

ensuing career cannot be overemphasized. I thank him for the education, advice,

and for giving me opportunities that would otherwise not be possible. A special

thanks to Dr. Rick Lind for his technical insight and encouragement. I express my

appreciation and gratitude to Dr. Carl Crane, Dr. Thomas Burks, and Dr. Dapeng

Oliver Wu, for lending their knowledge and support. It is a great priviledge to work

with such far-thinking and inspirational individuals. All that I have learnt and

accomplished during the course of my thesis would not have been possible without

their dedication.

I especially thank Dr. Nick Gans for his invaluable guidance and support

during the last semester of my research. I thank all of my colleagues who helped

during my thesis research: Siddhartha Mehta, Guoqiang Hu, Darren Aiken, Sanjay

Solanki, and Vilas Chitrakaran.

Most importantly I would like to express my deepest appreciation to my

parents, my sister, and Amrita. Their love, understanding, patience and personal

sacrifice made this dissertation possible.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ....... iv

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

CHAPTER

ABSTRACT ................... ....... ....... viii

1 INTRODUCTION ..................... ........ 1

2 LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF
A PARACATADIOPTRIC VISION SYSTEM ....... ......... 7

2.1 Introduction . . . . . . . 7
2.2 Projection ................... ........... 8
2.3 Range and Motion Identification .......... ....... .. 11
2.3.1 Objective ............. .... .......... 11
2.3.2 Estimator Design and Error System ....... ....... 12
2.4 Analysis ...................... ......... 13
2.5 Numerical Simulation ................... .... 16
2.6 Conclusion .................... ........ 19

3 LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF
A NONAFFINE PERSPECTIVE DYNAMIC SYSTEM ........ 20

3.1 Introduction ................... ....... 20
3.2 Nonaffine Projection .. ..... ........... .... 21
3.3 Range and Motion Identification .......... ....... .. 24
3.3.1 Objective ............. .... .......... 24
3.3.2 Estimator Design and Error System ....... ....... 25
3.4 Analysis ...................... ......... 27
3.5 Numerical Simulation ................... ..... 30
3.6 Conclusion .................... ........ 33

4 HARDWARE IN THE LOOP ........... ............. 35

4.1 Introduction ................... ....... 35
4.2 Hardware Setup ................... ........ 37
4.3 Image Processing ................... ....... 38
4.3.1 Feature Point Detection and Tracking ............. 40









4.4 Multi-view Photogrammetry . . . ... . 42
4.4.1 Euclidean Reconstruction . . . . ... 43
4.4.2 Camera, Screen, and Virtual Scene Geometry . ... 47
4.4.2.1 Problem Statement. . . . 47
4.4.2.2 Virtual Scene Geometry . . . ... 47
4.4.2.3 Camera Geometry. . . . 49
4.4.2.4 Camera to Screen Geometry . . .. 50
4.5 Socket Communication ....... .. ....... .... .. 54
4.6 Conclusion. ....... .. .... ........ .. ... 55

5 CONCLUSION ...... ... .. .. ... .. ........ .. 56

5.1 Summary of Results .......................... 56
5.2 Recommendations for Future Work. . . . 57

APPENDIX

A LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF
A PARACATADIOPTRIC VISION SYSTEM . . . .... 59

B LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF
A NONAFFINE PERSPECTIVE DYNAMIC SYSTEM ......... 62

C HARDWARE IN THE LOOP ......................... 66

C.1 Program on the Server Machine ................. .. 66
C.2 Program on the Image Processing Machine .............. 81
C.3 Program to Compute Screen-Camera Homography ......... 117

REFERENCES ................................... 120

BIOGRAPHICAL SKETCH ........................... 123














LIST OF FIGURES
Figure page

2-1 Euclidean point projected onto paraboloid mirror and then reflected to
an orthographic camera. .................. . ... 8

2-2 Estimation error of auxiliary signals (a) el(t) (b) t2(t) and (c) 63(t) in
[pixels/sec]. .................. ........ ....... 17

2-3 Mismatch between y4(t) and 4(t) . . . . . 17

2-4 Estimation error of auxiliary signals in the presence of noise (a) el(t) (b)
62(t) and (c) 3e(t) in [pixels/sec]. ................ .... 18
2-5 Mismatch between y4(t) and y4(t) in the presence of noise. . ... 19

3-1 A paraboloid imaging surface centered at [0,0,0]T. . . . 21

3-2 Estimation error of auxiliary signals (a) el(t) (b) 62(t) and (c) 63(t) in
[pixels/sec]. .................. ........ ....... 31

3-3 Mismatch between y4(t) and y4(t). .................. ... 32

3-4 Estimation error of auxiliary signals in the presence of noise (a) el(t) (b)
62(t) and (c) e3(t) in [pixels/sec]. ................ .... 32
3-5 Mismatch between y4(t) and y4(t) in the presence of noise. . ... 33

4-1 Overview of HILS facility. .................. ..... 36

4-2 Modular display in the HILS facility at the University of Florida. ...... ..38

4-3 Supporting workstation in the HILS facility at the University of Florida. 39

4-4 Moving camera looking at a reference plane. . . ...... 42

4-5 Camera, projector plane and virtual scene geometry. . . ... 46

4-6 (a) Screen image of 1280x 1024 resolution. .. . . 52

4-7 (b) Camera image of 640x480 resolution....... . . 52

4-8 Illustration of virtual reality camera's instrinsic parameters. ...... ..54















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION FOR AFFINE
AND NON-AFFINE 3D VISION SYSTEMS

By

Sumit Gupta

May 2006

Chair: Dr. Warren E. Dixon
Major Department: Mechanical and Aerospace Engineering

Many applications require interpreting Euclidean motion of features of a

3-dimensional (3D) object through 2D images. We examined determination of

the Euclidean coordinates of the features of a 3D object undergoing general affine

motion for a central paracatadioptric imaging system. The nonlinear estimator we

examined asymptotically determines the Euclidean motion and range information

from a single camera, provided that some observability conditions are satisfied, and

that the Euclidean motion parameters are known. I proposed techniques which

were developed through a Lyapunov-based design and stability analysis. Simulation

results show the performance of the state estimator.

However, unlike image systems based on a planar image surface (or spherical

or ellipsoidal surfaces), the dynamic system resulting from projecting a general

3D imaging surface is not guaranteed to maintain an affine form. Because of the

nonaffine form, existing range-identification observers cannot be directly used.

Therefore we developed a nonlinear state estimator that can be applied to a

nonaffine vision system to determine the range and Euclidean coordinates of an

object feature without using linear methods.









As with many vision-based estimation and control strategies, our results were

based on feature point tracking. Feature point's tracking is an essential capability

of many coordinated guidance, navigation, and control applications. The hardware-

in-the-loop simulation (HILS) facility at the University of Florida provides a rapid

prototyping testbed for simulating such applications. The facility centers around

a virtual environment to investigate visualization and computational technologies

and assists in simulating vision-based feedback controllers. We described the HILS

testbed, examined the theoretical concepts behind the visual serving algorithms,

and discussed solutions to problems faced while developing the control loop.















CHAPTER 1
INTRODUCTION

Autonomous vehicle/robotic guidance, navigation and control applications typ-

ically require interpretating of the Euclidean motion of features of a 3-dimensional

(3D) object through 2D images that are projected from the 3D feature. Determin-

ing the Euclidean motion of an object through the image projection is challenging,

because the distance from the camera to the object along the focal length (i.e.,

the range to the object) is an unmeasurable time-varying signal that appears non-

linearly in the projected image-space. Although the problem of determining the

Euclidean coordinates of a moving object from its 2D images is inherently non-

linear, typical results are based on linearization based methods such as extended

Kalman filtering ([5], [10]). Classical structure from motion (SFM) results, ([18],

[31]), are examples of such linearization-based methods.

Several researchers have recently investigated the problem of determining

the unknown states (i.e., the Euclidean coordinates) when the motion parameters

(feature velocities) are known. For example, a discontinuous observer was devel-

oped [17] to exponentially identify range information of features from successive

images of a camera where the object model is based on known skew-symmetric

affine motion parameters. Motivated to generalize the object motion beyond the

skew-symmetric form [17], Chen and Kano developed a new discontinuous observer

[6] that exponentially forces the state observation error to be uniformly ultimately

bounded (UUB) for known motion parameters. In comparison to the UUB result

[6], a continuous observer was constructed [8] to asymptotically identify the range

information for a general affine system with known motion parameters (i.e., result

[8] eliminated the skew-symmetric assumption and yielded an asymptotic result









with a continuous observer). Ma et al. ([20], [21]) developed a state estimation

strategy for affine systems with known motion parameters where only a single

homogeneous observation point is provided (i.e., a single image coordinate).

All of the aforementioned results assumed that a conventional planar imaging

surface were used. Using a planar imaging surface is restrictive for some appli-

cations because of limitations in the field of view (FOV). To improve the FOV,

researchers proposed using spherical, elliptical, or paraboloid imaging surfaces

[22]; rotating imaging systems [1]; fish-eye lenses [32]; catadioptric lenses [25]; or a

cluster of cameras [28]. A vision system retains an affine form [22] for some of these

imaging surfaces (e.g., planar, spherical, or ellipsoidal). That is, the projection

of the affine Euclidean motion of an object onto some imaging surfaces yields an

affine dynamic system. In previous results, the affine form of the vision system was

transformed into the nonlinear system expressed by:


I = wT(X, U)X2 + 0(X1,U) (1 1)

x2 = g(xl, x, u)

y = xl


where w(xi, u) and g(xu, x2, u) are nonlinear functions of their arguments. Once the

vision system is written as in equation (1-1), the identifier based observer (IBO)

proposed by Jankovic and Ghosh [17] can be applied directly to determine the

object range and motion [22].

A catadioptric system combines reflective (catoptric) and refractive (dioptric)

elements (i.e., a camera and a mirror) [16]. Catadioptric systems with a single

effective viewpoint are classified as central catadioptric systems. Central catadiop-

tric systems are desirable because they yield pure perspective images [12]. Baker

and Nayar [3] derived the complete class of single-lens single-mirror catadioptric









systems (e.g., paraboloid mirror under orthographic projection) that satisfy the

single-viewpoint constraint.

Catadioptric systems provide a larger FOV in a manner that is better than

alternative technologies. For example, a rotating camera system has a reduced

effective bandwidth, has moving parts, and requires extra care to be taken to elim-

inate blur as the acquired images are stitched together to construct a panoramic

scene. For many applications, the cost of a cluster of cameras is inhibitive when

compared to a catadioptric system with a similar FOV. Moreover, the viewpoints of

all the cameras must coincide for a cluster of cameras to generate pure perspective

images (which is a nontrivial calibration obstacle).

Catadioptric systems also exhibit several limitations. In general, the coordi-

nates of an object are projected onto a mirror and then onto a camera lens. For

cameras using a lens that yields a perspective projection, alignment of lens and

mirror must be calibrated for the distance between them. Paracatadioptric systems

are a special kind of central catadioptric system constructed with a paraboloid

mirror and an orthographic lens. Using the orthographic lens reduces the alignment

requirements; and hence, simplifies calibration of the system, and computation

of pure-perspective images [24]. Compared to other technologies that extend the

FOV, another limitation of catadioptric systems is that using a curved mirror

warps the image. This distorted image mapping presents a challenging obstacle for

reconstructing the Euclidean coordinates of observed feature points.

Our contribution was to develop a nonlinear estimator, [8], to extract range

information and Euclidean coordinates from a paracatadioptric system. We used a

Lyapunov-based analysis to prove that the 3D Euclidean coordinates of an object

moving with general affine motion are asymptotically identified.

However, a projected image from a general 3D imaging surface is not guaran-

teed to maintain an affine form. Motivated by the benefits of an improved FOV,









Ma et al. suggested that range identification could be achieved using a linear

approximation-based observer for a paraboloid imaging system whose focal point

and vertex coincide. The resulting state estimation of the original nonlinear system

is produced from a sequence of approximate linear, time-varying observers [22].

Gupta et al. [14] constructed a nonlinear observer, based on [8], to asymptoti-

cally identify the range for the system considered [22]. The IBO [22] (or most of

the existing nonlinear observers) can not be directly applied for projections on a

paraboloid imaging system. Ma et al. [22] suggested that range identification could

be achieved for a paraboloid image surface using a linear approximation-based

observer where the state estimation of the original nonlinear system is carried out

by a sequence of approximate linear time-varying observers [21].

I developed a nonlinear estimator that can be used to identify range and the

Euclidean coordinates from a nonaffine imaging system without the use of linear

approximations [22]. The structure of the nonlinear observer is based on a previous

observer [8]; however, new observability conditions are imposed due to the nonaffine

form of the projection resulting from the paraboloid image surface. A Lyapunov-

based analysis is used to prove the range and the 3D coordinates of an object

moving with general affine Euclidean motion (and nonaffine image dynamics) are

asymptotically identified. Numerical simulation results are provided that illustrate

the performance of the estimator.

We considered the affine Euclidean motion of an object feature [8]

.i a11 a12 a3 x1 bi

2 a21 a22 a23 X2 + b2 (12)

a3 a31 a32 a33 3 b3

where x(t) = [xl(t), x2(t), X3(t)]T e R3 denote the unmeasurable Euclidean

coordinates of an object feature along the X, Y, and Z axes of an inertial reference






5


frame, respectively, and the Z axis is colinear with the optical axis of the camera.

In (1-2), the parameters ai,j(t) E R and bi(t) Vi,j = 1,2,3 denote known motion

parameters [6], [30]. The affine motion dynamics introduced in (1-2) are expressed

in a general form that describes an object motion that undergoes a rotation,

translation, and linear deformation [30]. It is assumed that the known motion

parameters ai,j(t) and bi(t) Vi, = 1,2, 3 introduced in (1-2) are bounded functions

of time and are second order differentiable.

