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INCOHERENT OPTICAL PROCESSING:
A TRISTIMULUS-BASED APPROACH
by
RICHARD FRANKLIN CARSON, B.S. in E.E,
A THESIS
IN
ELECTRICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
Approved
August, 1982
//fc.J''^ - ACKNOWLEDGMENTS
T C^crV'^^
I am extremely grateful to Dr. John F. Walkup and
Dr. Thomas F. Krile for their guidance, assistance, and
encouragement throughout the course of this research. The
critique and suggestions from Dr. Marion 0. Hagler and Dr.
Thomas Newman have also been very helpful. My student
colleagues, especially David Nelson and Fernando Bermudez,
have assisted in many ways. I gratefully acknowledge the
financial support of the Air Force Office of Scientific
Research under grant 79-0076. This thesis is dedicated
to my fiancee, Carol Coggin, for her sustaining support
and encouragement.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS i
ABSTRACT iii
LIST OF FIGURES iv
LIST OF TABLES viii
1. INTRODUCTION I 1.1 Motivation I 1.2 The Complex Superposition Integral 2 1.3 The Structure of the Thesis 3
2. TRISTIMULUS THEORY AND COLOR TELEVISION 4 2.1 Color Matching and Three-Space Representations . . . 4 2.2 Models for Tristimulus Systems 10 2.3 The N.T.S.C. Tristimulus Systems and
Color Television 13
3. COMPLEX NUMBER ADDITION AND MULTIPLICATION 25 3.1 Complex Number Multiplication 25 3.2 Complex Number Addition 36 3.3 Range Limitations and Separability of Coordinates . . 41
4. EVALUATING THE COMPLEX SUPERPOSITION INTEGRAL 47 4.1 The Sampled-Form Approximation 47 4.2 A System Architecture 56 4.3 Problems to be Investigated 65
5. AN EXPERIMENTAL SYSTEM 68 5.1 System Components 68 5.2 Characterization, Linearity, and Range Tests . . . . 79 5.3 Complex Number Mathematics 100
6. CONCLUSION 114
APPENDIX A. EQUIVALENT 1-D EQUATIONS FROM SECTION 4.2 . . . . 117
APPENDIX B. A REAL-TIME SPATIAL LIGHT MODULATOR 120
APPENDIX C. INSTRUMENTATION AND TRANSFORM ELECTRONICS . . . . 125
APPENDIX D. CHARGE-COUPLED DEVICE IMAGING CAMERAS 139
FOOTNOTES 143
SELECTED BIBLIOGRAPHY 148
ii
ABSTRACT
Optical systems which make use of incoherent illumination
provide potential benefits in noise immunity over systems using
coherent illumination. Because incoherent systems are linear with
respect to intensity, however, complex number operations must be
synthesized. The tristimulus (three-color) methods used in color
television provide vector-space models that allow the complex
number operations of polar-form multiplication and rectangular-form
addition to be achieved, using incoherent illumination. When these
operations are combined in a hybrid (optical-electronic) processor
that uses color television equipment, general space-variant optical
systems can be approximated by a sampled-form approach.
Ill
LIST OF FIGURES
Figure Page
2-1. Additive and Subtractive Color Matching Systems . . . . 5
2-2. Spectral Responses for Color Matching Functions r, g, and b 5
2-3. Relative Spectral Irradiance (Power) of Standard
Illuminants A, B and C 8
2-4. The (R,G,B) Color Solid 8
2-5. Spectral Responses for Color Matching Functions
X, y, and z 12
2-6. CIE 1931 (x,y) Chromaticity Diagram 12
2-7. The Intensity, Hue, Saturation (I,H,S) Color Space . . 14
2-8. Spectral Responses for Typical Television Phosphors . . 16
2-9. N.T.S.C. Color Range Shown on the (x,y) Chromaticity
Diagram 16
2-10. The (Y,I,Q) Color Space 18
2-11. A (R -Y, B -Y) Constant Intensity Plane 18
n n '' 2-12. The (R ,G ,B ) Color Solid in the (R ,G ,B )
Color Space 20 2-13. The (R ,G ,B ) Color Solid in the (Y, R -Y, B -Y)
Color Space 20
2-14. Typical Color Television Camera Block Diagram 21
2-15. Vidicon Camera Tube Response 23
2-16. Color CRT Response 23
3-1. Constant Y Plane in the (Y, R -Y, B -Y) Color Space . . 27
n n ^ 3-2. Vector Addition on a Constant Y Plane 27 3-3. Number Lines Represented by the B -Y and R -Y
Color Signals ^. . . . ? 28 iv
Figure Page
3-4. Angle Scaling on the B -Y and R -Y Number Lines . . . . 30 n n
3-5. Angle Addition Using a Beamsplitter 30
3-6. Angle Addition Example 33
3-7. Multiplication of Complex Number Moduli 35
3-8. Alternate Multiplication Scheme 35
3-9. The Complex Number Plane Expressed on a Constant Y
(R -Y. B -Y) Plane 37 n n
3-10. Complex Number Addition Scheme 39
3-11. A Complex Number Addition Example 39
3-12. Color Ranges of the (R ,G ,B ) Space for Constant
Y Values ? .". 43
3-13. Physical Representation for the (Y, R -Y, B -Y) Space). 45
4-1. Samples in the Output (x) Line 49
4-2. Samples in Both the Output (x) Line and the
Input (O Line 49
4-3. Array Mappings of Pixels 51
4-4. Pixel Arrays Split into Separate Amplitude and
Phase Angle Arrays 51
4-5. Summation in 1-D Form Using a Cylindrical Lens . . . . 52
4-6. Sampling h in the Output Plane 54
4-7. Pixel Arrays for 2-D Processing 55
4-8. Pixel Arrays Split into Amplitude and Phase Angle . . . 55
4-9. Complex Summation in 2-D Form 57
4-10. Block Diagram of the Complete System Architecture . . . 59
4-11. Polar-to-Rectangular Transform Block Diagram 63
4-12. Summation by Sub-Sections 67 V
Figure Page
5-1. The Experimental System 69
5-2. Oscilloscope Traces of Single Scan Lines 71
5-3. Response of the Black & White (B&W) Camera 73
5-4. Spectral Responses of the Beamsplitters 73
5-5. Details of the N.T.S.C. Primary Simulator 74
5-6. Color Temperature vs. Voltage for Tungsten Lamps . . . 74
5-7. Transform, Electronics, and Vector Display 78
5-8. Block Diagram of Transform and Display Electronics . . 78
5-9. Noise Characteristics of a Single Horizontal Scan . . . 80
5-10. Color Signal Range for Initial Adjustment of
Color Camera #1 83
5-11. Luminance (Y) Signal Response for Camera #1 87
5-12. Color Space for Camera #1 87
5-13. Photographs of the Experimental System 92
5-14. Color Signal Range for the Cameras in System Form . . . 97
5-15. Matrix Models for the Complete Experimental System . . 99
5-16. Polar-to-Rectangular Transform Circuitry 103
5-17. Sampled Form and Input of the R -Y Signal 103
5-18. Range of Values for Transformed Color Signals 104
5-19. Single Horizontal Scans for the Product and
Transform Signals 107
5-20. Double-Pixel Display for Addition Testing 109
5-21. Using a Lens to Perform Pixel Addition 112
B-1. Use of an Optically Addressed Liquid Crystal Light
Valve (LCLV) for Pixel-by-Pixel Multiplication . . . . 122 vi
Figure Page
C-1. System Block Diagram 127
C-2. Synchronization Electronics for the B&W Camera . . . . 129
C-3. Display Electronics 130
C-4. Scan Line Selection Electronics 131
C-5. PoIar-to-RectanguIar Transform Electronic Block
Diagram 132
C-6. Timing Diagram for Transform Electronics 134
C-7. Transform Electronics Circuit Details 136
C-8. 8-Quadrant Polar-to-Rectangular Transform using
Acousto-Optic (A-0) Cells 137
D-1. Input-Output Response of a CCD Imaging Camera 142
D-2. Spectral Response of a CCD Imaging Array 142
vii
LIST OF TABLES
Table Page
5-1. Chromaticity Coordinate Matching for the
N.T.S.C. Phosphor Primaries 76
5-2. Matrix Match Test for Camera #1 84
5-3. Color Space Tests for Various Y Values 86
5-4. Addition Test Results for Camera #1 89
5-5. Matrix Match Test for Camera #2 91
5-6. Matrix Test for the Entire System 95
5-7. Polar-to-Rectangular Transform Tests 106
5-8. Addition with a Lens for Camera //2 110 B-1. Specifications and Typical Performance Levels of
a Liquid Crystal Light Valve 124
Vlll
CHAPTER 1.
INTRODUCTION
The speed and bandwidth capabilities inherent in optical
processing systems have many potential benefits for applications
involving high-speed data processing. Optical processing schemes
also have the advantage of offering parallel processing capabilities,
which allow for large amounts of data to be handled in a short
amount of time. Numerous methods for optical data processing
systems have been proposed, using both coherent and incoherent
illumination. In this thesis, a method is presented for evaluating
the general complex superposition integral, using incoherent light.
This processing scheme uses the tristimulus (i.e. three primary
color) theory and methods which form the basis for color television.
I.l Motivation
Incoherent optical processing provides several advantages over
processing techniques which use coherent light. First, noise factors
such as input noise, optical system noise, and output detector noise
are, in general, greatly reduced for systems that use incoherent 2
illumination. Also, incoherent light provides potentially simpler
and less expensive electro-optic data interfacing schemes that can
use self-luminous and diffusely reflecting objects such as television 3 4
monitors as sources and data inputs. ' Additionally, incoherent
systems are less sensitive to spatial alignment errors and system
aberrations because they carry no phase information. This lack of
phase information is due to the fact that incoherent light is a
linear mapping with respect to intensity, which is the squared
modulus of the amplitude of a light wave. Thus, the input and
response functions for single channel incoherent systems must be
nonnegative. It follows that incoherent optical systems have the
disadvantage of only being able to represent real, nonnegative
values without the use of more than one channel.
In order to represent bipolar or complex-valued data and opera
tions, special techniques must be used with multichannel systems.
Several such techniques have been developed, including the use of
spatially separated channels, segmentation of channels, spatial
frequency carrier modulation, and the use of color channels to 8 9
represent and operate on complex-valued data. ' In the method
presented here, the vector-space and separability properties of
tristimulus coordinate systems are exploited to represent complex
numbers. Tristimulus methods, which use vector combinations of
three primary stimuli or colors, were originally developed to
represent a range of various colors and intensities of light in
color vision models. These methods were extended to facilitate
the development of color television technology. Thus, color
television monitors and cameras are used as sources and detectors
in the processing scheme presented here.
