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The multiaperture optical (MAO) system based on the apposition principle

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The multiaperture optical (MAO) system based on the apposition principle
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Lin, Shih-Chao, 1955-
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English
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xiii, 140 leaves : ill., photos ; 28 cm.

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Subjects / Keywords:
Binocular vision ( jstor )
Concentration ratio ( jstor )
Cylinders ( jstor )
Eyes ( jstor )
Geometric angles ( jstor )
Insects ( jstor )
Lighting ( jstor )
Micrometers ( jstor )
Optics ( jstor )
Point sources ( jstor )
Synthetic apertures ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Shih-Chao Lin.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AFL1803 ( NOTIS )
19879193 ( OCLC )
AA00004814_00001 ( sobekcm )

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THE MULTIAPERTURE OPTICAL (MAO) SYSTEM
BASED ON THE APPOSITION PRINCIPLE












By

SHIH-CHAO LIN


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



UNIVERSITY OF FLORIDA


1988




THE MULTIAPERTURE OPTICAL (MAO) SYSTEM
BASED ON THE APPOSITION PRINCIPLE
By
SHIH-CHAO LIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988


To my parents
Mr. Chung-Liang Lin
&
Mrs. Yang Li-Shuang Lin
sax*


ACKNOWLEDGMENTS
The author would like to express his deepest
appreciation and gratitude to Dr. Richard T. Schneider, the
chairman of his supervisory committee, for his guidance and
support in this research and for the faith and friendship he
showed toward the author throughout this academic endeavor.
His deepest appreciation also goes to Dr. Edward E. Carroll
for his guidance, encouragement and friendship. Sincere
thanks are also extended to the other members of the
supervisory committee, Dr. William H. Ellis, Dr. Tom I-P
Shih and Dr. Gerhard Ritter.
Special note should be made of the valuable support
from Dr. Neil Weinstein. Dr. Weinstein not only gave the
author moral support but also help the author overcome his
language problem.
The author will be forever indebted to his parents, Mr.
Chung-Liang Lin and Mrs. Yang Li-Shuang Lin for their
unfailing faith and encouragement. Without them the author
would never have been able to complete all his academic
endeavors.
in


TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT xii
CHAPTER
1 INTRODUCTION 1
1.1 Introduction 1
1.2 Vision Systems 2
1.3 Literature Survey on Multiaperture
Optical Systems 5
2 THE INSECT EYES 7
2.1 Optical Part 7
2.1.1 Cornea 9
2.1.2 Crystalline Cone 9
2.1.3 Crystalline Tract 10
2.2 Sensory Part 10
2.2.1 The Sensor 10
2.2.2 The Nerve Connection 14
2.3 Image Formation 14
3 OPTICAL STUDIES 16
3.1 Light Horn 16
3.2 Optics of Light Horns 18
3.2.1 Experimental Setup 18
3.2.2 The Ray Tracing Program 23
3.3 Results of the Optical Studies 25
IV


V
3.3.1 Vertical Cross Section 25
3.3.2 Wall Pattern 41
3.3.3 Horizontal Cross Section 55
3.4 Applications 58
3.4.1 Vertical Cross Section 58
3.4.2 Wall Pattern 65
3.5 Summary and Discussion 95
4 LIGHT COLLECTION OF THE LIGHT HORN 105
4.1 Intensity Concentration 106
4.2 Comparison of Light Horns and Lenses . 112
5 THE MULTIAPERTURE OPTICAL SYSTEM DESIGN .... 116
5.1 The System Design 116
5.1.1 MAO Mask 118
5.1.2 Detector Board 120
5.1.3 Memory Board and Processor Board. 124
5.2 The Performance of the MAO Device 125
5.3 Discussions 130
6 SUMMARY AND CONCLUSIONS 132
REFERENCES 138
BIOGRAPHICAL SKETCH
140


LIST OF TABLES
Table Page
3.1 Dimensions of experimental devices 21
3.2 The distance-polar angle relation of light
horn used 54
3.3 Matrices of point sources at different location 62
3.4 Matrix of a point source image on the 5X5
light horn array 66
vi


LIST OF FIGURES
Figure Page
2.1 The ommatidium of insect eye.
a) Photopic eye; b) Scotopic eye 8
2.2 The cross section of rhabdom
a) Photopic eye; b) Scotopic eye 12
2.3 The microvilli 13
3.1 Schematic of the MAO device eyelet 17
3.2 The experimental setup 19
3.3 Three different modes of generating the
pattern. a) Vertical cross section;
b) Wall of cylinder; c) Horizontal slice . 22
3.4 The diagram of the law of reflection 24
3.5 Photo, vertical cross section image pattern,
parallel beamparallel to axis 26
3.6 Two dimensional ray tracing for light horn;
parallel beamparallel to axis 28
3.7 Photo, vertical cross section image pattern,
parallel beamparallel to axisyellow
side right of axis, red side left of axis. 29
3.8 Photo, vertical cross section image pattern,
parallel beam5 degrees off axis 31
3.9 Photo, vertical cross section image pattern,
parallel beam11 degrees off axis 32
3.10 Computational pattern, vertical cross section
image pattern, parallel light source,
parallel to axis 34
3.11 Computational pattern, vertical cross section
image pattern, parallel light source,
5 degrees off axis 35
vi 1


VI11
3.12 Computational pattern, vertical cross section
image pattern, parallel light source,
11 degrees off axis 37
3.13 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, parallel
to axis 38
3.14 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, 5 degrees
off axis 39
3.15 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, 11 degrees
off axis 40
3.16 Vertical cross sections across cylinder;
a) 33 units b) 48 units c) 86 units away
from the exit aperture 42
3.17 Ray pattern impinging on cylinder wall for
object on axis 44
3.18 Photo, wall pattern, parallel light source,
parallel to axis 45
3.19 Ray pattern impinging on cylinder wall for
object one degree off axis 46
3.20 Ray pattern impinging on cylinder wall for
object two degrees off axis 47
3.21 Distance of center of bands from "Zero" band
vs. the off-axis angle 49
3.22 Angular resolution as a function of
polar angle 51
3.23 Angular resolution vs. light horn length. ... 53
3.24 The computational pattern which would appear
on the horizontal slice 56
3.25 Photo, pattern on horizontal slice 57
3.26 A 5 X 5 detector array arrangement 59
3.27 Vertical cross section pattern projected on a
5X5 detector array and its possible
matrices 60


IX
3.28 Computational pattern for multiple light horns
arranged on a 3 X 3 array 63
3.29 Computational pattern for multiple light horns
arranged on a 5 X 5 array 64
3.30 Fiber shaped detector 67
3.31 The correlation of the image information
among three adjacent eyelets 69
3.32 Detector tube with ring shape detectors .... 70
3.33 Intensity distribution on the cylinder wall;
object on the axis, 1000 units distance ... 72
3.34 Intensity distribution on the cylinder wall,
object 0.34 degree off axis 73
3.35 Intensity distribution on the cylinder wall,
object 0.52 degree off axis 74
3.36 Characteristic band position of the objects,
object 1000 units away from entrance
aperture 75
3.37 The triangular grid 76
3.38 The object space covered by the eyelet
system 77
3.39 The image pattern created by the Addition
method, object point at node (1,1) 80
3.40 The image pattern created by the Addition
method, object point at node (5,5) 81
3.41 The image pattern created by the Addition
method, object point at node (1,26) 82
3.42 The image pattern created by the Addition
method, object point at node (17,17) 83
3.43 The image pattern created by the Multiplication
method, object point at node (1,1) 86
3.44 The image pattern created by the Multiplication
method, object point at node (5,5) 87
3.45 The image pattern created by the Multiplication
method, object point at node (1,26) 88


X
3.46 The image pattern created by the Multiplication
method, object point at node (17,17) 89
3.47 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,1) 91
3.48 The image pattern created by the Cross
Correlation with Addition method,
object point at node (5,5) 92
3.49 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,26) 93
3.50 The image pattern created by the Cross
Correlation with Addition method,
object point at node (17,17) 94
3.51 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,1) 96
3.52 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (5,5) 97
3.53 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,26) 98
3.54 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (17,17) 99
3.55 The image pattern created by the Multiplication
method, object points at nodes (1,1) and
(17,17) 100
3.56 The image pattern created by the Multiplication
method, object points at nodes (1,26) and
(17,17) 101
4.1 Concentration ratio curves of conical light
horns with different cone angles for an
on-axis point source at 1000 units
distance 107
4.2 The concentration ratio of a parabolic light
horn at different distances away from
the exit aperture with an on-axis point
source at 1000 units away from entrance . 108


XI
4.3 Comparison of concentration ratios between
a light horn and a lens 110
4.4 The concentration ratio curves of the
off-axis point source Ill
4.5 The light collection of a lens 114
5.1 The multiaperture optical (MAO) device 117
5.2 Arrangement of detectors on optic RAM.
a) detector location; b) detector in use
(marked by "X") 121
5.3 Detector arrangement underneath the light
hole 123
5.4 Performance of the system using the mask
with cylindrical holes 126
5.5 Performance of the system using the mask
with conical holes 127
5.6 The result pattern of Figure 5.4 after
the clean-up 128
5.7 The result pattern of Figure 5.5 after
the clean-up 129


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the degree of Doctor of Philosophy
THE MULTIAPERTURE OPTICAL (MAO) SYSTEM
BASED ON THE APPOSITION PRINCIPLE
By
Shih-Chao Lin
Apri1, 1988
Chairman: Dr. Richard T. Schneider
Major Department: Nuclear Engineering Sciences
Automation freed mankind from repeated boring labor
and/or labor requiring an instantaneous response. When
applied as robotics it could even free mankind from
dangerous labor such as handling radioactive material. For
a robot or an automated system a vision device has proven to
be an important element.
Almost all artificial vision systems are similar in
design to the human eye with its single large lens system.
In contrast, the compound eye of an insect is much smaller
than the human eye. Therefore, it is proposed to imitate
the insect eye in order to develop a small viewing device
useful in robotic design.
The basic element of the multiaperture optical system
described here is a non-imaging light horn. The optical
xi i


XI 1 1
studies on the non-imaging light horn (a simulated insect
eye eyelet) have been done and show that this device may
produce images when several horns are used together in an
array. The study also shows that with several non-imaging
devices the position of an object point light source can be
determined very easily.
One possible realization of multiaperture optical
system design based on the apposition principle is proposed
and discussed. The multiaperture optical system proposed is
a small, low cost device with digital image processing.


Chapter 1
INTRODUCTION
1.1 Introduction
Machines freed mankind from labor which required great
strength or long endurance. Automation freed mankind from
repeated boring labor and/or labor requiring instantaneous
response. Robotics finally frees mankind from dangerous
labor and from the kinds of tasks of which humans are not
capable or willing to endure because they require special
ized abilities, albeit, with low levels of intelligence.
Early machines needed only very primitive sensors or no
sensor at all. Automation needed more sophisticated
sensors. A photocell rather than an electrical contact may
have been used. Logically one may expect then that robotics
will require even more sophisticated sensorsone of them
being vision rather than just the detection of light.
One such vision system may be a television camera
attached to suitable digitizing equipment. The digitizing
equipment is required since the information provided by the
TV camera is not viewed by a human observer. Instead, it
has to be interfaced with specialized intelligence which has
been programmed into the main computer of the robotic
1


2
system. This constitutes the major difference between
automation and robotics. Therefore, the question arises,
what kind of vision system is easiest to interface with a
computer?
Images can be easily evaluated by a computer if they
are made available in the form of digitized picture elements
(pixels). Therefore, vision systems with digitization
ability are preferred when a computer is chosen to do the
recognition process.
There seem to be two principal preferred designs for
vision systems to be found in nature. These are (1) the
single lens eye found in vertebrates and (2) the
multiaperture eye found in arthropods (the "insect eye").
Most optical instruments have been patterned after the
vertebrate eye. This raises the question: is this design
superior to the insect eye or only considered to be so by
optical designers?
It is the objective of this dissertation to investigate
whether the insect eye is indeed generally inferior to a
single lens eye, or if there are special applications where
the insect eye may be more suitable than the single lens
eye. If the second case is true, the proof should be
offered in the form of a description of such a superior
device.
1.2 Vision Systems
The difference between the two vision systems consists
in the number of apertures used to form the image. For this


3
reason one needs to investigate the difference between
single aperture optics (SAO) and multiple aperture optics
(MAO). This dissertation is concerned with some of the
unique features of multiple aperture optics.
Therefore, let us describe the insect eyes, albeit in
somewhat oversimplified terms. In the overall MAO system
there could be three possible ways of extracting an image
from an object. These three possible ways are as follows:
1. Each eyelet collects only one pixel and the
resulting overall image is a mosaic (the so-called
apposition eye).
2. The lens of each eyelet projects a fairly large
image onto the retina and all these images are
superimposed precisely (the so-called superposition
eye) .
3. Each lens projects a small image onto the retina,
the individual images do not overlap and together
they form the total image.
For the artificial multiaperture optics system, one
cannot possibly worry about focusing of many eyelets or
adjusting them so that all of the individual images are
superimposed correctly. Therefore, the second option, the
superposition eye, is not a desirable choice. Similarly,
the focusing and the overlapping problems of the small
individual images are the reasons to reject the third
option. The first option, the apposition eye, removes the
focusing problem but raises the question of whether or not


4
acceptable resolving power can be achieved. The answer to
this question is one of the major subjects of this disser-
tation. From behavioral studies it is known that
insects have remarkably acute vision and are capable of
resolving even small, rapidly moving distant objects at low
ambient light levels. For the designer, who intends to
build a camera based on the design principles of the insect
eye, it would be very helpful to know how such "super
resolution" is achieved. In this dissertation an answer is
proposed.
In an effort to understand the optics of the insect
eye, a simple model consisting of a hollow cone with a
reflecting wall attached to a non-reflecting cylindrical
section was selected for analysis. This cone, similar to
the crystalline cone of the insect eye, is suggested as a
model for the eyelet of MAO devices. In Chapter 3 the
studies of the geometrical optics of these artificial
multiaperture optical elements will be discussed. The light
concentration of these optical elements will also be
discussed in Chapter 4.
If one were to assume that each eyelet acquires only
one pixel, one would have to conclude (using a conventional
approach) that the insect has a very poor resolution system.
The question that arises is: why would anybody want a small
camera which has poor resolution? The answer is that maybe
one does not want to take pictures which are intended to be
viewed by a human observer with this camera. Instead, it


5
could be a camera which recognizes objects and reports the
presence of the object to the main computer of a vehicle or
a robot. For example, if such a camera were the size of a
postage stamp, it could be fitted into the "hand" of a robot
and could make the task of picking up certain objects much
easier. If the recognition scheme could be hardwired
into the detector array, the restrictions on the motion of
robot would be reduced. Therefore, the recognition cannot
be too complex. Also any preprocessing by optical means
would be very beneficial.
1.3 Literature Survey on Multiaperture Optical Systems
An early insect eye model was studied by Schneider and
2
Long. They constructed an insect eye model with 100
eyelets. Each individual eyelet consisted of two lenses,
one aperture stop and an optical fiber bundle. The end of
each fiber was attached to one photosensitive detector which
was connected to an amplifier in order to obtain signals
which were strong enough for analysis. A computer was used
to study the resulting signals. A computer program
reconstructed the image pattern and the image was displayed
on a video terminal. Although it was an early model of a
multiaperture optical device, Schneider and Long were able
to conclude that the multiaperture optical system could have
inherent digitization and large field of view abilities.
They concluded that the multiaperture optical device can
have "small depth of the structure," which means a thin
device with a large field of view.


6
3
Kao has presented this first generation mechanical
insect eye in much detail. He discussed three different
models: a one-eyelet, a seven-eyelet and a 89-eyelet model.
The computer system used here was an HP-85 microcomputer
with an HP6942A multiprogrammer analog-to-digital converter.
With this system, the image was converted to a digital
pattern and analyzed. The recognition technique was
discussed in his studies.
The multiaperture optical systems in the earlier
studies were quite primitive. They were large in size and
the mechanisms were not much different from the human eye
and optical function studies were not done. Basically,
those models combined several shrunken single lens eyes into
a large array to form a semi-compound eye.
The present study builds on these earlier works. Two
insect eye models were built and studied. Computer ray
tracing programs were developed to simulate the path of
light in the insect eye. Unexpected patterns were obtained
which showed how insect eyes may produce an image. The
optics study led to the multiaperture optical device which
is discussed in Chapter 5. Finally, the studies will be
summarized and discussed in Chapter 6.