As with many vision-based estimation and control strategies, the results

in Chapter Two and Three are based on feature point tracking. Feature point

tracking is a standard task of computer vision with numerous applications in

navigation, motion understanding, surveillance, scene monitoring, and video

database management. Hence, there is a definite need for a rapid prototyping

testbed where visual serving techniques can be simulated. The hardware-in-the-

loop-simulation (HILS) facility at the University of Florida provides such a testbed

to simulate vision based control strategies. The complete control loop at the HILS

facility is described in the following steps:

1. A virtual reality simulator renders a virtual environment on the projector

screens.

2. Images are captured by a camera viewing the screens and passed into an

image processing workstation through a analog to digital converter using a firewire.

3. Image processing and computer vision algorithms identify feature points

from the images obtained, determine the relationship between current and reference

frames and accordingly generate control commands.

4. Sockets are used to communicate the generated control commands into the

virtual reality simulator in order to control motion of the virtual environment.






6


In my thesis, I also describe the testbed and the theoretical concepts behind

image processing algorithms, and discuss solutions to the problems faced while

developing the control loop.















CHAPTER 2
LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF A
PARACATADIOPTRIC VISION SYSTEM

2.1 Introduction

Autonomous vehicle/robotic guidance, navigation and control applications

typically require the interpretation of the Euclidean motion of features of a 3-

dimensional (3D) object through 2D images that are projected from the 3D feature.

The research described in this chapter is focused on determining the Euclidean

coordinates of features of a 3D object moving with affine motion dynamics using

central catadioptric systems. A catadioptric system combines reflective (catoptric)

and refractive (dioptric) elements (i.e. a camera and a mirror) [16]. Catadioptric

systems with a single effective viewpoint are classified as central catadioptric

systems. Central catadioptric systems are desirable because they yield pure

perspective images [12]. Specifically, in this result we consider a paracatadioptric

system (i.e., paraboloid mirror combined with an orthographic lens) derived by

Baker and Nayar in [3].

The contribution of the current result is the development of a nonlinear

estimator, based on [8], to extract the range information and the Euclidean

coordinates from a central paracatadioptric system. Specifically, a Lyapunov-based

analysis is used to prove that the 3D Euclidean coordinates of an object moving

with general affine motion are asymptotically identified. Numerical simulation

results are provided to illustrate the performance of the observer.

















1^ y- z


Y/ f

Figure 2-1: Euclidean point projected onto paraboloid mirror and then reflected to
an orthographic camera.

2.2 Projection
The projection of a point x(t) onto a paraboloid mirror with its focus at the
origin (Figure. 2-1) can be described as follows [13]:

YA Yi Y2 Y3 j= x1 x2 3(21)

where f E R denotes the constant known distance between the focal point and the
vertex of the paraboloid, and L(x) E R is defined as

L = -x3 + xX + X + x (2-2)

Because the projection from the paraboloid mirror to the camera is orthographic
in nature (i.e., reflected light rays are parallel to the optical axis), yi(t) and y2(t)
correspond to the measured pixel coordinates of the camera. Also, because the
paraboloid is rotationally symmetric, y3(t) is computed from the measured pixel
coordinates as
y3 Y f (23)
4f









To facilitate subsequent development, the auxiliary signal y4(t) E R is defined as

2f
4 (2-4)
L

and contains the unknown range information. Substituting (2-4) into (2-1), I

obtain the following system

y = ',i (2-5)

Taking the time derivative of (2-5) and utilizing (1-2), the following can be

determined:

(a31i2 + 1' Y2 +a331Y)3
1i = ali + i- + a+ i33 + -(a3--Y 212a331y + ) i (2-6)
2f

12 = a21yl + ',,',, + a:23Y3 + + 92 (2 7)
2f
(a3iyiy3 + :'/'/: + :/)
3 = a311 + :'/ + a33y3 + 1 +93 (2-8)
2f
where (2-6) through (2-8) signify the projected dynamics of the object feature onto

the paracatadioptric system.

In (2-6) through (2-8), the unmeasurable signals g(t) A [g1(t), 92(t), g3(t)]T e

R are defined as

9 = y4b + Qoy (2-9)

where Qo(t) E R is defined as

b3 Y'11 + Y2X2 + Y3X3
L 2f(L + 3) (210)

To facilitate the later development, the dynamics in (2-6) through (2-8) can be

rewritten as

[ yi U 3 ] = 1+g (211)

where i (t) E R3 denotes a matrix of measurable and known signals.









Assumption 2-1: In contrast to the systems examined in [14] and [22], the

development in this result is based on the more general (and more practical)

assumption that the focal point is not at the vertex of the paraboloid. Moreover,

the focal point is not a vanishing point (i.e. f E Co).

Assumption 2-2: The image-space feature coordinates yl(t), y2(t) are bounded

functions of time; hence, from (2-3) y3(t) L,.

Assumption 2-3: The object feature is not a vanishing point (i.e. L LC

therefore y4(t) 7 0). I assume that L 7 0 (i.e. xl, x2 7 0 simultaneously); hence,

the object feature does not intersect the optical axis of the imaging system. Since

L 7 0, (2-4) can be used to conclude that y4(t) LC.

Assumption 2-4: The object must translate in at least one direction (i.e.

bl, b2, b3 7 0 simultaneously).
From (3-9), the signal y4(t) containing the range information can be defined as

2 (y291 y192)2 + (Y39 ylg3)2 + ('I:I,- ,- :)2
(y2bl yib2)2 + (3bi yib3)2 + (3b2 y2b3)2

Remark 2-1: Assumptions 2-1 through 2-4 are standard assumptions that are

practically properties of the physical system rather than assumptions.

Remark 2-2: Based on Assumptions 2-1 through 2-4, the expressions given in (2

6) through (3-31) can be used to determine that y(t), l((t), and g(t) E Lo. Given

that these signals are bounded, Assumptions 2-1 through 2-4 and the development

in the appendix can be used to prove that


119() I < 1 I )11 < 2 ()11 3 (2-13)


where (1, (2 and (3 E R denote known positive constants.









2.3 Range and Motion Identification

2.3.1 Objective

The objective of this result is to extract the Euclidean coordinate information

of the object feature from its projection onto the paracatadioptric system. From
(2-5) and the fact that yl(t), y2(t) and y3(t) are measurable, if y4(t) could be iden-
tified then the complete Euclidean coordinates of the feature can be determined.

To achieve this objective, an estimator is constructed based on the unmeasurable
image-space dynamics for y(t). To quantify the objective, a measurable estimation

error, denoted by e(t) A [ei(t), e2(t), e3(t)]T e R3, is defined as follows:

e = y- (2-14)

where y(t) A [y(t), y2(t), 3 (t)]T e IR3 denotes a subsequently designed estimate.
An unmeasurable' filtered estimation error, denoted by r(t) A [el(t), c2(t),

e3 (t)]T R3, is also defined as
r = + ae (215)

where a R3x3 denotes a diagonal matrix of positive constant gains al, a2,

a3 e R. The error systems in (2-14) and (2-15) are defined based on the goal to
prove that the projected dynamics given in (2-6) through (2-8) can be identified

(i.e., that g(t) can be identified). If g(t) can be identified, the fact that y(t) are
measurable can be used along with (2-12) to compute y4(t).



1 The filtered estimation signal is unmeasurable due to a dependence on the un-
measurable terms gl(t), g2(t), 93(t)-









2.3.2 Estimator Design and Error System

Based on (3-31) and the subsequent analysis, the following estimation signals
are defined:

[1 y2 Y3 ]= 1 + (2-16)

where g(t) A [gi(t), g2(t), 93(t)]T E R3 denotes a subsequently designed estimate for

g(t). The following error dynamics are obtained after taking the time derivative of
e(t) and utilizing (3-31) and (2-16):

e = g (2-17)

Based on the structure of (2-15) and (2-17), g(t) is designed as follows [8]:

g = -(ks + a)g + %sgn(e) + ak,e (2-18)

where ks, 7 e R3x3 denote diagonal matrices of positive constant estimation gains,
and the notation sgn(.) is used to indicate a vector with the standard signum
function applied to each element of the argument. After using (2-15), (2-17) and

(2-18), the following expression can be obtained:

r = r ksr 7sgn(e) (2-19)

where (t) A [ ql 2 q3 CR3 is defined as


q = g + (k, + a) g. (2-20)

Remark 2-3: Based on (2-13) and (2-20), the following inequalities can be
developed:



where (4 and 5 E R denote known positive constants.









Remark 2-4: The structure of the estimator in (2-18) contains discontinuous

terms; however, as discussed in [8], the overall structure of the estimator is continu-

ous (i.e., g(t) is continuous).

Remark 2-5: Considering (2-12), the unmeasurable signal y4(t) can be identified

if g(t) approaches g(t) as t oo (i.e., yi(t), y2(t) and y3(t) approach yi(t), y2(t)

and y3(t) as t -- oo) since the parameters bi(t) Vi = 1, 2, 3 are assumed to be

known, and yl(t), y2(t) and y3(t) are measurable. After y4(t) is identified, (2-5)

can be used to extract the 3D Euclidean coordinates of the object feature (i.e.

determine the range information). To prove that g(t) approaches g(t) as t oo,

the subsequent development will focus on proving that Ie(t)| 0 and ||e(t)| 0

as t -- oc based on (2-14) and (2-17).

2.4 Analysis

The following theorem and associated proof can be used to conclude that the

observer design of (2-16) and (2-18) can be used to identify the unmeasurable

signal y4(t).

Theorem 2-1: For the paracatadioptric system in consideration, the unmeasurable

signal y4(t) (and hence, the Euclidean coordinates of the object feature) can be

asymptotically determined from the estimator in (2-16) and (2-18) provided

the elements of the constant diagonal matrix 7 introduced in (2-18) are selected

according to the sufficient condition


1i > C4 + 1-5 (2-22)
ai

Vi = 1,2, 3 and (4, (5 are defined in (2-21).

Proof: Consider a non-negative function V(t) R as follows (i.e., a Lyapunov

function candidate):

V = r (2-23)
2









After taking the time derivative of (2-23) and substituting for the error system

dynamics given in (2-19), the following expression can be obtained:

iV = -rksr + (e + ae)T (r 7sgn(e)). (2-24)

After integrating (2-24) and exploiting the fact that


i -* sgn({) = ||il V(4 e R,

the following inequality can be obtained:
t 3 t
V(t) < V(to) (rT (o) ksr (o)) doZ+ ai e (o)I (|, (()| 7i) do+Xi (2-25)
i 1

where the auxiliary terms Xi(t) E R are defined as
/t t
Xi = i (-) qi (o-) do 7i ~ (o) sgn(ei ())do (2-26)
=to ito

Vi = 1, 2, 3. The integral expression in (2-26) can be evaluated as


Xi = ei (0.) i () (o) i (o) do- 7 |ei (o) Io (2-27)
to
= ei (t) rTi (t) ei (,o) ii (o) do 3i ei (t)| ei (to) li (to) + 7i |e (to)

Vi = 1, 2, 3. Substituting (2-27) into (2-25) and performing some algebraic

manipulation yields


to
V(t) < V(to) (rT (o-) kr (o)) do-+ X4 + (o

where the auxiliary terms X4(t), (o E R are defined as


X4 = eai Ci (u) (u) () + i \I i (0) -7 7) du + 1 C| 10i (t) 7i)
i=1 to =1i
3
Co = (to) i (to) + i e (to)
i-i









Provided 7 Vi = 1,2, 3 are selected according to the inequality introduced in
(2-22), X4(t) will always be negative or zero; hence, the following upper bound can
be developed:
t
V(t) < V(to) \(rT (o) kr (o)) do + (o (2-28)

From (3-27) and (3-38), the following inequalities can be determined:

V(to) + o > V(t) > 0;

hence, r(t) E L,. The expression in (3-38) can be used to determine that


J(r (o) kr (o)) do < V (to) + Co to

By definition, (2-29) can now be used to prove that r(t) E 2. From the fact
that r(t) E LC, (2-14) and (2-15) can be used to prove that e(t), e(t), y(t), and

y(t) ~,. The expressions in (2-16) and (2-18) can be used to determine that
g(t) and g(t) E L,. Based on (2-13), the expressions in (2-19) and (2-20) can be
used to prove that rl(t), il(t), r(t) E L.. Based on the fact that r(t), r(t) E C
and that r(t) E 2, Barbalat's Lemma [27] can be used to prove that I||(t)| 0 as
t -- o; hence, standard linear analysis can be used to prove that I|e(t)ll 0 and

I|e(t)ll 0 as t oo. Based on the fact that Ile(t)ll 0 and I|e(t)ll 0 as t oo,
the expression given in (2-14) can be used to determine that yl(t), y2(t) and y3(t)
approach yl(t), y2(t) and y3(t) as t -- oo, respectively. Therefore, the expression in
(2-17) can be used to determine that g(t) approaches g(x) as t -- oo. The result
that g(t) approaches g(x) as t -- oo, the fact that the parameters bi(t) Vi = 1, 2, 3
are assumed to be known, and the fact that the image-space signals y1(t), y2(t) and

y3(t) are measurable can be used to identify the unknown signal y4(t) from (2-12).
Once y4(t) is identified, the complete Euclidean coordinates of the object feature
can be determined using (2-5).









2.5 Numerical Simulation

In this section, numerical simulation results are provided to illustrate the

performance of the range identification observer for the paracatadioptric system.

The object feature is assigned the following affine motion dynamics [8]:

I1 -0.2 0.4 -0.6 Xl
'2 = 0.1 -0.2 0.3 X2

X3 0.3 -0.4 0.4 x3
T
+ 0.5 0.25 0.3

with the initial Euclidean coordinates:
~T T
Xi(0) X2(0) X3(0) = 0.4 0.6 1

By arbitrarily letting f = the expressions in (2-1), (2-2), and (2-4) can be used

to determine that
yi(to) = yi(to) = 1.72

2(to) = y(to) = 2.58

y3(t0) = y3(t) = 4.29

y4(t0) = 4.29.
The estimates for g(x) were initialized as follows:


.l(to) = 1 2(t)= 1 3(t)= 1.