1.2 The Complex Superposition Integral
The purpose of this investigation is to explore a tristimulus-
based method of using incoherent light to represent and evaluate
the two dimensional (2-D) complex superposition integral of the
form.
f(x,y) = //h(x,y;C,n)g(^,n)dcdn (I-l)
For one dimension (1-D), the integral takes on the form:
f(x) = /h(x,Og(Od^ (1-2)
where h and g are both potentially complex functions. The integral
represents a space-varianc operation in that the complex impulse
response, h, varies both with the input ( ,n) and the output (x,y),
and does not just depend on the differences (x-^) and (y-n) as is the
case with the 2-D convolution integral that represents space-invariant
operations. ' The input function, g, is only a function of the
input variables (^,n), as discussed in Section 4.1.
These integrals are important because they can be used to
represent the most general types of optical systems, including 3-D
information processing, radar ambiguity functions, nonunity
magnification, Fourier transforms, and other integral transforms.
A number of techniques have been developed for implementing space-
variant operations. The approach described here makes use of
sampling techniques that have been developed for both the input, and
the spatial variation of the system's impulse response. A complete
discussion of the application of this sampled-function approach
appears in Section 4.1.
1.3 The Structure of the Thesis
Because this work is dependent on the vector-space properties
of tristimulus color representations, a detailed introduction to
tristimulus theory, methods, and systems appears in Chapter 2.
Chapter 3 describes the ways in which the tristimulus systems of
color television may be used to represent, multiply, and add complex
numbers. Chapter 4 presents a system architecture which is based
on tristimulus methods, and may be used to evaluate the superposition
integrals of Eqs. (1-1) or (1-2) in sampled form. In Chapter 5, the
results of an experimental system are discussed, while Chapter 6
includes conclusions and recommendations. The four appendices at
the end of this thesis contain, respectively, applicable 1-D versions
of equations in Section 4.2, a discussion of applicable spatial
light modulators and alternatives for their use for multiplication
of complex number moduli, the schematics applicable to the experi
ments of Chapter 5, and a discussion of the suitability of CCD
(charge-coupled device) imaging array cameras to replace the vidicon
cameras that are currently in use to demonstrate tristimulus methods.
CHAPTER 2
TRISTIMULUS THEORY AND COLOR TELEVISION
Tristimulus systems are used to reproduce colors in terms of
sets of three stimuli or primary colors. By this method, known as
"color matching," these primaries form a basis for three-dimensional
color solids that describe a naturally occurring range of colored
light. Based on the 1931 standards of the C L E . (Commission
Internationale de I'Eclairage), a number of tristimulus systems
have been derived, using the method of color matching. Of special
interest is the N.T.S.C. (National Television Systems Committee) 12
standard system on which television in the United States is based.
2.1 Color Matching and Three-Space Representations
Tristimulus theory is based on the assumption that an arbitrary
color can be matched by appropriate combinations of the primary
stimuli. In additive systems such as color television, red, green,
and blue light sources are projected in different combinations to
reproduce colored light as shown in Fig. 2-Ia. In subtractive
systems such as color printing, a white light source passes through
combinations of cyan (blue-green), magenta, and yellow filters as 13
shown in Fig. 2-Ib. Only the additive system will be considered
in detail here, as it forms the basis for the color television system
which is used in this work.
Additive color matching is based on three laws known as "Grass-14
mann's Laws of Additive Color Mixture" (Grassmann, 1853) :
Law #1. Four colors Q, R, G_, B (in vector notation) are always
linearly dependent; i.e., there exist scalar multipliers
Q, R, G, and B such that Q^ + R^ + GG + B^ = 0 where
Q, R, G, and B are not all zero.
Law #2. The color resulting from the mixture of color vectors
Q and Q is the same as that resulting from a mixture of
Q and Q 7 if Q "matches" Q2 (as a vector addition of
Colored Light
(a) Additive System
Yellow Filter
^ Cyan \ Filter
Colored Light
'VA^->-
Magenta Filter
(b) Subtactive System
Figure 2-1. Additive and Subtractive Color Matching Systems.
0.4 _
0.3 -
Relative Sensitivity 0.2
0.1
0
-0.1
700 \ (nra)
Figure 2-2. Spectral Responses for Color Matching Functions r,p,, and b
primaries as in Law #1), although spectral energy
distributions {P (X)dX} and {P2(A)dX} (X is the wave
length of the color stimuli) corresponding, respectively,
to Q- and Q^ may be different.
Law y/3. A continuous change in spectral energy distribution {P(X)dX}
of the color stimulus of color Q results in a continuous
change of , including the possibility that Q remains
constant.
In a tristimulus system, the three primary stimuli are
determined by wavelength-dependent color matching functions that
represent the responses of receptors with spectral sensitivities
S-(X), s (X), s (X), where X is the wavelength. For the system
defined by the C L E . these sensitivities are given by color matching
functions r(X), g(X), E(X) shown in the spectral sensitivity curves 1 ft
of Fig. 2-2. According to Grassmann's second law, the three color
matching functions can be integrated over wavelength to yield the
tristimulus values R, G, B for any color-stimulus function, (()(X):
R = k/(})(X)f(X)dX
G = k/({)(X)g(X)dX (2-1)
B = k/(|)(X)b(X)dX
where k is a scalar constant.
For an illuminated object, the color-stimulus function, (|)(X) is
given by the spectral reflectance p(X), the spectral irradiance
factor 3(X), or the spectral transmittance T(X) of the object when
it is illuminated by a source with spectral power distribution s(X).
Thus, (})(X) is obtained as one of the following:
<J)(X) = p(X)s(X)
(J)(X) = B(X)s(X) (2-2)
<|)(X) = T(X)s(X)
The tristimulus values R, G, and B are, of course, dependent
on the spectral power distribution of the source. The C L E . has
developed standards for sources or illuminants so that tristimulus
values can be reproduced correctly. The power spectrums of standard
illuminants A, B, and C are shown in Fig. 2-3. Illuminant A
approximates a tungsten filament lamp, illuminant B is equivalent
to noon sunlight, and illuminant C is equivalent to average daylight.
When compared visually, illuminant A appears slightly yellowish,
while illuminant C appears slightly bluish..
The three "chromaticity coordinates" r, g, and b for an
illuminated object are defined as dimensionless ratios:
— R _ G , _ B 9_'5"\ ^ ~ R+G+B ^ ~ R+G+B " R+G+B ^ ^
where R, G, and B are defined by Eq. (2-1). Since r+g+b = 1.0, the
value of any single unknown coordinate can be obtained by knowing the 18 values for the other two.
As shown in Grassmann's Laws, no two primary colors can be added
to match the third. It follows that the three primaries form a 19
linearly independent set of vectors as expressed by the following:
RR + GG + BB 7 0 (2-4)
Assume that scalars R, G, and B do not all equal zero. Thus, the
linearly independent vectors R., G_, and B form a set of basis vectors 20
for a three-dimensional vector space. Any color vector Q which
falls within the limits of the space can be represented as a vector
addition of the three primaries, as indicated in Fig. 2-4, and
described by:
^ = RR + GG + B^ (2-5)
Also shown, the three-dimensional space enclosed by unit values of
8
200
Relative Spectral Irradiance
100
400 500 600 Wavelength X
700 (nm)
Figure 2-3. Relative Spectral Irradiance (Power) for Standard Illuminants A, B, and C
B
Figure 2-4. The (R,G,B) Color Solid.
R
the three primary vectors is known as a "color solid."
Grassmann's second and third laws imply that different sets of
primaries yield different tristimulus values that are linearly 21
related. In terms of a three-dimensional vector space, this
corresponds to a change of basis. It follows then, that a single
color vector Q has different representations with respect to
different bases. For example, basis may be changed to a new
basis 3 by the matrix transformation:
B = [A]3 (2-6)
22 where [A] is a square matrix of the same dimension as .
For the (R,G,B) color space to be changed to a color space
defined by the basis vectors X, Y_, and Z, the linear relationship
between the spaces is expressed by:
Y = a .R + a G + a^jB
Z = a R + a22G + a^^^
(2-7)
where the values for R, G, B and X, Y, Z are scalar multipliers as
in Eq. (2-5).
Coefficients {a ,} are the amounts of the X, Y and Z primaries i,l
required to match the color vector with elements R=l, G=0, B=0; i^^ 2^
are the amounts needed to match the vector with R=0, G=l, B=0, etc.23
The matrix form of the equations is given by:
X
Y
Z
= [A]
R
G
B_
where [A] =
1 ^11 ^12 ^13
^21 ^22 ^23
^31 ^32 ^33J
(2-8)
To go from the (X,Y,Z) space to the (R,G,B) space, the inverse
equation is used:
10
R
G
B
= [A] -1
X
Y
Z
(2-9)
The conversions between bases are very useful for describing
physical tristimulus systems, since a given color can be expressed
in terms of any tristimulus system simply by a change of the basis
vectors.
2.2 Models for Tristimulus Systems
Using matrix transformations, a number of tristimulus systems
have been derived from the original (R,G,B) system. The C L E .
(X,Y,Z) system of 1931 is of particular interest here, as it allows
changes in luminance to be separated from changes in color.
Fig. 2-2 shows that the color matching functions of the (R,G,B)
system are negative for some wavelengths. The result is that the 24
tristimulus values obtained for the system may be negative. To
overcome this and other drawbacks, the C L E . made a change in the
basis vectors to obtain a system that would contain only positive
tristimulus values. The resulting (X,Y,Z) system has a set of color 25
matching functions that are shown in Fig. 2-5. The tristimulus
scalar values for primaries X, Y , and Z that form the basis set for
this system are given by:
X = k/(t)(X)x(X)dX
Y = k/(f)(X)y(X)dX
Z = k/(|)(X)z(X)dX
(2-10)
Now k is chosen so that variations of the amounts of X and Z affect
the color of the match to a particular object but leave any difference
in luminance unchanged. It follows that changes in Y represent 26 27
changes in luminance of an object. * These conditions are met
when:
11
• = ^ Tmw "' (2-^^)
where s(X) is the spectral power distribution given in Eq. (2-2).
The separability of luminance from color will be discussed in detail
in Sec. 2.3.
The chromaticity coordinates associated with the (X,Y,Z) system
are expressed as dimensionless quantities:
— X _ Y _ Z /oiox
^ ~ X+Y+Z ^ " X+Y+Z ^ " X+Y+Z u-i^;
Since x+y+z = I, the entire x,y,z color solid may be conveniently
described by an x,y slice of the chromaticity space. Such a slice,
known as a chromaticity diagram, is shown in Fig. 2-6. The range of
visual colors lies within the curved figure contained in the diagram.