CHAPTER 2
THE INSECT EYES
Since the insect eye is used as a model for the MAO
device, it is helpful to review the anatomy of the insect
eye here briefly. According to Chapman/ most adult insects
have a pair of compound eyes bulging out, one on each side
of the head. This provides for a wide field of view,
essentially in all directions. Each compound eye has up to
10,000 eyelets which are known as ommatidia. Each
ommatidium is believed to be a non-imaging optical system.
Each ommatidium consists of approximately 30 cells, is
about tens of micrometers in diameter, and is hundreds of
micrometers in length. Functionally speaking, the
ommatidium consists of two parts (as shown in Figure 2.1):
an optical part and a sensory part. The optical part
collects the light and forms the special pattern for
recognition. The sensory part analyzes the pattern and is
capable of perceiving the image of the scene.
2.1 Optical Part
The basic optical system of the ommatidium consists of
two lenses, (1) a cornea (which is a biconvex lens) and (2)
a conical optical element. Some insects also have a
7


8
Cornea
Crystalline.
Cone
Crystalline
Tract
Rhabdom
(a)
(b)
Figure 2.1 The onmatidium of insect eye.
a) Photopic eye; b) Scotopic eye


9
wave-guide-like crystalline tract at the end of the conical
crystalline cone.
2.1.1 Cornea
4
Chapman states that the cornea of the insect eye
consists of two corneagen cells, usually forming a biconvex
corneal lens at the outer end of the ommatidium. The lens
is transparent and colorless. It is also a cuticular
surface, often thick and solid, which can protect the soft
tissue of the insect eye.
5
According to Meyer-Rochow, the diameter of the corneal
lens of most insects falls between 25 and 35 micrometers.
Unlike the diameter, the thickness of the cornea varies
drastically from 4% up to 20% of the total length of the
ommatidium.
2.1.2 Crystalline Cone
The crystalline cone of the ommatidium usually consists
of four cells, known as the Semper cells. The cone is
transparent with a surface like a paraboloid. Hausen^
studied the optical properties of the crystalline cone. He
concluded that the cone has a length of about 42 pm. The
distal end is slightly curved. Measurements in his studies
showed that the index of refraction can be approximated as a
parabolic function. At the center axis, the index has the
highest value of about 1.50, while at the edge it has a
value of 1.383. From Snells Law, one can easily see that
this change in the index of refraction would cause the light
at the edge of the crystalline cone to be totally reflected


10
and not transmitted to the neighboring eyelets. This is
like the gradient-index lenses which are now being manu
factured. Thus, the light going into the crystalline cone
is either transmitted to the rhabdom or reflected back out.
2.1.3 Crystalline Tract
Most entomologists believe that the two main categories
of the insect eyes can be described as either the
apposition eye or the superposition eye. Goldsmith and
Bernard^' believed that the more suitable names for
these classifications are photopic and scotopic eyes,
respectively. As discussed by them, the scotopic (super
position or clear zone) eyes are capable of adaptation to
variations in light intensity. Therefore, the scotopic eye
is also known as a light-adapted eye.
The crystalline tract occurs only in the scotopic eyes.
It is located between the crystalline cone and the rhabdom.
The optical function of the tract is like that of a wave
guide. It has an important role on the adaptations of the
scotopic eye to the light.
2.2 Sensory Part
2.2.1 The Sensor
Goldsmith and Bernard7, reported that for the
photopic eyes, there is a hose-like structure attached at
the exit of the crystalline cone. This attached structure
is the sensory element and is called the rhabdom. The
scotopic eye is similar to the photopic eye with the


11
exception that there is a wave-guide, the crystalline tract,
positioned between the cone and the rhabdom.
The rhabdom consists of retinula cells. Normally there
are seven or eight retinula cells in an ommatidium. Near
the ommatidial axis, the retinula cells are differentiated
to form the rhabdomeres. Therefore, most of the eyelets
contain seven or eight rhabdomeres. Hence, the sensory
part of the ommatidium is called the rhabdom. A cross
section through the rhabdom of the photopic eye is shown in
Figure 2.2.a and the cross section of the scotopic eye is
shown in Figure 2.2.b. For the scotopic eyes, pigment is
located close to the exit of crystalline cone at low
intensity levels and moved halfway down the crystalline
7
tract at higher light levels. Goldsmith and Bernard also
state that the pigment granules within retinular cells 1 to
6 (see Figure 2.2.b) migrate laterally to the rhabdomeres,
when in the light-adapted state, but the pigment granules do
not migrate within the two central cells.
In the rhabdomere the light sensitive elements are the
microvillithese are tiny tubes having typically a diameter
7
of less than one micrometer. Layers of these tubes are
oriented in an alternating crossing pattern, more or less
perpendicular to the longitudinal axis of the rhabdomere as
indicated in Figure 2.3. Mazokhin-Porshnyakov* concluded
that the visual pigments are disposed on the surface of the
tubules.


Figure 2.2 The cross section of rhabdom. a) Photopic eye
b) Scotopic eye.


13
i Microvilli i
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Retinular
cell
Figure 2.3 The microvilli


14
2.2.2 The Nerve Connection
As it was seen in Figure 2.2, six rhabdomeres are
arranged around a seventh (and possibly an eighth) at the
7
center. According to Goldsmith and Bernard, each retinular
cell has one nucleus and one axon. The six outer retinular
cells synapsed in a single cartridge, and in the case of
the scotopic eye these six cells do migrate laterally in the
light-adapted state. The one or two central cells have
different connections than the surrounding six cells.
2.3 Image Formation
The mosaic theory, which is believed to be the theory
g
governing insect vision, was proposed by Mller in 1826.
The mosaic theory assumed that each eyelet of the insect eye
is only capable of a limited field of view which does not
overlap with the field of the adjacent eyelets which make up
the compound eyes. Each ommatidium contributes only one
point out of the total image pattern. Hence, under the
mosaic theory, one eyelet of the facet eye is a non-imaging
optical device.
As mentioned earlier, the rhabdom consists of seven or
9
eight rhabdomeres as the sensing elements. Kuiper observed
the rhabdomeres by illuminating the rhabdom of Apis
mel1ifera. He found that only the individual rhabdomeres
were illuminated, but not the rhabdom as a whole unit.
Furthermore, the arrangement of the rhabdomeres in the
rhabdom is in a symmetrical, radial pattern. This indicates
that each eyelet of an apposition eye could contribute more


15
than just one pixel to the total image although this
ommatidium could still be a non-imaging optical element.


CHAPTER 3
OPTICAL STUDIES
3.1 Light Horn
In order to understand the function of the ommatidium
as an optical element, models were built to simulate it.
Like the ommatidium, each model (seen in Figure 3.1)
consists of a light horn (a hollow cone with reflecting
walls)simulating the crystalline coneattached to a
non-reflecting cylindrical sectionsimulating the rhabdom
of the insect eye.
As mentioned in Chapter 2, the crystalline cone of the
insect is transparent but has a refractive index larger than
the surrounding medium which causes the light entering the
crystalline cone to be internally totally reflected. In the
models, total reflection was replaced by regular (specular)
reflection. The simpler of the two designs, the cone with
reflecting walls, was selected in this study because it can
be manufactured more easily.
Light horn studies were started long ago. Initially,
they caught the attention of optical scientists because
light horns seem to circumvent the Second Law of Thermo-
16


Figure 3.1 Schematic of the MAO device eyelet


18
dynamics. Obviously, one would think that by making the
exit aperture small enough, one should be able to obtain an
illumination density larger than the intensity of the light
source. However this is not the case, as will become clear
in the analysis presented below.
In the modern age, the light horn was studied as a
possible concentrator for solar energy. Williamson^
commented that although the light horn can be designed to
transmit images, its principal usefulness is found in the
transmission of the maximum amount of energy rather than the
possible image forming potential of the light horn. Welford
and Winston^ concluded that the "ideal concentrator" would
have walls shaped like a rotation parabola, similar to the
conical wall of an insect eye.
3.2 Optics of Light Horns
In the present dissertation the emphasis is on the
image formation rather than on the energy concentration
phenomena. Light horn optics were studied in two parts:
experimentally and theoretically (ray tracing).
3.2.1 Experimental Setup
Figure 3.2 shows the experimental setup. It consists
of a source of parallel light, a photographic shutter, a
light horn and a film holder (or camera). In contrast to
normal picture taking techniques, there is no lens between
the light source and the image.
In the experimental setup two different light horns
were used, one with a large cone angle and the other with a


X
Light Source
Figure 3.2 The experimental setup


20
small cone angle. The dimensions of these light horns are
shown in Table 3.1. Since only the optical properties are
of interest here, all the dimensions are quoted in units
relative to the radius of the entrance aperture (i.e., one
unit represents the entrance aperture radius of the light
horn). (Note: For different size light horns, dimensions
will all scale linearly.)
Because of the difficulty in manufacturing a parabolic
light horn, only conical light horns were used in the
experiment (However, the parabolic light horn was studied by
ray tracing).
The patterns generated could be observed in three
different modes (see Figure 3.3) depending on where the
camera (detector) was placed. The images were:
(1) on vertical cross sections of the cylinder (at
various distances from the exit of the light horn),
(2) on the walls of the cylinder,
(3) on horizontal slices through the cylinder (at
various elevations).
For the vertical cross sections, the film is held perpen
dicular to the axis of the light horn. For the wall pattern
the film can be wrapped around the cylinder or the image
pattern can be observed directly on the interior surface of
the cylinder. In this dissertation, the wall pattern is
determined by taking pictures of the side of the frosted
glass cylindrical tube. Photos of these three positions of
the camera were used to support the results of ray tracing.


21
Table 3.1
Dimensions of experimental devices
Entrance Aperture Radius:
Exit Aperture Radius:
Light Horn Length:
Cylinder Radius:
Cylinder Length:
Device 1
52.83 mm
11.18 mm
154.94 mm
Device 2
9.18 mm
7.35 mm
174.50 mm
9.50 mm
1219.20 mm


Figure 3.3 Three different modes of generating the pattern.
a) Vertical cross section; b) Wall of cylinder;
c) Horizontal slice


23
3.2.2 The Ray Tracing Program
For light horns of the sizes that are of interest here,
diffraction effects play only a minor role and therefore
this study was only concerned with geometrical optics. The
inner surface of the light horn is assumed to be a totally
reflective surface and to obey the law of reflection. For
computer ray tracing, it is useful to write a program based
on the vector form (by components) of this law of
reflection. Welford and Winston1' gave the vector equation
of the law of reflection as
r = r. 2(ii-£.)n (3.1)
r i i
where r. is the unit vector of the incident ray, ?r is the
unit vector of the reflected ray and n is the normal vector
of the surface (see Figure 3.4).
When Welford and Winston" studied the non-imaging
concentrator, they indicated that some rays were returned
back out through the entrance aperture if the incident rays
have too large an angle with the optical axis. For large
cone opening angle light horn, not all of the incident rays
passed through the exit aperture of the light horn and only
part of the light beam contributed to the final pattern.
Therefore for a viewer observing in front of the light horn
entrance aperture, although there was no light source behind
the light horn, he who can still observe the shining
reflection. Mazokhin-Porshnyakov' mentioned that on the
surfaces of many insect eyes a "wandering" spot is found.


24
Figure 3.4 The diagram of the law of reflection


25
This "wandering" spot is a black spot with a shining
background and its location changes with different direction
of observation. It is so-called the "pseudopupil." The
partly reflected and partly passed through of the light
beams could be the reason why the pseudopupil is found in
the insect eyes.
The vector method can be used to calculate the
trajectories of the light beams and the resulting image on
the film or the detectors. The study was restricted to the
following conditions: 1. The light horn was of conical
shape and hollow. 2. The exit aperture of the light horn
has a smaller radius than the entrance aperture.
3.3 Results of the Optical Studies
3.3.1 Vertical Cross Section
When a parallel incoherent light source was mounted
paraxially with the axis of the light horn, an interesting
pattern was found on the image side (shown in Figure 3.5).
The image pattern in this case is a set of concentric rings
around a central disk. It was unexpected that these
concentric rings should have very sharp edges and that there
is no light between the ringsAs will be shown below,
these concentric rings are not a diffraction pattern but can
be explained with geometrical optics alone. Nevertheless,
it is most astonishing that an empty cone should produce
such a sharp structure.
In order to understand the reason why the parallel beam
could construct this sharp image-like structure, a two


26
Figure 3.5 Photo, vertical cross section image pattern,
parallel beamparallel to axis


27
dimensional ray tracing is shown in Figure 3.6. It shows an
individual light horn and four mirror images. Assume a
parallel light beam (or a point light source at infinity)
enters the light horn parallel to the optical axis. The
rays bordered by lines 1-1 is the non-reflected light beam
which forms the central disk of the image pattern. The
light beam bordered by lines 1-2, is reflected once and
projected to the opposite side of the axis of the light
horn. The light bordered by lines 1-2 and l-2 forms an
annulus which constitutes the first bright ring. Similarly,
the rays between 2-3 and 2-3 form the second bright ring.
This is why there is a sharp edge and a dark space between
the center disk and the first ring.
To prove that this interpretation is correct, a color
filter was added to the white light source. It consisted of
two halves, a yellow and a red one, whereby the dividing
line was located on a diameter of the light horn aperture.
The result is shown in Figure 3.7. Although, in this
dissertation the color is not shown, the grey tone of this
picture still indicates the result. As predicted from the
interpretation above, the center disk is divided into a
yellow half (right half) and a red half (left half), the
dividing line being the diameter of the center disk. The
first ring is also divided into a yellow half and a red half
along the same dividing line. In contrast to the center
disk, the right side of the first ring is red. For the


28
Figure 3.
6 Two dimensional ray tracing for light horn;
parallel beam--parallel to axis


29
Figure 3.7 Photo, vertical cross section image pattern,
parallel beamparallel to axisyellow side
right of axis, red side left of axis


30
second ring, the colors are reversed again, yellow to the
right and red to the left.
To study the effect when the light source is not on the
axis of the light horn, the parallel light source was made
to have 5 degrees with the axis of the light horn. Figure
3.8 shows the image in this case. The central disk is
displaced slightly compared to the first case (Figure 3.5).
The rings of this case split into twin pairs and they are no
longer perfect rings. There is a sharp cutout which
occurred on the brighter part of the twins. It is believed
that this cutout was reflected to the opposite side to form
the "twin." Figure 3.9 shows the case when the light source
is moved still further off-axis (11 degrees). The central
disk almost disappears. The first ring and its twin turn
into crescents and appear on the same side. The central
disk disappears when the angle is larger than 12 degrees.
While these results are certainly unexpected and
interesting, one can also draw some practical conclusions
from them. If one placed one detector at the center of the
light horn and one at the edge of the first ring, the
combination of the illumination on the detectors could be
used to detect the off-axis angle of the point source at
infinity, at least for 0, 5 and 12 degrees. For example, if
both detectors show strong illumination, the object must be
off-axis by less than 5 degrees. If just the center
detector detects the brightness of the source and the
detector on the wedge does not detect the source, then the


31
Figure 3.8 Photo, vertical cross section image pattern,
parallel beam5 degrees off axis


32
Figure 3.9 Photo, vertical cross section image pattern,
parallel beam11 degrees off axis


33
object is in between 5 and 12 degrees. If both detectors
see nothing, then this shows that the source is off axis by
more than 12 degrees.
Therefore, although the light horn is a non-imaging
optical device, it is still capable of more than just simply
detecting the presence or absence of an object within its
field of view (FOV).
The above study was done with a light horn of certain
specific dimensions (light horn 1, see Table 3.1). To make
sure that a generic effect was discovered, a second light
horn with different dimensions was used. Similar results
were obtained. The only differences between results of the
two light horns are the size and the number of rings. The
number of rings depends on the length of the light horn and
the light horn opening angle. In general, only the
unreflected light rays and singly reflected rays are
important; the doubly reflected rays appear only at large
polar angles where poor resolution destroys their
usefulness. The relationship between the maximum cone
length and the cone angle opening resulting in only one ring
and the central disk is shown as equation 2.
L = A. tan(2a) cot(a)/(tan(a) + tan(2a)) (3.2)
Figure 3.10 shows the results of the ray tracing for
light which is parallel to the axis of the light horn. As
can be seen, the obtained pattern agrees very nicely with
the photograph (Figure 3.5). Figure 3.11 shows the case of


34
Figure 3.10 Computational pattern, vertical cross
section image pattern, parallel light
source, parallel to axis


35
Figure 3.11 Computational pattern, vertical cross
section image pattern, parallel light
source, 5 degrees off axis


36
parallel light 5 degrees off axis. Again it agrees with the
photograph (Figure 3.8), even the wedge in the first ring
shows up exactly the same. Figure 3.12 (corresponding to
the photograph, Figure 3.9) shows the case which is 11
degrees off axis. Notice the similarity with the experi
mental result; i.e., the central disk almost disappears.
The parabolic light horn was also studied mathe
matically to examine the vertical cross section pattern.
The ray tracing result of the light source in front of the
light horn at the axis with the vertical cross section taken
at three units down the cylinder is shown in Figure 3.13.
Similar to the cone-shaped light horn (Figure 3.5), the
image of parabolic light horn consists of a central
disk and one ring-like annulus. The difference of these two
(Figures 3.10 and 3.13) is the cone shape light horn could
have more than one rings but the parabolic light horn
has only one ring-like annulus. In both cases the ring or
the ring-like annulus all have a sharp edge at the inner
boundary. When the light source is 5 degrees off axis
(Figure 3.14), similar to Figure 3.11, the center disk moved
aside and the ring-like annulus changed its shape. For the
light source located at 11 degree of axis, the image pattern
is shown in Figure 3.15. Again it is similar to Figure
3.12. This indicates that the parabolic light horn and the
cone-shaped light horn have similar properties.
In the cases discussed above, the patterns were taken
at a fixed location, the outlet of the light horn. The


37
Figure 3.12 Computational pattern, vertical cross
section image pattern, parallel light
source, 11 degrees off axis


38
***
i is. *-
.> jT
.V. i"1

Figure 3.13 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, parallel to axis


39
Figure 3.14 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, 5 degrees off axis


40
Figure 3.15 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, 11 degrees off axis


41
question arises: what if the location of the sensors was
moved as in the scotopic eye? Figure 3.16 shows typical
patterns of the number 1 light horn (see Table 3.1) at three
different vertical cross sections along the length of the
cylinder when the object point is at infinity on the axis.
The image is approximately in focus in the case shown in
Figure 3.16.(a), which is 33 units down the cylinder from
the exit aperture. (Again, one unit is the radius of the
entrance aperture.) Larger and larger rings are produced
further down the axis (e.g., Figure 3.16.(b) is at 48 units)
until the reflected rays begin to impinge on the cylinder
wall as shown in Figure 3.16.(c), at 86 units. From Figure
3.16, one can draw the conclusion that when the sensor
location is moved further away from the light horn, the size
of the ring becomes larger and larger, but the size of the
central disk stays almost the same. It is also found that
the intensity of the light on the rings is higher when the
film location moves closer to the light horn. This might be
a reason why the pigment granules of the scotopic eye move
closer to the crystalline cone and migrate to the center
under conditions of darkness.
3.3.2 Wall Pattern
When the light beam passes through the light horn, it
projects an image on the wall of the attached cylinder. The
image pattern of this case is a three dimensional pattern.
To simplify the analysis one may imagine that the cylinder


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(a) (b) (c)
Figure 3.16 Vertical cross sections across cylinder;
a) 33 units b) 48 units c) 86 units away
from the exit aperture.