After adjusting the observer gains as

ks = diag{50, 50, 50} a = diag{15,15,15}

7 = diag{1, 1, 1} x 10-5

the resulting mismatch between gl(x) and gi(t), g2(x) and g2(t), and g3(x) and

g3(t) (i.e., e(t), 62(t), and 63(t) respectively) is depicted in Figure. 2-2. The
mismatch between y4(t) and y4(t) is provided in Figure. 2-3. The y4(t) term is




























0 2 4 6 8 10
(b)
5

0

-5 -
0 2 4 6 8 10
(c)
Time [sec]


Figure 2-2: Estimation error of auxiliary signals (a) ti(t) (b) 82(t) and (c) 83(t) in
[pixels/sec].


Time [sec]


Figure 2-3: Mismatch between y4(t) and y4(t).














_5
0 2 4 6 8 10
(a)



-5-----
0 2 4 6 8 10
(b)



-51----
0 2 4 6 8 10
(c)
Time [sec]

Figure 2-4: Estimation error of auxiliary signals in the presence of noise (a) e (t)
(b) e2(t) and (c) 63(t) in [pixels/sec].


obtained from numerical integration of the y4(t) term, while the estimated value is

obtained by replacing g(x) with g(t) in (3-19).

Additive-white-gaussian-noise (AWGN) was injected into the measurable

image-space signals y (t), y2(t) via the awgn() function in MATLAB, while

maintaining a constant signal-to-noise-ratio of 20. Without changing any of the

other simulation parameters, the mismatch between gl(x) and gi(t), g2(x) and

g2(t), and g3(x) and g3(t) (i.e., 1(t), e2(t), and 63(t) respectively) is provided in

Figure. 2-4, while the mismatch between y4(t) and y4(t) is provided in Figure. 2-5.

The results depicted in Figures. 2-2 through 2-5 indicate that the proposed

observer can be used to identify range, and hence, the Euclidean coordinates of an

object feature moving with affine motion dynamics projected onto a paracatadiop-

tric system provided the observability conditions are satisfied. These results are

comparable to the results obtained in [8] for a planar image surface.







19




4

3

2



0





S-4
-2 -

-3 -


0 2 4 6 8 10
Time [sec]


Figure 2-5: Mismatch between y4(t) and y4(t) in the presence of noise.


2.6 Conclusion

The range and the Euclidean motion of an object feature undergoing general

affine motion are determined for a central paracatadioptric system via a nonlinear

estimator. The nonlinear estimator was proven, via Lyapunov-based analysis, and

numerically demonstrated to asymptotically determine the range and Euclidean

coordinates of features of a 3D object moving with known motion parameters, for

a paracatadioptric system















CHAPTER 3
LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF A
NONAFFINE PERSPECTIVE DYNAMIC SYSTEM

3.1 Introduction

Consider a 3-dimensional (3D) object undergoing affine motion in Euclidean

space. A vision system retains the affine form for some imaging surfaces (e.g.,

planar, spherical, or ellipsoidal). However, projection of the affine Euclidean motion

of an object onto some imaging surfaces is not guaranteed to yield an affine form

in general. L. Ma in [22], states that the IBO (or most of the existing nonlinear

observers) can not be directly applied for projections that do not maintain an affine

form. In [22], Ma et.al. suggested that range identification could be achieved for a

nonaffine projection from a paraboloid image surface using a linear approximation-

based observer where the state estimation of the original nonlinear system is

carried out by a sequence of approximate linear time-varying observers originally

proposed in [21]. The vision system under consideration, as described in [22]

assumes that the focal point and the vertex of the paraboloid imaging surface

coincide to produce the nonaffine transformation. However, in this result, the

physical construction issue is disregarded, and focus is on the range and motion

identification problem via observation on a paraboloid imaging surface.

The contribution of the result in this chapter is the development of a nonlinear

estimator that can be used to identify range information and Euclidean coordinates

from a nonaffine projection resulting from the paraboloid image system described in

[22], without the use of linear approximations. A Lyapunov-based analysis is used

to prove the range and the 3D coordinates of an object moving with general affine

Euclidean motion (and nonaffine image dynamics) are asymptotically identified.











I X4 X2rX3 IT


z




Y

Figure 3-1: A paraboloid imaging surface centered at [0,0,0].

Numerical simulation results are provided that illustrate the performance of the
estimator.

3.2 Nonaffine Projection

The projection of the Euclidean coordinates onto a paraboloid imaging surface

(see Figure. 3-1) is defined as follows:

Xp = xZl3/L, Yp = x2x3/L, Zp = x~/L (3-1)

where Xp(t), Yp(t), Zp(t) E R denote the projected coordinates of xi(t), x2(t), x3(t),

and L E R is defined as

L = x + x2. (3-2)

The image-space coordinates, denoted by y(t) E R4, can then be defined as follows:


y Yi 2 3 Y4 = Xp Yp Zp 1/L (3-3)

where yl(t) and y2(t) are the measured pixel coordinates from the camera, y3(t) is

computed from the measured pixel coordinates as


Y3 = Y1 + Y4)


(3-4)









and y4(t) contains the unknown range information [22]. The projection onto the

paraboloid can be expressed by the following nonaffine perspective dynamic system

(PDS) [22]:

2 3
Yi
i = a1yl + ,, + a3y3 + a31- + a32 + a33 2all- (3-5)
y3 y3 y3
9 2
-2(a12 + 21) 2a22j12 2a13y 2a23yy2 +
y3 y3

2 9
2 = a21iy + '1/ + a23y3 + 3 a3 a32 + : 2anl (3-6)
Y3 Y3 Y3
2 3
2(a12 + a21) 1 21229 2a13yy ,2 2:,2 + f2
Y3 Y3


y3 = 231Y1 + 2"7,:Q.!/2/ + 2a33y3 2aly 2, :..i, (3-7)

2(a12 + a21)yly2 2a13yly3 2r.z'/l9/3 + f3


2a1_ ,/ ,/ i 2. ;
Y4 2,1 ,:/_, / 2bly4Y5 / (3-8)
y3 y3
2a13yly4 2(a12 + a21) -Y211,/ 1Y6
Y3

where fi(yi, y5, Y6, Y7), f2 (Y2, Y5, Y6, Y), f3(Y3, y5, Y6, Y1) E R are unmeasurable

signals defined as

T T
f fl f2 f3 = 1 Y5 Y6 Y7 j (3-9)


In (3-9), Q(yl, y12, Y3) E R3x3 denotes the following matrix of known and measur-

able signals:

(b3 2bly1) -2b2yl bi
1 -2b1y2 (b3- 21,1) b2 (3-10)

-2blY3 -2b2y3 2b3









and the unmeasurable auxiliary signals Y5(yi, y3, Y4), Y6(Y2, Y3, Y4), Y7(Y3, Y4) R are
defined as

Ys5 y6 7 = L [ 1 X2 x3 (311)

yl Y Y-- y2-- V'/
Y V y3 I
To facilitate the subsequent development, the PDS in (3-5) through (3-7) can be
rewritten as
[ 3] =] 2+f (3-12)

where Q2 (l, Y2, Y3) R3 denotes a vector of measurable and known signals.

Assumption 3-1: The image-space feature coordinates yl(t), y2(t) and y3(t) are

bounded functions of time.

Assumption 3-2: The object feature motion avoids the degenerate case where the

feature intersects the image plane (i.e., x3(t) 7 0).

Assumption 3-3: The object feature does not collide with the imaging surface

(i.e., L > Zp). Hence, from (3-1) and (3-3) y4(t) E Co. Moreover, the object
feature is not a vanishing point since LE Loo.

Assumption 3-4: The matrix Qi(y1, Y3) introduced in (3-10) is invertible

provided

b3 # 0 (3-13)

and

b 2 2l.,1.:I, + b y3 2bib3Y1 + b y3 0. (3-14)

After utilizing (3-4), the condition in (3-14) can be written as


(yi k1)2 + (y2- k2)2 0 (3-15)

where kl(t) and k2(t) E R are auxiliary terms defined as

bib3 b2b3
kl = bb k2 = bb (3-16)
b2 + b2 b2 + b2









Geometrically, the observability condition in (3-15) indicates that the projection

of the object feature cannot intersect the point (kl, k2). For the special case when

bl = b2 = 0, (3-14) reduces to (3-13). Based on (3-15) and (3-16) the observability

condition in (3-13) is equivalent to the physical property given in Assumption 3-3.

That is if b3 = 0, then (3-1) through (3-4) indicate that x3(t) = 0.

Remark 3-1: Assumptions 3-1 through 3-4 are standard assumptions that are

practical properties of the physical system rather than assumptions.

Remark 3-2: Based on Assumptions 3-1 through 3-4, the expressions given in

(3-5) through (3-8) and (3-19) can be used to determine that ;' (t), Q1(yi, y2, Y3),

and f(yi, y2, Y3, Y5, Y6, Y7) E C Vi = 1,2, ...4. Given that these signals are bounded,

Assumptions 3-1 through 3-4 can be used to determine that f(.) and f(.) Lo.

3.3 Range and Motion Identification

3.3.1 Objective

The objective in this paper is to identify the unmeasurable state y4(t) of the

PDS described by (3-5) through (3-12). From (3-11) and the fact that yl(t),

y2(t) and y3(t) are measurable, if y4(t) could be identified then the complete
Euclidean coordinates of the feature can be determined. To achieve this objective,

an estimator is constructed based on the unmeasurable image-space dynamics

for y(t). To quantify the objective, a measurable estimation error, denoted by

e(t) E R3, is defined as follows:

e -- eIi 2 C3 Y- 1 Y2i 2 Y3 3 (3 17)









where (t) E R Vi = 1, 2,3 denotes a subsequently designed estimator. An
unmeasurable' filtered estimation error, denoted by r(t) E R3, is also defined as

r= r r2 r3 = + ae (3-18)

where a E R3x3 denotes a diagonal matrix of positive constant gains.
The error systems in (3-17) and (3-18) are defined based on the goal
to prove that the PDS in (3-5) through (3-7) can be identified (i.e., that

f(yi, 2, Y3, Y5, Y6, y7) can be identified). If f(yl, Y2, Y3, ys, y6, Y7) can be identi-
fied, and assuming the conditions in (3-13) and (3-14) are satisfied, then (3-9)
can be used to identify Y5(Yi, Y3, Y4), Y6(Y2, Y3, Y4), and Y7(Y3, Y4). If Y5(Yi, Y3, Y4),

y6(y2, Y3, Y4), and Yz(y3, Y4) can be identified, then (3-11) can be used to compute
y4(t) as follows:
Y3(Y5 + y2 + y72)
,y4- = 2 (3 19)
Yi + vi + v3

3.3.2 Estimator Design and Error System

Based on the PDS in (3-12), the estimation objective, and the subsequent
analysis, the following estimation signals are defined
rT
1 y 2 =3 2 + f(t) (3-20)

where f(t) A [fi(t), f2(t), f3(t)]T E R3 denotes a subsequently designed estimate

for f(yi, Y2, Y3, Y5, Y6, Y7) introduced in (3-9). The following error dynamics are
obtained after taking the time derivative of e(t) and utilizing (3-12) and (3-20):

e = f f. (3-21)



1 The filtered estimation signal is unmeasurable due to a dependence on the un-
measurable terms fi(yi, Y5, Y6, Y7), f2(y2, Y5, Y6, Y7), f3(Y3, Y5, Y6, y7).









The time-derivative of (3-18) can be expressed as

f ff + (f ) (3-22)

after utilizing (3-21) and the time derivative of (3-21). Based on the structure of

(3-22) and the subsequent analysis, f(t) is designed as follows [8]:

f = -(k, + a)f + 7sgn(e) + ak,e (3-23)

where ks, 7 E R3x3 denote diagonal matrices of positive constant estimation gains,

and the notation sgn(.) is used to indicate a vector with the standard signum
function applied to each element of the argument. After substituting (3-23) into

(3-22) and then adding and subtracting the terms ksf (y, Y2, Y3, Y5, Y6, Y7), the
following expression can be obtained:

t = q ksr 7sgn(e) (3-24)

where q(t) q 92 q3 R3 is defined as

A f + (k + a) f. (3-25)

Remark 3-3: The structure of the estimator in (3-23) contains discontinuous

terms; however, as discussed in [8], the overall structure of the estimator is continu-
ous (i.e., f(t) is continuous).

Remark 3-4: Once y4(t) is identified, the complete 3D Euclidean coordinates

of the object feature can be determined using (3-3) and (3-11). Provided the
observability conditions given in (3-13) and (3-14) are satisfied y4(t) can be

identified if f(t) approaches f(t) as t -- oo (i.e., yl(t), y2(t) and y3(t) approach

yl(t), y2(t) and y3(t) as t -- oo) since the parameters bi(t) Vi = 1,2,3 are
assumed to be known, and yi(t), y2(t) and y3(t) are measurable. To prove that f(t)









approaches f(t) as t -- oo, the subsequent development will focus on proving that

|Ie(t)ll 0 and Ile(t)ll 0 as t -- oc based on (3-17) and (3-21).
3.4 Analysis

The following theorem and associated proof can be used to conclude that the

observer design of (3-20) and (3-23) can be used to identify the unmeasurable state

y4(t) (i.e., the object feature range and motion can be determined) provided the
observability conditions in (3-13) and (3-14) are satisfied.

Theorem 3-1: Given the PDS in (3-5) through (3-11), the unmeasurable

state y4(t) (and hence, the Euclidean coordinates of the object feature) can be

asymptotically determined from the estimator in (3-20) and (3-23) provided

the elements of the constant diagonal matrix 7 introduced in (3-23) are selected

according to the sufficient conditions

> 1 + I 2 > 2 + 2 (3-26)

73 > | T31+ Ir3

where rq(t) is defined in (3-25), and the observability conditions introduced in

(3-13) and (3-14) are satisfied.