The figure is bounded on its curved portion by the locus of spectral
(single wavelength) colors, where the wavelengths are shown on the
figure. (The straight line that connects the ends of the locus
corresponds to the visual limits of the purple colors, which are
not associated with any single wavelengths.) The chromaticity
coordinates for these monochromatic stimuli are given by:
/A^N _ x(X")
x(X ) = =-x(XO + y(X') + i(x-)
y^^^^ = (XO + y(A') + z(XO ^ ' ^
.... Z(XO z(^ ) = 1=: x(X') + y(XO + z(X')
where X^ is the wavelength associated with a given spectral color,
while x(X'), y(X'), z(X') are the color matching functions at that u 28 wavelength.
In order to more clearly illustrate the separability between
luminance and color, it is useful to consider the cylindrical (I,H,S)
coordinate space shown in Fig. 2-7. This space, proposed as a
Relative Power
1 s
1 ,0
0.5
0
zCk)
yM ^ x(X)
— ~ — — ~ 12
400 500 600 Wavelength X
700
Figure 2-5. Spectral Responses for Color Matching Functions x, y, and z
0.8
0.6
0.4
0.2
A520
k 500
M80
460 \ ^
560 2^
5 8 0 ^ ^
V 6 0 0
\
0.0 0.2 0.4 x
0.6 0.8
Figure 2-6. CIE 1931 (x,y) Chromaticity Diagram (wavelengths marked on the spectral locus are in nm).
13
representation for image processing, corresponds very closely to the
Munsell color system, which describes color in the psychological 29 terms of hue, brightness (or intensity), and saturation.
By definition, hue is the attribute of visual (color) perception
which has given rise to traditional color names such as blue, green,
yellow, etc. Saturation is the attribute which permits a judgment
to be made on the proportion of a pure chromatic color in the total 30
perception. Any totally desaturated color is white or gray, while
monochromatic (spectral) colors are at maximum saturation. Brightness
(used interchangeably with luminance or intensity, here) is related
to the perceived intensity that is radiating or reflecting from a
colored object. Evjen though these terms are psychological rather
than objective, they will be used freely to describe what the
observer would perceive while experimenting with color television
signals.
2.3 The N.T.S.C. Tristimulus Systems and Color Television
Because this work is so closely linked to color television tech
nology, the primary color system established by the N.T.S.C (National
Television Systems Committee) is of great importance. Equally
important are the N.T.S.C. systems for the transmission of television
pictures. These systems are arranged much like the (I,H,S) system
of Fig. 2-7, as they separate a luminance signal from two chrominance
signals.
The N.T.S.C. receiver primary color coordinate system is based
on a set of the three (red, green, and blue) primary phosphors in a
color CRT (Cathode Ray Tube) that have spectral responses similar to 31
those shown in Fig. 2-8. The N.T.S.C. primary color unit vectors,
R , G , and B are described by the following x,y,z chromaticity n n n coordinates:
X = .670 y = .330 z = .000 for R • _n
X = .210 y = .710 z = .080 for G (2-14) n
X = .140 y = .080 z = .780 for B n
A Intensity Axis 14
Constant Saturation
Constant Intensity
Constant Hue
(a) Intensity, Hue, Saturation (I,H,S) Color Space.
Saturation
(b) A constant Intensity Slice of the (I,H,S) Color Space. Figure 2-7. The Intensity, Hue, Saturation (I,H,S) Color Space .
15
32 where x+y+z = 1.0. These positions appear on the x,y chromaticity
diagram as shown in Fig. 2-9, where the range of colors that can be
reproduced by the N.T.S.C. tristimulus system appears inside the 33
triangle. These primaries yield a matrix conversion to a (R ,G ,B ) n n n
tristimulus space: n n n
R n
G n
B - n J
=
1.910
- . 9 8 5
_ . 0 5 8
- . 5 3 3
2 .00
- . 1 1 8
- . 2 8 8
- . 0 2 8
.896.
X
Y
Z.
(2-15)
where R , G , and B are scalars associated with vectors R , G , and o/ n n n _n _n
B . n
Q = R R + G G + B B — n n n n n n
(2-16)
For ease of transmission and compatability with monochrome
receivers, the (R ,G ,B ) primary system is converted to another n n n
tristimulus system. In this new system, the following criteria are
met:
1. One signal (Y) must be proportional to luminance.
2. Subcarrier (color) signals (two in number) do not affect
luminance.
3. The two subcarrier signals become zero for white (referred
to the given definition of white corresponding to illuminant
C of Fig. 2-3).
The result is a color space that is analogous to the (I,H,S) space
shown in Fig. 2-7. One signal is arranged to be proportional to
luminance while the two (color) subcarriers are modulated in
quadrature. The unipolar voltages corresponding to the R , G , and B signals are transformed to the unipolar (Y) luminance voltage n
signal while the color subcarrier signals (I and Q for n-phase and
quadrature) are bipolar. The resulting (Y,I,Q) tristimulus space
is shown in Fig. 2-10. Note its similarity to the (I,H,S) space of
Relative Power
16
Blue
/
/
/
/ / ' \
Red
/ .^^
400 500 600 Wavelength X
700 (nm)
Figure 2-8. Spectral Responses for Typical Television Phosphors
Figure 2-9. NTSC Color Range shown on the (x,y) Chromaticity Diagram (wavelengths marked are in nm).
17
Figure 2-7. The matrix conversion from the (R , G , B ) space to n n n
the (Y,I,Q) space given by:
299
596
211
.587
-.274
-.523
R n
n L\i
(2-17)
Note that the Y signal is the same Y value that defines changes in 37 —-
luminance in the (X,Y,Z) system.
The conversion from the (R ,G ,B ) voltages to the (Y,I,Q)
voltages is done electronically inside the color television camera.
There is an intermediate step, however, where the (R ,G ,B ) signals n n n
are converted to the luminance signal Y, and the two color difference signals, R -Y and B -Y. The R -Y and B -Y axes simply correspond to
n n n n ~Q a rescaling and rotation of the I and Q axes as shown in Fig. 2-11. The equations describing this transformation are: 39
R -Y n B -Y •- n -•
1 0
0 .956
0 -1.106
0
.621
1.703
Y
I
.QJ (2-18)
By substituting and multiplying Eq. (2-17) and Eq. (2-18) the relation
between the (R ,G ,B ) system and the (Y,R -Y,B -Y) system is found: n n n " n n
so t
Y
R -Y n B -Y •-n -
hat:
T 0
0
0
.956
-1.106
0
.621
1.703_
.299
.596
_ 211
.587
-.274
-.523
.114
-.322
.3I2_
R n G
B" ' n
(2-19)
R -Y n B -Y n
LI
299
701
300
.587
-.587
-.588
.114
-.114
.887
R n
(2-20)
Y A
--+1.0
Luminance
-1.0 -1.0
•Hue
18
Figure 2-10. The (Y,I,Q) Color Space.
R^-Y /I.14
+1.0 < ^ B-Y /2.03
n
Fieure 2-11. A (R -Y, B -Y) Constant Intensity Plane. " n n
19
n Equation (2-20) now J^fines a theoretical linear mapping from the (R^,
jj» j ) color generation space of a color television monitor to the
(Y» ^n~^' n~^^ jdetection signal space of a theoretical color camera.
The inverse is given by:
R n G n B - n-
=
1.000
1.000
l.OOI
1.000
-.509
.0004
.000
-.194
.992
Y
R -Y n B -Y ^ n -•
(2-21)
It is important to consider both the restrictions on color signal
range and the restrictions on luminance in the N.T.S.C. tristimulus
color systems. It has already been shown, as in Fig. 2-9, that not
all visible^colors can be reproduced by the N^T.S.C primaries. Also
shown, the (R ,G ,B ) system cannot have a negative stimulus, which n n n
establishes the (R ,G ,B ) signals in a color camera as positive. In n n n
normalized form, the R , G , and B signals may each be considered n n n
to range from zero to one. The tristimulus color solid is then a 40
cube in (R ,G ,B ) space as shown in Fig. 2-12. In the (Y, R_-Y, n n n n B -Y) space this cube becomes a parallelepiped, as shown by Fig. 2-13.
For different colors, different ranges of luminance can be obtained,
as the color and the luminance must always be inside the color solid.
This will be examined for specific cases in relation to number
scaling for incoherent processing in Chapter 3. It will also be
studied experimentally.
In a typical television camera, diagrammed in Fig. 2-14, imaging 41
tubes are used as image pickup devices. A colored filter is
placed in front of each tube to match its color response to the
chromaticity coordinates of the R , G , and B primaries. In a •' n n n
single tube color camera, the filters are in front of the three
electrodes in the tube that yield color channels. The signal
voltages from the camera tubes are amplified and passed through a
cross-connected voltage divider to obtain the desired matrix
conversion. In most cameras there exists considerable latitude
Magenta
R =1 n
20
Yellow
Figure 2-12. The (R ,G ,B ) Color Solid in the (R ,G ,B ) Space, n n n n n n Yellow
Green White
Figure 2-13. The (R ,G ,B ) Color Solid in the (Y,R -Y,B -Y) Space n n n n n
Gamma-
r Compensated Signals
21
R n
R
1 Imaging Tubes
Colored Filters
n
B n r-
^_^ .IIB
Gamma Compensating Amps
__ Red
f. _ _ Green
_ _ Blue
Image Gathering-Optics
Y Matrix
n
_> 59G — > n
I
.30R 1 n
~.-^^ Y=.3R +.59G +.11B I n n n
1
n ' -> Add B
I
R -Y n
Figure 2-14. Typical Color Television Camera Block Diagram.
22
in the adjustments that can be made to the matrix. The resulting
color difference signals are used to obtain the subcarrier signals
(I and Q) which are quadrature modulated and combined with the Y
signal into a composite signal. The composite signal is then
transmitted.
In a typical television receiver, the composite signal is
received, demodulated, and the reverse matrix transform is performed.
With the signals now converted back to (R ,G ,B ) space, the red, n n n
green and blue phosphors in the color CRT are stimulated in appropriate
proportion in small "triads" on the face of the CRT to give a color
picture display. If there is a direct cable connection between the
camera and receiver without radio frequency modulation, the receiver
is called a monitor.
In this work, the camera-monitor system will be treated, to the
extent possible, as a linear system. In fact, however, there are many
nonlinearities present both in cameras and in monitors. In the
camera, the pick-up tubes typically have response curves as shown 42
in Fig. 2-15. The response of the phosphors in a color monitor 43 is also nonlinear as shown in Fig. 2-16.