43
has been cut along the top and then unrolled and flattened.
The radius of the cylinder is taken to be 1.0 unit.
The wall pattern of an on-axis object point source at
infinity appears to be a uniform band around the cylinder
wall superimposed on a weak and uniform background. Figure
3.17 shows the computational results in this pattern of rays
impinging on the wall. The uniform background, produced by
the unreflected rays, is not shown on this plot. Here the
abscissa represents the distance down the cylinder while the
ordinate represents the angle around the axis. Figure 3.18
is the luminous wall pattern of the light source observed
down the cylinder. To have the image shown clearly on the
cylinder wall, a frosted glass tube was attached to the exit
aperture of the light horn, the simulate cylindrical part of
the non-imaging device. (The center of the pattern is on
the bottom of the tube and the photograph was taken from the
side.) In this case, the tube used to take the photos of
the wall pattern was too long to remain complete rigid and
it bent at the non-supported part where the band should
occur. Therefore the photo showed only part of the band.
The gap in the band is caused by the laser. Exclude the
nonuniform phenomena caused by the material and laser beam,
both the experimental photo and the computational result
showed that they agreed with each other.
The wall pattern from the computational results for an
object point one degree off the optical axis is shown in
Figure 3.19. A somewhat distorted band is found between 34


cylinder
44
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It
XaWV
If
Jii
.W.w

P
AVA .
f#
55*00:
Ml
I I I I I I I I
I
0
Distance from light horn exit
(10 units per scale)
I
Figure 3.17 Ray pattern impinging on cylinder wall
for object on axis


45
Figure 3.18
Photo, wall pattern, parallel light source,
parallel to axis


cylinder
46
I
I
0
Distance from light horn exit
(10 units per scale)
Ray pattern impinging on cylinder wall for
object one degree off axis
Figure 3.19


47
and 47 units down the cylinder, and the unreflected rays
form a more intense background, producing the horizontal
elliptical pattern at the center. Although the band of this
case is no longer uniformly distributed, the band location
and the band width were still clearly shown. Figure 3.20
is the wall pattern for an object point two degrees off the
optical axis.
Comparing Figures 3.19 and 3.20, one can clearly see
that when the object point moves off-axis, the band location
moves closer to the light horn exit aperture. This
indicates that the distance of the band could be a measure
of the polar angle of the object point. Of course this
measurement has to be taken from a reference point. The
band caused by an on-axis object, as shown in Figure 3.17,
could be the reference or "zero" band. The width of the
bands produced on the cylinder wall is a measure of the
polar angular resolution of the device, while the distance
of a certain observed band from the "zero" band is a measure
of the polar angle of a certain object point. Figure 3.21
plots the distance of the center of various bands from the
"zero" band as a function of off-axis polar angle for
several different light cones. The center point (Bc) and
the width (B^) of the "zero" band can be found by the
following equations (symbols as in Figure 3.1):
B =
c
[ (Dc + Ag/2 + A./2) cot (2a + 7.) L ]/2
(3.3)


48
0 Distance from light horn exit
(10 units per scale)
Figure 3.20 Ray pattern impinging on cylinder wall
for object two degrees off axis


49
Figure
Distance of center of 1
vs. the off-axis angle


50
Bf = [L (A. Ag) cot (2a + y.)/ 2 ] / 2 (3.4)
where D is the diameter of the cylinder and 7. is the angle
between the incident ray and the axis of the light horn.
The distance, D, between the "zero" band and an off-axis
o
angle band (one reflection only) would be
Db = (Dc + A./2 + Ag/2) (cot (2a) cot(2a + 7j)/2. (3.5)
In each case, the radius of the entrance aperture of the
light horn is 1.0 unit.
If two object points are present at different polar
angles, they may be distinguished if their respective bands
do not overlap much. The band width, B^, may be thus
converted to polar angles and the angular resolution plotted
as a function of polar angle as in Figure 3.22. The value
plotted is the full-width of the bands converted to a polar
angle, in radians. The angular resolution improves as the
light horn exit aperture approaches the size of the entrance
aperture. The fraction (F) of the total light incident on
the horn which is reflected to form bands is
F = (Rq2 Rj2) / Rfl2 (3.6)
where R^ is the entrance aperture radius, and Rj is the exit
aperture radius. Hence, the resolution improves at the
expense of the efficiency as one might expect.


51
Figure 3.22 Angular resolution as a function of
polar angle


52
The unreflected light patterns (for example, the
central elliptical patterns in Figures 3.19 and 3.20) could
conceivably be used to distinguish objects at different
azimuthal angles. However, for the geometries studied here
the azimuthal angular resolution would be very poor. More
precise azimuthal angular information can be obtained by
cross correlation between several eyelets. This will be
discussed later.
As far as the polar angular resolution is concerned it
is of interest to see how it varies with the length of the
light horn when the entrance and exit apertures are kept
constant. Figure 3.23 shows the results for an entrance
aperture radius of 1.0 unit, an exit aperture radius of 0.8
unit, cylinder radius of 0.8 unit, and various horn lengths
from 5.0 to 60.0 units. The comparison is made for an
object point one degree off axis. Optimum resolution
appears at a length of about 20.0 units.
The polar angular resolution of the device using wall
patterns is thus limited by (1) the width of the bands, (2)
the presence of background from the unreflected rays, and
(3) the distortion of the bands for off-axis object points.
Nevertheless, a number of angular bands may be
distinguished. For example, using light horn device 2, the
data indicate that this device might be fitted with sensor
rings inside the cylinder as shown in the Table 3.2.
If this horn-cylinder combination were used as a simple
collimator, its light acceptance would be characterized by


53
Figure 3.23 Angular resolution vs. light horn length


54
Table 3.2 The distance-polar angle relation of light
horn used
Distance from light horn Polar angle range
(units of entrance radius) (mi11iradians)
80.
to
72.
0.0
to
1.5
72.
to
61.
1.5
to
4.5
61.
to
49.
4.5
to
8.0
49.
to
42.
8.0
to
13.5
42.
to
29.
13.5
to
24.0
29.
to
15.
24.0
to
50.0


55
the half-angle of the 1 x 19 unit collimator, i.e. about
0.05 radian. Thus, use of the reflection bands on the
cylinder wall enables finer resolution and possible imaging
within a single collimator tube. If the light horn with an
exit aperture of 0.95 unit were used, about 20 distinguish
able angular ranges would be obtained. However, the
reflecting surface area would drop from 36 percent of the
total entrance aperture to about 10 percent and the length
of the cylinder would need to be increased from about 150
units to over 300 units.
3.3.3 Horizontal Cross Section
As we have seen, both the vertical cross section and
the wall pattern could be used to improve the resolution of
the non-imaging optical device. It is interesting to know
what the horizontal cross section (seen as in Figure 3.3.c)
pattern would be and whether it could be used to improve
resolution or not. Such a pattern was computed for the
plane located 1.0 unit above the optical axis and is shown
in Figure 3.24 (the object point was on axis). In this case
the pattern appears as a parabolic band on a uniform
background. Figure 3.25 shows the photograph pattern
corresponding to the calculation shown in Figure 3.24 where
the observed fine structure is caused by the laser and
should be considered as an artifact. Of course the
photographs show more detail than the ray tracings predict,
which is to be expected since multiple reflections and
diffraction effects were ignored. Although the parabolic


one unit
Figure 3.24 The computational pattern which would appear on
the horizontal slice


57
Figure 3.25 Photo, pattern on horizontal slice


58
band can be used (similar to the wall pattern band) to
define the polar angle, the location of the parabolic band
changes too dramatically as the elevation of the plane
varies. Therefore, it is suggested not to use this
information, since the wall pattern can do the job nicely
already.
3.4 Applications
3.4,1 Vertical Cross Section
As discussed in Section 3.3.1, the vertical cross
sectional pattern of a point (or parallel) light source
produced by light horn is a disk and several rings. If this
pattern is projected on a 5 X 5 detector array, as shown in
Figure 3.26, the number of rays falling on each detector,
in connection with a threshold setting, can be used to
identify the location of the object point. Figure 3.27
(a,b,c) shows the overlapping pattern of the 0, 5 and 11
degrees cases, respectively. The detector responses can be
used to analyze the edge of an object boundary. If the
detector response is set to be "1" when the number of rays
is larger than the threshold setting and to be "0" when it
is less than the threshold setting, the response of the
detector array is used to construct the matrix which
indicates the location of the point object (or edge of an
object). Figure 3.27 (d,e,f) includes the possible matrices
that correspond to Figure 3.27 (a,b,c) respectively.
For this application the detector array does not
necessarily have to be exactly a 5 X 5 array; more detectors


59





Figure 3.26 A 5 X 5 detector array arrangement


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0 VQ\Q,.
on
o
a.
b.
0 1110
11111
11111
11111
0 1110
(0 degree)
10 10 0
0 1111
0 110 0
0 1111
10 10 0
(5 degrees)
10 111
0 1111
0 10 0 0
0 1111
10 111
f.
(11 degrees)
Figure 3.27 Vertical cross section pattern projected on a
5X5 detector array and its possible matrices


61
would give better resolution. As discussed earlier, the
higher order rings are produced by several reflections. The
intensities of the higher order rings are not as large as
the intensity of the first ring and therefore might not be
sufficient for detection. Consequently, it is suggested
that the length of the light horn be chosen so that only one
ring and the disk be produced in the zero degree case. In
this case a 3 X 3 detector array or, (as in the rhabdomeres
of the insect eye), six on a circle and one or two at the
center is sufficient to resolve a relatively small polar
angle difference especially when the field of view is made
small. Also, the depth of the device could be held to a
minimum when the number of rings is minimal. By doing so,
not only the size of device could be reduced but also the
amount of information to be analyzed is reduced. Table 3.3
lists the matrices of a point source at different locations
which could be used to determine the polar angle of the
object point.
Having the results of the single light horn optical
study available, it is desirable to expand the study to
multiple light horns. Figure 3.28 shows the resulting
computational pattern of a centered object point in front of
9 light horns arranged on a 3 by 3 array. In this case, the
displacement of the light horn apices were set to be four
units (two times the entrance diameter). Figure 3.29 shows
the pattern of 25 light horns. A 3 by 3 detector array is
placed behind each light horn, the resulting matrix is shown


62
Table 3.3 Matrices of point sources at
different location
0
10
20
30
60
90
1
1
1
1
1
1
0
1
0
1
0
1
1
0
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
1
1
1
0
1
1
1
1
0
1
0
0
0
10
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
1
1
0
0
0
1
0
0
0
20
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
1
1
0
1
0
0
0
0
1
0
0
0
30
0
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
60
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
0
1
1
0
1
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
90
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
(Note
The
point
sources
is
at
1000
units
away 1
from
the
1ight horn.)


63
Figure 3.28 Computational pattern for multiple light
horns arranged on a 3 X 3 array


64
Figure 3.29
Computational for multiple light horns
arranged on a 5 X 5 array


65
in Table 3.4. It can be seen in Figure 3.27 that some of
the rays strike the detector array of the neighboring light
horns if the off-axis angle is not equal to zero. Thus a
cylinder-like divider, similar to the pigment cells of
scotopic compound eye, is suggested to prevent the
overlapping of the image patterns of neighboring light
horns.
3.4,2 Wall pattern
As described earlier, the light sensitive elements of
the insect eye are the layers of the microvilli which divide
the cylindrical shaped rhabdom into a multitude of pixels;
albeit in the direction of the cylinder axis rather than
perpendicular to the axis as one would expect for a focal
plane array. The microvi11i-contained rhabdom is a long
cylinder rather than just the couple layers of microvilli
shown in Figure 2.3. It suggests a possible application of
the wall patterns. The fiber shaped detectors which was
12
designed by Schneider (shown as Figure 3.30) could be a
detector for this application.
Based on the studies of the wall pattern, the position
of the singly reflected band can be used as a measure of the
off-axis polar angle for object points in the field of view
of the light horn. However, it seems unlikely that more
information than just the polar angle can be determined with
a single horn-cylinder combination. At any specific polar
angle, the patterns are quite insensitive to the azimuthal


66
Table 3.4
Matrix of <
light horn
a point
array.
; source
image
on the
1
0
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
1
0
0
1
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
0
0
1
1
1
1
1
0
1
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
0
0
0
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
1
1
0
1
1
1
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
0
0
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
0
0
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
1
1
0
1
1
1
0
0
0
0
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
0
0
0
1
1
0
1
1
1
1
1
0
0
1
0
0
1
1
1
1
1
0
1
0
0
1
1
1
0
0
1
0
1
0
1
0
0
1
1
1
0
0
0
0
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0
0
1


67
Signal Wire
Detector
Figure 3.30 Fiber shaped detector


68
angle. The construction of an image would require at least
three adjacent horn-cylinder combinations with overlapping
fields of view. Figure 3.31 shows the two dimensional area
which is defined by overlapping the fields of view of three
adjacent light horns. With a honeycomb shaped arrangement
the light horns could cover all of object-space.
If ring shaped detectors could be made and arranged to
form a detector tube as shown in Figure 3.32, these
detectors would not only detect the presence of light but
would act as number of single channel analyzers. An
individual single channel analyzer would measure the
intensity of light which strikes on the inner surface of the
ring shaped detector. The total output of the detector tube
is then an intensity distribution along the length of the
wall of the cylinder. The intensity distributions (of the
three adjacent eyelets) may be correlated to reconstruct the
image perceived by the three eyelet system.
Since a detector cannot distinguish the singly
reflected beam from other light beams, all of the light
beamsinstead of just the singly reflected beamswould
contribute to the intensity distribution of the wall
pattern. However, the length of the horn element may be
chosen to eliminate all rays reflected more than n times,
where n is any arbitrary integer, n = 0, 1, 2, . .
Through ray tracing, the intensity distributions (along
the length of the cylinder wall) at different polar angles
were created and several of them are shown in Figures


Object area seen by
Figure 3.31 The correlation of the image information
among three adjacent eyelets


Figure 3.32 Detector tube with ring shape detectors


71
3.33-35. (Note: The light horns used here are 20 units long
with a one degree cone angle. The object plane is assumed
to be 1000 units away from the entrance aperture of these
horn-cylinder combinations.) Among these intensity
distributions, it is found that the maximum is located at
different longitudinal positions for different polar angles.
Similar to the position of the singly reflected rays, the
location of the peak depends also on the off-axis polar
angle. This relation between the peak positions and the
polar angles of the objects is shown in Figure 3.36. The
intensity at this characteristic position, the position
where the peak is located, indicates the possible
contribution to the image pattern from the particular polar
angle. Therefore this characteristic position is also used
to construct the image.
In order to simplify the analysis, both the object and
image spaces were put on the identical triangular grid, each
sampled in identical two-dimensional arrays with the node
point arrangement as shown in the Figure 3.37. Although
three eyelets could be arranged to see an area larger than
the triangular area formed by the axes of the three eyelets,
for simplicity the three eyelets were arranged to have
overlapping fields of view only to such a degree to just
cover this triangular area. Any object area other than that
covered by this triangle may be covered by other eyelets in
the array (see Figure 3.38). Hence the image space is also
limited to this triangular area.


Intensity
Figure 3.33 Intensity distribution on the cylinder wall;
object on the axis, 1000 units distance


Intensity
0.200
0.100-
0.000
0
Figure
Distance from exit aperture (units)
H
50
.34 Intensity distribution on the cylinder wall;
object 0.34 off axis.
u>


Intensity
Figure 3,35 Intensity distribution on the cylinder wall;
object 0,52 off axis.


Band position on the cylinder (units)
Figure 3.36 Characteristic band position of the objects,
objects 1000 units from entrance aperture.


76
(51,1)
(17,17)
(1,26)
Figure 3.37 The triangular grid


77
Figure 3.38 The object space covered by the eyelet
system.


78
A source point in object space produces illuminated
pattern on the detection surfaces of the three horn-cylinder
combinations. Since the point sources at any location on
the ring centered at the optical axis have only one unique
polar angle resulting only one set of illuminated pattern.
It is not possible to locate the position of the object
point directly since its relation to the polar angle is
ambiguous. When constructing an image from these detector
patterns, an object point (i,j), of intensity I.. is mapped
bj
into an irregular area (point spread function) around image
point (i, j) This point spread function is not spatially
invariant (isoplanatic). However, one can pick an image
point (i,j) and calculate the probability that the tube
patterns are caused by a source point at the corresponding
object point (i,j). The resulting probability distribution
on the image space is then an "image" of the point (i,j) of
the object space.
Four methods were used to construct the image patterns:
Addition, Multiplication, Cross Correlation with Addition
(CCA) and Cross Correlation with Multiplication (CCM). All
of them use the same grid system and the definition of these
operations are discussed below.
For a known object point the intensity distributions on
1 2 3
the three eyelets, f f and f can be easily found,
because the polar angle to the three eyelets can be calcu
lated. To construct the image pattern, all the nodes on the
triangle were scanned. One can then search for the


79
characteristic position of the image point by checking the
intensity distribution along the detector tube in order to
determine the contribution of light from the object point to
this particular point. Therefore, when tracing the point
source on the image space, this intensity is placed on the
particular ring which is related to this particular polar
angle. Thus, for every image node point (i,j) the relative
1 2 3
polar angles, 0 0 0 and the characteristic
1 J bj 1J
1 2 3
positions, Lc> L £, Lc to the light horns were found.
The intensity valuesfn(L,") where n stands for 1, 2 or 3
at the characteristic positions were extracted from the
distribution curves. Since the intensity on the node point
stems from various eyelets, it is reasonable to apply a
superposition law to the intensities and therefore a
reasonable way to reconstruct the image has been found. The
sum of the intensities at the characteristic positions,
fn(L"), from the distributions of all three eyelets were
assigned to the node and then the intensity of the image
point (i,j) could be expressed as
IiBa?e= f1 ( L 1)
i.J c'
+ f2
+ f3(L;).
(3.7)
This Addition method was used to construct the
intensity patterns created by projection of all the nodes
into the object space and these patterns can be used as a
template to identify the position of the objects. Figures
3.39-42 are four intensity patterns which represent the


80
POINT SOURCE AT NODE (1,1)
Addition method
Figure 3.39 The image pattern created by the Addition
method, object point at node (1,1)


81
POINT SOURCE AT NODE (5,5)
>1.06 =>1.48
>1.69
>1.90
=2.12
Addition method
Figure 3.40
The image pattern created by the Addition
method, object point at node (5,5)


82
POINT SOURCE AT NODE < 1 26 )
>1.03 =>1.44
>1.64
>1.85
-2 .05
Addition method
Figure 3.41 The image pattern created by the Addition
method, object point at node (1,26)


83
POINT SOURCE AT NODE < 17 # 17 )
>0,78 >1,09 11 >1,24
>1.40
=1.55
Addition method
Figure 3.42 The image pattern created by the Addition
method, object point at node (17,17)


84
possible image of an object point. Due to the fact that it
is impractical to indicate the exact value of the intensity
at the various nodes on the figures, these intensities are
shown in ranges which are characterized by different
patterns. Only the intensity values larger than half of the
maximum value are shown on the figures. The intensity
values of less than half of the maximum do not make a
significant contribution to the total image. Besides, the
position of the intensity value of half the maximum
indicates the resolution of the system on that particular
object point. From these patterns it can be seen that the
highest intensity value occurs as an area which includes the
position of the object point. Thus one can conclude that
the Addition method gives good results for identifying the
location of the object point. However, one must remember
that the intensity distribution is not a delta function,
but rather a point spread function with a finite area.
Hence, the relationship between Ln and is only
c i, j
appoximate. The detector intensity fn represents only the
probabi1ity that there is a source point at the object
point (i,j).
Although the Addition method generated a maximum at the
location of the object point, the intensity differences
between the point and its neighbors were found to be small
or even zero and therefore indistinguishable. Source points
closer together than about one third of the side of the
triangle would probably not be distinguishable. Therefore,