Proof: Consider a non-negative function V(t) R as follows (i.e., a Lyapunov

function candidate):

V ^r1r (3-27)
2
After taking the time derivative of (3-27) and substituting for the error system

dynamics given in (3-24), the following expression can be obtained:

V = -rT kr + (e + aO)T (q ;sgn(e)). (3-28)

After integrating (3-28) and exploiting the fact that


si sgn(Qi)= l~il Vji E IR,








the following inequality can be obtained:

V(t) < V(to) (r (c) kr (c)) du + a, e1 (o-)| (1 (o-)| 1) do + Xi (3-29)
to ito
+02 f 62()1 (T2(7)- 72) d+X2+"a3 e3(0)l ( 3(0) 73)d +X3

where the auxiliary terms Xi(t), 2(t), X3(t) E R are defined as
t t
Xi A i (o-) (o-) duo- 1 (o-) sgn(e (o))do- (3-30)
to to

X2 A 2 (o-) 2 (o) 72 e2 (o-) sgn(e2 (o-))do- (3-31)
to to

X3 (A ) 3 3 () o-- 3 (o-) sn(e3 (o-))do-. (3-32)
to to
The integral expressions in (3-30) through (3-32) can be evaluated as

Xi = i (-) (-) d -f ci (o-) (o-) do- 71 (o-) (3 33)

= e (t) 7i (t) e (o) i (o) do 71 ei1 (t) e1 (to) 17i (to) + 71 e (to)
to

X2 = 2 (t) '72 (t) 2- (o-) 72 () o- 72 I (t)- 2 (to) T/2 (to) + 72 (to) (3-34)

X3 = 3 (t) 3 (t) 63 (o) 3 (o) do-- 7 3 (t) 3 (to) 3 (to) + 3 3 (to) | (3-35)

Substituting (3-33) through (3-35) into (3-29) and performing some algebraic
manipulation yields


to
V(t) < V(to)- (rT (o-) kr (o-)) do- + X4 + (o

where the auxiliary terms X4(t), o E R are defined as

X4 a a1 (1 1 (O)I (7i) +I i (o-)I 71) do (3-36)

+ (t ( (t)o (t) ( (t) (t ( (t)
+ 02 f 62(07)1 (\rT2(a)1 + I 2(a)1 72) d,7 + 03 f 63(07)1 (1(17)1 + I73(17)1 -73) d,

1 (t)\ (I (t)\I 71) + |62 (t)\ (|T2 (t) 72) + 1C3 ()| (3 ()| 73)









0 Co -e1 (to) T1i (to) + 71i |1 (to)| (3-37)

e2 (to) T/2 (to) + 72 62 (to) 3 (to) T13 (to) + 73 e3 (to)

Provided the diagonal entries of 7 are selected according to the inequalities
introduced in (3-26), X4(t) will always be negative or zero; hence, the following
upper bound can be developed:

t
v(t) < V(to) (T (o-) kr (o-)) do- + Co. (3-38)

From (3-27) and (3-38), the following inequalities can be determined:

V(to) + o > V(t) > 0;

hence, r(t) E L,. The expression in (3-38) can be used to determine that

S(r (o) kr (o)) d < V (to) + Co to

By definition, (3-39) can now be used to prove that r(t) E 2. From the fact
that r(t) E C,, (3-17) and (3-18) can be used to prove that e(t), e(t), y(t), and

y(t) C,. The expressions in (3-20) and (3-23) can be used to determine that
f(t) and f(t) E C,. Assumptions 3-1 through 3-4 can be used to determine that

f(.) and f(-) E LC. Based on the facts that f(.), f(.) E L, the expressions in
(3-24) and (3-25) can be used to prove that l(t), il(t), r(t) E LC. Based on the
fact that r(t), r(t) E LC and that r(t) E 2, Barbalat's Lemma [27] can be used to
prove that |r(t)|| -- 0 as t -- oo; hence, Lemma 1.6 of [7] can be used to prove that

I|e(t)ll 0 and I|e(t)ll 0 as t oo.
Based on the fact that Ile(t)ll 0 and I||(t)l -+ 0 as t -+ oo, the expression
given in (3-17) can be used to determine that yl(t), y2(t) and y3(t) approach yl(t),

y2(t) and y3(t) as t -- oo, respectively. Therefore, the expression in (3-21) can be
used to determine that f approaches f as t -- oo. If the observability conditions









given in (3-13) and (3-14) are satisfied, then the result that f(t) approaches f(t)

as t oo, the fact that the parameters bi(t) Vi = 1, 2,3 are assumed to be known,

and the fact that the image-space signals yi(t), y2(t) and y3(t) are measurable can

be used to identify the unknown Euclidean parameter y4(t) from (3-19). Once

y4(t) is identified, the complete Euclidean coordinates of the object feature can be

determined using (3-9) through (3-11).

3.5 Numerical Simulation

In this section, numerical simulation results are provided to illustrate the

performance of the range identification observer given a paraboloid imaging surface.

The object feature is governed by the following affine motion dynamics:

i1 -0.2 0.4 -0.6 .1
'2 = 0.1 -0.2 0.3 X2

3 0.3 -0.4 0.4 x3
T
+ 0.5 0.25 0.3

with the following initial Euclidean coordinates
T T
xi(0) x2(0) x3(0) = 0.4 0.6 1T.

Based on (3-1), the relationship in (3-1) through (3-3) can be used to determine

the following initial conditions in the image-space


yi(to) = yi(to) = 0.769

y2(to) = y(to) = 1.150

3(to) = y3(t) = 1.920

y4(t) = 1.920

The estimates for f(t) were initialized as follows:


fl(too)= f2(t) = f3(to) =













-2 8 10
0 2 4 6 8 10
(a)



-2
0 2 4 6 8 10
(b)



0 2 4 6 8 10
(c)
Time [sec]

Figure 3-2: Estimation error of auxiliary signals (a) el(t) (b) e2(t) and (c) 3(t) in
[pixels/sec].


After adjusting the observer gains as


= dig{10, 10,10} a = diag {8, 8, 8}

7 = diag{1, 1, 1} 10-5

the resulting mismatch between fi(t) and fi(t), f2(t) and f2(t), and f3(t) and f3(t)

(i.e., e(t), e2(t), and 63(t) respectively) is provided in Figure. 3-2. The mismatch

between y4(t) and y4(t) is provided in Figure. 3-3. The y4(t) term is obtained

from numerical integration of the y4(t) term, while the estimated value is obtained

by replacing f(t) with f(t) in (3-9), solving for ys(t), y6(t), and y7(t), and then

utilizing equation (3-19).

Additive-white-gaussian-noise (AWGN) was injected into the measurable

image-space signals yi(t), y2(t) via the awgn( function in MATLAB, while

maintaining a constant signal-to-noise-ratio of 20. Without changing any of the

other simulation parameters, the mismatch between fi(t) and fi(t), f2(t) and f2(t),

and f3(t) and f3(t) (i.e., e1(t), e2(t), and 63(t) respectively) is provided in Figure.

3-4, while the mismatch between y4(t) and y4(t) is provided in Figure. 3-5.



































AL


Time [sec]


Figure 3-3: Mismatch between y4(t) and y4(t).


0

-2
0 2 4 6 8 10
(a)



-1
-2
-2 ----- i ---- --- ---.............i --------


0 2 4 6 8 10
(b)
1

-1.... .......

0 2 4 6 8 10
(c)
Time [sec]


Figure 3-4: Estimation error of auxiliary signals
(b) 42(t) and (c) 63(t) in [pixels/sec].


in the presence of noise (a) tl(t)


3.


























0 2 4 6 8 10
Time [sec]

Figure 3-5: Mismatch between y4(t) and y4(t) in the presence of noise.


The results depicted in Figure. 3-2 through Figure. 3-5 indicate that the

proposed observer can be used to identify the range and hence, the Euclidean

coordinates of an object feature moving with affine motion dynamics and a

nonaffine PDS, provided the observability conditions are satisfied. These results

are comparable to the results obtained in [8] which applied the range identification

observer to a planar imaging surface.

3.6 Conclusion

The range and the Euclidean coordinates of an object undergoing general

affine motion are determined for a paraboloid imaging system. Unlike image

systems that are based on a planar image surface (or spherical or ellipsoidal

surface), the perspective dynamic system resulting from projected image of a

general 3D imaging surface is not guaranteed to maintain an affine form as shown

in [22].

A nonlinear state estimator is developed for the nonaffine projection resulting

from the paraboloid image surface whose focal point and the vertex coincide (see

Figure. 3-1), without the use of linear approximations. The nonlinear estimator






34


was proven and numerically demonstrated to asymptotically determine the range

information from a single camera provided some observability conditions are

satisfied and that the Euclidean motion parameters are known. Although the

system discussed disregards the physical construction issue, this result indicates

that the technique applicable to catadioptric systems (discussed in the previous

chapter) is also applicable to general 3D vision systems that do not maintain an

affine form.















CHAPTER 4
HARDWARE IN THE LOOP

4.1 Introduction

The hardware-in-the-loop-simulation (HILS) facility at the University of

Florida provides a rapid prototyping testbed to simulate vision based guidance,

navigation and control applications. The development of the HILS involved a

three fold process hardware setup, software development and integration of their

components.

The hardware required for the HILS facility can be subdivided into the

following five main components: (1) virtual reality simulator; (2) modular display;

(3) supporting workstation; (4) interface; and (5) image processing workstation.

Much of the software developed in the HILS is based on image processing and

computer vision methods, primarily feature point identification and multi-view

photogrammetry technique.

The contributions discussed in this chapter include the real-time identification

of features and the implementation of a multi-view photogrammetry technique to

facilitate visual servo control applications. An overview of the complete control

loop at the HILS facility is illustrated in Figure. 4-1 and described in the following

steps:

1. The virtual reality simulator renders the visual environment database on the

projector screens (modular display).

2. Images are captured by a camera and passed to the image processing

workstation through an analog to digital converter using a firewire.










CENTER
LEFT RIGHT















SIMULATOR


CONTROL



VISION PROCESSING
WORKSTATION

Figure 4-1: Overview of HILS facility.


3. Image processing and computer vision algorithms identify feature points

from the images obtained, determine the relationship between current and reference

frames and accordingly generate control commands.

4. Sockets are used to communicate the generated control commands into the

virtual reality simulator in order to control motion of the virtual environment.

This chapter is organized in the following manner. In Section 4.2, a description

of the hardware setup of the HILS facility is presented. In Section 4.3, the image

processing algorithm for real-time feature detection and tracking is discussed.

Implementation of a multi-view photogrammetry technique is illustrated in Section

4.4. In Section 4.5, the use of sockets for communicating control information

between workstations is expressed, and concluding remarks are provided in Section

4.6.
4.6.









4.2 Hardware Setup

The hardware for the virtual environment consists of the following five main

components: (1) virtual reality simulator; (2) modular display; (3) supporting

workstation; (4) interface; and (5) image processing workstation. Each component

is necessary to construct a virtual environment for supporting visualization and

computational technologies.

The first component is the virtual reality simulator which generates the data

associated with the virtual environment. The virtual reality simulator utilizes

MultiGen-Pardigm-Vega Prime software package, an extensible COTS tool available

for the creation and deployment of visual simulation, urban simulation, and general

visualization applications. Such generation includes construction of the complete

3D space of the virtual environment. Most importantly, the simulator accounts for

the position and orientation to properly describe the image being viewed. The HILS

facility is setup to generate the urban visual environment of University of Florida

in real-time. The setup includes four computers for rendering the image on the

projector screens. One computer is the master (server), and the remaining three act

as slaves and are responsible for generating the virtual scene on each of the three

projector screens.

The second component is the modular display. The display consists of a set

of three screens as depicted in Figure. 4-2, and projectors that render images

of the virtual database onto the screens. Each screen presents a different aspect

of the virtual environment. The third component is the supporting workstation

as illustrated in Figure. 4-3. This workstation consists of machines which allow

access to the virtual reality simulator and the displays. The machines are standard

off-the-shelf computers running Windows operating systems.

The fourth component is the interface. This interface allows the virtual envi-

ronment to communicate with external components associated with other facilities.

































Figure 4-2: Modular display in the HILS facility at the University of Florida.


Purpose of the interface is to exchange data with other computational facilities and

virtual environments so a massive virtual laboratory can be constructed.

The fifth segment consists of the image processing equipment such as the

camera setup, image processing workstation and their interfacing equipment. A

separate machine running Windows is used to run the image processing algorithms.

The camera viewing the display screens is connected to the image processing

workstation using an analog to digital converter and a firewire for high speed video

transmission.

4.3 Image Processing

Many vision-based estimation and control strategies require real-time detection

and tracking of features. The most important segment of the image processing

algorithm (see Appendix C) is real-time identification of features required for

coordinated guidance, navigation and control in complex 3D surroundings such as

urban environments. The algorithm is developed in C/C++ using Visual Studio 6.0




















































Figure 4-3: Supporting workstation in the HILS facility at the University of
Florida.









and Visual Studio .NET. Intel's Open Source Computer Vision Library is utilized

for real-time implementation of most of the image processing functions. Also,

GNU Scientific Library (GSL) is used, which is a numerical library for C and C++

programming. GSL provides a wide range of mathematical routines for most of the

computation involved in the algorithms.

4.3.1 Feature Point Detection and Tracking

As stated in [26], no feature-based vision system can work unless good features

can be identified and tracked from frame to frame. Two important issues need

to be addressed; a method for selecting good and reliable feature points; and a

method to track the feature points frame to frame. These issues are discussed in

detail in [29] and [26], where the Kanade-Lucas-Tomasi (KLT) tracker is promoted

as a solution.