To model nonlinearities in a color television system the concept
of a component's "gamma" is employed. The gamma associated with a
component's response is the exponent of a number describing the
slope of the transfer characteristic at a specified voltage level.
By the same definition, the value for gamma also describes the
slope of a log-log response curve. Thus, a gamma of one defines a
linear response. A working value for the gamma of the vidicon tube
which is commonly used in color cameras is around 0.65, while the 45 gamma of the color CRT is usually adjusted to around 2.2. The
matrix conversion of Eq. (2-20) for a color camera then becomes:
44
R ^/^-Y n
B ^^-Y L-n
299 .587 .114
701 -.587 -.114
-.300 -.588 .887
R 1/Y n
I/Y n
B 1/Y n
(2-22)
Output Voltage (Normalized)
Input Light Intensity (Normalized)
Figure 2-15. Vidicon Camera Tube Response.
Output Light Intensity
/
t t
Input Voltage
Figure 2-16. Color CRT Response.
24
The inverse matrix for a color monitor is changed similarly. Because
this research required transforms on the signals from a color camera
before they are passed on to the receiver, gamma will have to be
corrected, as much as possible, to a value of one.
As shown in Fig. 2-14, this correction can be done electronically
in the camera. Electronic adjustments may also be made in the CRT
monitor. Adjusting the brightness or "cutoff" controls translates
the curves of Fig. 2-16 while adjustment of the gains of the
individual red, green, and blue guns in the CRT will change the
slope of the curves.
CHAPTER 3
COMPLEX NUMBER ADDITION AND MULTIPLICATION
As shown in Chapter 2, a range of colors at different luminance
levels may be described as vectors in a three-dimensional tristimulus
vector space. It follows that these vectors, when appropriately
represented by signals from a color television camera, may be used
in various combinations to represent complex numbers and associated
complex mathematical operations. The two operations of interest here
are complex number multiplication and complex number addition.
Tristimulus methods for achieving these operations will be discussed
and examples of matrix transforms between the (R ,G ,B ) color n n n
generation space and the (Y, R -Y, B -Y) detection space will be n n
presented for each operation. Scaling, dynamic range limitations,
and separability of bases will also be discussed.
3.1 Complex Number Multiplication
The complex multiplication of two pixels may be performed most
easily when those pixels represent complex numbers in polar form.
The choice of a polar-form representation follows from the fact that,
while complex (vector) addition follows directly from tristimulus 47
methods, complex multiplication does not. If the two polar-form numbers are expressed as A/9 (for Ae**^) (for Be-*^) the product of
the complex multiplication of the numbers is described by:
C ^ = A/^ X B ^ = AB/e+(l) (3-1)
In order to do complex addition, the moduli or amplitudes, A
and B, must be expressed and multiplied completely independently of
the angles [Q_ and /^, which must be added. Since A, B, and their
product, C, are positive real quantities, it follows that the unipolar
Y luminance signal from the television camera should be used to
represent both the moduli of the complex numbers to be multiplied
and the modulus of their product. One of the two bipolar color
25
26
difference signals could then be used to represent the positive and
negative values of the angles of the complex numbers. Angle addition
then takes place along the two-hue number line which is defined by
either of the color difference signals (B -Y or R -Y) when they are n n ^
operated in a constant luminance (Y) plane in the (Y, R -Y, B -Y) n n
tristimulus space. These procedures will now be explained in detail with the aid of figures. The addition of the angles will be described
first.
When a constant luminance value is maintained, the three-
dimensional (Y, R^-Y, B -Y) space is reduced to a two-dimensional
slice as in Fig. 3-1. In this "constant luminance" or "constant
intensity" plane, each color may be expressed as a vector combination
of the B^-Y and R^-Y color difference signals as shown in Fig. 3-2,
where
Q = (R -Y) R -Y + (B -Y) B -Y (3-2) — n n n n
where (R -Y) and (B -Y) are scalars while R -Y and B -Y are the n n n n
bipolar unit vectors shown in Fig. 3-2. If only one of the two
signals is used (while the other signal is zero), this plane may be
further reduced to a single two-hue number line as shown in Fig. 3-3.
Along this line, positive number values are represented by various
saturations of one hue, while negative values are represented by
the complementary hue. A complementary hue is one that, when
combined with an equal saturation of some other hue (its complement),
yields the desaturated colors white, gray, or black (depending on
the luminance values at which those colors appear). For the number
line defined by the R -Y color difference signal, increasing n
saturations of the hue magenta correspond to increasing positive
signal values while increasing saturations of green correspond to
increasingly negative values of the signal. For the number line
defined by the B -Y signal, positive values are represented by a
blue hue while negative values correspond to the complementary
27
Planar Slice (Y=0.5)
Figure 3-1. Constant Y Plane in the (Y,R -Y,B -Y) Color Space n n
-1
+1 '
(R -Y) n
-r
. R -Y ^ n
Q
(B -Y) n
<
+1 V
—T > B -Y n
Figure 3-2. Vector Addition on a Constant Y Plane.
Increasing Saturation
Yellow White Blue
->. B -Y n
-1.0 0.0 +1.0
-^ Increasing Saturation
Green/Cyan
-1.0
White
0.0
Red/Magenta
-> R -Y n
+1.0
Figure 3-3. Number Lines Represented by the B^-Y and R^-Y Color Signals .
28
29
yellow hue. For either number line, a value of zero for the color
signal is equivalent to a reference value for white or gray,
depending on the value for Y.
For angle addition, each of the angles 9 and <j), indicated in
Eq. (3-1), must be scaled so that any angle in the four quadrants
of a complex number plane may be expressed. Since R -Y and B -Y are n n
each bipolar, a good choice for scaling is to express the positive
and negative maximum values of the signal as ranging from +7r radians
to -IT radians, respectively as shown in Fig. 3-4a. Thus, either
of the color difference signals may be chosen for the angle addition
operation.
Angle addition may be accomplished by using a 50/50 beamsplitter,
as shown in Fig. 3-5. In this configuration, a spectrally flat beam
splitter is used so that all colors are attenuated equally. Thus,
the intensity of each beam is attenuated by a factor of 2. It
follows that one-half of the light or one-half of the stimuli from
each of the sources is lost while passing through this "ideal"
beamsplitter. The result is that the Y signal remains at the constant
value that has been determined by each of the two sources when they
operate individually at a constant intensity (Y). Since all stimuli
have been attenuated equally, the color signals are reduced to one-
half of the original values that would be determined by each of the
two sources. For example, if source 1 and source 2 of Fig. 3-5 are
operated individually at the following points when not attenuated
by the beamsplitter;
r R -Y
n B - Y j
^ n -•
0.500
0.000
0.500
R -Y n B -Y •- n -'
0.500
0.000
ig.250_
(3-3)
then the camera will detect the following values when these sources
are operated together through the beamsplitter:
30
Ye
V
-IT
11 ow White
0 .0
Blue
+ r ^V^
Gre'en/Cyan White Red/Magenta
• > R - Y n
- I T 0.0 +rr
(a) Scaling (in Radians) for Single Angles
Yellow White Blue
^ B ^ - Y
-27T 0.0 +27T
Green/Cyan White
-2TT 0.0 (b) Scaling (in Radians) for Angle Sums
Blue
^ R -Y n
+27T
Figure 3-4. Angle Scaling on the B -Y and R -Y Number Lines.
Each Signal Attenuated by a factor of 2.
^
r •' Source L. #1. ^
• '
//
' 1
1 / ^
' ,.—-
Source //2.
-50/50 Bear
Constant Y
Figure 3-5. Angle Addition Using a Beamsplitter
31
R R -Y n B -Y L-n -• 1+2
0.500
0.000
0.375
(3-4)
Thus the B -Y signal is equal to one-half of the value from source I
plus one-half of the value from source 2.
As shown by Fig. 3-4b, the scale for the summed angle must be
doubled to accommodate all possible angle sums. Thus, the value of
B -Y in Eq. (3-4) is one-half of the sum of the two sources, but that
attenuation is negated when the scale is doubled, as will be shown
by the following example. Included in the example are transform
matrices and scaling.
The complex number values, 9 and cj) of Eq. (3-1) are assumed to
be 9 = -7r/6 radians and (f) = +57r/6 radians. The B -Y color signal is n
assumed to be scaled such that a value of B -Y = I corresponds to n
Ti radians and B -Y = -I corresponds to -TT for each individual angle. n
The chosen Y value will be unity for both angles. The vector
representations for 9 and (j) are given by: Y
R -Y n
B -Y •-n -
=
9
1.000
0 .000
j - 0 . 1 6 7
• *
Y
R -Y n
B -Y ' -n -
=
*
1.000
0 .000
_ . 8 3 3
9 = Tr/6 (j) = 5TT/6
(3-5)
and are indicated in Fig. 3-6. Using the matrix formation of Eq.
(2-21), the values for the (R ,G ,B ) stimuli of the color monitors n n n
of Fig. 3-6 are given by:
R n
G n
B n
=
9
1.000
1.032
0 .834
R n
n
n <})
1.000
.838
1.8311
(3-6)
When the stimuli from sources I and 2 are added by the 50/50 spectrall V
32
flat beamsplitter of Fig. 3-6, the result at the color camera has
the form:
R n
n B n
= L
Y=9+(()
R n
n B n 9
R" n
G n B _n
=
-e-
I.OOO
.935
1.334
(3-7)
The result of Eq. (3-7) is transformed back into the (Y, R -Y, n
B^-Y) detection space as in Eq. (2-20) to yield the signals from the
color camera in Fig. 3-6 for "H, the sum of the two angles. Now the
\ - ^ signal is scaled such that B -Y = 1 yields a sum angle of 2
radians, while B^-Y = -I corresponds to -2-n. The results are given
by the relationship:
R -Y n
B -Y L n -J
4*
1.000
0.000
0.334
(3-8)
where:
»f = 9+c|) = - T T / 6 + 57T/6 = 477/6
(B -Y)„, = . 3 3 4 X 277 ~ 477/6 n 4'
(3-9)
Note that R -Y is again zero, so the angle addition takes place along
the B -Y line only. (It could optionally take place along the R -Y
line, also.) According to Eqs. (3-5) through (3-9), the desired
addition of angles is achieved (within the truncation errors of the
calculator), by the constant Y addition scheme of Fig. 3-6. Note
also that the resultant Y is still at the same value as for the two
original sources described by Eq. (3-5). Thus, the optical path for
angle addition is called a "constant Y" path in Fig. 3-5 and in Fig.
3-6.