85
the Multiplication method is also suggested. The Multipli
cation method finds the possible intensities at the node
point from all three intensity distributions then calculates
the product of those three values and assigns this product
value to the node point and this could be expressed as
I13?6 = f!(L 1) f2(L 2) f3(L 3) .
i, j c (3.8)
Figures 3.43-46 show the results of the Multiplication
method. Although there is still no sharp image point in
some of the cases, the intensity ratios are much larger and
the location of the object point is more distinct, such as
at the node point (1,1) in Figure 3.43. Source points
closer together than about one-fifth of the side of the
triangle would probably not distinguishable.
As mentioned, the intensity distributions along the
cylinder wall that result from a given object point can be
found when the point object position is known. One can also
construct the intensity distribution functions along the
cylinder, 0n(r,z), for each light horn where r stands for
the distance of the image point from the axis of the light
horn n, (n=l, 2 or 3), and z is the distance from exit
aperture. Thus, the integral (with respect to z) of the
product of the functions fn(z) and 0n(r,z) is the
probability, Pn(r) of the image point occurring on the ring
of the radius r centered at the axis of light horn n. For
the intensity distribution was generated by the ring shaped
detector, it is a discrete function instead of continuous


86
POINT SOURCE AT NODE (1,1)
> 2.3/i = > 3.2x II > 3.6X ¡ > 4.1 y. M 4.5x
Multiplication nethod
Figure 3.43 The image pattern created by the
Multiplication method, object point
at node (1,1)


Full Text
UNIVERSITY OF FLORIDA
3 1262 08556 7849


THE MULTIAPERTURE OPTICAL (MAO) SYSTEM
BASED ON THE APPOSITION PRINCIPLE
By
SHIH-CHAO LIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988

To my parents
Mr. Chung-Liang Lin
&
Mrs. Yang Li-Shuang Lin
ft
ft
m
m
£
ii
&
â– k
±
£
sax*

ACKNOWLEDGMENTS
The author would like to express his deepest
appreciation and gratitude to Dr. Richard T. Schneider, the
chairman of his supervisory committee, for his guidance and
support in this research and for the faith and friendship he
showed toward the author throughout this academic endeavor.
His deepest appreciation also goes to Dr. Edward E. Carroll
for his guidance, encouragement and friendship. Sincere
thanks are also extended to the other members of the
supervisory committee, Dr. William H. Ellis, Dr. Tom I-P
Shih and Dr. Gerhard Ritter.
Special note should be made of the valuable support
from Dr. Neil Weinstein. Dr. Weinstein not only gave the
author moral support but also help the author overcome his
language problem.
The author will be forever indebted to his parents, Mr.
Chung-Liang Lin and Mrs. Yang Li-Shuang Lin for their
unfailing faith and encouragement. Without them the author
would never have been able to complete all his academic
endeavors.
in

TABLE OF CONTENTS
PAGE
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT xii
CHAPTER
1 INTRODUCTION 1
1.1 Introduction 1
1.2 Vision Systems 2
1.3 Literature Survey on Multiaperture
Optical Systems 5
2 THE INSECT EYES 7
2.1 Optical Part 7
2.1.1 Cornea 9
2.1.2 Crystalline Cone 9
2.1.3 Crystalline Tract 10
2.2 Sensory Part 10
2.2.1 The Sensor 10
2.2.2 The Nerve Connection 14
2.3 Image Formation 14
3 OPTICAL STUDIES 16
3.1 Light Horn 16
3.2 Optics of Light Horns 18
3.2.1 Experimental Setup 18
3.2.2 The Ray Tracing Program 23
3.3 Results of the Optical Studies 25
IV

V
3.3.1 Vertical Cross Section 25
3.3.2 Wall Pattern 41
3.3.3 Horizontal Cross Section 55
3.4 Applications 58
3.4.1 Vertical Cross Section 58
3.4.2 Wall Pattern 65
3.5 Summary and Discussion 95
4 LIGHT COLLECTION OF THE LIGHT HORN 105
4.1 Intensity Concentration 106
4.2 Comparison of Light Horns and Lenses . . . 112
5 THE MULTIAPERTURE OPTICAL SYSTEM DESIGN .... 116
5.1 The System Design 116
5.1.1 MAO Mask 118
5.1.2 Detector Board 120
5.1.3 Memory Board and Processor Board. . 124
5.2 The Performance of the MAO Device 125
5.3 Discussions 130
6 SUMMARY AND CONCLUSIONS 132
REFERENCES 138
BIOGRAPHICAL SKETCH
140

LIST OF TABLES
Table Page
3.1 Dimensions of experimental devices 21
3.2 The distance-polar angle relation of light
horn used 54
3.3 Matrices of point sources at different location . 62
3.4 Matrix of a point source image on the 5X5
light horn array 66
vi

LIST OF FIGURES
Figure Page
2.1 The ommatidium of insect eye.
a) Photopic eye; b) Scotopic eye 8
2.2 The cross section of rhabdom
a) Photopic eye; b) Scotopic eye 12
2.3 The microvilli 13
3.1 Schematic of the MAO device eyelet 17
3.2 The experimental setup 19
3.3 Three different modes of generating the
pattern. a) Vertical cross section;
b) Wall of cylinder; c) Horizontal slice . . 22
3.4 The diagram of the law of reflection 24
3.5 Photo, vertical cross section image pattern,
parallel beam—parallel to axis 26
3.6 Two dimensional ray tracing for light horn;
parallel beam—parallel to axis 28
3.7 Photo, vertical cross section image pattern,
parallel beam—parallel to axis—yellow
side right of axis, red side left of axis. . 29
3.8 Photo, vertical cross section image pattern,
parallel beam—5 degrees off axis 31
3.9 Photo, vertical cross section image pattern,
parallel beam—11 degrees off axis 32
3.10 Computational pattern, vertical cross section
image pattern, parallel light source,
parallel to axis 34
3.11 Computational pattern, vertical cross section
image pattern, parallel light source,
5 degrees off axis 35
vi 1

vi i i
3.12 Computational pattern, vertical cross section
image pattern, parallel light source,
11 degrees off axis 37
3.13 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, parallel
to axis 38
3.14 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, 5 degrees
off axis 39
3.15 Computational pattern of the parabolic
light horn, vertical cross section image
pattern, parallel light source, 11 degrees
off axis 40
3.16 Vertical cross sections across cylinder;
a) 33 units b) 48 units c) 86 units away
from the exit aperture 42
3.17 Ray pattern impinging on cylinder wall for
object on axis 44
3.18 Photo, wall pattern, parallel light source,
parallel to axis 45
3.19 Ray pattern impinging on cylinder wall for
object one degree off axis 46
3.20 Ray pattern impinging on cylinder wall for
object two degrees off axis 47
3.21 Distance of center of bands from "Zero" band
vs. the off-axis angle 49
3.22 Angular resolution as a function of
polar angle 51
3.23 Angular resolution vs. light horn length. ... 53
3.24 The computational pattern which would appear
on the horizontal slice 56
3.25 Photo, pattern on horizontal slice 57
3.26 A 5 X 5 detector array arrangement 59
3.27 Vertical cross section pattern projected on a
5X5 detector array and its possible
matrices 60

IX
3.28 Computational pattern for multiple light horns
arranged on a 3 X 3 array 63
3.29 Computational pattern for multiple light horns
arranged on a 5 X 5 array 64
3.30 Fiber shaped detector 67
3.31 The correlation of the image information
among three adjacent eyelets 69
3.32 Detector tube with ring shape detectors .... 70
3.33 Intensity distribution on the cylinder wall;
object on the axis, 1000 units distance ... 72
3.34 Intensity distribution on the cylinder wall,
object 0.34 degree off axis 73
3.35 Intensity distribution on the cylinder wall,
object 0.52 degree off axis 74
3.36 Characteristic band position of the objects,
object 1000 units away from entrance
aperture 75
3.37 The triangular grid 76
3.38 The object space covered by the eyelet
system 77
3.39 The image pattern created by the Addition
method, object point at node (1,1) 80
3.40 The image pattern created by the Addition
method, object point at node (5,5) 81
3.41 The image pattern created by the Addition
method, object point at node (1,26) 82
3.42 The image pattern created by the Addition
method, object point at node (17,17) 83
3.43 The image pattern created by the Multiplication
method, object point at node (1,1) 86
3.44 The image pattern created by the Multiplication
method, object point at node (5,5) 87
3.45 The image pattern created by the Multiplication
method, object point at node (1,26) 88

X
3.46 The image pattern created by the Multiplication
method, object point at node (17,17) 89
3.47 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,1) 91
3.48 The image pattern created by the Cross
Correlation with Addition method,
object point at node (5,5) 92
3.49 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,26) 93
3.50 The image pattern created by the Cross
Correlation with Addition method,
object point at node (17,17) 94
3.51 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,1) 96
3.52 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (5,5) 97
3.53 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,26) 98
3.54 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (17,17) 99
3.55 The image pattern created by the Multiplication
method, object points at nodes (1,1) and
(17,17) 100
3.56 The image pattern created by the Multiplication
method, object points at nodes (1,26) and
(17,17) 101
4.1 Concentration ratio curves of conical light
horns with different cone angles for an
on-axis point source at 1000 units
distance 107
4.2 The concentration ratio of a parabolic light
horn at different distances away from
the exit aperture with an on-axis point
source at 1000 units away from entrance . . . 108

XI
4.3 Comparison of concentration ratios between
a light horn and a lens 110
4.4 The concentration ratio curves of the
off-axis point source Ill
4.5 The light collection of a lens 114
5.1 The multiaperture optical (MAO) device 117
5.2 Arrangement of detectors on optic RAM.
a) detector location; b) detector in use
(marked by "X") 121
5.3 Detector arrangement underneath the light
hole 123
5.4 Performance of the system using the mask
with cylindrical holes 126
5.5 Performance of the system using the mask
with conical holes 127
5.6 The result pattern of Figure 5.4 after
the clean-up 128
5.7 The result pattern of Figure 5.5 after
the clean-up 129

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the degree of Doctor of Philosophy
THE MULTIAPERTURE OPTICAL (MAO) SYSTEM
BASED ON THE APPOSITION PRINCIPLE
By
Shih-Chao Lin
Apri1 , 1988
Chairman: Dr. Richard T. Schneider
Major Department: Nuclear Engineering Sciences
Automation freed mankind from repeated boring labor
and/or labor requiring an instantaneous response. When
applied as robotics it could even free mankind from
dangerous labor such as handling radioactive material. For
a robot or an automated system a vision device has proven to
be an important element.
Almost all artificial vision systems are similar in
design to the human eye with its single large lens system.
In contrast, the compound eye of an insect is much smaller
than the human eye. Therefore, it is proposed to imitate
the insect eye in order to develop a small viewing device
useful in robotic design.
The basic element of the multiaperture optical system
described here is a non-imaging light horn. The optical
xi i

XI 1 1
studies on the non-imaging light horn (a simulated insect
eye eyelet) have been done and show that this device may
produce images when several horns are used together in an
array. The study also shows that with several non-imaging
devices the position of an object point light source can be
determined very easily.
One possible realization of multiaperture optical
system design based on the apposition principle is proposed
and discussed. The multiaperture optical system proposed is
a small, low cost device with digital image processing.

Chapter 1
INTRODUCTION
1.1 Introduction
Machines freed mankind from labor which required great
strength or long endurance. Automation freed mankind from
repeated boring labor and/or labor requiring instantaneous
response. Robotics finally frees mankind from dangerous
labor and from the kinds of tasks of which humans are not
capable or willing to endure because they require special¬
ized abilities, albeit, with low levels of intelligence.
Early machines needed only very primitive sensors or no
sensor at all. Automation needed more sophisticated
sensors. A photocell rather than an electrical contact may
have been used. Logically one may expect then that robotics
will require even more sophisticated sensors—one of them
being vision rather than just the detection of light.
One such vision system may be a television camera
attached to suitable digitizing equipment. The digitizing
equipment is required since the information provided by the
TV camera is not viewed by a human observer. Instead, it
has to be interfaced with specialized intelligence which has
been programmed into the main computer of the robotic
1

2
system. This constitutes the major difference between
automation and robotics. Therefore, the question arises,
what kind of vision system is easiest to interface with a
computer?
Images can be easily evaluated by a computer if they
are made available in the form of digitized picture elements
(pixels). Therefore, vision systems with digitization
ability are preferred when a computer is chosen to do the
recognition process.
There seem to be two principal preferred designs for
vision systems to be found in nature. These are (1) the
single lens eye found in vertebrates and (2) the
multiaperture eye found in arthropods (the "insect eye").
Most optical instruments have been patterned after the
vertebrate eye. This raises the question: is this design
superior to the insect eye or only considered to be so by
optical designers?
It is the objective of this dissertation to investigate
whether the insect eye is indeed generally inferior to a
single lens eye, or if there are special applications where
the insect eye may be more suitable than the single lens
eye. If the second case is true, the proof should be
offered in the form of a description of such a superior
device.
1.2 Vision Systems
The difference between the two vision systems consists
in the number of apertures used to form the image. For this

3
reason one needs to investigate the difference between
single aperture optics (SAO) and multiple aperture optics
(MAO). This dissertation is concerned with some of the
unique features of multiple aperture optics.
Therefore, let us describe the insect eyes, albeit in
somewhat oversimplified terms. In the overall MAO system
there could be three possible ways of extracting an image
from an object. These three possible ways are as follows:
1. Each eyelet collects only one pixel and the
resulting overall image is a mosaic (the so-called
apposition eye).
2. The lens of each eyelet projects a fairly large
image onto the retina and all these images are
superimposed precisely (the so-called superposition
eye) .
3. Each lens projects a small image onto the retina,
the individual images do not overlap and together
they form the total image.
For the artificial multiaperture optics system, one
cannot possibly worry about focusing of many eyelets or
adjusting them so that all of the individual images are
superimposed correctly. Therefore, the second option, the
superposition eye, is not a desirable choice. Similarly,
the focusing and the overlapping problems of the small
individual images are the reasons to reject the third
option. The first option, the apposition eye, removes the
focusing problem but raises the question of whether or not

4
acceptable resolving power can be achieved. The answer to
this question is one of the major subjects of this disser-
tation. From behavioral studies it is known that
insects have remarkably acute vision and are capable of
resolving even small, rapidly moving distant objects at low
ambient light levels. For the designer, who intends to
build a camera based on the design principles of the insect
eye, it would be very helpful to know how such "super
resolution" is achieved. In this dissertation an answer is
proposed.
In an effort to understand the optics of the insect
eye, a simple model consisting of a hollow cone with a
reflecting wall attached to a non-reflecting cylindrical
section was selected for analysis. This cone, similar to
the crystalline cone of the insect eye, is suggested as a
model for the eyelet of MAO devices. In Chapter 3 the
studies of the geometrical optics of these artificial
multiaperture optical elements will be discussed. The light
concentration of these optical elements will also be
discussed in Chapter 4.
If one were to assume that each eyelet acquires only
one pixel, one would have to conclude (using a conventional
approach) that the insect has a very poor resolution system.
The question that arises is: why would anybody want a small
camera which has poor resolution? The answer is that maybe
one does not want to take pictures which are intended to be
viewed by a human observer with this camera. Instead, it

5
could be a camera which recognizes objects and reports the
presence of the object to the main computer of a vehicle or
a robot. For example, if such a camera were the size of a
postage stamp, it could be fitted into the "hand" of a robot
and could make the task of picking up certain objects much
easier. If the recognition scheme could be hardwired
into the detector array, the restrictions on the motion of
robot would be reduced. Therefore, the recognition cannot
be too complex. Also any preprocessing by optical means
would be very beneficial.
1.3 Literature Survey on Multiaperture Optical Systems
An early insect eye model was studied by Schneider and
2
Long. They constructed an insect eye model with 100
eyelets. Each individual eyelet consisted of two lenses,
one aperture stop and an optical fiber bundle. The end of
each fiber was attached to one photosensitive detector which
was connected to an amplifier in order to obtain signals
which were strong enough for analysis. A computer was used
to study the resulting signals. A computer program
reconstructed the image pattern and the image was displayed
on a video terminal. Although it was an early model of a
multiaperture optical device, Schneider and Long were able
to conclude that the multiaperture optical system could have
inherent digitization and large field of view abilities.
They concluded that the multiaperture optical device can
have "small depth of the structure," which means a thin
device with a large field of view.

6
3
Kao has presented this first generation mechanical
insect eye in much detail. He discussed three different
models: a one-eyelet, a seven-eyelet and a 89-eyelet model.
The computer system used here was an HP-85 microcomputer
with an HP6942A multiprogrammer analog-to-digital converter.
With this system, the image was converted to a digital
pattern and analyzed. The recognition technique was
discussed in his studies.
The multiaperture optical systems in the earlier
studies were quite primitive. They were large in size and
the mechanisms were not much different from the human eye
and optical function studies were not done. Basically,
those models combined several shrunken single lens eyes into
a large array to form a semi-compound eye.
The present study builds on these earlier works. Two
insect eye models were built and studied. Computer ray
tracing programs were developed to simulate the path of
light in the insect eye. Unexpected patterns were obtained
which showed how insect eyes may produce an image. The
optics study led to the multiaperture optical device which
is discussed in Chapter 5. Finally, the studies will be
summarized and discussed in Chapter 6.