The algorithm developed for the HILS facility (see Appendix C) is based on

the KLT tracker that selects features which are optimal for tracking. The basic

principle of the KLT is that a good feature is one that can be tracked well, so

tracking should not be separated from feature extraction. As stated in [29], a good

feature is a textured patch with high intensity variation in both x and y directions,

such as a corner. By representing the intensity function in x and y directions by gx

and gy, respectively, we can define the local intensity variation matrix as


2
Z = g Y (4-1)
2
[gigy gy

A patch defined by a 7 x 7 pixels window is accepted as a candidate feature if in

the center of the window both eigenvalues of Z, exceed a predefined threshold. Two

large eigenvalues can represent corners, salt and pepper textures, or any other pat-

tern that can be tracked reliably. Two small eigenvalues mean a roughly constant

intensity profile within a window. A large and a small eigenvalue correspond to a









unidirectional texture pattern. The intensity variations in a window are bounded

by the maximum allowable pixel value, so the greater eigenvalue cannot be arbi-

trarily large. In conclusion, if the two eigenvalues of Z are A1 and A2, we accept a

window if



in(Ai, A2) > A (4-2)

where A is a predefined threshold.

As described in [26], feature point tracking is a standard task of computer

vision with numerous applications in navigation, motion understanding, surveil-

lance, scene monitoring, and video database management. In an image sequence,

moving objects are represented by their feature points detected prior to tracking

or during tracking. As the scene moves, the patterns of image intensities changes

in a complex way. Denoting images by I, these changes can be described as image

motion:



I(x, y,t + 7) = I(x ((, y,t, ), (x, y,t, )). (4-3)

Thus a later image taken at time t + T (where T represents a small time interval)

can be obtained by moving every point in the current image, taken at time t, by a

suitable amount. The amount of motion 6 = (T, ) is called the displacement of the

point at m = (x, y). The displacement vector 6 is a function of image position m.

Tomasi in [26], states that pure translation is the best motion model for tracking

because it exhibits reliability and accuracy in comparing features between the

reference and current image, hence


6 =d











Object Plane


Reference Location


Current Location


Figure 4-4: Moving camera looking at a reference plane.


where d is the translation of the feature's window center between successive frames.

Thus, the knowledge of translation d allows reliable and optimal tracking of the

feature points windows.

In the algorithm developed at the HILS facility (see Appendix C), the method

employed for tracking is based on the pyramidal implementation of the KLT

Tracker described in [4].

4.4 Multi-view Photogrammetry

The feature points data obtained during motion through the virtual environ-

ment is used to determine the relationship between the current and a constant

reference position as shown in Figure. 4-4. This relationship is obtained by deter-

mining the rotation and translation between corresponding feature points on the

current and reference image position. The rotation and translation components

relating corresponding points of the reference and current image is obtained by

first constructing the Euclidean homography matrix. Various techniques can then









be used (e.g., see [10], [33]) to decompose the Euclidean homography matrix into

rotational and translational components.

4.4.1 Euclidean Reconstruction

Consider the orthogonal coordinate systems, denoted F and F* that are

depicted in Figure. 4-4. The coordinate system T is attached to a moving camera.

A reference plane 7 on the object is defined by four target points Oi Vi = 1, 2, 3,

4 where the three dimensional (3D) coordinates of Oi expressed in terms of F and

F* are defined as elements of mi (t) and mJ E R3 and represented by


(4-4)


(4-5)


The Euclidean-space is projected onto the image-space, so the normalized coordi-

nates of the targets points mi (t) and mf are defined as

mi xi ; 1
mi =- 1 (4
Zi Zi Zi

mrn* ,_ 1 (4


under the standard assumption that zi (t) and z4 > e, where e denotes an arbitra:

ily small positive scalar constant.

Each target point has pixel coordinates that are acquired from the moving

camera, expressed in terms of 7, denoted by ui (t), ; (t) E R, and are defined as

elements of pi (t) E R3 as follows:


a T
Pi = ui 1


n() A ) ., (t) Zi

T
iA ;- i z* ]


6)


7)


(4-8)


The pixel coordinates of the target points at the reference position is expressed in

terms of T* (denoted by u7, v E R) and are defined as elements of p7 E R3 as


r-









follows:
T
U vf 1 (4-9)

The pixel coordinates pi (t) and p7 are related by the following global invertible

transformation (i.e., the pinhole model) to the normalized task-space coordinates

mi (t) and mn respectively:


Pi = Ami (4-10)

P7 = AmJ

where A is the intrinsic camera calibration matrix.

The constant distance from the origin of F* to the object plane T along the

unit normal n* is denoted by d* E R and is defined as


d* A n*T (4 11)


The coordinate frames 7 and F* depicted in Figure. 4-4 are attached to the

camera and denote the actual and reference locations of the camera. From the

geometry between the coordinate frames, mf can be related to m i(t) as follows


mi = Xf + Rm-. (4-12)


In (4-12), R (t) E SO(3) denotes the rotation between 7 and T*, and Xf (t) E R3

denotes the translation vector from 7 to T* expressed in the coordinate frame F.

By utilizing (4-6), (4-7), and (4-11), the expressions in (4-12) can be written as

follows
mi = a (R + Xhn*T) m7.
(4 13)
H

In (4-13), xh (t) E R3 denotes the following scaled translation vector


Xh = (4-14)
d*









and ai(t) denotes the depth ratio defined as


~*
a = (4-15)
zi
After substituting (4-10) into (4-13), the following relationships can be developed

pi = ai (AHA-1) p-
pH 1p (4 16)
G

where G (t) = [gij(t)], Vi,j = 1,2,3 R3x3 denotes a projective homography

matrix. After normalizing G(t) by g33(t), which is assumed to be non-zero without

loss of generality, the projective relationship in (4-16) can be expressed as follows:


Pi = .. ,::Gnp* (4-17)

where Gn e R3 denotes the normalized projective homography. From (4-17), a set

of 12 linearly independent equations given by the 4 target point pairs (p',pi (t))

with 3 independent equations per target pair can be used to determine G(t) and

ai(t)g33(t). Based on the fact that intrinsic camera calibration matrix A is assumed
to be known, (4-16) and (4-17) can be used to determine g33(t)H(t). Various

techniques can then be used (e.g., see [10, 33]) to decompose the product g33(t)H(t)

into rotational and translational components. Specifically, the scale factor g33(t),

the rotation vector R(t), the unit normal vector n*, and the scaled translation

vector denoted by xh(t) can all be computed from the decomposition of the product

g33(t)H(t). Since the product ai(t)g33(t) can be computed from (4-17), and g33(t)
can be determined through the decomposition of the product g33(t)H(t), the depth

ration ai(t) can also be computed.

































jVm


m* -------
.. RC TC

a
/ L /C




-----. ---
it





Rc, Tc *,'/T R,, Ts



Figure 45: Camera, projector plane and virtual scene geometry.









4.4.2 Camera, Screen, and Virtual Scene Geometry

4.4.2.1 Problem Statement

In the HILS facility, the camera is viewing a screen onto which a virtual scene

is projected. The camera processes images of the screen in order to control the

motion of the virtual scene using the multi-view geometry technique discussed in

the previous section. However the multi-view photogrammetry technique cannot be

directly applied for the HILS system. This is due to the fact that a homography

exists between the on screen current image and the on screen goal image, but what

we are given is the camera views of the on screen images instead. Thus, there exists

an additional homography between the points on the screen and the points on the

camera image. Determining this additional homography allows us to recover the

homography between the current and goal on screen images.

The HILS is modelled as if the virtual environment is a true 3D scene which

the physical camera is viewing. However, the camera does not look at the 3D scene

directly. The camera views points of 3D objects that are projected onto a 2D plane.

Geometry between the projector, camera and virtual scene is illustrated in Figure.

4-5. The camera is rigidly connected to the projection plane and any change

of scene on the projector plane can be modelled as the projector plane moving

through the virtual environment.

4.4.2.2 Virtual Scene Geometry

The virtual environment is projected onto a virtual image plane which is

reproduced on a physical display screen. We model this projection of the virtual

world onto the screen by the pinhole projection model with some focal point fs.

Reference frame F* is attached at f, which is typical for camera projection as seen

in Figure. 4-5. We refer to F~ as the goal frame. A point ps lies on the virtual

camera's image plane T, having coordinates fm* for the virtual camera's goal frame.

Motion of the virtual scene results in the virtual camera undergoing a virtual






48


rotation R, and translation Ts to a new pose Fs. Hence ps now has coordinates frs

in the virtual camera's new frame 5,.

The relationship between the goal and new frame is expressed as



m, = Rm + Ts. (4-18)

Denoting d* as the distance between the virtual camera and T, and n* as the unit

vector normal to T in the camera frame, we obtain the following relationship:


d- = nTim. (4-19)


After combining (4-18) and (4-19) we can express rns as




ns = Rsn* + (n in) (4-20)
mZ +^O m:. (4-21)
ins = (Rs + T n ). (4-21)


We substitute T*d and normalize min and rns by the following equations:


fins 2*s TS
m,= -; m, = Tsd = (4-22)
Zs Zs C d
Using (4-21) and (4-22), we can rewrite rns as:




nm = (Rs + T*n T)m
zs

m, = Hsm (4-23)
zs

where Hs = (R + TI:,.T). Hs is the virtual Euclidean homography matrix

which can be determined using (4-23). Hence, the virtual screens rotation and

translation information (i.e., Rs, T*d and n*) can be recovered through the Faugeras

decomposition method described in [9].










4.4.2.3 Camera Geometry

A physical camera is looking at a screen onto which a 2D image is projected

from 3D virtual scene; hence, there exists an additional homography between the

points on the screen and the points on the camera image. Recovering this homog-

raphy allows us to recover the homography between the current and goal virtual

scene images. As seen in Figure. 4-5, a reference frame FT is attached to the cam-

era at the focal point. A point on the screen represented by Ps has coordinates mc

in the real camera's reference frame. A homography exists between any two planes

that provides a bijective map. Thus, there is a constant homography between the

real camera image plane and the screen. The camera-to-screen Euclidean homogra-

phy matrix is represented by Hcs E R3x3 is and it maps i to m~5 by the following

equations:



m* = Hcms and m, = Hcss. (4-24)

The normalized coordinates of ps in the camera frame are defined as



m, =- (4-25)
Zc

where zc is the 3D Euclidean coordinate of ps in the camera frame, under the

standard assumption that Zc > e, where e denotes an arbitrarily small positive

scalar constant. After substituting (4-22) through (4-24) in (4-25) we can rewrite

Tnc as

z*
mc = zcHcs Hstm (4-26)
Zs
Z* (H5_ 1 (4 27)
= zcHcs HS Hcs m (4-27)
z \Z21 )

= z z HcsHns' (4-28)
Z( Zs

= Zc z Hemn (4-29)
Zc Zs









where Hc is the camera-to-camera Euclidean homography matrix, which maps

points in the goal camera frame to points in the current camera frame.

4.4.2.4 Camera to Screen Geometry

The Euclidean coordinates m, and mc cannot be determined a priori. Rather,

we can determine the pixel coordinates, denoted by ps and Pc, of points on the

screen and on the camera image, respectively. The pixel coordinates are related to

m, and mc as follows:


Ps = Asms (4-30)

pc = Acme, (4-31)


where Ac and As are intrinsic camera calibration matrices of the real and virtual

camera, respectively. The real camera calibration matrix Ac can be determined

through typical camera calibration techniques whereas the virtual camera cali-

bration matrix, denoted by As, can be determined from the settings of the virtual

reality simulator.

Equations (4-23) and (4-24) are altered as


ps = Gsps and pc = GcsPs, Pc = GcspJ (4-32)


where Gs is the virtual projective homography matrix, and Gcs is the camera-to-

screen projective homography matrix. These matrices can be expressed as


Gs = AsHsA 1 and Gcs = AcHcsA 1. (4-33)


When solving for Gs or Gs we generally find normalized solutions



GsN = G and GcsN (4-34)
gs33 9cs33
where gs33 and gcs33 are the bottom right element of the Gs and Gcs matrices,

respectively.









After combining (4-28), (4-30) and (4-31), we get


pc = Zc zs AcHC, H, H,-'A'p (4-35)
Zc Zs

After substituting (4-33) and (4-34) in (4-35), we can rewrite pc as




Pc = z ((cs33GcsNAs)Hs (As .gcs3GcN)p (4 36)
Zc zg
Zc Zsgs33 G- l (4 37)
GcsNGsNGsN Pc (437)

tc as

= acaGcNPc (4-38)


where GcN is the normalized projective camera-to-camera homography matrix.

Given GcN, GcsN, and As, we can solve for Hs as



Hs = gA-'s GYGsNGcsNAs. (4-39)

We can solve for GcsN given the pixel coordinates of at least four corresponding

points pc in the camera image of points ps on the screen image using a calibration

technique. The calibration technique replaces typical camera calibration, and

knowledge of Ac is no longer required. A routine was developed for solving GcsN

using a calibration technique. An image of 30 squares is displayed on the screen

(see Figure. (4-6)). The pixel locations of each corner point on the screen (ps's) is

extracted by hand. A camera image of the screen is captured (see Figure. (4-7)),

and the pixel locations of all visible corner points (pc's) are extracted by hand too.

This gives up to 120 point correspondences. Estimation of the coordinates of the

points pc is subject to sensor noise and is a random process as well. To eliminate

the effects of noise, GcsN is estimated as the solution to linear equations using

RANSAC [11] in MATLAB (see Appendix C). RANSAC is an algorithm for robust














Figure 4-6: (a) Screen image of 1280x 1024 resolution.


1 E mE


Figure 4-7: (b) Camera image of 640x480 resolution.


HM
H.









fitting of models in the presence of uncertainty and data outliers. It returns a

"best guess" of GcsN that fits Pc and Ps, but eliminates any points that appear to

be corrupted or inconsistent with the solution obtained (i.e., outliers). RANSAC

itself is governed by a random process. To counter this, RANSAC is run many

times (100 1000 times). Any points that are estimated to be outliers more often

than a specified threshold (e.g. 20% of the time) are completely removed from

consideration, and the process is repeated. This is repeated perhaps three or four

times. On the final run, all solutions for GcsN are saved and the mean GcsN is kept

as our estimate. The most recent trial for the setup at the HILS facility gave a

mean and standard deviation for GcsN of



0.6500 -0.0296 -83.2262 0.0016 0.0007 0.6298

1/(GcsN)= 0.0072 0.6366 -61.7221 (GcsN) = 0.0004 0.0013 0.3714

-0.0000 -0.0001 1.0000 0.0000 0.0000 0.0000

The constant calibration matrix of the virtual camera As can be determined from

settings in the Vega Prime software responsible for rendering the visual database

on the projectors. Vega Prime has a "field of view" setting, with a default value

of 45. A field of view of 450 means that the focal length of the camera is equal to

half the size of the image surface. This is illustrated in Figure 4-8. The number of

pixels in the vertical and horizontal directions is a setting of the image rendering

program. In the case of the HILS facility, the screen's resolution is 1280 x 1024.