33
9= -77/6 R -Y B^-Y
n
1.000 0 .000 - . 1 6 7
Color Camera
=
9
1.000 1.032 0.834
(j)= +577/6
Y R -Y B^'-Y
n _
¥ G"
B " L_n J
=
-e-
=
<|)
1.000 0 .000 0 .833
1.060 0.838 1.833
+ |_nj (})
Y R -Y B -Y n n
[Y R -Y B'^-Y
n _J
=
^
1.000 0 .000 0.334 1— —1
w h e r e \p=Q+<^ and (B -Y) ,= y i e l d s 277.
= 477/6
1.0
Figure 3-6. Angle Addition Example.
34
The moduli or amplitudes of the two complex numbers introduced
in Eq. (3-1) are multiplied on a separate optical path as shown in
Fig. 3-7. In this arrangement, the light from the constant Y path
is sampled by the beamsplitter to form a separate optical path.
Again, the beamsplitter must be spectrally flat, so that all colors
remain at a constant Y value. The back-to-back filters in the
separate optical path are also spectrally flat. Thus, the intensity
transmittances of the filters, normally expressed as x (X) and T^(X),
are now expressed as T, independent of X. T and x , constant for
all colors, are now scaled to be proportional to the amplitudes of
the polar form complex numbers to be multiplied. A and B are then
expressed as numbers between 0 and 1.0, so X- and x„ can be scaled
to express A and B directly. For example, if A = 0.5 and B = 0.9,
then X and x„ are scaled as X- = 0.5 (50% intensity transmittance)
while X = 0.9 (90% transmittance). The result follows:
C = AB = X X = 0.5 X 0.9 = 0.45 (3-10)
The result may be detected by a wavelength integrating detector
such as a black-and-white television camera. The detector must be
spectrally compensated so that different colors are weighted equally.
Some arbitrary maximum value of Y is scaled as a maximum output of
one, so the accuracy of the multiplication is dependent on the dynamic
range of the detector. Of course, the detector output must be linear
with respect to the input intensity, or it must be linearized by
electronic means.
An alternate scheme is shown in Fig. 3-8, where a separate
source is used instead of sampling the light from the constant Y
path as in Fig. 3-7. This approach has an advantage in that the
constant Y path is no longer disturbed by possible aberrations or
spectral changes induced by the insertion of a beamsplitter. The
separate path for multiplication in turn, is not affected by any
changes in Y that may erroneously occur in the constant Y path.
Mirror
Intensity 1=1.0 K
B.S.
Constant Y Path for Angle Addition
t ± JL
To Color Camera
I=C= AxB= T xx2= 0.45
— -»• Y = 1= 0.45 P
Figure 3-7. Multiplication of Complex Number Moduli
\
Intensity 1=1.0
Constant Y Source
T = A= 0.5
I=C= AxB= 0.45
•M—
X2= B= 0.9
Figure 3-8. Alternate Multiplication Scheme.
B&W Camera
I
Y = I P
= 0.45
36
Also, the requirement for the spectrally flat filters and detector
may be relaxed by using a source with a relatively narrow bandwidth.
By the separate manipulation of color as in Fig. 3-6, and
intensity or luminance as in Figs. 3-7 or 3-8, complex polar form
multiplication may be achieved. The need for separate optical paths
for the two operations, and proper scaling for color addition will
be discussed in Sec. 3.3.
3.2 Complex Number Addition
Complex numbers may be added in rectangular form by vector
additions on a constant Y slice of the (Y, R -Y, B -Y) color space. n n
As with the addition of angles on a number line, complementary hues
represent positive and negative values. For complex addition, the
two-hue number lines of the B -Y and R -Y signals are used to n n ®
represent the real and imaginary lines of the complex number plane.
As with the angle addition of Sec. 3.1, the sum must be scaled
according to the number of vectors that are to be added.
A constant Y slice of the (Y, R -Y, B -Y) space is again shown n n
in Fig, 3-9. When the R -Y line represents the bipolar imaginary
number line of the complex number plane and the B -Y line represents
the bipolar real number line, the complete four-quadrant complex
number plane may be represented, where a given complex number is
represented as a vector on the plane. The real and imaginary (Re
and Im) components of the complex number values fall between +1 and
-1. The operation of rectangular-form complex number addition may
be represented as in Fig. 3-10. In this scheme, each complex number
is represented by a pixel on a display such as a color CRT. A vector
value in the B -Y, R -Y plane, representing a vector in the complex n n
number plane, is assigned to each of the pixels on the display.
These pixels are summed optically in two spatial dimensions by a
lens as shown in Fig. 3-10. The lens must have minimal chromatic
aberration for good results.
If an entire array of pixels to be summed is focused down to
37
j l . O
- 1 . 0
3 -Y»Re
H 1.0
-.—-,' fnr ^rrr'-^^r Pl^^n?: ^
38
the size of a single pixel, the intensity of the summation pixel
is increased by the same factor as the number of pixels to be
summed. (Assume each pixel is the same size.) Thus, a spectrally
flat attenuator can be used to compensate for this intensity build-up.
Since this is equivalent to a scalar multiplication of the stimuli,
the complex vector equivalent to the sum of the pixels is also scaled
by one over the number of pixels to be summed.
As an example, consider the addition of the four rectangular-
form complex numbers of Eq. (3-II). This example is illustrated in
Fig. 3-II.
75 + jl + .5 - j.25 + -.25+J.5 + -.25+J.75 =
a + b + c + d
= .75 + j2 - (3-11) e
In terms of the (Y, R -Y, B -Y) representation, R -Y = B -Y = ±1 n n n n
corresponds directly to Im = Re = ±1, where Im and Re are the real
and imaginary parts of a given complex number. The vector values
for a through d with Y set at a nominal value of unity are given
by:
Y
R -Y n
B -Y _ n _
=
a
I..06
1.00
J).75
»
Y
R -Y n
B -Y ^ n -
=
b
~ 1 . 0 0
- 0 . 2 5
J) .50
Y
R -Y n
B -Y _ n _
c
""l .O(f
0 .50
zP-25_
• »
Y "
R -Y n
B -Y _n __ d
" i .od
0.75
zP-25.
(3-12)
In the (R ,G ,B ) monitor of Fig. 3-11, these values, when transformed n n n
as in Eq. (2-21), are found to be:
Constant Y 39
R --H n
G --H n B --H n
Color Monitor
Summing Lens
Attenuator
Figure 3-10. Complex Number Addition Scheme
Color Monitor
(Y=1.0 f
B -Y= 0.19
= 0.f5"
R -Y= 0.5 n
e^ = 2.0 Im
Color Camera
At Attenuator:_ e= i2;(a+b+c+d) Y=1.0 for each pixel
Figure 3-11. A Complex Number Addition Example (See Eq. 3-11 to 3-16)
40
R n
n B n
R n
n B n
2 .00
0 .35
1.75
• >
R n
G n
B _ n "-"-•b
i.5o' 0 .79
p.75_
• >
R n
G n
B
0.75
1.03
1.50
1.75
0.67
0.75
(3-13)
When these are added together and attenuated by a spectrally flat
device at a factor of one over the total number of pixels, the
(R ,G ,B ) representation has the form: n n n
R~ n
G n
B
=k
e [
R" n
G n
L\
+
a
R" n
G n
B _n
+
b
R " n
G n
B Ln
+
c
R~ n
G n
B _ n,
d
^
1.50
0 . 7 1
1-19
(3-14)
When reconverted by the (Y, R -Y, B -Y) detection space that • n n
describes the output of a color television camera, the result is
given by:
R -Y n B -Y Ln _. U -I Q
LOO
0.50
O.I9J (3-15)
In order to accommodate all possible sums, the resultant e vector
is now scaled such that R -Y = B -Y = ±1 corresponds to complex number n n
components Im = Re = ±4, respectively. Equation (3-15) then yields
the desired rectangular-form complex number from Eq. (3-10):
e„ = 0.75 Re
e^ = 2.00 Im
(3-16)
41
A comparison of Eq. (3-12) and Eq. (3-15) shows that the Y value
remains constant at unity.
One drawback to this method is immediately apparent. When a
large number of pixels are to be summed, the dynamic range becomes
compressed by the number of pixels to be summed. For example, if
there are many complementary numbers cancelling each other in the
summations, the result may be a very small value. This value is
attenuated by a factor of I/N, where N is the number of pixels to
be summed. When N is very large, that value could, in a physical
system, be down in the noise range of the detector. Even at larger
values, the accuracy of the sum will be limited as a function of the
number of pixels to be summed and as a function of the dynamic range
of the detector. This is analogous to the "bias build up" problems
that are encountered in multiplex holography or in bipolar incoherent 48, 49
processing schemes. * This problem and some alternative
solutions will be discussed in later chapters.
3.3 Range Limitations and Separability of Coordinates
The reader may have noticed two apparent contradictions in the
analyses of Sections 3.1 and 3.2. First, in Eqs. (3-6), (3-7), (3-12),
and (3-13), the stated limit that R , G , and B must each be between n n n
0 and I.O (see Fig. 2-10 and Fig. 2-11), has been exceeded. Second, while R , G , and B form a set of independent, orthogonal vectors,
n n n the matrix transform, Eq. (2-19) yields a set of vectors which are
not orthogonal with respect to R , G , and B . The result is that -JL -JL -JL
physically, the luminance signal, Y, may not be manipulated without
some effect on the color signals. This implies the physical necessity
for the separate optical path of Figs. 3-7 and 3-8.
Since the values of R , G , and B must each be between 0 and n n n
1.0, certain limits are placed on the ranges of color difference signals, R -Y and B -Y, at a given Y value. These limits are
n n
equivalent to those set by the color solid mapping of Fig. 2-12,
but are more clearly pictured as a set of planes for arbitrary
42
constant Y values as in Fig. 3-12. In each of these illustrations,
the maximum ranges of the color difference signals are found by
choosing the Y value to be used, then calculating the values for
the R -Y and B -Y signals that may be obtained without exceeding
the limits on R , G , and B . The optimum value to use for the n n n
constant Y is that value which yields the maximum range for B -Y
and R -Y, thus maximizing the dynamic range for the complex or
bipolar addition operation. It is helpful if this range is also
symmetric about zero so that the negative maximum is the same as
that for positive values. This condition is shown in Fig. 3-12 for
Y = 0.5, where R -Y = B -Y = 0.5 at their positive and negative n n *
maximum values. The plots in Fig. 3-12 were found by testing the limits on R , G , and B for a given value of Y and various values
n n n of R -Y and B -Y. For angle additions as in Eqs. (3-3) and (3-4),
n n the maximum values of R -Y = ±0.5 correspond to (|) = 9 = ±77 and
n
^ = ±271, respectively. For two-dimensional vector summations such
as Eqs. (3-12) or (3-14), those maximum values for R -Y and B -Y
correspond to complex components Im or Re, respectively, at the
maximum value of ±1.00 for the terms of the summation.