CHAPTER 2
THE INSECT EYES
Since the insect eye is used as a model for the MAO
device, it is helpful to review the anatomy of the insect
eye here briefly. According to Chapman/ most adult insects
have a pair of compound eyes bulging out, one on each side
of the head. This provides for a wide field of view,
essentially in all directions. Each compound eye has up to
10,000 eyelets which are known as ommatidia. Each
ommatidium is believed to be a non-imaging optical system.
Each ommatidium consists of approximately 30 cells, is
about tens of micrometers in diameter, and is hundreds of
micrometers in length. Functionally speaking, the
ommatidium consists of two parts (as shown in Figure 2.1):
an optical part and a sensory part. The optical part
collects the light and forms the special pattern for
recognition. The sensory part analyzes the pattern and is
capable of perceiving the image of the scene.
2.1 Optical Part
The basic optical system of the ommatidium consists of
two lenses, (1) a cornea (which is a biconvex lens) and (2)
a conical optical element. Some insects also have a
7

8
Cornea-
Crystalline
Cone
Crystalline
Tract
Rhabdom-
(a)
(b)
Figure 2.1 The onmatidium of insect eye.
a) Photopic eye; b) Scotopic eye

9
wave-guide-like crystalline tract at the end of the conical
crystalline cone.
2.1.1 Cornea
4
Chapman states that the cornea of the insect eye
consists of two corneagen cells, usually forming a biconvex
corneal lens at the outer end of the ommatidium. The lens
is transparent and colorless. It is also a cuticular
surface, often thick and solid, which can protect the soft
tissue of the insect eye.
5
According to Meyer-Rochow, the diameter of the corneal
lens of most insects falls between 25 and 35 micrometers.
Unlike the diameter, the thickness of the cornea varies
drastically from 4% up to 20% of the total length of the
ommatidium.
2.1.2 Crystalline Cone
The crystalline cone of the ommatidium usually consists
of four cells, known as the Semper cells. The cone is
transparent with a surface like a paraboloid. Hausen^
studied the optical properties of the crystalline cone. He
concluded that the cone has a length of about 42 pm. The
distal end is slightly curved. Measurements in his studies
showed that the index of refraction can be approximated as a
parabolic function. At the center axis, the index has the
highest value of about 1.50, while at the edge it has a
value of 1.383. From Snell’s Law, one can easily see that
this change in the index of refraction would cause the light
at the edge of the crystalline cone to be totally reflected

10
and not transmitted to the neighboring eyelets. This is
like the gradient-index lenses which are now being manu¬
factured. Thus, the light going into the crystalline cone
is either transmitted to the rhabdom or reflected back out.
2.1.3 Crystalline Tract
Most entomologists believe that the two main categories
of the insect eyes can be described as either the
apposition eye or the superposition eye. Goldsmith and
Bernard^' believed that the more suitable names for
these classifications are photopic and scotopic eyes,
respectively. As discussed by them, the scotopic (super¬
position or clear zone) eyes are capable of adaptation to
variations in light intensity. Therefore, the scotopic eye
is also known as a light-adapted eye.
The crystalline tract occurs only in the scotopic eyes.
It is located between the crystalline cone and the rhabdom.
The optical function of the tract is like that of a wave¬
guide. It has an important role on the adaptations of the
scotopic eye to the light.
2.2 Sensory Part
2.2.1 The Sensor
Goldsmith and Bernard7, reported that for the
photopic eyes, there is a hose-like structure attached at
the exit of the crystalline cone. This attached structure
is the sensory element and is called the rhabdom. The
scotopic eye is similar to the photopic eye with the

11
exception that there is a wave-guide, the crystalline tract,
positioned between the cone and the rhabdom.
The rhabdom consists of retinula cells. Normally there
are seven or eight retinula cells in an ommatidium. Near
the ommatidial axis, the retinula cells are differentiated
to form the rhabdomeres. Therefore, most of the eyelets
contain seven or eight rhabdomeres. Hence, the sensory
part of the ommatidium is called the rhabdom. A cross
section through the rhabdom of the photopic eye is shown in
Figure 2.2.a and the cross section of the scotopic eye is
shown in Figure 2.2.b. For the scotopic eyes, pigment is
located close to the exit of crystalline cone at low
intensity levels and moved halfway down the crystalline
7
tract at higher light levels. Goldsmith and Bernard also
state that the pigment granules within retinular cells 1 to
6 (see Figure 2.2.b) migrate laterally to the rhabdomeres,
when in the light-adapted state, but the pigment granules do
not migrate within the two central cells.
In the rhabdomere the light sensitive elements are the
microvilli—these are tiny tubes having typically a diameter
7
of less than one micrometer. Layers of these tubes are
oriented in an alternating crossing pattern, more or less
perpendicular to the longitudinal axis of the rhabdomere as
indicated in Figure 2.3. Mazokhin-Porshnyakov* concluded
that the visual pigments are disposed on the surface of the
tubules.

Figure 2.2 The cross section of rhabdom. a) Photopic eye
b) Scotopic eye.

13
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Figure 2.3 The microvilli

14
2.2.2 The Nerve Connection
As it was seen in Figure 2.2, six rhabdomeres are
arranged around a seventh (and possibly an eighth) at the
7
center. According to Goldsmith and Bernard, each retinular
cell has one nucleus and one axon. The six outer retinular
cells synapsed in a single cartridge, and in the case of
the scotopic eye these six cells do migrate laterally in the
light-adapted state. The one or two central cells have
different connections than the surrounding six cells.
2.3 Image Formation
The mosaic theory, which is believed to be the theory
g
governing insect vision, was proposed by Müller in 1826.
The mosaic theory assumed that each eyelet of the insect eye
is only capable of a limited field of view which does not
overlap with the field of the adjacent eyelets which make up
the compound eyes. Each ommatidium contributes only one
point out of the total image pattern. Hence, under the
mosaic theory, one eyelet of the facet eye is a non-imaging
optical device.
As mentioned earlier, the rhabdom consists of seven or
9
eight rhabdomeres as the sensing elements. Kuiper observed
the rhabdomeres by illuminating the rhabdom of Apis
mel1ifera. He found that only the individual rhabdomeres
were illuminated, but not the rhabdom as a whole unit.
Furthermore, the arrangement of the rhabdomeres in the
rhabdom is in a symmetrical, radial pattern. This indicates
that each eyelet of an apposition eye could contribute more

15
than just one pixel to the total image although this
ommatidium could still be a non-imaging optical element.

CHAPTER 3
OPTICAL STUDIES
3.1 Light Horn
In order to understand the function of the ommatidium
as an optical element, models were built to simulate it.
Like the ommatidium, each model (seen in Figure 3.1)
consists of a light horn (a hollow cone with reflecting
walls)—simulating the crystalline cone—attached to a
non-reflecting cylindrical section—simulating the rhabdom
of the insect eye.
As mentioned in Chapter 2, the crystalline cone of the
insect is transparent but has a refractive index larger than
the surrounding medium which causes the light entering the
crystalline cone to be internally totally reflected. In the
models, total reflection was replaced by regular (specular)
reflection. The simpler of the two designs, the cone with
reflecting walls, was selected in this study because it can
be manufactured more easily.
Light horn studies were started long ago. Initially,
they caught the attention of optical scientists because
light horns seem to circumvent the Second Law of Thermo-
16

Figure 3.1 Schematic of the MAO device eyelet

18
dynamics. Obviously, one would think that by making the
exit aperture small enough, one should be able to obtain an
illumination density larger than the intensity of the light
source. However this is not the case, as will become clear
in the analysis presented below.
In the modern age, the light horn was studied as a
possible concentrator for solar energy. Williamson^
commented that although the light horn can be designed to
transmit images, its principal usefulness is found in the
transmission of the maximum amount of energy rather than the
possible image forming potential of the light horn. Welford
and Winston^ concluded that the "ideal concentrator" would
have walls shaped like a rotation parabola, similar to the
conical wall of an insect eye.
3.2 Optics of Light Horns
In the present dissertation the emphasis is on the
image formation rather than on the energy concentration
phenomena. Light horn optics were studied in two parts:
experimentally and theoretically (ray tracing).
3.2.1 Experimental Setup
Figure 3.2 shows the experimental setup. It consists
of a source of parallel light, a photographic shutter, a
light horn and a film holder (or camera). In contrast to
normal picture taking techniques, there is no lens between
the light source and the image.
In the experimental setup two different light horns
were used, one with a large cone angle and the other with a

X
Light Source
Figure 3.2 The experimental setup

20
small cone angle. The dimensions of these light horns are
shown in Table 3.1. Since only the optical properties are
of interest here, all the dimensions are quoted in units
relative to the radius of the entrance aperture (i.e., one
unit represents the entrance aperture radius of the light
horn). (Note: For different size light horns, dimensions
will all scale linearly.)
Because of the difficulty in manufacturing a parabolic
light horn, only conical light horns were used in the
experiment (However, the parabolic light horn was studied by
ray tracing).
The patterns generated could be observed in three
different modes (see Figure 3.3) depending on where the
camera (detector) was placed. The images were:
(1) on vertical cross sections of the cylinder (at
various distances from the exit of the light horn),
(2) on the walls of the cylinder,
(3) on horizontal slices through the cylinder (at
various elevations).
For the vertical cross sections, the film is held perpen¬
dicular to the axis of the light horn. For the wall pattern
the film can be wrapped around the cylinder or the image
pattern can be observed directly on the interior surface of
the cylinder. In this dissertation, the wall pattern is
determined by taking pictures of the side of the frosted
glass cylindrical tube. Photos of these three positions of
the camera were used to support the results of ray tracing.

21
Table 3.1
Dimensions of experimental devices
Entrance Aperture Radius:
Exit Aperture Radius:
Light Horn Length:
Cylinder Radius:
Cylinder Length:
Device 1
52.83 mm
11.18 mm
154.94 mm
Device 2
9.18 mm
7.35 mm
174.50 mm
9.50 mm
1219.20 mm

Figure 3.3 Three different modes of generating the pattern.
a) Vertical cross section; b) Wall of cylinder;
c) Horizontal slice

23
3.2.2 The Ray Tracing Program
For light horns of the sizes that are of interest here,
diffraction effects play only a minor role and therefore
this study was only concerned with geometrical optics. The
inner surface of the light horn is assumed to be a totally
reflective surface and to obey the law of reflection. For
computer ray tracing, it is useful to write a program based
on the vector form (by components) of this law of
reflection. Welford and Winston** gave the vector equation
of the law of reflection as
r = r. — 2(ii-£.)n (3.1)
r i i
where r. is the unit vector of the incident ray, ?r is the
unit vector of the reflected ray and ii is the normal vector
of the surface (see Figure 3.4).
When Welford and Winston** studied the non-imaging
concentrator, they indicated that some rays were returned
back out through the entrance aperture if the incident rays
have too large an angle with the optical axis. For large
cone opening angle light horn, not all of the incident rays
passed through the exit aperture of the light horn and only
part of the light beam contributed to the final pattern.
Therefore for a viewer observing in front of the light horn
entrance aperture, although there was no light source behind
the light horn, he who can still observe the shining
reflection. Mazokhin-Porshnyakov* mentioned that on the
surfaces of many insect eyes a "wandering" spot is found.

24
Figure 3.4 The diagram of the law of reflection

25
This "wandering" spot is a black spot with a shining
background and its location changes with different direction
of observation. It is so-called the "pseudopupil." The
partly reflected and partly passed through of the light
beams could be the reason why the pseudopupil is found in
the insect eyes.
The vector method can be used to calculate the
trajectories of the light beams and the resulting image on
the film or the detectors. The study was restricted to the
following conditions: 1. The light horn was of conical
shape and hollow. 2. The exit aperture of the light horn
has a smaller radius than the entrance aperture.
3.3 Results of the Optical Studies
3.3.1 Vertical Cross Section
When a parallel incoherent light source was mounted
paraxially with the axis of the light horn, an interesting
pattern was found on the image side (shown in Figure 3.5).
The image pattern in this case is a set of concentric rings
around a central disk. It was unexpected that these
concentric rings should have very sharp edges and that there
is no light between the ringsAs will be shown below,
these concentric rings are not a diffraction pattern but can
be explained with geometrical optics alone. Nevertheless,
it is most astonishing that an empty cone should produce
such a sharp structure.
In order to understand the reason why the parallel beam
could construct this sharp image-like structure, a two

26
Figure 3.5 Photo, vertical cross section image pattern,
parallel beam—parallel to axis

27
dimensional ray tracing is shown in Figure 3.6. It shows an
individual light horn and four mirror images. Assume a
parallel light beam (or a point light source at infinity)
enters the light horn parallel to the optical axis. The
rays bordered by lines 1-1’ is the non-reflected light beam
which forms the central disk of the image pattern. The
light beam bordered by lines 1-2, is reflected once and
projected to the opposite side of the axis of the light
horn. The light bordered by lines 1-2 and l’-2’ forms an
annulus which constitutes the first bright ring. Similarly,
the rays between 2-3 and 2’-3’ form the second bright ring.
This is why there is a sharp edge and a dark space between
the center disk and the first ring.
To prove that this interpretation is correct, a color
filter was added to the white light source. It consisted of
two halves, a yellow and a red one, whereby the dividing
line was located on a diameter of the light horn aperture.
The result is shown in Figure 3.7. Although, in this
dissertation the color is not shown, the grey tone of this
picture still indicates the result. As predicted from the
interpretation above, the center disk is divided into a
yellow half (right half) and a red half (left half), the
dividing line being the diameter of the center disk. The
first ring is also divided into a yellow half and a red half
along the same dividing line. In contrast to the center
disk, the right side of the first ring is red. For the

28
Figure 3.
6 Two dimensional ray tracing for light horn;
parallel beam--parallel to axis

29
Figure 3.7 Photo, vertical cross section image pattern,
parallel beam—parallel to axis—yellow side
right of axis, red side left of axis

30
second ring, the colors are reversed again, yellow to the
right and red to the left.
To study the effect when the light source is not on the
axis of the light horn, the parallel light source was made
to have 5 degrees with the axis of the light horn. Figure
3.8 shows the image in this case. The central disk is
displaced slightly compared to the first case (Figure 3.5).
The rings of this case split into twin pairs and they are no
longer perfect rings. There is a sharp cutout which
occurred on the brighter part of the twins. It is believed
that this cutout was reflected to the opposite side to form
the "twin." Figure 3.9 shows the case when the light source
is moved still further off-axis (11 degrees). The central
disk almost disappears. The first ring and its twin turn
into crescents and appear on the same side. The central
disk disappears when the angle is larger than 12 degrees.
While these results are certainly unexpected and
interesting, one can also draw some practical conclusions
from them. If one placed one detector at the center of the
light horn and one at the edge of the first ring, the
combination of the illumination on the detectors could be
used to detect the off-axis angle of the point source at
infinity, at least for 0, 5 and 12 degrees. For example, if
both detectors show strong illumination, the object must be
off-axis by less than 5 degrees. If just the center
detector detects the brightness of the source and the
detector on the wedge does not detect the source, then the

31
Figure 3.8 Photo, vertical cross section image pattern,
parallel beam—5 degrees off axis

32
Figure 3.9 Photo, vertical cross section image pattern,
parallel beam—11 degrees off axis

33
object is in between 5 and 12 degrees. If both detectors
see nothing, then this shows that the source is off axis by
more than 12 degrees.
Therefore, although the light horn is a non-imaging
optical device, it is still capable of more than just simply
detecting the presence or absence of an object within it’s
field of view (FOV).
The above study was done with a light horn of certain
specific dimensions (light horn 1, see Table 3.1). To make
sure that a generic effect was discovered, a second light
horn with different dimensions was used. Similar results
were obtained. The only differences between results of the
two light horns are the size and the number of rings. The
number of rings depends on the length of the light horn and
the light horn opening angle. In general, only the
unreflected light rays and singly reflected rays are
important; the doubly reflected rays appear only at large
polar angles where poor resolution destroys their
usefulness. The relationship between the maximum cone
length and the cone angle opening resulting in only one ring
and the central disk is shown as equation 2.
L = A. tan(2a) cot(a)/(tan(a) + tan(2a)) (3.2)
Figure 3.10 shows the results of the ray tracing for
light which is parallel to the axis of the light horn. As
can be seen, the obtained pattern agrees very nicely with
the photograph (Figure 3.5). Figure 3.11 shows the case of

34
Figure 3.10 Computational pattern, vertical cross
section image pattern, parallel light
source, parallel to axis

35
Figure 3.11 Computational pattern, vertical cross
section image pattern, parallel light
source, 5 degrees off axis

36
parallel light 5 degrees off axis. Again it agrees with the
photograph (Figure 3.8), even the wedge in the first ring
shows up exactly the same. Figure 3.12 (corresponding to
the photograph, Figure 3.9) shows the case which is 11
degrees off axis. Notice the similarity with the experi¬
mental result; i.e., the central disk almost disappears.
The parabolic light horn was also studied mathe¬
matically to examine the vertical cross section pattern.
The ray tracing result of the light source in front of the
light horn at the axis with the vertical cross section taken
at three units down the cylinder is shown in Figure 3.13.
Similar to the cone-shaped light horn (Figure 3.5), the
image of parabolic light horn consists of a central
disk and one ring-like annulus. The difference of these two
(Figures 3.10 and 3.13) is the cone shape light horn could
have more than one rings but the parabolic light horn
has only one ring-like annulus. In both cases the ring or
the ring-like annulus all have a sharp edge at the inner
boundary. When the light source is 5 degrees off axis
(Figure 3.14), similar to Figure 3.11, the center disk moved
aside and the ring-like annulus changed its shape. For the
light source located at 11 degree of axis, the image pattern
is shown in Figure 3.15. Again it is similar to Figure
3.12. This indicates that the parabolic light horn and the
cone-shaped light horn have similar properties.
In the cases discussed above, the patterns were taken
at a fixed location, the outlet of the light horn. The

37
Figure 3.12 Computational pattern, vertical cross
section image pattern, parallel light
source, 11 degrees off axis

38
• • •&' ‘'I»'; •
■•■i i:::.
•••.> Tjr
’.V. i*»‘ ■ ■ •
Figure 3.13 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, parallel to axis

39
Figure 3.14 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, 5 degrees off axis

40
Figure 3.15 Computational pattern of the parabolic light
horn, vertical cross section image pattern,
parallel light source, 11 degrees off axis

41
question arises: what if the location of the sensors was
moved as in the scotopic eye? Figure 3.16 shows typical
patterns of the number 1 light horn (see Table 3.1) at three
different vertical cross sections along the length of the
cylinder when the object point is at infinity on the axis.
The image is approximately in focus in the case shown in
Figure 3.16.(a), which is 33 units down the cylinder from
the exit aperture. (Again, one unit is the radius of the
entrance aperture.) Larger and larger rings are produced
further down the axis (e.g., Figure 3.16.(b) is at 48 units)
until the reflected rays begin to impinge on the cylinder
wall as shown in Figure 3.16.(c), at 86 units. From Figure
3.16, one can draw the conclusion that when the sensor
location is moved further away from the light horn, the size
of the ring becomes larger and larger, but the size of the
central disk stays almost the same. It is also found that
the intensity of the light on the rings is higher when the
film location moves closer to the light horn. This might be
a reason why the pigment granules of the scotopic eye move
closer to the crystalline cone and migrate to the center
under conditions of darkness.
3.3.2 Wall Pattern
When the light beam passes through the light horn, it
projects an image on the wall of the attached cylinder. The
image pattern of this case is a three dimensional pattern.
To simplify the analysis one may imagine that the cylinder

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•ill'
N>
(a) (b) (c)
Figure 3.16 Vertical cross sections across cylinder;
a) 33 units b) 48 units c) 86 units away
from the exit aperture.