Thus, assuming that the pixels have equal height and width, As is computed as


640 0 640

As= 0 640 512

0 0 1

Thus, after determining GN(t), GcsN, and As, we can solve for H,(t) using (4-39).










Focal length = 640 pix



512 pix-. z



450 / Focal
I / length
I //
I /
I /
Focal point


Figure 4-8: Illustration of virtual reality camera's instrinsic parameters.


It is interesting to note that H,(t) cannot be solved directly from (4-39) because

of lack of gcs33. Hence, we use GcsN to map pc to screen points s = GcsNPc and

p* = G 1 The point p^ will not generally be in proper homogenous coordinates

due to the missing scale factor. We can correct this by renormalizing


PsN 7
Ps (3)

At this point, we can solve for GCN using ps and p*, and then recover normalized

Hs using (4-40) as

HsN = Al'GsNAs. (4-40)

4.5 Socket Communication

The rotation and translation information obtained from multi-view photogram-

metry technique is required to maneuver the virtual scene on the projector screens.

Hence, control commands generated at the image processing workstation are com-

municated to the main server workstation. This communication is set up using

sockets. Sockets use TCP/IP protocol and specified port number for sending and

receiving data between computers. The sockets program developed communicate

data between C/C++ programs since the programs on both workstations are writ-

ten in C/C++ (see Appendix C). They are based on the client-server architecture









where the image processing workstation acts as a client which is sending data to

the main server. The data to be communicated is converted to a string and stored

in a buffer of specified size before being sent. The reverse is done at the receiv-

ing computer where the control commands are utilized by the virtual database

rendering program to control the motion of the virtual scene.

4.6 Conclusion

In this chapter, we discuss development of a testbed for real-time identification

and tracking of features required for some visual servo control methods. The

development included descriptions of efficient algorithms utilizing image processing

and multi-view photogrammetry techniques. Feature point identification and

tracking, and homography decomposition can be achieved at 20-25 Hz. In this

chapter, we also discuss the solutions to problems faced during the development of

the testbed.















CHAPTER 5
CONCLUSION

5.1 Summary of Results

We discuss determination of range and the Euclidean coordinates of an object

feature undergoing general affine motion for a central paracatadioptric system via

a nonlinear estimator. The nonlinear estimator was proven, via Lyapunov-based

analysis, and numerically demonstrated to asymptotically determine the range

and Euclidean motion coordinates for an object moving with known affine motion

dynamics for a paracatadioptric system.

However, unlike image systems that are based on a planar image surface (or

spherical or ellipsoidal surface), the dynamic vision system resulting from a general

3D surface's projected image is not guaranteed to maintain an affine form. In

Chapter Three, the nonlinear state estimator developed for the paracatadioptirc

system is shown to be applicable for the nonaffine projection resulting from the

paraboloid image surface as described in [22]. The nonlinear estimator was proven

and numerically demonstrated to asymptotically determine the range information

and Euclidean coordinates from a single camera provided some observability

conditions are satisfied and that the Euclidean motion parameters are known.

This thesis also describes my efforts in the development of a HILS visualization

facility in Chapter Four. Chapter Four describes the development of a testbed for

identification of features required for many visual servo control algorithms. The

Kanade-Lucas-Tomasi algorithm is incorporated with multi-view photogrammetry

techniques. Almost real-time (20-25 Hz) implementation of the "closed loop"

was achieved, and this facility can now be utilized as a testbed for implementing

developed vision based controllers.









5.2 Recommendations for Future Work

Our results prove that the range and Euclidean coordinates of an object

feature undergoing general affine motion with known Euclidean motion parameters

can be determined. The result would be strengthened if we can identify range and

Euclidean coordinates of an object feature without knowledge of motion parameters

a prior.

In Chapters Two and Three radial distortion effects of 3D imaging surfaces

and camera lens could be taken into consideration in the model of the vision sys-

tem. As stated in [3], two factors combine to cause blur in catadioptric systems:

(1) the finite size of the lens aperture, and (2) the curvature of the mirror. Inaccu-

racies in computation caused by defocus blur can be accounted for by implementing

methods for simultaneous computation of the defocus blur.

Multi-view photogrammetry techniques works under the assumption that

the four feature points are coplanar points. Image segmentation and texture

recognition techniques can be used to recognize different planes which would

help in defining the region of interest of the feature detection algorithm to ensure

feature points are coplanar. This would also make the tracker more consistent

and reliable to intensity variations in the scene. Another issue to be addressed

is to make sure that the points selected are not collinear. Also, the condition of

the four feature points required to be coplanar and collinear can be eliminated by

implementing the eight point algorithm proposed in [15], where the feature points

don't have to satisfy the mentioned constraints.

Indexing of the feature points being tracked would enable recognizing feature

points and assist in storing and retrieving them, even when feature points are no

longer in view. This additional information may be required in other 3D motion

determination techniques, e.g. structure from motion.









In multi-view photogrammetry techniques, decomposition of the homography

matrix does not ensure a unique solution. Additional information is obtained

by slightly moving the camera and comparing the current normals to normals of

the previous image which helps in determining a unique solution of the normal.

Corresponding unique rotation and translation can hence be obtained. However

this method still does not guarantee the right solution. Hence, more reliable

method can be implemented to obtain the additional information to determine the

unique solution during homography decomposition. Also, quaternion representation

can be used instead of angle-axis representation to get rid of any singularities.

Implementing socket communication between the image processing program

in C/C++ and MATLAB/ Simulink would allow using accurate dynamics of any

autonomous system modelled in MATLAB/ Simulink.

Further efficient methods can be used for image capturing from the firewire. A

ring buffer mechanism along with genlock techniques can be used instead of using a

callback function for each frame.














APPENDIX A
LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF A
PARACATADIOPTRIC VISION SYSTEM

To prove that g1i(-), g2('), 93() ,C the time derivative of (3-9) is determined

as follows:

91 = Y4bl + y4bl + P ilo + 1YO (A-1)

g2 = .' + 'i/1 + y2QO + y2o (A-2)

g3 = *'/1I: + 'll': + y3Qo + y3Qi (A-3)

where
a31y1+ i :;' ~ 33Y3 (A
y4 = + Y4o (A-4)
L

Sb3 b3L (L + 3) (Yi11i + Y2-2 + Y3X3) (A
L L2 2f(L + x3)2
YX1+ y + y I 2 + + y2 + Y3X3 + Y3X3
2f(L + x3)

xxi + x2.2 LX3
L= (A-6)
L + X3
; 1= + 1x + + L.3 L3_ (x111 + x2.2 L3S)(L + .3) (A7)
L + X3 (L + X3)2
a. = &iixi + aii' + a2x2 + a12x2 + i :' + ,i :; + b1 (A-8)

.2 = a21X1 + a2 1 + a22x2 + a22x2 + a23x3 + a23x3 + b2 (A-9)

;3 = a31 + a31i' + a32X2 + a32-2 + a33X3 + a33X3 + b3. (A-10)

The facts that y(t) E L can be used along with Assumptions 2-1 through 2-4 to

conclude from (A-1) through (A-10) that g(, gi(-), 93(') E L,.









To prove that /1(-), g2( ), 93(') E oo, the time derivative of (A 1) through
-(A-3) can be determined as follows:


91 = ~i4b1 + 2W/4b1 + y4bl + jj1Q0 + 2y1o + y 1o (A-11)

j2 = ;4/i2 + 2,' 1, + y462 + y2o + 2y,20o + y2o (A-12)

g3 = ; /I': + 2.i 1: + .;I/,: + y3Qo + 2y30 + y3Qo. (A-13)

In (A-11) through (A-13)


ih = a&1yl + ai~li + i3 + ,,, I" + a13y3 + a13s3 (A-14)
a31Yi + 2a31yl li + '' :1192 + a33Y1Y3
2f
+ a32(y 2 + Y162) + a33 (1Y3 + Y913)
+ + .1
2f


y2 = a21yl + a21~1 + ', '~3 + 'I," + a23Y3 + a23j3 (A 15)
Sa31Y1y2 + a31(~1y2 + Y12) +
2f
2,I:4)!.)i.) + 4 ::,/i,/: + a33(," ,,: + :)
S+2f+ .2



y3 = ayl + a31a 1 + :'':/' + ,:'/' + ::/: (A-16)
Sa 31yy3 + a3l(1iy3 + y13) + :
+ a333 +3
2f
a32(," i + + i-)+ 1:: + 2'::j./:.//:j
+ 2f+ g3
2f


Y4 = 4- + (A-17)

(a31iy1 :-"/+'l :/:) L
L2









b3 b3L b3L + b3L) 2b3L2 (18)
Ro = + (A-18)
L L2 L2 L3
1J2l + 21iy + ylZi + .:; ;2 + 2j2-'2
2f(L + x3)

Y2+2 + j3X3 + 2y33 + 33 1 (L +
2f(L + X3) 2f(L + x3)2
(Y1 1 + y 2j + Y2'2 + Y3x3 + y3x3) (L + 63)
2f(L + x3)2
(L + 3) ( 1i + Y1 i + Y2X2 + Y2X;2 + Y3X3)
2f(L + x3)2
(L + 63)Y3 3 (L + X3) (Y1iX + Y2X2 + Y3X3)
+ +
2f(L + x3)2 2f(L+ x3)2
(L + -63)2 (yi1i + Y2X2 + y3x3)
f(L + X3)3


1l = ainix + 2a&'li + an.il + 512x2 + 2&12'2 (A-19)

+ a12'2 + "' :' i + 2' : ; K + ,: i ,7 + bi


x2 = a2 X + 2&21 ~1 + a21xi + a22x2 + 2&22-2 (A-20)

+ a22x2 + a23x3 + 2&23'3 + a23x3 + b2


X3 = a31X1 + 2a31i'1 + a31x1 + a32x2 + 2a322 (A-21)

+ a32, 2 + 33X3 + 2&33-3 + a33 3 + b3.

The expressions given in (A-1) through (A-21), Assumptions 2-1 through 2-5,
and the facts that, y(t), g(.) E o can now be used to prove that gi(-), g) ,

3(.) E LC.














APPENDIX B
LYAPUNOV-BASED RANGE AND MOTION IDENTIFICATION OF A
NONAFFINE PERSPECTIVE DYNAMIC SYSTEM

To prove that fi(),f2(), 3(') 0o, the time derivative of equation (3-9) is
determined as follows


fi = b3y5 + b3N5 2biiy5 2blyii5 2biiys5 (B-1)

22Y16 -2b2j/16 2b2y1,/6 + biy7 + bi/7


f2 = -2bi,',,,, 2b1,,,', 2b1,,_ ,-, + b3Y6 + b3N6 (B-2)

2,,,, ,. 21, t,,,. -- 21,,,,,. + b2Y7 + b27


3 = -2biy3y5 2bi13y5 2bly315 2/'l,_:,. (B-3)

21,,,t:,;1,. 21.,,1:,/,. + 263Y7 + 2b3T7-

The facts that I((t) E L can be used along with Assumptions 3-1 through 3-4 to
conclude from (B-1) through (B-3) that fi(-) to f3() E L,.
To prove that fi(), f2(') and f3(') E C, the time derivative of (B 1) through

(B-3) can be determined as follows:

fi = b3y5 + 2b3N5 + b3D5 2byys5 4by1 4li y105 4b1i1s5 (B-4)

2blyli5 2bljjy5 2b2y1y6 41',,/1y6 41'l, ,. .. 2b2jly6

41,_,/it ,,. 2b2yljj6 + blY7 + 2bi7 + bi7












+ b3Y6 + 2b3y6 + b3i6


21, i,/i ,.


41,', 1,. 21,'./1:,. + b2y7 + 2b2+7 + b2j7


f3 = -26bi3Y5 4b1i3y5 4I,'l:'/.-,- 2blyj35 4b1i35 2by3jj5


+ 2b3y7 + 4'.:1;. + 2b37.