In an actual television camera, many different values may be
used to scale output signals. Also, the gains of the R , G , and B
color channels may be changed electronically. It follows that the
matrix transform between (R ,G ,B ) and (Y, R„-Y, B -Y) may be changed n n n n n
from that shown in Eq. (2-20). Thus, while the forms of the examples
presented in this chapter are correct, the actual matrix values for
a physical system may be changed.
Another difference between the theoretical examples presented
here and the operation of a physical system is the fact that the
values for Y, and those for R -Y or B -Y may not be physically n n
manipulated in an independent fashion, as will be shown in the
following discussion.
The separate optical path for the complex amplitude multiplica
tion "synthesizes" an orthogonality between Y, which is used to
R -Y Y=.05 +1 -T- n
-1
-1
f ^ +1 . B -Y i n
43
Y=.125 +1 R -Y
-r- n
-t
-1.
:3?i^n-^
Y=.25 + 1 T R -Y n
-1
-I
B -Y n
+1
Y=.375 +l-r R -Y n
-1
-1
B -Y n
+1
Y=.5 +l-rV^
-1
-1
B -Y n
+1
Y=.625 +l-r n
-f
-1
B -Y n
+]
Y=.75 +1T V ^
-1
-1
•+T B -Y n
^ ^ - II K. — 1 Y=.825 + 1 T n
^
-1
-1
+1
B -Y _Q
Figure 3-12. Color Ranges of the (R ,G ,B ) Space for Constant Y Values.
44
multiply amplitudes, and the color signal used to add angles. For
clarification, again consider the operation of a neutral density
(spectrally wide-band) attenuator, beamsplitter, or aperture stop
on a camera. Recall that it attenuates all primary stimuli
(Rj »G »B ) equally, so it corresponds to a scalar multiplication
as in Eq. (3-7) or Eq. (3-14). Thus, it attenuates the color
difference signals and the Y signal. The result, for a physical
system, is the color space of Fig. 3-13. Provided the Y value is
still below the electronic saturation level of the detector where
the color signals break down, the greater the value for Y is, the
greater the magnitude of the color signals will be, up to the level
where Y is no longer attenuated, (Y = I). In order to change Y
without changing B -Y or R -Y, any filter that attenuates Y would n n '
have to enhance the saturation of the color associated with the
initial value of Y, or else the generated values for the color
difference signals would have to be enhanced, a_ priori, according
to the attenuation to be assigned to a given pixel. The general
color-enhancing filter of the first case is not realizable, while
the a_ priori enhancement of colors would require extra processing.
The usable dynamic range is also greatly reduced in all dimensions,
since the usable range of signals would have to fall within the
cylinder inside the solid of Fig. 3-13. By using the separate
optical path for manipulation of Y and carrying out all additions
on a constant Y path, as in Fig. 3-5, these problems can be avoided.
Considered in terms of statistical processes, the covariance of unit vectors Y and B -Y may be used to define the inner product of
- ^ 50 the (Y, B -Y) part of the (Y, R -Y, B -Y) vector space. ^ n n n
E {Y, B -Y} = <Y, B -Y> (3-17) — n n
When this inner product is zero, the vectors are orthogonal,
geometrically and statistically. Since a tristimulus space is
an inner product space, the axioms of inner products require the
45
Saturation Point for Y
Green
Usable Range with Compensa
Figure 3-13. Physical Representation for the (Y,R^-Y,B^-Y) Space
46
following condition for orthogonality: 52
<Y, B -Y> = <Y, B^-Y> = <Y, B > - <Y, Y> = 0 (3-18)
Eq. (3-18) holds if and only if:
<Y, B > = <Y, Y> (3-19) — Q
By the same axioms, Eq. (3-19) holds if and only if:
B = Y (3-20)
Eq. (3-20) can not be true, however, as shown by Eq. (2-20) and (2-21)
Thus, Y_ and B -Y are not orthogonal. (The same steps apply to R -Y.)
Statistically, this implies that, while Y and B -Y (or R -Y) may be
uncorrelated (recall that they must be linearly independent for Y_,
R -Y, and B -Y to form a vector space), in physical systems they n n
still have the relation that is illustrated by Fig. 3-13. The
physical relation on which this quantitative discussion is based
will be discussed in Chapter 5.
CHAPTER 4
EVALUATING THE COMPLEX SUPERPOSITION INTEGRAL
The superposition integral, introduced in Chapter I, may be
evaluated in sampled form by using the methods developed in Chapter
3. A system architecture is presented here, based on those methods,
that will evaluate the sampled form integral as a combination of
complex multiplications and summations. This architecture is
designed to make the most of the parallel processing capabilities
of optical processing for both 1-D and the 2-D superposition
integrals. It should be understood, however, that the system is
presented in "idealized" form and that, as an analog system,
accuracy and performance are dependent on the characteristics of
the individual components in the system. There are many practical
problems that may be associated with components such as television
monitors, cameras, etc., including nonlinearities and dynamic range
limitations. Some of these problems have been introduced already
and will be investigated in terms of system performance here and in
Chapter 5.
4.1 The Sampled-Form Approximation
In its I-D form, the superposition integral appears as:
s(x) = /h(x;^)g(C)d^ (4-1)
where the function h is the complex, space-variant function (the 53
point-spread function for the 2-D case). This function varies
both with the output, x, and the input, E,. Function g is the input
of the system, which varies only with the input variable, E,.
A Fourier transform of fi can be taken with respect to x to
yield a spatial transfer function:
H (f ;C) =J'[h(x;0] (A-2) X X X
47
48
where f^ is the spatial frequency associated with x. The Fourier
transform of h with respect to ^ yields the variation spectrum:
H^(x;u) =^^[h(x;C)] (4-3)
where u is the frequency associated with C. If the line spread
function h varies sufficiently slowly with x, then the spatial
transfer function may be considered to be bandlimited:
H (x;u) = 0 for |f| < W. for all f (4-4) X L
Here, 2 W^ is the spatial frequency bandwidth. If the point spread
function is band limited, with variation bandwidth 2 W , then: u
H_(x;u) = 0 for lul < W for all u (4-5)
indicating the degree to which h varies slowly with ^. If the
Fourier transform of g is also bandlimited with bandwidth W ,
G^(u) =3r^[g(5)] = 0 for |u| < W^ (4-6)
then both h and g may be sampled in E, with minimum sampling rate
2(W + W-.) as given by the Whittaker-Shannon sampling theorum,
while h may be additionally sampled in x with minimum sampling rate 54
2(W + W^). This rate may be reduced by the use of a low-pass
kernal approach.
This sampling technique is illustrated in Fig. 4-1 where h(x;^)
and g(^) are shown for sampled values of x. In this figure, the
curves represent continuous changes in B, of both the amplitude and
the phase of h and g, when they are expressed in complex polar form.
Note that g(^) remains the same for all values of x.. Now g(C) and
h(x;^) are sampled along E, as shown in Fig. 4-2 to yield sampled
values g(m) and h(i,m), where i and m are the indices for sample
values X, and E, . When these are displayed as a matrix of pixels, 1 m
h(l;0
x^^ i=l
49
Sampled x Mapping
X, i=2
h(2;?)
h(n;C)
x^^ i=3
Figure 4-1. Samples in the Output (x) line (curves represent amplitude and phase).
Sampled X
Mapping
h(l;l) h(l;2)
i=l g(l)
h(2;l)
i=2
i=I
8(M)
jr-g(M)
h(I;M)
g(M)
g(2) Sampled E,
Figure 4-2. Samples in both the Output (x) line and the Input (") Line (samples are in amplitude and phase).
50
shown in Fig. 4-3, each horizontal line of pixels h represents a
new value for each index i (in x), while each vertical line of
pixels represents a new value of g or h for each index m (in E,) .
Note that the sampled values for g are displayed as a series of
vertical columns.
The pixel arrays represented by g and h are divided into
separate arrays for magnitude (modulus) and phase as in Fig. 4-4.
Array A(i,m) is multiplied, pixel-by-pixel with B(m) while (|)(i,m)
is added, pixel-by-pixel with 9(m). Thus, the kernal of the
integral in Eq. (4-1) is expressed in sampled form as a set of
multiplications:
h(x;C)g(^) =i>fi(i;m)g(m), i=I,2,...I; m=l,2,...M (4-7)
where h and g are expressed in polar form. Here, indices i and m
vary with array values as in Fig. 4-3 or 4-4.
When the kernel of Eq. (4-1) is expressed in sampled form as
in Eq. (4-7), the integration over E, may be approximated by a
summation over index m. Mathematically, this corresponds to
Eq. (4-1) being approximated by:
M s(i) = Eh(i;m)g(m), i=l,2,...I (4-8)
m
M is the maximum sample index value. Eq. (4-1) has now been
reduced to a form that can be evaluated by complex multiplica
tions and additions (summations). Thus, the operations described
in Chapter 3 are the only operations required to approximate the
superposition integral.
In a physical system, the operation of suiranation in Eq. (4-8)
can be accomplished as shown in Fig. 4-5, where a cylindrical lens
is used to obtain a set of line-by-line summations of pixels. In
Fig. 4-5, the pixel-by-pixel products of Eq. (4-7) are displayed as
8(2)
iay^ i +x
8(?)=> g(m)
h(l;iL ^<''% h ( 2 ; l V - ^ ^ ^
Mapping
1 1 1 1 ' I I
M I ! M •
4-M-H
I ' l l
I I I i i ! I M I
tr
• ^ zr H ! ; I I
t I t I I I t
+H-I I i
h(x;e)=> h(i;m)
Figure 4-3. Array Mappings of Pixels (each pixel represents amplitude and phase).
51
B(l)_rl ^"^
t
A(2;l)^^i,lh'iii ' 11111
+-1^ -rf-r -rt-
n-I 1 i I
-M-t-
-M-
! I i t I *
-f-H-
(a) Amplitude Arrays
Q < 1 ) - ^ 9(2)* • •
"X
(b) Phase Angle Arrays
g(m)=^ B(m)/9(m)
^(2;2):
<J)(i;2)
+-
X
h(i;m)=» A(i;my(j) (i ;m)
Figure 4-4. Pixel Arrays Split into Separate Amplitude and Phase -Ajigle Arrays.