43
has been cut along the top and then unrolled and flattened.
The radius of the cylinder is taken to be 1.0 unit.
The wall pattern of an on-axis object point source at
infinity appears to be a uniform band around the cylinder
wall superimposed on a weak and uniform background. Figure
3.17 shows the computational results in this pattern of rays
impinging on the wall. The uniform background, produced by
the unreflected rays, is not shown on this plot. Here the
abscissa represents the distance down the cylinder while the
ordinate represents the angle around the axis. Figure 3.18
is the luminous wall pattern of the light source observed
down the cylinder. To have the image shown clearly on the
cylinder wall, a frosted glass tube was attached to the exit
aperture of the light horn, the simulate cylindrical part of
the non-imaging device. (The center of the pattern is on
the bottom of the tube and the photograph was taken from the
side.) In this case, the tube used to take the photos of
the wall pattern was too long to remain complete rigid and
it bent at the non-supported part where the band should
occur. Therefore the photo showed only part of the band.
The gap in the band is caused by the laser. Exclude the
nonuniform phenomena caused by the material and laser beam,
both the experimental photo and the computational result
showed that they agreed with each other.
The wall pattern from the computational results for an
object point one degree off the optical axis is shown in
Figure 3.19. A somewhat distorted band is found between 34

cylinder
44
Wk
>ÁwV
%
%
.W.'.v
Ü
,í^<-
m
m
■•■000:
Hi
i i i i i i i i i i i i i i i i
o
Distance from light horn exit
(10 units per scale)
I
Figure 3.17 Ray pattern impinging on cylinder wall
for object on axis

45
Figure 3.18
Photo, wall pattern, parallel light source,
parallel to axis

cylinder
46
PC
CvJ
I
I
0
Distance from light horn exit
(10 units per scale)
Ray pattern impinging on cylinder wall for
object one degree off axis
Figure 3.19

47
and 47 units down the cylinder, and the unreflected rays
form a more intense background, producing the horizontal
elliptical pattern at the center. Although the band of this
case is no longer uniformly distributed, the band location
and the band width were still clearly shown. Figure 3.20
is the wall pattern for an object point two degrees off the
optical axis.
Comparing Figures 3.19 and 3.20, one can clearly see
that when the object point moves off-axis, the band location
moves closer to the light horn exit aperture. This
indicates that the distance of the band could be a measure
of the polar angle of the object point. Of course this
measurement has to be taken from a reference point. The
band caused by an on-axis object, as shown in Figure 3.17,
could be the reference or "zero" band. The width of the
bands produced on the cylinder wall is a measure of the
polar angular resolution of the device, while the distance
of a certain observed band from the "zero" band is a measure
of the polar angle of a certain object point. Figure 3.21
plots the distance of the center of various bands from the
"zero" band as a function of off-axis polar angle for
several different light cones. The center point (Bc) and
the width (B^) of the "zero" band can be found by the
following equations (symbols as in Figure 3.1):
B =
c
[ (Dc + Ag/2 + A./2) cot (2a + 7.) - L ]/2
(3.3)

48
0 Distance from light horn exit
(10 units per scale)
Figure 3.20 Ray pattern impinging on cylinder wall
for object two degrees off axis

49
Figure
Distance of center of 1
vs. the off-axis angle

50
Bf = [L - (A. - Ag) cot (2a + y.) / 2 ] / 2 (3.4)
where D is the diameter of the cylinder and 7. is the angle
between the incident ray and the axis of the light horn.
The distance, D, , between the "zero" band and an off-axis
o
angle band (one reflection only) would be
Db = (Dc + A./2 + Ag/2) (cot (2a) - cot(2a + 7j)/2. (3.5)
In each case, the radius of the entrance aperture of the
light horn is 1.0 unit.
If two object points are present at different polar
angles, they may be distinguished if their respective bands
do not overlap much. The band width, B^, may be thus
converted to polar angles and the angular resolution plotted
as a function of polar angle as in Figure 3.22. The value
plotted is the full-width of the bands converted to a polar
angle, in radians. The angular resolution improves as the
light horn exit aperture approaches the size of the entrance
aperture. The fraction (F) of the total light incident on
the horn which is reflected to form bands is
F = (Rq2 - Rj2) / Rfl2 (3.6)
where R^ is the entrance aperture radius, and Rj is the exit
aperture radius. Hence, the resolution improves at the
expense of the efficiency as one might expect.

51
Figure 3.22 Angular resolution as a function of
polar angle

52
The unreflected light patterns (for example, the
central elliptical patterns in Figures 3.19 and 3.20) could
conceivably be used to distinguish objects at different
azimuthal angles. However, for the geometries studied here
the azimuthal angular resolution would be very poor. More
precise azimuthal angular information can be obtained by
cross correlation between several eyelets. This will be
discussed later.
As far as the polar angular resolution is concerned it
is of interest to see how it varies with the length of the
light horn when the entrance and exit apertures are kept
constant. Figure 3.23 shows the results for an entrance
aperture radius of 1.0 unit, an exit aperture radius of 0.8
unit, cylinder radius of 0.8 unit, and various horn lengths
from 5.0 to 60.0 units. The comparison is made for an
object point one degree off axis. Optimum resolution
appears at a length of about 20.0 units.
The polar angular resolution of the device using wall
patterns is thus limited by (1) the width of the bands, (2)
the presence of background from the unreflected rays, and
(3) the distortion of the bands for off-axis object points.
Nevertheless, a number of angular bands may be
distinguished. For example, using light horn device 2, the
data indicate that this device might be fitted with sensor
rings inside the cylinder as shown in the Table 3.2.
If this horn-cylinder combination were used as a simple
collimator, its light acceptance would be characterized by

53
Figure 3.23 Angular resolution vs. light horn length

54
Table 3.2 The distance-polar angle relation of light
horn used
Distance from light horn Polar angle range
(units of entrance radius) (mi11iradians)
80.
to
72.
0.0
to
1.5
72.
to
61.
1.5
to
4.5
61.
to
49.
4.5
to
8.0
49.
to
42.
8.0
to
13.5
42.
to
29.
13.5
to
24.0
29.
to
15.
24.0
to
50.0

55
the half-angle of the 1 x 19 unit collimator, i.e. about
0.05 radian. Thus, use of the reflection bands on the
cylinder wall enables finer resolution and possible imaging
within a single collimator tube. If the light horn with an
exit aperture of 0.95 unit were used, about 20 distinguish¬
able angular ranges would be obtained. However, the
reflecting surface area would drop from 36 percent of the
total entrance aperture to about 10 percent and the length
of the cylinder would need to be increased from about 150
units to over 300 units.
3.3.3 Horizontal Cross Section
As we have seen, both the vertical cross section and
the wall pattern could be used to improve the resolution of
the non-imaging optical device. It is interesting to know
what the horizontal cross section (seen as in Figure 3.3.c)
pattern would be and whether it could be used to improve
resolution or not. Such a pattern was computed for the
plane located 1.0 unit above the optical axis and is shown
in Figure 3.24 (the object point was on axis). In this case
the pattern appears as a parabolic band on a uniform
background. Figure 3.25 shows the photograph pattern
corresponding to the calculation shown in Figure 3.24 where
the observed fine structure is caused by the laser and
should be considered as an artifact. Of course the
photographs show more detail than the ray tracings predict,
which is to be expected since multiple reflections and
diffraction effects were ignored. Although the parabolic

one unit
Figure 3.24 The computational pattern which would appear on
the horizontal slice

57
Figure 3.25 Photo, pattern on horizontal slice

58
band can be used (similar to the wall pattern band) to
define the polar angle, the location of the parabolic band
changes too dramatically as the elevation of the plane
varies. Therefore, it is suggested not to use this
information, since the wall pattern can do the job nicely
already.
3.4 Applications
3.4,1 Vertical Cross Section
As discussed in Section 3.3.1, the vertical cross
sectional pattern of a point (or parallel) light source
produced by light horn is a disk and several rings. If this
pattern is projected on a 5 X 5 detector array, as shown in
Figure 3.26, the number of rays falling on each detector,
in connection with a threshold setting, can be used to
identify the location of the object point. Figure 3.27
(a,b,c) shows the overlapping pattern of the 0, 5 and 11
degrees cases, respectively. The detector responses can be
used to analyze the edge of an object boundary. If the
detector response is set to be "1" when the number of rays
is larger than the threshold setting and to be "0" when it
is less than the threshold setting, the response of the
detector array is used to construct the matrix which
indicates the location of the point object (or edge of an
object). Figure 3.27 (d,e,f) includes the possible matrices
that correspond to Figure 3.27 (a,b,c) respectively.
For this application the detector array does not
necessarily have to be exactly a 5 X 5 array; more detectors

59
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡ â–¡
â–¡ â–¡ â–¡ â–¡ â–¡
Figure 3.26 A 5 X 5 detector array arrangement

•m m
•••** ____ 1 Mj,
□"□’..GK-crq
.
â–¡3
•mj
□ ■¡pspsTQ' e):' .■
0 □VQ'.DkÚ
on
o
a.
b.
0 1110
11111
11111
11111
0 1110
(0 degree)
10 10 0
0 1111
0 110 0
0 1111
10 10 0
(5 degrees)
10 111
0 1111
0 10 0 0
0 1111
10 111
f.
(11 degrees)
Figure 3.27 Vertical cross section pattern projected on a
5X5 detector array and its possible matrices

61
would give better resolution. As discussed earlier, the
higher order rings are produced by several reflections. The
intensities of the higher order rings are not as large as
the intensity of the first ring and therefore might not be
sufficient for detection. Consequently, it is suggested
that the length of the light horn be chosen so that only one
ring and the disk be produced in the zero degree case. In
this case a 3 X 3 detector array or, (as in the rhabdomeres
of the insect eye), six on a circle and one or two at the
center is sufficient to resolve a relatively small polar
angle difference especially when the field of view is made
small. Also, the depth of the device could be held to a
minimum when the number of rings is minimal. By doing so,
not only the size of device could be reduced but also the
amount of information to be analyzed is reduced. Table 3.3
lists the matrices of a point source at different locations
which could be used to determine the polar angle of the
object point.
Having the results of the single light horn optical
study available, it is desirable to expand the study to
multiple light horns. Figure 3.28 shows the resulting
computational pattern of a centered object point in front of
9 light horns arranged on a 3 by 3 array. In this case, the
displacement of the light horn apices were set to be four
units (two times the entrance diameter). Figure 3.29 shows
the pattern of 25 light horns. A 3 by 3 detector array is
placed behind each light horn, the resulting matrix is shown

62
Table 3.3 Matrices of point sources at
different location
0
10
20
30
60
90
1
1
1
1
1
1
0
1
0
1
0
1
1
0
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
1
1
1
0
1
1
1
1
0
1
0
0
0
10
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
1
1
0
0
0
1
0
0
0
20
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
1
0
1
1
0
1
0
0
0
0
1
0
0
0
30
0
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
1
0
0
1
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
60
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
0
1
1
0
1
1
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
90
0
1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
(Note
The
point
sources
is
at
1000
units
away 1
from
the
1ight horn.)

63
Figure 3.28 Computational pattern for multiple light
horns arranged on a 3 X 3 array

64
Figure 3.29
Computational for multiple light horns
arranged on a 5 X 5 array

65
in Table 3.4. It can be seen in Figure 3.27 that some of
the rays strike the detector array of the neighboring light
horns if the off-axis angle is not equal to zero. Thus a
cylinder-like divider, similar to the pigment cells of
scotopic compound eye, is suggested to prevent the
overlapping of the image patterns of neighboring light
horns.
3.4,2 Wall pattern
As described earlier, the light sensitive elements of
the insect eye are the layers of the microvilli which divide
the cylindrical shaped rhabdom into a multitude of pixels;
albeit in the direction of the cylinder axis rather than
perpendicular to the axis as one would expect for a focal
plane array. The microvi11i-contained rhabdom is a long
cylinder rather than just the couple layers of microvilli
shown in Figure 2.3. It suggests a possible application of
the wall patterns. The fiber shaped detectors which was
12
designed by Schneider (shown as Figure 3.30) could be a
detector for this application.
Based on the studies of the wall pattern, the position
of the singly reflected band can be used as a measure of the
off-axis polar angle for object points in the field of view
of the light horn. However, it seems unlikely that more
information than just the polar angle can be determined with
a single horn-cylinder combination. At any specific polar
angle, the patterns are quite insensitive to the azimuthal

66
Table 3.4
Matrix of <
light horn
a point
array.
; source
image
on the
1
0
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
1
0
0
1
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
0
0
1
1
1
1
1
0
1
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
0
0
0
1
0
0
0
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
1
1
0
1
1
1
0
0
0
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
0
0
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
0
0
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
0
0
0
1
1
1
0
1
1
0
0
0
0
0
1
1
0
1
1
1
0
0
0
0
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
0
1
1
0
1
1
1
1
1
0
1
1
0
0
0
1
1
0
1
1
1
1
1
0
0
1
0
0
1
1
1
1
1
0
1
0
0
1
1
1
0
0
1
0
1
0
1
0
0
1
1
1
0
0
0
0
0
1
1
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
1
0
0
0
1

67
Signal Wire
Detector
Figure 3.30 Fiber shaped detector

68
angle. The construction of an image would require at least
three adjacent horn-cylinder combinations with overlapping
fields of view. Figure 3.31 shows the two dimensional area
which is defined by overlapping the fields of view of three
adjacent light horns. With a honeycomb shaped arrangement
the light horns could cover all of object-space.
If ring shaped detectors could be made and arranged to
form a detector tube as shown in Figure 3.32, these
detectors would not only detect the presence of light but
would act as number of single channel analyzers. An
individual single channel analyzer would measure the
intensity of light which strikes on the inner surface of the
ring shaped detector. The total output of the detector tube
is then an intensity distribution along the length of the
wall of the cylinder. The intensity distributions (of the
three adjacent eyelets) may be correlated to reconstruct the
image perceived by the three eyelet system.
Since a detector cannot distinguish the singly
reflected beam from other light beams, all of the light
beams—instead of just the singly reflected beams—would
contribute to the intensity distribution of the wall
pattern. However, the length of the horn element may be
chosen to eliminate all rays reflected more than n times,
where n is any arbitrary integer, n = 0, 1, 2, . . .
Through ray tracing, the intensity distributions (along
the length of the cylinder wall) at different polar angles
were created and several of them are shown in Figures

Object area seen by
Figure 3.31 The correlation of the image information
among three adjacent eyelets

Figure 3.32 Detector tube with ring shape detectors

71
3.33-35. (Note: The light horns used here are 20 units long
with a one degree cone angle. The object plane is assumed
to be 1000 units away from the entrance aperture of these
horn-cylinder combinations.) Among these intensity
distributions, it is found that the maximum is located at
different longitudinal positions for different polar angles.
Similar to the position of the singly reflected rays, the
location of the peak depends also on the off-axis polar
angle. This relation between the peak positions and the
polar angles of the objects is shown in Figure 3.36. The
intensity at this characteristic position, the position
where the peak is located, indicates the possible
contribution to the image pattern from the particular polar
angle. Therefore this characteristic position is also used
to construct the image.
In order to simplify the analysis, both the object and
image spaces were put on the identical triangular grid, each
sampled in identical two-dimensional arrays with the node
point arrangement as shown in the Figure 3.37. Although
three eyelets could be arranged to see an area larger than
the triangular area formed by the axes of the three eyelets,
for simplicity the three eyelets were arranged to have
overlapping fields of view only to such a degree to just
cover this triangular area. Any object area other than that
covered by this triangle may be covered by other eyelets in
the array (see Figure 3.38). Hence the image space is also
limited to this triangular area.

Intensity
Figure 3.33 Intensity distribution on the cylinder wall;
object on the axis, 1000 units distance

Intensity
0.200
0.100-
0.000 —
0
Figure
Distance from exit aperture (units)
H
50
.34 Intensity distribution on the cylinder wall;
object 0.34° off axis.
u>

Intensity
Figure 3,35 Intensity distribution on the cylinder wall;
object 0,52° off axis.

Band position on the cylinder (units)
Figure 3.36 Characteristic band position of the objects,
objects 1000 units from entrance aperture.