In (B-4) through (B-6)


y4
s = 1 -
V Y+


y6 = Y2 +
V Y3


'/'[ / Y3 Y1Y4Y3 y3
-y yV 2 y4
2 y3 94 2 j3 C4


2y3 V Y4


2y2


yL3
SY4


(!/:!./4 + Y3Q4)
Y7 =



i1 = allyl + ajy1i + ;, + i,, '/1 + al3y3 + a1393


y3(a3y1y + 2a31Y111)


as31y Y3


Y3( :"/IL /2 + a32(iy + Y12))


S:,'IY2Y3 2 (y3(
2 + a33Yl + asA -
Y3
2y3( ''.!,ii + a12(2yi yiy2 + i/I/))


a2l(2yl12 + '/,'/.11)


2+ ,,2,, ) :
.3
U/ ', I


2(a13y 2 + 2,, :,1,,1)


j1yi5 + 3a iiy2i)


auyys3)


2,, / / ,:) [ Y3 2!,, ,
+ 2 2
Y3 [ Y3
2( ,,,',,i ,i, + a22 (yiy + 2;l,, /', ))
Y3

2(a23al y2 + a23 (iy2 + yly2)) + fi


- 2bli ,.:,.-, 4bl,).,-, 2bly2 f/5


(B-6)


(B-7)


(B-8)


(B-9)



(B-10)


f2 = 2b i .-, 4b ;il..-, 4b i, I)/.-,


(B-5)


2i ,,1.,,1. .- 4 ',' i, ,,. 4 ,' ,. ,,.,


21 '/ /_ 11/:










i2 = a21Yl + a21~1 + ',,, + ',,', + a233 + a23i3 (B-11)
+ Y3 :21i2 + 2 1.:-,/ 11*'2) 1i ,1I: a31yy 2 + a3l( iy2 + y1y2)
+ +
Y3 Y3
a31Yi' / 3 (222 ; -/ /) a22Yl 3
2 + ,::,,, ''- 2 2

2a^ Lylyl + a12(2y l,';_,,', ,';?,")_ 2 __ a211 yly J
33 Y Y3
Y3a21(2'1 al2(2yi + + 2,' i3 21)1
Y302 (21;/ 11 ,', + ,;, ;)- 1 ,,, 3 [( 1 '" "' + all(;/"/' + 2yly2l1)
+ -2
Y 3 Y3
a- '1 2 l + 2 :/) 2(al3Y1Y2 + a013(1Y2 1+ Y12)) 2+ 2
y3


/3 = 2(&a31 + a031 ) + 2( :,,, + :,,;) + 2( *::,: + :,) (B-12)

2(iy~ + 2aIjyii) 2(i&2y1y2 + a12 ('12 + y112))

2(n21Y 2+ a21 ( 2 + i, + .I,)) 2('',. + 2-7 ;.19,)

2( 3y 13 + a 13(1Y3 + y1 13) 2(,i.,:,./2./: 3 + a23((.1 1: + 11..'/:)) + f3


9 1 *9
I ,ll/i[ + all (2y1z 1y4 + ,,;0 a,1 i, ) (
14 = -2 (B -13)
y3 y j
I /3 1
F. 9 9. \
-2 + 2-


S &12Y1'l/ / + a12 1'/V 1' + a 12l13 (I 11 /1) al21[11 1,3
3Y3 Y3

&2lYl' 1 + a2l ,i,' [ a2yy3( (_ 'IQ + '/ 11 1) a2y111,'_' 3
-2 +
Y3 Y33

2( L, :,i,4, + ab (,l;3 + ,li;)) 2(,, :+ ,, + a23 ( L'/ + ; '/)

2(blY4Y5 + b1(1/415 + Y4/5)) 2(b2Y4Y6 + b2(l/46 + N46))









Y5 = Yl + 4

Y 3 11y4 + [
+1( /Y3
2 I Y4 Y3


1 f Y3 / i./3!./4
+ 1 2 +
2 V Y4 Y3
+ 3 -y4y3 Y4 /3
4V 3


6 = Y2 +
V Y3


(B-14)


(!/.'/4 y4 y3)

3V Y4
Y4


F ,2 y/3 /
Yi / /:
L 4


y/-3//4- Y4/3)
V Y4


+1 +Y2 "y; +
-2 IV Y4 Y3
+ _3( ii4 41 3)
YV 4 Y3
-I / ,; L,


-Y2


. ?4 3, / (/ /Y) 1
Y3Y4 3 -- 2
2y3 4 3


. y4y3
2y4


(B-15)


(B-16)


V Y4

S3/2/3

V Y4
E S (y"ii


3 /4
3Y4) 2I-3
2y3,


4 2y


2 + 2
2 34
y3 3y


1
7= (.=- '/3:/4
2 I1
+ !-/3!/4) 2
2 / -!/


Y4+ /:3/4 + '/'/4
2 V 4Y3) I.3


The expressions given in (B-4) through (B-17), Assumptions 3-1 through 3-4, and

the facts that, y(t), f(Y ys, ye, ) E ALx can now be used to prove that fi(), f2()

f3(') e C.


+ (
V 4


(B-17)


\. v I u uj















APPENDIX C
Hardware In The Loop

C.1 Program on the Server Machine

// This program render the virtual database on the projector screen

// using Vega Prime 2.0. The program receives control information

// through sockets to manuever the virtual scene.

#include

#include

#include

#include

#include "vuAllocTracer.h"

#include "vuDistributed.h"

#include

#include

#include

#include

#include

#include "vuSocketTCP.h"

#include "vulmageLoader.h"

#include "vulmageFactory.h"

#include "vsgi_bmp.h"

#include

#include

#include

#include









#include
#include
#define N 18
#define M 4
#define PI180 .0174-.:;-' 1l
#define VSB PORT 9898
#define NETWORK ERROR -1

#define NETWORK OK 0
vuAllocTracer tracer(true, true);
void ReportError(int, const char *);

//###############################
// # To run this distributed rendering sample, use the #

// # Distributed Rendering Utilites to setup a master and #

// # a slave system #
//###############################
class myApp : public vpApp, public vpKernel::Subscriber

{
public:

myApp() : //m_pMotionUFO(NULL),
m_pMotionGame(NULL),

m_pObserver(NULL)

{};
myApp()

{
//if (m_pMotionUFO) { m_pMotionUFO->unref(); }
if (m_pMotionGame) { m_pMotionGame->unref();}
if (m_pObserver) { m_pObserver->unref(); }









if (m_pPlane) { m_pPlane->unref(); }

};
virtual void run();

virtual void setSockets();

double C[N][1];

double D[N][1];

double x[N][1];

double xO[N] [1];

double U,V,W,vn,ve,vh,No,Ea,He,k;

int configure()

{
// pre-configuration
// configure vega prime system first

vpApp:: configure();
m pMotionGame = new vpMotionGame ;

assert (m_pMotionGame != NULL);

m_pMotionGame->ref();

// Make sure the observer exists

m_pObserver = vpObserver::find(" myObserver");

assert(m_pObserver != NULL);

m_pObserver->ref();

m_pPlane = vpObject::find("plane");

assert(m_pPlane != NULL);

m_pPlane->ref();

start recv=0;

vx=0;vy=0;vz=0;wx=0;wy=0;wz=0;

Tx=0;Ty=0;Tz=0;Tx=0;Ty=0;Tz=0;









m_pObserver- >setPosition(2200,2809,150);

m_pObserver->setOrientation(0,0,0);

// Turn off the slaves motion strategy so it can be positioned

if( vuDistributed::getMode() == vuDistributed::MODE_SLAVE )

{
m_ pObserver->set Strategy(NULL);

k=l;

}
return vsgu::SUCCESS;

}
virtual void onKeyInput(vrWindow::Key key, int mod)

{
k=0;

switch (key)

{
case vrWindow::KEY z:

wy = .1;

break;

case vrWindow::KEY c:

wy= -.1;

break;

case vrWindow::KEY x:

wy= 0;

break;

case vrWindow::KEY s:

case vrWindow::KEY S:

setSockets();









start recv=1;

break;

case vrWindow::KEY w:

case vrWindow::KEY W:

m_pObserver->getPosition(&Tx,&Ty,&Tz);

m_pObserver->getOrientation(&Rz, &Rx, &Ry);

break;

case vrWindow::KEY v:

case vrWindow::KEY V:

m_pObserver->setPosition(6897.260 ,1319.363 ,646.864);

m_pObserver->setOrientation(239.727 ,272.676 ,0.000);

break;

default:

vpApp::onKeyInput (key, mod);

break;

k=l;





protected:
*

* Handler of the non-latency critical event. Print out selected

* input sources. This is only an example that based on Saitek

* Cyborg 3D Rumble Force joystick. Please refer to your joystick's

* manual for the available input sources.

*/

void notify(vpKernel::Event event, const vpKernel *service)

{









#ifdef WIN32
system("cls");

#endif

}
private:

//vpMotionUFO* m_pMotionUFO;

vpMotionGame* m pMotionGame;

vpObserver* mpObserver;

vpObject* m pPlane;

vuMatrix< double> Twcl;

vuMatrix< double> Rwcl;

vuMatrix< double> Tclc2;

vuMatrix< double> Rclc2;

int amount, nret, startrecv;

double Tx,Ty,Tz,Rx,Ry,Rz, vx,vy,vz,wx,wy,wz;

SOCKET theClient;

SOCKET listeningSocket;

};
void ReportError(int errorCode, const char *whichFunc)

{
char errorMsg[92]; // Declare a buffer to hold

// the generated error message

ZeroMemory(errorMsg, 92); // Automatically NULL-terminate the string

// The following line copies the phrase, whichFunc string, and integer

errorCode into the buffer

sprintf(errorMsg, "Call to %s returned error %d!", (char *)whichFunc,

errorCode);








MessageBox(NULL, errorMsg, "socketIndication", MB_OK);

}
void myApp::setSockets()

{
if( vuDistributed::getMode() == vuDistributed::MODE_MASTER )

{
WORD sockVersion;
WSADATA wsaData;
sockVersion = MAKEWORD(1, 1); // We'd like Winsock version 1.1

// We begin by initializing Winsock
WSAStartup(sockVersion, &wsaData);
listeningSocket = socket(AF_INET, // Go over TCP/IP
SOCK_STREAM, // This is a stream-oriented socket
IPPROTO_TCP); // Use TCP rather than UDP
if (listeningSocket == INVALID_SOCKET)

{
nret = WSAGetLastError(); // Get a more detailed error
ReportError(nret, "socket()"); // Report the error with our custom
function
WSACleanup(); // Shutdown Winsock
printf("error creating listen socket\n");
return ;

} // Return an error value
// Use a SOCKADDR_IN struct to fill in address information
SOCKADDR IN serverInfo;
serverInfo.sin family = AFINET;
serverInfo.sin addr.s addr = INADDR ANY; // Since this socket









// is listening for connections, any local address will do
serverInfo.sin port = htons(6262); // Convert integer 8888 to

//network-byte order and insert into the port field
// Bind the socket to our local server address
nret = bind(listeningSocket, (LPSOCKADDR)&serverInfo, sizeof(struct
sockaddr));
if (nret == SOCKET_ERROR)

{
nret = WSAGetLastError();
ReportError(nret, "bind()");

WSACleanup();
printf(" NETWORK_ERROR 1");
return ;

}
// Make the socket listen
nret = listen(listeningSocket, 10); // Up to 10 connections may

//wait at any one time to be accept()'ed
if (nret == SOCKET_ERROR)

{
nret = WSAGetLastError();
printf("%d \n",nret);
Sleep(2000);
WSACleanup();
return ;

}
theClient = accept (listeningSocket,
NULL, // Address of a sockaddr structure (see explanation below)









NULL); // Address of a variable containing size of sockaddr struct
if (theClient == INVALID_SOCKET)

{
nret = WSAGetLastError();
ReportError(nret, "accept()");
printf("error accepting socket\n");
WSACleanup();
return ;

}
printf("3 Client Connected\n");//output to see how far it gets, delete
when happy

}
}
void myApp::run()

{
uint frameNum;
m _pMotionGame->setSpeed(0.1);
m_ pMotionGame->setRateLook(5.0);
amount = 1;
struct DRTest

{
double xl, yl, zl, hi, pi, rl;

};
struct DRTest drBuffer;
m_ pObserver-> set StrategyEnable (true);
m_ pObserver->setStrategy(m_ pMotionGame);
char buffer[24];// On the stack









((int*)buffer)[0] =0; ((int*)buffer)[1]=0; ((int*)buffer)[2]=0;

((int*)buffer)[3] =0;((int*)buffer)[4]=0; ((int*)buffer)[5]=0;

int gotdat=l, quit = 0;

int frame = 0, tmp=0;

// rendering loop

while ( (frameNum = vpKernel::instance()->beginFrame()) > 0 && quit


//printf("-");

got_dat=0;

int myframenum = frameNum;

tmp=0;

if( vuDistributed::getMode()


vuDistributed::MODE_MASTER )


if(start _recv:


nret = recv(theClient,

buffer,

24, // Complete size of buffer
MSG_PEEK);

if(nret != 0)

start recv=2;


}
if(start


recv:


printf("recv\n");

nret = recv(theClient,









buffer,

24, // Complete size of buffer

0);
got dat=l;

}
if (nret == SOCKET_ERROR)

{
// Get a specific code
// Handle accordingly
printf(" NETWORK_ERROR 2: ");

nret = WSAGetLastError();
printf("%d \n",nret);

Sleep(2000);

quit=1;

}
else if (nret != 0) // if(nret==24)//if(got_dat)

{
m_ pObserver->getPosition(&Tx,&Ty,&Tz);

m_ pObserver->getOrientation(&Rz, &Rx, &Ry);

//note that observer frame is not the same as vision system camera
frame!

//need to change order

//vision system:
//+Tx is camera right, +Ty is camera down, +Tz is camera forward

//+wx is pitch //+wy is yaw +wz is roll
//observer:

//+x is camera right, +y is camera forward, +z is camera up









Twcl.makeTranslate(Tx,Ty,Tz);

//in order of yaw, pitch, roll (z, x, y in observer frame)
Rwcl.makeRotate(Rz,Rx,Ry);
Rwcl.postMultiply(Twcl);
vx = (double) ((int*)buffer) [0]/1000;

vy = (double) ((int*)buffer) [2] /1000;
vz = -(double)((int*)buffer) [1]/1000;
wz = -(double)((int*)buffer) [5]/1000;

wy = (double) ((int*)buffer)[3]/1000;
wx = (double) ((int*)buffer)[4]/1000;
Tclc2.makeTranslate(vx,vy,vz);

//in order of yaw, pitch, roll (z, x, y in observer frame)
Rclc2.makeRotate(wz,wx,wy);
Rclc2.postMultiply(Tclc2); //makes a proper transform
Rclc2.postMultiply(Rwcl) ;
Rclc2.getTranslate(&Tx,&Ty,&Tz);
Rclc2.getRotate(&Rz, &Rx, &Ry);
m_pObserver->setPosition(Tx,Ty,Tz);
m_pObserver->setOrientation(Rz, Rx, Ry);

}//end got data

}
if (tmp == 100)
quit=1;
m_pObserver->getPosition(&drBuffer.x1,&drBuffer.yl,&drBuffer.zl);
m_pObserver->getOrientation(&drBuffer.hl,&drBuffer.pl,&drBuffer.rl);
vuDistributed::sync( (char*) &drBuffer, sizeof(drBuffer), vuDistrib-
uted::SYNC_LABEL_USER, vuDistributed::WAIT_ SLAVES );









// If a machine is a slave, take data from the struct and position the
observer.
if( vuDistributed::getMode() == vuDistributed::MODESLAVE )

{
m_pObserver->setPosition(drBuffer.xl ,drBuffer.yl ,drBuffer.zl);
m_pObserver->setOrientation(drBuffer.hl, drBuffer.p ,drBuffer.rl);

}
vpKernel: :instance()->endFrame();

}
// Shutdown Winsock
closesocket (theClient);
closesocket (listeningSocket);
WSACleanup();
unconfigure();

}
int main(int argc, char *argv[])

{
//initialize vega prime
vp: initialize(argc, argv);
vuDistributed: :set SyncEnable( vuDistributed:: SYNC_LABELPRE
SWAPBUFFERS, true );
vuDistributed: :setSyncEnable( vuDistributed::SYNC_LABELPOST
SWAPBUFFERS, false);
// ACF synchronization across DR slaves. Setting to "true" will send the
// master's version of an ACF to the slaves (slaves will ignore their local

// copy). Setting to "false" will force slaves to use their local copy of
// the ACF passed into ::define().









// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNC_LABEL_ACF, true );

// Time synchronization across DR slaves. Setting to "true" will give every

// system the same frame time. Setting to "false" will let each system

// use its system's clock for time.

// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNCLABELTIME, true );

// Window messages synchronization. Setting to "true" will send the

// master's messages to all the slaves. Setting to "false" will prevent

// sending the master's window messages to the slaves.

// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNCLABELMESSAGES,

true );

// Virtual Texture synchronization. Setting to "true" will force all

// systems to page virtual texture at the same time. Setting to "false"

// will cause systems to page virtual texture independently.

// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNCLABEL_VT, true );

// Synchronize paging. Setting to "true" will force all systems to page

// texture and geometry at the same time. Setting to "false" will cause

// systems to page texture and geometry independently.

// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNCLABELPAGING, true



// Synchronize input devices. Setting to "true" will make each system

// react as if the master's input device is connected to the slave.

// Setting to "false" will not pass input device updates from the master









// to the slaves.

// Default is "true."

vuDistributed::setSyncEnable( vuDistributed::SYNC_LABEL_INPUT, true );

// Synchronize Level of Detail Stress. Setting to "true" will ensure all

// LOD changes are synchronized. Setting to "false" let each system

// determine independently when LOD's should change.

// Default is "true."

vuDistributed: :set SyncEnable( vuDistributed:: SYNC LABEL

LOD_STRESS, true);

// User defined synchronization. Setting to "true" will enable user

// defined synchronization. Setting to "false" disables all user

// defined synchronization. In this example, disabling user defined sync

// result in different lowest random numbers for each system.

// Default is "true."

vuDistributed: :setSyncEnable( vuDistributed::SYNCLABEL USER, true );

// create a vpApp instance

myApp *app = new myApp;

//load acf file

if (argc <= 1)

app->define("vpinput _user.acf");

else

app->define(argv [1]);

// configure my app

app- >configure();

// runtime loop

app->run();

// unref my app instance









app->unref();

Sleep(2000);

// shutdown vega prime

vp::shutdown();

return 0;

}
C.2 Program on the Image Processing Machine

//Image processing project of the HILS

// Feature Point Tracking, Homography Decomposition and Socket Communi-

cation

#include

#include

#include

#include

#include

#include "cv.h"

#include "highgui.h"

#include "video stream.h"

#include "feature tracking.h"

#include

#include

#include

#include

#include

#include

#include

#include









#include

#include

#include

#include

#include
using namespace std;

#ifdef _DEBUG

#define new DEBUG NEW

#undef THIS FILE
static char THIS_FILE[] = _FILE

#endif

#define USESOCKETS

SOCKET theSocket;

char sockbuffer[24];

// this definition allows the filter graph to be remotely viewed using

// the graph edit utility (comment to remove this functionality

// #define REGISTER_FILTERGRAPH

// this definition enables the rendering of the frame grabber output

#define RENDER OUTPUT

// global variables -video streaming

HWND ghApp=0;// display window handle

IGraphBuilder *pGraph = NULL; // graph builder

IMediaControl *pControl = NULL; // graph control

IMediaEventEx *pEvent = NULL; // event handler

ICaptureGraphBuilder2 *pBuild = NULL; // capture graph

IVideoWindow *pVidwin = NULL; // video window

IBaseFilter *pSmarttee = NULL; // smart tee filter









IBaseFilter *pGrabberF = NULL; // sample grabber filter

ISampleGrabber *pGrabber = NULL; // callback interface

PLAYSTATE g_psCurrent = Stopped; // state of playback

DWORD g dwGraphRegister=0; // name of filter graph

AMMEDIATYPE mmediaType; // streamed media format

int mediatype = 0; // used in callback function

int firstframe = 0; // used in callback function

CSampleGrabberCB CB; // frame grabber

IplImage *img_prev; // stored previous image

IplImage *frame scaled;

IplImage *cur frame;

int frame count = 0,nret;

BYTE *pData;// Pointer to the actual image buffer

CvSize size, scaledsize;

VIDEOINFOHEADER vih;

int addremove_pt=0;

CvVideoWriter* savetestvideo;

// THE HOMOGRAPHY DECOMPOSITION ALGORITHM IS USING 4

POINTS FOR HOMOGRAPHY DECOMPOSITION

Matrix<3,4> pi,ps,psn,mi_star;// current frame feature point coordinates

Matrix<3,4> pi_star,ps_star,psn_star;// previous frame feature point

coordinates

Matrix<3,3> Gn, Hn, G_csn;// homography and projective homography

matrices

Matrix<3,3> R_bar;// rotation matrix

Vector<4> alpha_g33;// (zi_star/zi)/g33

Vector<3> x h bar;// translation vector









Vector<3> nstarActual,n_starl,n_star2;// normal unit vector (taken as

z-axis)

Vector<3> rpy;

Matrix<3,3> As;

Matrix<3,3> invAs;

double dum, dest = 1;

CvPoint new_pt;

void sleep( clock_t wait );

void on_mouse( int event, int x, int y, int flags, void* param )

{
if( !frame_scaled )

return;

if( frame_scaled->origin )

y = frame scaled->height y;

if( event == CV_EVENT_LBUTTONDOWN )

{
new_pt = cvPoint(x,y);

add removept = 1;

}

}
// returns a vector of RPY angles

// corresponding to the rotational part of the homogeneous transform TR.

void tr2rpy(Matrix<3,3> &R Vector<3> &rpy)

{
float sp, cp;

if ( fabsf( R(1,1) ) < le-5 && fabsf( R(2,1) < le-5))

{








rpy(1) =0;
rpy(2) = atan2( R(3,1), R(1,1) );
rpy(3) = atan2( -R(2,3), R(2,2));
}else

{
rpy(1) = atan2( R(2,1), R(1,1) );
sp = sin(rpy(1));
cp = cos(rpy(1));
rpy(3) = atan2( -R(3,1), cp*R(1,1)+sp*R(2,1) );
rpy(2) = atan2( sp*R(1,3) cp*R(2,3), cp*R(2,2) sp*R(1,2));

}
//converting radians to degrees
rpy=rpy*(180*7/22);
}
void ReportError(int errorCode, const char *whichFunc)

{
char errorMsg[92];// Declare a buffer to hold
// the generated error message
ZeroMemory(errorMsg, 92);// Automatically NULL-terminate the string
// The following line copies the phrase, whichFunc string, and integer error-
Code into the buffer
sprintf(errorMsg, "Call to %s returned error %d!", (char *)whichFunc, error-
Code);
MessageBox(NULL, errorMsg, "socketIndication", MB_OK);

}
// main program
void main(











cvInitSystem( 0,NULL );

remove("final homography output.txt");

remove ("feature_ point s.txt ");

remove("homography function output.txt");

createHGWorkspace(4);

//initilize variables

// Camera Caliberation Matrix

G csn = 0.6507, -0.0282, -82.4109, 0.0085, 0.6380, -61.7579, 0, -0.0001, 1;

As= 640, 0, 640, 0, 640, 512, 0, 0, 1;

invAs = inverse(As);

// Establishing socket communication

#ifdef USESOCKETS

WORD sockVersion;

WSADATA wsaData;

int nret;

sockVersion = MAKEWORD(1, 1);

// Initialize Winsock as before

WSAStartup(sockVersion, &wsaData);

// Store information about the server

LPHOSTENT hostEntry;

in addr iaHost;

////////////////////////////////////
iaHost.saddr = inet_addr("192.168.211.12"); //change IP address

accordingly

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII









hostEntry = gethostbyaddr((const char *)&iaHost, sizeof(struct in_addr),
AF_INET);

if (!hostEntry)

{
nret = WSAGetLastError();

ReportError(nret, "gethostbyaddr()");// Report the error as before

WSACleanup();

return;

}
theSocket = socket(AF_INET,// Go over TCP/IP

SOCK_STREAM,// This is a stream-oriented socket
IPPROTO_TCP);// Use TCP rather than UDP

if (theSocket == INVALID_SOCKET)

{
nret = WSAGetLastError();

ReportError(nret, "socket()");

WSACleanup();

return ;

}
// Fill a SOCKADDR_IN struct with address information

SOCKADDR IN serverInfo;
serverInfo.sin family = AFINET;

// At this point, we've successfully retrieved vital information about the
server,

// including its hostname, aliases, and IP addresses. Wait; how could a


single









// computer have multiple addresses, and exactly what is the following
line doing?

// See the explanation below.
serverInfo.sinaddr = *((LPIN_ADDR)*hostEntry->h_ addr_list);

///////////////////////////
serverInfo.sin_port = htons(6262);// Change to network-byte order and
//insert into port field

////////////////////////////
// Connect to the server
nret = connect(theSocket,

(LPSOCKADDR)&serverInfo,
sizeof(struct sockaddr));

if (nret == SOCKET_ERROR)

{
nret = WSAGetLastError();

ReportError(nret, "connect()");
WSACleanup();
return ;

}
#endif
MSG msg={0};
INSTANCE hInstance = NULL;

int nCmdShow = 1;

WNDCLASS wc;
cvNamedWindow( "OpenCV", 1);

cvSetMouseCallback( "OpenCV", on_mouse, 0 );

//initialize the COM library.









HRESULT hr = Colnitialize(NULL);
if (FAILED(hr))

{
printf("ERROR Could not initialize COM library");
return;

}
// register the window class (dshow render window)
ZeroMemory(&wc, sizeof wc);
wc.lpfnWndProc = WndMainProc;
wc.hInstance = hInstance;
wc.lpszClassName = CLASSNAME;
wc.lpszMenuName = NULL;
wc.hbrBackground = (HBRUSH)GetStockObject(BLACK_BRUSH);
wc.hCursor = LoadCursor(NULL, IDC_ARROW);
wc.hIcon = LoadIcon(hInstance,
MAKEINTRESOURCE(IDI_VIDPREVIEW));
if(! RegisterClass (&wc))

{
printf("failed to register window class\n");
CoUninitialize);
exit (1);

}
// create the main window. The WS_CLIPCHILDREN style is required.

ghApp = CreateWindow(CLASSNAME, APPLICATIONNAME,
WS_OVERLAPPEDWINDOW I WS_CAPTION |
WSCLIPCHILDREN, CWUSEDEFAULT,
CW USEDEFAULT, DEFAULT VIDEO WIDTH,








DEFAULT_VIDEO_HEIGHT, 0, 0, hInstance, 0);
if(ghApp)

{
// create DirectShow graph and start capturing video
hr = CaptureVideo();
if (FAILED (hr))

{
CloseInterfacesQ);
DestroyWindow(ghApp);

}
else

{
// don't display the main window until the DirectShow preview graph has
been created.
ShowWindow(ghApp, nCmdShow);
// this loop handles changes to the render window (close, change size,
etc.)
while(GetMessage(&msg,NULL,0,0))

{
TranslateMessage(&msg);
DispatchMessage(&msg);

}
}
}
// destroy OpenCV preview window
cvDestroyWindow(" OpenCV");
// release COM








CoUninitialize ;
cvReleaseVideoWriter(&savetestvideo);
releaseHGWorkspace();
cvReleaseImage( &frame_scaled );

//cvReleaseImage( &cur_frame );

}/

//*****Image Capturing Fucntions****
// function for capturing the video from the first available capture device
HRESULT CaptureVideo(

{
HRESULT hr; // error checking
IBaseFilter *pCap = NULL; // video capture filter.
// intialize graph structure
hr = GetInterfaces();
if (FAILED(hr))

{
printf("failed to get interfaces\n");
return hr;

}
// attach the filter graph to the capture graph
hr = pBuild->SetFiltergraph(pGraph);
if (FAILED(hr))

{
printf("failed to set capture filter graph\n");
return hr;

}
ICreateDevEnum *pDevEnum = NULL; // device enumerator




Full Text

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4.4Multi-viewPhotogrammetry......................42 4.4.1EuclideanReconstruction....................43 4.4.2Camera,Screen,andVirtualSceneGeometry.........47 4.4.2.1ProblemStatement..................47 4.4.2.2VirtualSceneGeometry...............47 4.4.2.3CameraGeometry..................49 4.4.2.4CameratoScreenGeometry.............50 4.5SocketCommunication.........................54 4.6Conclusion.... ............................55 5CONCLUSION.... ............................56 5.1SummaryofResults..........................56 5.2RecommendationsforFutureWork..................57 APPENDIX ALYAPUNOV-BASEDRANGEANDMOTIONIDENTIFICATIONOF APARACATADIOPTRICVISIONSYTEM................59 BLYAPUNOV-BASEDRANGEANDMOTIONIDENTIFICATIONOF ANONAFFINEPERSPECTIVEDYNAMICSYSTEM.........62 CHARDWAREINTHELOOP........................66 C.1ProgramontheServerMachine....................66 C.2ProgramontheImageProcessingMachine.. ............81 C.3ProgramtoComputeScreen-CameraHomography..........117 REFERENCES............. ......................120 BIOGRAPHICALSKETCH............................123 vi

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PAGE 26

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PAGE 27

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