52
CNl
d
CO a o
6
<30
•H
s we
)CD
- d <u
nH
o-B CO
iiO d
•H a a CO
CO X S
CD d
H4
CO a
•H
d •H iH > ^
U
CO
60 d
•H
e M O
Q I
d •H d o
CO
d CO
i n I
OJ
_ SO
53
complex rectangular-form complex numbers. Since i is represented
by rows of pixels and m is represented by columns, the cylindrical
lens yields a summation in m, as required by Eq. (4-8).
The 2-D superposition integral is expressed as:
s(x,y) = //h (x,y; C,Ti)g( ,n)dedn (4-9)
Applying the same sampling techniques as for the I-D integral,
Eq. (4-9) may be approximated as a 2-D summation of products:
s(i,j) = EZh(i,j;m,k)g(m,k) (4-10) mk
where (i,j) are sample points (indices) in the (x,y) output plane
while (m,k) are sample points in the (C»n) input plane.
The sampling technique for 2-D is demonstrated in Fig. 4-6,
where the impulse response and the input are shown by contour lines
that represent continuously changing values of complex amplitude
and phase. The impulse response is shown in a form where h(x,y;C»n)
has been sampled in x and y to yield the form h(i,j;C,ri), where i
and j represent the indices of the sampled values of x and y.
Next, the functions h and g are sampled as shown in Fig. 4-7.
Here, each square represents a sample in E, and ri, taken from the
"smooth" mapping of Fig. 4-6. Each of these squares may then be
represented by one pixel for phase and another for magnitude of the
associated complex number, as in Fig. 4-8. Now, A(i,j;m,k) and
B(m,k) are multiplied while (J)(i,j;m,k) is added with 9(m,k) to
yield the 2-D complex product:
h(i,j;m,k)g(m,k) = A(i,i :m,k)B(m,k)/(j)(i,j ;m,k) + 9(m,k) (4-II)
The (i,j) index pairs are represented at different points in
time, while, for each pair, m and k are represented by the horizontal
54
>Xl
d
CO
X d
<U
e CO en
CO
J X -klOLT
W2
A u;*
S CO > %
X 0)
4-t . - I CO
en •H V-i
o ; OX) u - i
d CO
4-J d a 4-1 d o 0)
4->
d
• H
ClO
d •H t H
eu 6 CO
CO I
<r <u u d 00 •H
•H-t-
n
•H-
-M-
i-f
' ' I
! I I I I
H M I ' ' I i I ' I
tu. - t - 1 - ^
I I t ' M
i « t t i I I I I
t-Hl-
-t^^-1-M i l t i 1 i j I
8(^;n)=^ g(m;k)
55
h(i,j ;C,n)=* h(i,j;m,k)
A New Array Appears for Each New (i,j) value.
Figure 4-7. Pixel Arrays for 2-D Processing (each pixel represents both magnitude and phase angle).
m
I "" 1
1
1 1 1
1 t 1 1
1
1 n
ixi ixi 11... 1 ' ! t 1 j { I j i 1 1 1 1 t 1 1 T
I 1 1 1 1 1 I !
1 1 1 M 1 I 1 1 1 1 1 1 1
1 1 I 1 1 1 1 1 ! I 1 1 I I
1 1 1 1 1 I 1 1 ! 1 1 1 1 1 I 1 1 ; 1 ! 1 1 1
f.
B(m,k)
i' m
f n *-f-r -H-H-) ' M )
r-r-1—r-
•Hi-t
-H--^-r
-U--H-
A(i,j ;m,k) (for each value of of (i,j)).
(a) Amplitude Arrays
M
• T " ^ T i h \\ ^
* I '
i 1
TT 1 1 II - * - ( - ! - 1
! 1 ! ' ! M i l ' 1 •
• 1 1 1 " 4--r
n i-tT^ i ' '
r 1 ' ' 1 ' L i M 1 '
i i ±-fi-9(m,k)
(b) Phase Angle Arrays
g(m,k)= B(m,k)/9(m,k)
4 n • i — I M ! ' I I i I !
J — i - ; - t — t -
-t-r -t-r
I I ' '
! I I I I )
ttm 4.4 > I I +^-r +-»—f-1—p-•t-f-
4—i-
i^H -t-T
XT
(()(i,j ;m,k)
h(i,j;m,k)= A(i,j ;m,k)^^^^j^injc)_
Figure 4-8. Pixel Arrays Split into Amplitude and Phase Angle.
56
and vertical spatial arrangements of Figs. 4-7 through 4-9. Thus,
for each point in time a new array for A and (^ would appear in Fig.
4-8, while B and 9 would be constant for all points in time. It
follows that sample points (i,j) in x and y are represented in a time
dimension, while sample points (m,k) in E, and n are represented in
space.
Since the summations of Eq. (4-10) are over variables E, and n,
when those variables are represented spatially in sampled form, a
lens may be used as in Fig. 4-9 to spatially sum the arrays of pixels
that represent the complex number products:
h(i,j;m,k)g(m,k) (4-12)
where the products are expressed in rectangular form and one product
array appears at each point in time for each (i,j) sample point. It
follows that one evaluation of § in Eq. (4-10) is also done for each
point in time. The number of samples that are required in (x,y) and
the number of samples that are needed in ( ,ri) can be determined by
extending the I-D bandwidth criteria of Eqs. (4-2) through (4-6) to
2-D.
4.2 A System Architecture
The methods of Chapter 3 may be applied to the calculations
described in Section 4.1 to design a system for evaluating the
complex superposition integral, using incoherent light. The system
is illustrated in detail in Fig. 4-10. Refer back to Fig. 4-10
throughout this section. Note that all equations in this section
and in Fig. 4-10 are given for the 2-D case. Where applicable, the
equations for the 1-D case are shown in-Appendix A", numbered to
match the equivalent 2-D equations that appear in the text.
For data to be input into this system, the desired complex-
valued input function g and impulse response fi must be sampled to
yield the pixel arrays of Fig. 4-3 (for 1-D) or Fig. 4-6 (for 2-D).
57
i H <U X
•H P-i
X (U
rH CX
s o a d
•H
g O
tM
>-l CO
jcn
o
Q I
d •H
d o
•H iJ CO
I d
CO
X 0)
rH
e o u
CJN I
>d-
0)
d 00
•H
58
Each of these pixel arrays must be separated into one array for
complex number moduli as in Fig. 4-4a (1-D) or Fig. 4-8a (2-D) and
another array for phase angles as in Fig. 4-4b or Fig. 4-8b. Thus,
h and g are expressed as:
h(i,j;m,k) = A(i, j ;m,k)/(f)(i, j ;m,k)
g(m,k) = B(m,k)/9(m,k) ^^'^^^
The arrays associated with phase angles (J) and 9 are encoded, by
appropriate scaling, into a representation proportional to the R -Y n
signal line of the NTSC (Y, R -Y, B -Y) signal space. Y is expressed
as a constant value, while B -Y = 0 for the arrays corresponding to
both <{) and 9. Note again that the array values expressed in Fig. 4-10
are in the form corresponding to the 2-D integral. Here, (i,j) are
determined by sample points displayed in time. Thus, one value of
h(i,j;m,k) appears for each time that the system completes a summation
in (m,k). It follows that the arrays representing h change for each
new value of (i,j), while the arrays for g do not change until all
values of (i,j) have been evaluated. Points representing (m,k) are
arranged spatially with values of m representing columns of pixels in
the optical paths of the system, while the k values represent rows as
shown in Fig. 4-7. For the I-D case, the time-space arrangement
expressed in Fig. 4-10 is simply replaced by a spatial arrangement
where i represents rows of pixels while m represents columns of pixels
as in Fig. 4-3.
4.2.1 Interface and Encoding
The arrays of pixels, arranged as explained above, are input
from a computer or data interface in scaled polar form and are encoded
electronically, as shown in Fig. 4-10, by the matrix shown in Eq. (2-20).
(Note that dotted path lines in Fig. 4-10 represent electronic signal
paths, while solid lines represent optical paths.) If Y is chosen as
a nominal constant value of 0.5, then the theoretical maximum
59
— 5C
t/3 -r-
X 1--J iJ — u
II
>-I
o
B U H ',^ =it
e
II
E*
> • I
I—
7=^
"3 "3 C
u 01
I I I I ±
C 1 OS ' - - 1
01 u u v n) JJ u^ 3 ^ C OJ E -> c c
•o ^
1
I
1
y«—V
^ « k
E • # i
• r - ) •
— ^ i ^
I c
^ u Z OJ
SM OS
u to
o
u - I E
00 o
ro If)
0 ta 01 u
» M
E*
I
CQ
I
•^ r -
E"
<
=!t:
^ U aj
n '- 1 0
-0
^
«»'
c; ac
^ j
M
^ M
r3 C
— « b .
) iC 3£
E E
II ^ y * v
^ • k
E , ^
•^ - ^ ^ E
CM <r>
> 1
cc ^—'
(U 1- u a; n •u 14^
~ >-C CJ E '-' 0 c U « 1
^
-^ j i i
•> E —' CkO
o II
/'^ >• 1
C
^ t 1 1
— — —.
•J}
^ u CC
c
cc
—1
1 s ^ CO
—.' OJ
.•.J
•^i rr 3
-0
— •
CO
— > u -H 01 H
:l u OJ c;
^
n
I J
Si " I
i c CO 01
z o w C/3
< II
E'
1
nji
c I
> •
I
II
I
I ui
c
— r: u > u C O O
— — E C IJ fl ^J
i J H !LJ =a=
- ! 7 = ^ L'. I
u c
E
I I I I 1
T
XI x
S - i ^ r^
>-i
d 4 - 1
a (U
o u
< B (U
4 - 1
cn
CO
(U 4 - 1 (U
. H
a, e o u (U
J=
U-l
o
s CO M CO CO
o o
.—I CQ
I <r
OJ u d CO
I :
60
symmetric range of signals extends from R -Y = -0.5 to R -Y = +0 5 n n
as in Fig. 3-12. Thus, when (j) and 9 extend from -IT to +7T (radians),
the following scale factor is used:
(R^-Y)^ (i,j;m,k) = 0.5(t)(i,j;m,k)
TT
(R -Y) (m,k) n L
= 0.59(m,k) (4-14)
ir
The outputs of the encoding matrices now appear as values in
the NTSC (R , G , B ) color space. They drive the (R , G , B ) outputs n n n n n n
of color monitors //I and #2, which are assumed to be linear with
respect to the input, or are linearized by electronic compensation.
The pixel arrangements discussed earlier are preserved as shown:
Y. =0.5 (constant)
(R -Y)_(i,j;m,k) n 1
(B^-Y). = 0 ^ n 1 — Y = 0.5 (constant")
(R -Y).(m,k) n z
(B^-Y) = 0 '— n /.