76
(51,1)
(17,17)
(1,26)
Figure 3.37 The triangular grid

77
Figure 3.38 The object space covered by the eyelet
system.

78
A source point in object space produces illuminated
pattern on the detection surfaces of the three horn-cylinder
combinations. Since the point sources at any location on
the ring centered at the optical axis have only one unique
polar angle resulting only one set of illuminated pattern.
It is not possible to locate the position of the object
point directly since its relation to the polar angle is
ambiguous. When constructing an image from these detector
patterns, an object point (i,j), of intensity I.. is mapped
bj
into an irregular area (point spread function) around image
point (i, j) . This point spread function is not spatially
invariant (isoplanatic). However, one can pick an image
point (i,j) and calculate the probability that the tube
patterns are caused by a source point at the corresponding
object point (i,j). The resulting probability distribution
on the image space is then an "image" of the point (i,j) of
the object space.
Four methods were used to construct the image patterns:
Addition, Multiplication, Cross Correlation with Addition
(CCA) and Cross Correlation with Multiplication (CCM). All
of them use the same grid system and the definition of these
operations are discussed below.
For a known object point the intensity distributions on
1 2 3
the three eyelets, f , f and f , can be easily found,
because the polar angle to the three eyelets can be calcu¬
lated. To construct the image pattern, all the nodes on the
triangle were scanned. One can then search for the

79
characteristic position of the image point by checking the
intensity distribution along the detector tube in order to
determine the contribution of light from the object point to
this particular point. Therefore, when tracing the point
source on the image space, this intensity is placed on the
particular ring which is related to this particular polar
angle. Thus, for every image node point (i,j) the relative
1 2 3
polar angles, 0 . 0 . 0 . and the characteristic
1 •J bj 1»J
1 2 3
positions, Lc> L £, Lc» to the light horns were found.
The intensity values—fn(L,") where n stands for 1, 2 or 3—
at the characteristic positions were extracted from the
distribution curves. Since the intensity on the node point
stems from various eyelets, it is reasonable to apply a
superposition law to the intensities and therefore a
reasonable way to reconstruct the image has been found. The
sum of the intensities at the characteristic positions,
fn(L,”), from the distributions of all three eyelets were
assigned to the node and then the intensity of the image
point (i,j) could be expressed as
IiBa?e= f1 ( L 1)
i.J c'
+ f2
+ f3(L;).
(3.7)
This Addition method was used to construct the
intensity patterns created by projection of all the nodes
into the object space and these patterns can be used as a
template to identify the position of the objects. Figures
3.39-42 are four intensity patterns which represent the

80
POINT SOURCE AT NODE (1,1)
Addition method
Figure 3.39 The image pattern created by the Addition
method, object point at node (1,1)

81
POINT SOURCE AT NODE (5,5)
>1.06 = >1.49
>1.69
>1.90
=2.12
Addition method
Figure 3.40
The image pattern created by the Addition
method, object point at node (5,5)

82
POINT SOURCE AT NODE í 1 , 26 )
>1.03 = >1.44
>1.64
>1.85
=2.05
Addition method
Figure 3.41 The image pattern created by the Addition
method, object point at node (1,26)

83
POINT SOURCE AT NODE < 17 # 17 )
>0,78 Ü >1,09 if >1,24
>1.40
=1.55
Addition method
Figure 3.42 The image pattern created by the Addition
method, object point at node (17,17)

84
possible image of an object point. Due to the fact that it
is impractical to indicate the exact value of the intensity
at the various nodes on the figures, these intensities are
shown in ranges which are characterized by different
patterns. Only the intensity values larger than half of the
maximum value are shown on the figures. The intensity
values of less than half of the maximum do not make a
significant contribution to the total image. Besides, the
position of the intensity value of half the maximum
indicates the resolution of the system on that particular
object point. From these patterns it can be seen that the
highest intensity value occurs as an area which includes the
position of the object point. Thus one can conclude that
the Addition method gives good results for identifying the
location of the object point. However, one must remember
that the intensity distribution is not a delta function,
but rather a point spread function with a finite area.
Hence, the relationship between Ln and 0? . is only
c 1,J
appoximate. The detector intensity fn represents only the
probabi1ity that there is a source point at the object
point (i,j).
Although the Addition method generated a maximum at the
location of the object point, the intensity differences
between the point and its neighbors were found to be small
or even zero and therefore indistinguishable. Source points
closer together than about one third of the side of the
triangle would probably not be distinguishable. Therefore,

85
the Multiplication method is also suggested. The Multipli¬
cation method finds the possible intensities at the node
point from all three intensity distributions then calculates
the product of those three values and assigns this product
value to the node point and this could be expressed as
I1”3?6 = f!(L 1) f2(L 2) f3(L 3) .
i, j c (3.8)
Figures 3.43-46 show the results of the Multiplication
method. Although there is still no sharp image point in
some of the cases, the intensity ratios are much larger and
the location of the object point is more distinct, such as
at the node point (1,1) in Figure 3.43. Source points
closer together than about one-fifth of the side of the
triangle would probably not distinguishable.
As mentioned, the intensity distributions along the
cylinder wall that result from a given object point can be
found when the point object position is known. One can also
construct the intensity distribution functions along the
cylinder, 0n(r,z), for each light horn where r stands for
the distance of the image point from the axis of the light
horn n, (n=l, 2 or 3), and z is the distance from exit
aperture. Thus, the integral (with respect to z) of the
product of the functions fn(z) and 0n(r,z) is the
probability, Pn(r) of the image point occurring on the ring
of the radius r centered at the axis of light horn n. For
the intensity distribution was generated by the ring shaped
detector, it is a discrete function instead of continuous

86
POINT SOURCE AT NODE (1,1)
> 2.3* = > 3.2x 11 > 3.6* Ü¡ > 4.1 y. M - 4.5x
Multiplication method
Figure 3.43 The image pattern created by the
Multiplication method, object point
at node (1,1)

87
POINT SOURCE AT NODE (5,5)
> 6.9/S = > 9,?'/. ill >11.1/. H >12.4x m =13.8v.
Mu 11ip1icat io n nethod
Figure 3.44 The image pattern created by the
Multiplication method, object point
at node (5,5)

88
POINT SOURCE AT NODE < 1 , 26 )
Multiplication nethod
Figure 3.45 The image pattern created by the
Multiplication method, object point
at node (1,26)

89
POINT SOURCE AT NODE ( 17 , 17 )
li > 7.8/ m > 9.7x
>11.1 z H >12.5'/
=13.9/
Multiplication nethod
Figure 3.46 The image pattern created by the
Multiplication method, object point
at node (17,17)

90
function. Therefore, the integral here should actually be a
summation and the probability that a source point exists on
the ring of radius r is proportional to
N
Pn(r) * 2 f" 0° (r, z.) 1 = 1, 2, .... N. (3.9)
1 = 1
For every node point, the distances r " ., to the light
horn axes can be found for the light horns, and thus the
probabilities, Pn(r? .) of finding the object point on
1 • J
that node is found.
The Cross Correlation with Addition method assigns to a
particular node point of the image space the sum of the
probabilities seen by all three light horns at that node
point and the intensity on the node point is proportional to
I1Ba?e = P1 (r ! .) + P2 (r 2 .) + P3 (r 3 .)
x. j i.j i.r i» j
(3.10)
Scanning through all the nodes, the probability of finding
the image point on the image space was determined and put
together as the total image pattern. The resulting images
determined by this method are shown in Figures 3.47-50. As
one can see in these figures, the maximum value appears as a
small area of the image space. This means that the image
position is better defined than either the Addition or
Multiplication method is used.
Although the image point could be better defined when
using the CCA method to construct the image, the possibility

91
Object at ( 1 , i )
>0.89 = >1.25 ¡I >1.43
>1.61 9 =1.79
Cross correlation uith addition
Figure 3.47 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,1)

92
Object at ( 5 j 5 )
A.
/
/ â– .
/
>1.03
>1.44
>1.65
>1.85
-2.06
Cross correlation with addition
Figure 3.48 The image pattern created by the Cross
Correlation with Addition method,
object point at node (5,5)

93
Cross correlation with addition
Figure 3.49 The image pattern created by the Cross
Correlation with Addition method,
object point at node (1,26)

94
Object at ( 17 , 17 )
>0.73 = >1.02
>1.17
>1.31
â–  =1-46
Cross correlation uith addition
Figure 3.50 The image pattern created by the Cross
Correlation with Addition method,
object point at node (17,17)

95
exists that the middle value is spread out in a larger area
than in the Multiplication method. This would lead to bad
resolution. Therefore, an approach combining these two
methods, Multiplication and CCA, is suggested and it is the
Cross Correlation with Multiplication method. This method
determines the probabilities from all three light horns and
then assigns to a given node the product of these three
probabilities which are derived from this node of interest.
The intensity on the node point (i,j) is then proportional
to
jimage
i.j
Pl(r Í .)
i.J
-2, 2 . _3, 3 .
P (r . .) P (r . .)
i.j i.j
(3.11)
As one can see in Figures 3.51-54, the object point on the
node (1,1) and (1,5) can be easily distinguished which
indicates an resolution of 0.286 degree. This means that
the CCM method not only gives a more well-defined image but
also has better resolution than the other methods.
In order to examine whether the system can distinguish
between two different point objects or not, the distribution
functions were determined and the multiplication method was
used on them. Figures 3.55-56 show two of the resulting
image patterns (These are presented as contour plot for
clarity), where there are two object points present.
Although the two points could not be seen clearly on the
figures, one can distinguish that there are two points
present rather than just one point.

96
Object at ( i , 1 )
A
/ \
/ \
\
/
\
/
\
/
/
/
/
A
> 1.5x = > 2.0X 11 > 2.3’/. I > 2.&x n - 2.9'/.
Cross correlation with multiplication
Figure 3.51 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,1)

97
Object at ( 5 , 5 )
> 5.2* = > 7.3x
> 8.4*
> 9.4* H =10.4*.
Cross correlation with multiplication
Figure 3.52 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (5,5)

98
Object at ( i , 26 )
> 8.0VS = >11.3/ m >12.9/. m >14.5v. â–  =16.1/
Cross correlation with Multiplication
Figure 3.53 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (1,26)

99
Object at ( 17 , 17 )
> 5.7* = > 8.0* ¡1 > 9.2* II >10.3* Ü =11.5*
Cross correlation with multiplication
Figure 3.54 The image pattern created by the Cross
Correlation with Multiplication method,
object point at node (17,17)

100
Figure 3.55 The image pattern created by the
Multiplication method, object points
at nodes (1,1) and (17,17)

101
Figure 3.56 The image pattern created by the
Multiplication method, object points
at nodes (1,26) and (17,17)

102
3.5 Summary and Discussion
This study, which showed the imaging possibilities of a
long cylinder lined with light sensors and equipped with an
input light horn, is a simulation of the insect eye. The
patterns of (a) unreflected rays and (b) singly reflected
rays have been observed to encode a distant object-space in
a fairly simple fashion within the receiving cylinder.
When the vertical pattern is used, a single such device
could distinguish an object at different positions but not
two different objects at nearby positions. Although this
give better resolution than the mosaic theory (one pixel out
of one eyelet), the resolution is still poor.
When the wall pattern is used, a single such device can
distinguish objects at different polar angles, but not at
different azimuthal angles (when reflected rays are used).
The maximum polar angle from which useful information may be
obtained is about the half-angle of acceptance of the
truncated light horn (when used as a collimator, for
instance). The angular resolution may be optimized by
proper selection of the light horn length and may be
improved (at the expense of efficiency loss) by reducing the
half-angle of the conical light horn.
If the object to be detected resembles a point (an
infrared target, for example), a small number of relatively
large light horns may be superior to a single lens-detector
array combination. Therefore, the correlation methods were

103
studied. It was shown that the wall pattern of three
specifically arranged horn-cylinder devices can be
correlated to generate more information out of such system.
This study showed that the Cross Correlation with
Multiplication method could give the resolution down to
about 0.286 degree for the same light horn used which is one
order of magnitude better than the light horn used as a
collimator which gives only 4.7 degrees resolution.
Since the resolving power increases with decreasing
cone angle and glancing angles () are small, a device such
as described here could conceivably be used in the X-ray
region. A large number of such devices would constitute an
X-ray imaging system with a (relatively) large effective
area. The only requirement of using the horn-cylinder
combination as described in this chapter is total reflection
13
of the light beams. McCall noted that in order to have
total reflection occur on the metal surface the requirement
on the X-ray energy is E(keV) < 4/9^(degrees). Thus one can
adjust the X-ray energy or select a suitable angle for the
light horn device to have total reflection and a new X-ray
imaging system is born.
In order to build a vision device, the above results
suggest that (for multiaperture optics systems) a hole mask
which contains hundreds or thousands of conical holes having
a diameter larger than the size of an individual detector
need to be used. In addition, the above described phenomena
of non-imaging elements could be used to reconstruct the

104
object from the observed pattern to providing respectable
resolving power. Thus, one can conclude that even if
non-imaging optical elements (light horns) are used, the
total number of the eyelets does not need to be extremely
large and the eyelets could be of reasonable size.

CHAPTER 4
LIGHT COLLECTION OF THE LIGHT HORN
In Chapter 3 the geometrical optics of light horns was
developed. It is clear that the image pattern could be used
to determine the polar angle of an object point. However,
the light collection phenomena, i.e., the amount or
intensity of light collected, has not yet been discussed.
Hence, this chapter is dedicated to the study of light
collection by light horns.
The perfect light detector does not exist. All
detectors give rise to some response even when no signal is
present. In other words, the background noise of a detector
always exists. Therefore, if the amplitude of a signal is
similar to that of the background, the signal cannot be
detected. This demonstrates the need for a high efficiency
of light collection in any such optical device.
The size of the entrance aperture and the concentration
ratio—the ratio of output intensity to input intensity—
affects the amount of collected light striking the
detectors. Since the aperture of the light horn would be
small for an MAO device, whether or not the light horn
collects enough light to distinguish the central disk, rings
105

106
and the space in between the rings will depend mostly on the
concentration ratio.
4.1 Intensity Concentration
In order to understand the light concentration
phenomena of light horns, let us examine the case in which
the detector array is located on a vertical cross section
plane (see Figure 3.1). This plane could be at any distance
from the exit aperture of the light horn. Since this is a
concentration study for the detection of images, the amount
of light which strikes the detector is of the most interest.
Hence, unlike other concentrator studies,this light
concentration study is not restricted to cases where the
detectors are right at the exit aperture. The size of the
detector is fixed when the detector is chosen. The area of
interest is chosen to be the size of the exit aperture.
Figure 4.1 shows the curves of the concentration ratio
of several conical light horns exposed to a distant point
source (1000 units away from the entrance aperture). As we
can see, the concentration ratios of any one of the small
cone angle cases are almost constant from right at the exit
aperture up to about several units away from the exit
aperture. Beyond these ranges, the concentration ratio
falls dramatically and then levels off to a constant value.
One can see also in Figure 4.1, that as the cone opening
angle increases, the concentration ratio increases. Figure
4.2 shows the concentration ratio curve for a parabolic
shaped light horn. The concentration ratio curve have a

107
fj 10. ripgppp;
0
Í-4
a
o
•H
*
u
+j
c
0)
o
c
0„
o¿.
-i—— 4 i
A «
¿-v >v -r, « «
Ki
i
â– r> - â–  n
1 r
(units)
Distance of illuminated area from the exit aperture
Figure 4.1 Concentration ratio curves of conical light
horns with different cone angles for an
on-axis point source at 1000 units distance.

Concentration ratio
108
23,
Í
4 5
(units)
Distance of illuminated area from the exit aperture
Figure 4.2 The concentration ratio of a parabolic shaped
light horn at different distances away from
the exit aperture with an on-axis point
source at 1000 units distance.

109
similar shape to those in Figure 4.1, i.e., all of them show
two levels and a transition between them. The different
levels of the concentration ratios could be an explanation
for the movement of the pigment cell in the scotopic eye.
As mentioned in Chapter 2, the pigment of the scotopic eye
is located close to the exit of the crystalline cone at low
intensity levels and moves halfway down the crystalline
tract at higher light levels. A possible explanation is
that at higher light levels, the lower concentration ratio
is already sufficient to bring enough light to the pigment
cell at halfway down the crystalline tract to trigger the
nerve. However, under low intensity light levels, a higher
concentration ratio is needed to concentrate the light
sufficiently so that the pigment cells reach the threshold
for detection. Therefore, under low light intensity levels
the pigments are much closer to the exit aperture to aid in
the collection of the light.
Figure 4.3 shows the curve of the concentration ratio
verses the distance of the point source when the detector
plane is fixed on the exit aperture. The concentration
ratio of a light horn increases dramatically as the point
source moves away from the light horn and then it reaches a
plateau and remains constant as the point source approaches
infinity.
Figure 4.4 shows the concentration ratios of various
light horns as a function of the off-axis angle of the point
source. The concentration ratios are almost constant until

Concentration ratio
110
18.
(units)
Distance of point source from
entrance
aperture
Figure 4.3 Comparison of concentration ratios
a light horn and a lens.
between

Concentration ratio
111
(degree)
Off-axis angle of the point
source
The concentration ratio
off-axis point source.
Figure 4.4
curves of the

112
the off-axis angle reaches the limitation angle—angle at
which unreflected light totally disappears.
4.2 Comparison of Light Horns and Lenses
After the concentration ratio of the light horns was
studied, it was decided to compare light horns with lenses
in this respect. Therefore, the light concentration effect
of lenses is presented for comparison.
Since a lens and light horn do not have similar
geometries, in order to make a reasonable comparison between
them, the aperture of the lens, "A," is chosen to be the
same size as the entrance aperture of the light horn and the
power of the lens is chosen so that the focal length of the
lens, "f," is equal to the length of the light horn. A
detector, of the size of the exit aperture of the light
horn, is assumed to be located on the focus as the "exit
aperture" of the lens.
When a point source is at a distance from the lens, p,
larger than the focal length, there is a true image point
behind the lens. When the point source is closer to the
lens than the focal length, there is a virtual image point
in the front of the lens. Due to diffraction, the size of
the image point is actually an Airy disk of the diameter,
"D " which can be found from
a
D = 1.22 \f/A. (4.1)
9
The light beams could be taken as converging to the image
point "q" and then spreading out within a cone angle.

113
Figure 4.5 indicates how a lens would collect the light to
form an image, concentrating the light before the image
point then diverging the light after the image point.
The concentration ratio is found to be proportional to
the square of the object distance and inversely proportional
to the square of the focal length. When the object is at
infinity the image is on the focus of the lens and the
concentration ratio of the lens becomes a maximum. This
2
maximum value could be found as (A/D^) . The concentration
ratio curve of the lens assumed is shown in Figure 4.3 with
the concentration ratio curve of the comparable light horn.
When comparing the concentration ratio curve of the
light horn to that of the lens, one can see clearly that at
short distances, such as shorter than the focal length of
the lens, the concentration ratio of the light horn is
larger than the concentration ratio of the lens. Therefore,
one can conclude that as a concentrator of light, the light
horn is better than the lens when the object is at short
distances.
The detector response is only limited within a certain
range. Therefore, when the detectors are chosen, the range
of the object is defined. For a lens system the
concentration ratio changes drastically with object
distance. This means that at long distances the position of
the detector would have to achieve focus Also, at short
distances nothing could be done to achieve focus. For a

Point source Lens Focus
Figure 4.5 The light collection of a lens
114

115
light horn optical system, the concentration ratio is higher
than the lens at short object distances and it becomes
constant after the object distance exceeds several entrance
aperture radii. Hence, for a multiaperture optical system
one would like to use a light horn instead of a lens to
avoid having to fine tune the detector location or having to
change the power of the lens.

CHAPTER 5
THE MULTIAPERTURE OPTICAL SYSTEM DESIGN
5.1 The System Design
For an apposition eye system each eyelet does not
necessarily have to form a conventional image. The image of
an object point is not necessarily a point; it could be a
disk or a ring or something irregularly shaped, as long as
it fits on the detector or detectors dedicated to the
eyelet. However, the eyelet not only has to be small and
easily manufactured, but also has to have a high light
concentration power. The light horn which was mentioned
earlier fills these requirements nicely as long as it is not
too small, i.e., so that the diffraction effects can still
be neglected. Therefore, one would suggest that a multiple
aperture optical (MAO) device should consist of many
horn-cylinder eyelets with a detector array embedded.
A possible MAO device design is shown in Figure 5.1.
As can be seen, the device consists of the following
components: an MAO mask—a mask consists of hundreds or
thousand light horn, a detector board, a memory board and a
processor board (or a computer). The MAO mask contains the
light horns, possibly fitted with lenses. The detector
116

Field Lens
Figure 5.1 The multiaperture optical (MAO) device
117

118
board contains the detector array, while the memory board
and hardwired processor board are used for image processing
and pattern recognition. The functions of the memory board
and processor board may be carried out by a separate
computer.
5.1.1 MAO Mask
From the studies of the previous chapters, the light
horn is suggested to be the basic element of the artificial
insect eye. As stated, naturally occurring compound eyes do
contain up to 10,000 eyelets. Therefore, to simulate the
compound eye the light horns need to be bundled together to
form the multiple aperture optical device. When several
light horn devices are assembled together there will be some
overlapping of the fields of view and some blind spots not
covered by the total field of view. It is suggested that
the light horns be arranged on the surface of a hemisphere
such that the axis of neighboring light horns would have a
small angle between them thereby reducing the overlap. To
reduce the size of the blind spots among eyelets, the
horn-cylinder elements have to be as close as possible. The
best way to hold these many tiny light horns together with
such an arrangement is to build a mask with the desired
holes. In addition to the requirements above, the internal
walls of these cones should be totally reflective. This
could be done by properly selecting the mask material, by
coating the inner surface of light horn, or by filling the
light horn with a high index of refraction material.

119
Two different MAO masks were made, one with cylindrical
holes, the other with conical holes. So far, the technique
that was used for manufacturing those masks with conical
light horns could not make them all of the same shape.
Therefore, although it is preferable to have conical holes
instead of cylindrical holes for the light horn, the first
mask was made with cylindrical holes. (Note: It has been
proved by ray tracing that the cylindrical holes give a
similar disk and ring image pattern to the conical holes.)
This MAO mask, with a large number of cylindrical holes, was
made of glass. The diameter of each hole was 100
micrometers. The distance between the holes was about 0.5
diameters (150 micrometers from center to center). The mask
was bent with a radius of 75 mm so that the angle between
neighboring eyelets was about 0.02 degree. (Note: For real
insect eyes the angle between the axes of neighboring
eyelets is 1 to 2 degrees1'^. ) The mask had a diameter of
13 mm although only an area of about 2 mm X 5 mm (the size
of the detector array) was used.
A second MAO mask was built with conical light horns.
This second mask is made of PC 777, obtained from Advanced
Polymer Products, Gainesville, Florida. Due to the
manufacture problems, the conical holes were not all of the
same size and shape. Also, the mask was not bent to produce
an angle between the neighboring light horns. Although the
light horns are not exactly of the same dimension, the
typical light horn of this mask has an entrance aperture of

120
about 300 micrometers and an exit aperture of about 100
micrometers. The distance between two entrance holes
(center to center) is 350 micrometers. The cylinder part of
the light horn-cylinder combination was not fabricated, due
to the technical difficulty. Therefore, there is no
suitable divider to prevent interference between neighboring
light horns.
5,1.2 Detector Board
It would be next to impossible to place individual
detectors within the cylinders of the light horn mask.
Therefore, the detector board (see Figure 5.1) is simulated
by placing the most suitable detector array behind the mask
to detect the vertical cross-section patterns. The detector
array used is an IS32A optical RAM manufactured by Micron
Technology, Inc. It nominally contains two arrays each with
128 X 256 silicon detectors arranged on a 129 X 514 matrix
shown, in part, in Figure 5.2.(a). Sensitivity of the
detector is approximately 2 pJ/sq. cm at 900 nm. Each
detector is 6.4 X 6.4 micrometers in size and the distance
between rows is 6.8 micrometers (center to center). The
center to center distance between two nearest neighbors in a
given row is 8.8 micrometers. The somewhat unconventional
arrangement of detectors forces the user either to fill in
the missing pixels by guesses or omit certain detectors as
indicated in Figure 5.2.(b). Now only the detectors marked
with an X are used. This reduces the data array to 64 rows
of 128 detectors each. Now the distance (center to center)

121
00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00
(a)
xo xo xo xo xo xo xo xo xo xo
00 00 00 00 00 00 00 00 00 00
xo xo xo xo xo xo xo xo xo xo
00 00 00 00 00 00 00 00 00 00
xo xo xo xo xo xo xo xo xo xo
00 00 00 00 00 00 00 00 00 00
(b)
Figure 5.2 Arrangement of detectors on optic RAM
a) detector location; b) detector in use
(marked by "X")

122
between active detectors is 34.4 micrometers in the rows and
13.6 micrometers in the column. Each detector element can
be individually addressed although no use was made of this
feature during the present effort.
When using the mask with cylindrical holes, the one
where the diameter of each hole is 100 micrometers, each
hole should have a footprint containing 11 X 11 locations in
the 129 X 514 matrix. Figure 5.3 shows a 12 X 14 field.
The footprint of the round hole is drawn slightly larger in
order to account for the spreading image as a result of the
small gap between the exit aperture and the detector plane.
The detectors which are actually in place are marked with
"0’s" and "X’s." The detectors actually used are indicated
by the X’s. Counting the actual detectors, one finds that
there are 2 columns in the 514 direction and 6 rows in the
129 direction under the 11 X 11 footprint. However, the bit
map reveals that, on average 2.25 X 7 detectors are affected
by one hole, giving an aspect ratio of approximately 3:1.
Therefore, this is the actual footprint of the hole,
containing those detectors which are located directly
underneath the exit aperture. There are, of course, other
detectors which are underneath void spaces (the spaces
between holes). Since there is a small gap (0.05 mm)
between the hole exit and the detector plane, oblique rays,
namely those which have been reflected one or more times at
the walls of the individual holes will illuminate the

123
X
0
s
'sc
Fo
p
X
o
&
r
0
0
o
*p,
0
X
X
0
X
0
V
\
X
o
i
0
o
o
0
o
b
X
0
X
0
X
o
'
X
0
o
0
0
o
o
0
X
p
X
o
X
o
1
X
0
v
0
0
o
0
0
d
X
X
o
X
0
/
X
o
b,
o
o
0
¿
0
X
o
\
p
-X-
t)
X
o
o
o
o
o
o
o
Figure 5.3 Detector arrangement underneath the
light hole

124
detectors located under the void spaces. The wall of the
holes are dark glass, so the reflected rays will be of
diminished intensity. The computer program will let the
charges on the detectors accumulate (soak) for a selectable
exposure time and then read as being exposed, all those
values above a certain threshold value. Therefore, if the
exposure time is kept short only the detectors directly
underneath the holes will exceed threshold, while at a
longer exposure time some or all of the detectors underneath
void spaces will also exceed threshold. In other words, at
short exposure time a target will appear to be depicted by
individual dots (footprints) while at long exposure time
these dots will grow together to a homogeneously filled
area.
5.1.3 Memory Board and Processor Board
The memory board and the processor board were simulated
by an IBM personal computer. Since the detector array was a
RAM chip, the "ON" or "OFF" of each detector could be read
out either as a "1" or a "0," respectively. These "l’s" and
"0’s" arranged with respect to their location form a huge
matrix. This matrix is a bit map which could be used to
reconstruct the image pattern or to evaluate the image
pattern for recognition. When the development is done this
memory board and processor board could be hardwired onto the
detector board to reduce the size and the cost of the MAO
device.

125
5.2 The Performance of the MAO Device
The two masks which were discussed in Section 5.1.1
were used to set up the system for these studies. The
characters "H" and "N" were put in front of the system for
performance evaluation. Figures 5.4 and 5.5 show the
reconstructed image pattern of the two characters for the
cylindrical and conical hole cases, respectively. Although
the cross bar of the "H" and the slanted part of "N" appear
fatter than they should be, these patterns are still
recognizable. There are three factors which contribute to
this effect: first, the arrangement of the detectors used;
second, the pixel size of the monitor screen; and third, the
distortion due to the printer and printing software. This
problem could be solved by rearranging the detectors and/or
rewriting the software.
Based on the discussion in Section 3.4, the detector
arrangement underneath the light horns could be used to
construct matrices similar to those in Table 3.3. These
matrices could be used to clean up the bit map and construct
the new pattern for recognition. The results of this clean
up are shown in Figures 5.6 and 5.7. Although the character
"N" in the conical hole mask case is still not recognizable
by the human eye, the "H" looks much better.
As can be seen, the mask with cylindrical holes gives a
better pattern result for the human eye system. However,
one has to consider that the entrance apertures on the
conical hole mask are 300 micrometers, three times that of

126
Figure 5.4 Performance of the system using the mask
with cylinder holes

127
Figure 5.5 Performance of the system using the mask
with conical holes

128
Figure 5.6 The result pattern of Figure 5
the clean-up
4 after

129
Figure 5.7
The result pattern of Figure 5.5 after
the clean-up
*

130
the cylindrical hole mask. One should not expect to have
better results from this mask. The differences between
these two figures are really a demonstration that the
resolving power of the apposition eye depends on the FOV of
the individual eyelet.
5.3 Discussions
The performance of this system indicated that this MAO
system without the coating on the inner surface of the light
horn or the high refractive index filling is only suited for
the observation of close objects rather than distant
objects. When the light horn is without any inner surface
coating or high refractive index filling the light concen¬
tration ability which was discussed in Chapter Four no
longer holds true. When the object is far away the light
intensity at the entrance is rather low. Therefore, the
concentration ratio needs to be increased for the detector
to detect.
Although the table of matrices could help to clean up
the gross image, sometimes the modified image pattern is
still not sharp enough for image recognition. This is
because the contrast between the object and the background
is not great enough. This can be improved by removing the
glass plate which is between the detector array and the
mask. The internal reflection and the light displacement
due to refraction of the glass plate reduces the contrast.
Therefore, it is suggested that the glass plate be removed

131
from on top of the detector and the mask should be placed
right on top of the detector array.
It is believed that if the surface coating or the
filling of the light horn is done, the performance of the
MAO system could be improved. Of course, in addition to the
suggestions discussed above, the cylindrical pipe section of
the light horn-cylinder device needs to be built into the
mask to prevent interference between eyelets.

CHAPTER 6
SUMMARY AND CONCLUSIONS
Based on the studies in Chapters Two and Three, the
optical system of the insect eye is quite different from
that of the vertebrate eye, but the visual ability of
insects is no less than other animals. Although they are
two different optical systems, there is not any indication
of retarded evolution. A possible explanation is that both
eye types have evolved to about the same degree and the
difference between these two optical systems is due to their
different environmental requirements. For a human designer
of optical systems this indicates that there are two
different options in the application of the geometrical
optics.
Natural species which are equipped with multiaperture
eyes are small compare to the animals which are equipped
with vertebrate eyes. This indicates that the applications
employing multiaperture eyes must be advantageous in terms
of size, even though there might be some loss of resolution.
A small-sized vertebrate eye would require extremely short
focal lengths to be able to focus the light onto the retina
132

133
at such short distances. This in turn limits the diameters
of the lens. To achieve the ability to see larger object
with some acceptable resolution, a large number of lenses is
required. This leads naturally to faceted eyes and
multiaperture optics. The multiaperture eyes could have
some built-in image preprocessing ability. This indicates
that the insect eye can be useful as a model for
technological applications, especially robotics.
It was the goal of this dissertation to better
understand the optical function of insect eyes, leading to
the design of small vision devices. The basis of the
optical studies, the structure of the insect eye, was
described in Chapter Two. There are two types of insect
eyes, the superposition eye and the apposition eye (based on
the belief of most of the entomologists). The present study
concerns itself only with the latter type, which is accepted
as the less complex of the two. However, the optical
studies in this dissertation suggested that these two types
might have the same optical properties.
In the apposition eye each eyelet acquires only one
pixel of the overall image. Nevertheless, each eyelet
contains eight receptors. Obviously these detectors must be
used to define the one pixel in more detail. Hence, in the
apposition eye the optical system of an individual eyelet
need not be capable of image formation. In other words, an
object point does not necessarily need to be represented by

134
an image "point." Hence, Chapter Three was dedicated to the
study of the optical elements of the insect eye.
A horn-cylinder combination was used in these studies.
Total internal reflection was assumed for the inner surface
of the light horn. Surprisingly, such a light horn when
symmetrically illuminated forms a set of concentric rings
and a central disk with very sharp dark spaces in between.
An off axis light source produces decentered rings, a fact
which can be used to describe the acquired pixel in more
detail. The ray tracing results supported this experimental
observation. The program was also applied to a multitude of
light horns that contribute to a camera which was based on
the design of the insect eye. A table of matrices was
constructed for the detector response array with a specific
threshold setting. This matrix table provides the decision
criteria to determine the position of a point source for
recognition schemes.
The imaging pattern on the cylinder wall of the horn-
cylinder combination was also studied. With the pattern of
one such device, the polar angle of an object can be
determined. Furthermore, four correlation methods were
introduced which correlated the wall patterns of a triple
horn-cylinder combination. Through these correlation
methods the image patterns were reconstructed. The
reconstructed images indicated that by using these methods,
(especially the Cross Correlation with Multiplication

135
method), the resolution is about one order of magnitude
better than the same light horn used as a simple collimator.
In Chapter Four the light collecting ability of the
light horn was studied. It turns out that the concentration
ratio of the light horn is better than that of a lens of the
same aperture, if the object point is at a short distance.
For insects there is a need not only to perceive the distant
environment but also to see objects at very short distances
since this is the environment important for survival.
Therefore, the requirement to be able to see short distances
might be one of the reasons that nature opts for the multi-
aperture optical system in the insect vision system designs.
One possible artificial multiaperture optical system
was designed, constructed and tested. This system had a
light horn mask, a detector board, a memory board and a
processor board. One of the light horn masks used in this
study had entrance apertures of 300 micrometers and exit
apertures of 100 micrometers. The other mask used has
cylindrical holes which are 100 micrometers in diameter.
Each of these masks were placed on a 256 X 128 detector
array. A personal computer was used as the memory board and
the processor board. The images obtained by this device
were poorly resolved. However, they were modified with
software resulting in better images. These modified images
had cleaner edges which allowed for easier recognition.
Summarizing, the present effort was devoted exclusively
to the apposition eye. It is fair to say that the knowledge

136
of the optical function of the apposition eye was sub¬
stantially improved by this study. This present study also
explored the possibility of improving the resolving power of
the non-imaging light horn by using a detector array. This
led to the design of a multiaperture optical device.
Based on this study, one can conclude that for a close
vision device the light horn system of a multiaperture
optical device is better than a lens system. Furthermore,
the lens system is only good for a certain range of
wavelengths of light and the light horn has basically no
limit in that respect. Therefore, when the wavelength of
the light is beyond the limit of the lens (e.g. X-ray) one
would use the light horn instead.
The wall pattern methods seem very promising and
therefore methods of implanting detectors onto the wall of
the cylindrical pipe need to be studied.
A better technique to manufacture the MAO mask is also
needed. Because the masks that were manufactured could not
have all the light horns of the same size. Also, the
cylindrical part is not yet in place to act as a space
divider. For better internal reflection, the technique of
filling the cones with a high refractive index material or
coating them on the inner surface of the light horn with a
mirror finish should be developed. Also, the glass plate on
top of the detector chip needs to be removed to reduce the
contrast problem.

137
Although, the object recognition technique for the
multiaperture optical system is not discussed in this
dissertation, a recognition technique would make the
multiaperture optical system become a complete robot vision
system. The recognition technique suggested by Schneider
. 14
and Lin especially for the multiaperture optical system is
a possibility.

REFERENCES
1. G. A. Mazokhin-Porshnyakov, Insect Vision, Plenum
Press, New York, 1969.
2. R. T. Schneider and J. F. Long, "Mechanical Model
of the Insect Eye," Applications of Digital Image Processing
IV in Proc. SPIE, Pp. 359, 1982.
3. M. L. Kao, "Robot Vision using Multiaperture
Optics," University of Florida, Gainesville, dissertation,
1984.
4. R. F. Chapman, The Insects: Structure and
Function, American Elsevier Publishing Company, Inc., New
York, 1969.
5. V. B. Meyer-Rochow, "The Dioptric System in Beetle
Compound eyes," Chapter 12 of The Compound Eye and Vision of
Insects, edited by G. A. Horridge, Oxford University Press,
London, 1975.
6. K. Hausen, "Die Brechungsindices im Kristallkegel
der Mehlmotte Ephestia Kuhniella," J. Compound Physiol.,
Vol. 82, Pp. 365-378, 1973.
7. T. H. Goldsmith and G. D. Bernard, "The Visual
system of Insects," Chapter 5 of The Physiology of Insecta,
Academic Press, New York, 1974.
8. J. Miiller, Zur vergleichenden Physiologie des
Gesichtsinnes des Menschen und der Tiere, C. Cnobloch,
Leipzig, 1826.
9. J. W. Kuiper, "The Optics of the Compound Eye,"
Symp. Soc. Exp. Biol., Vol. 16, Pp. 58-71, 1962.
10. D. E. Williamson, "Cone Channel Condenser Optics,"
J. Opt. Soc. Am., Vol. 42, Pp. 712-715, 1952.
11. W. T. Welford and R. Winston, The Optics of
Nonimaging Concentrators, Light and Solar Energy, Academic
Press, New York, 1978.
138

139
12. R. T. Schneider, "High Efficiency Fiber-shaped
Detector," U. S. Patent, Pat. No. 4 585 937, 1986.
13. G. H. McCall, "X-ray Imaging in the Laser-Fusion
Program," X-ray Imaging, Proceedings of SPIE, Vol. 106, Pp.
2-7, 1977.
14. R. T. Schneider and S. C. Lin, "A Camera based on
the Principles of the Insect eye," IEEE Southeastcon ’87,
Vol. 1, Pp. 424-430, 1987.

BIOGRAPHICAL SKETCH
Shih-Chao Lin (also known as David) was born March 2,
1955, in Taipei, Taiwan. He received the Bachelor of
Science degree in physics from Fu-Jen Catholic University in
June of 1977. After two years serving in the ROTC program
of the Republic of China, he joined the Hua Shia Institute
of Technology as an instructor in physics. In May of 1983
he earned his Master of Engineering degree in mechanical
engineering (specializing in the fluid dynamics and heat
transfer) from Old Dominion University, Norfolk, Virginia.
David has been a research assistant under Professor
Schneider at the University of Florida since 1985. He is
also a Research Scientist at RTS Laboratories, Inc. Both of
these two endeavors involved studies of the multiaperture
optical systems.
140

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of/T)octor of Philosophy.
Richard T. Schneider, Chairman
Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Edward E. Carroll f '
Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
¿'TV1
i11iam H. Ellis
Associate Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Mechanical Engineering

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Gerhard Ritter
Professor of Computer and
//Information Sciences
This dissertation was submitted to the Graduate Faculty
of the College of Engineering and to the Graduate school,
and was accepted as partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
April 1988
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08556 7849




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