R^^(i,j;ni,k)
G^^(i,j;m,k)
iB j (i,j ;m,k2
R o(m,k) n2 G ^(m,k) nz B ^(m,k) _n2
(4-15)
4.2.2 Complex Multiplication
Each display is passed through the 50/50 beamsplitter of Fig.
4-10 to accomplish phase angle addition. Note that the array of
Monitor //2 is imaged onto the beamsplitter by a mirror, to compensate
for the "mirror image" that the beamsplitter reflects to the camera.
As explained in Section 3.2, each of the (R^, G^, B^) stimuli are
attenuated by a factor of two by the wideband beamsplitter:
R (i,j;m,k) nr
G^^(i,j;m,k)
B^^(i.j;ni,k)
ilR^Q(i,j;m,k) + iiR^^(i,j ;m,k)
ilG^Q(i,j;m,k) + i^G^^d, j ;m,k)
i$B^Q(i,j;m,k) + bB^^d, j ;m,k)
(4-16)
61
The color camera detects the result and transforms it back into the
(Y, R -Y, B -Y) signal space: n n ^ IT
(Rn-Y)y(i.j;m,k) = i^(R^-Y)^(i,j;m,k) + if(R -Y)2(i, j ;m,k) (4-17)
(B^-Y)^ = il(B^-Y)^ + hi^^-Y)^ = 0
Also explained in Section 3.2, R -Y is now scaled as (R -Y) = +0.5 n n "i
= ±211 radians, to accommodate all angle sums. This 2i\ scale factor,
however, is already introduced by the attenuation factor of the beam
splitter as shown by substituting Eq. (4-14) into Eq. (4-17):
(R^-Y)^(i,j;m,k) = ii{0.5(j)(i, j ;m,k) + 0.59(m,k)} IT TT
= 0.5((t)(i,j;m,k) + 9(m,k)) (4-18) 2-n
= 0.5'y(i,j;m,k) 27T
The moduli of the complex number samples of h and g are fed
from the interface into back-to-back variable transmittance filter
arrays (x- and T„ in Fig. 4-10). These devices are illuminated by
a source that is constant in Y over the entire spatial arrangement of
the pixels. The devices themselves must be able to display various
arrays of intensity transmittances that correspond, pixel-by-pixel, to
the (m,k) (or (i,m) for 1-D) spatial arrangement of color monitors #1
and #2. Thus, an electronically-controlled array of transmitting
pixels would be needed such that each pixel could be varied individu
ally in real-time or near real-time over a transmittance range from
0 to 1, for incoherent light. Such an advanced spatial light
modulator has not yet been perfected. Attempts are currently under-
way, however, to produce and evaluate such devices. Appendix
B contains a discussion of the possibilities and limitations of
applying the devices that are currently being developed.
62
Assuming that appropriate devices are available, the pixels
corresponding to moduli A(i,j;m,k) and B(m,k), (A(i;m) and B(m) for
I-D), can be scaled directly as a transmittance for each given pixel,
when A and B range between 0 and 1, as discussed in Section 3.1. Thus,
the per-pixel transmittances are expressed as:
T^(m,k) = B(m,k) ^ _ ^
T2(i,j;m,k) = A(i,j;m,k)
The spatial coordinates ((m,k) for 2-D or (i;m) for I-D) must
correspond to those detected by the color camera as in Eqs. (4-13)
through (4-18). This spatial synchronization requirement dictates
that the black and white camera of Fig. 4-10 must be in horizontal
and vertical synchronization with the color camera that detects phase
angle addition.
When a nominal signal level of 1.0 (units) is detected by the
black and white camera for A = B = T - = T = 1 . 0 , then with respect
to this signal level, the product of each pair of pixels follows
directly:
C(i,j;m,k) = A(i,j;m,k)B(m,k) (4-20)
The output signal of the camera is expressed as Y (i,j;m,k) (or
Y (i,m) for 1-D). P
4.2.3 Polar-to-Rectangular Transform
The bipolar (R -Y),„ electronic signal which represents the ^ n T
array of sum angles, H'(i,j;m,k), is combined with the unipolar Y
signal to do a polar-to-rectangular transform on the product pixels.
As shown in Fig. 4-11, the (R -Y)^ signal, scaled over the range
from -2-n to +27T at maximum voltage, is input into an 8-quadrant sine
and cosine transformer. The results of these transforms are multiplied by the Y signal from the black and white camera to complete the
p polar-to-rectangular transform. The output of this transformer is
63
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64
encoded into the R^-Y and B -Y outputs of the camera:
Y =0.5 (constant) 2-n
(R^-Y)*(i,j;m,k) = Yp(i,j;m,k)sin[ f^(R-Y)^(i, j ;m,k) ]
= Im{C(i,j;m,k)/H^(iij;m,k)} (4-21)
(B^-Y)*(i,j;m,k) = Yp(i, j ;m,k)cos[|^ (R-Y)^(i, j ;m,k) ]
= Re{C(i,j;m,k)/H'(i,j;m,k)}
Thus, for each pixel, the rectangular form of the pixel product is
displayed again in the (R , G , B ) space on the NTSC monitor. n n n
4.2.4 Complex Addition and Final Detection
A lens is then used to sum all of the pixels displayed on the
NTSC monitor of Fig. 4-12. As explained in Section 3.2, when these
pixels are condensed in 2-D to the size of a single pixel, the
intensity goes up by a factor of MxK where M is the total number of
pixel columns and K is the total number of pixel rows (assume each
pixel is the same size). Thus, when the intensity is attenuated back
to the original Y level (0.5 for this discussion), the stimuli are
attenuated by a factor of (MK). For the I-D summation of Fig. 4-5,
this factor is M, where M is the number of columns of pixels.
When the compensated resultant pixel is detected by the final
color camera, the complex summation, scaled by the compensator
attenuation factor, appears on the color signals of the final camera
in Fig. 4-10:
M K (R -Y)**(i,j)-Im{-^ E E C(i,j;m,k)/H'(i,j;m,k)}
m=I k=l
^ M K ( -"> (B -Y)**(i,j)«Re{-^ E E C(i, j ;m,k)/4'(i, j ;m,k) } " ^ m=l k=l
(For 2-D)
When (R -Y) and (B -Y) are scaled such that the values n n
65
(R^-Y)** = (B^-Y)** = ±0,5 correspond to ±(MK), then a scale factor is
included in the output signals of the final color camera.
0 5 ^ K . (R^-Y)**(i,j) = ^ I n i E E h(i,j;m,k)g(m,k)
m=I k=I
0 5 ^ ^ (B^-Y)**(i,j) = ^ R e E E h(i,j;m,k)g(m,k)
m=l k=l
(4-23)
When the proper compensation factor is used, the desired result is
obtained in complex polar form:
MK ^ • ^ " 075 ((V^^**^^'J^ " J(R^-Y)**(i,j))
M K = E E Re[h(i,j;m,k)g(m,k)] (4-24) m=I k=I
M K + j E E Im[h(i,j;m,k)g(m,k)]
m=I k=l
4.3 Problems to be Investigated
There are many practical problems inherent in the system archi
tecture of Fig. 4-10. Since the system is analog in nature, its
performance is dependent on the dynamic range and linearity of the
system components, especially the color cameras and display monitors.
These two factors, introduced in Chapters 2 and 3, will be the
subject of much of the experimental work discussed in Chapter 5.
The problem associated with a bias build-up in complex addition
was introduced in Section 3.2, and is a disadvantage which is inherent
in incoherent optical summations. The magnitude of the detrimental
effects on system accuracy that result from this build-up are deter
mined by the dynamic range or signal-to-noise ratio of the system.
These factors are especially important to the final color camera
because the detrimental effects of the build-up arise due to the fact
66
that large attenuation factors as in Eq. (4-18) may yield a result
that is down in the noise region of the camera as explained in
Section 3.2.
The bias build-up problem may be minimized and system performance
improved, at the expense of processing time, by displaying and
processing a smaller number of pixels at a time. The results of each
of these "sub" processes are then combined serially in another level
of summation. The form of this process can be illustrated by Fig.
4-I2a, where the results of each "sub" summation are stored into
another array. This array is then summed again, in another pass
through the summation section of the system as in Fig. 4-12b. The
more passes that are made with smaller numbers of pixels, the greater
the accuracy will be, but the processing time will also be greater.
This type of serial-parallel process could also be used in the
event that the bandwidth requirements of h and g dictate that more
pixels are needed than could be displayed on the system at a single
time. In this case, parts of h and g could be multiplied, summed,
and stored separately into the storage array. The stored values in
this array could then be summed in another pass through the summation
section. Again, there is a trade-off between accuracy and processing
time.
67
= Summation of Entire Pixel Array
(a) Mathematical Architecture
Complex Multiplication
Polar-to-
Rectangular • ^
-»
Display
T—
Sub-Array and Sum Storage
Control for Storage and Display
(b) System Block Diagram
Figure 4-12 Summation by Sub-Sections.
CHAPTER 5
AN EXPERIMENTAL SYSTEM
In order to investigate practical problems, actual physical
characteristics, and the feasibility of implementing a processor of
the type discussed in Chapter 4, an experimental system was con
structed, using commercially available color television cameras, a
monitor, incoherent light sources, optics, and electronic instrumenta
tion. The purpose of performing the experiments described herein was
twofold. First, the responses of the cameras and monitor were
studied for range and linearity. Second, the basic operations of
multiplication and addition of complex numbers were demonstrated.
The system described is a "proof-of-principle" experimental system
only and is not to be considered as an actual realization of the
processing system presented in Chapter 4, although it does realize
the various separate components of that system.
The experimental system is constructed as in Fig. 5-1 and includes
the same basic components as in Fig. 4-10. These components and their
use in system construction are first discussed, the results of
characterization experiments are shown, and finally, the basic opera
tions of multiplication and addition are demonstrated.
5.1 System Components
A brief introduction to each of the components of the experi
mental system follows. Included are the color television cameras,
the black and white (B & W) camera, the monitor, the optics, and
the light sources. The construction of the system and its signal
display instrumentation are also discussed.
The color television cameras used are model VC1-2100E single
tube vidicon cameras manufactured by Nippon Electric Corporation
(NEC). They were chosen for their low cost and especially for ease
of access to the Y, R -Y, and B -Y signals. Camera #1 uses a 25 ram n n
f/2.4 vidicon lens, manufactured by Ampex, while Camera //2 has a
68
Optical Signal
Electronic Signal q 1 69
Chromega B ?
Chromega A
r 55/45 Beamsplitter
0 I I I
F i l t e r Holder , //la i
F i l t e r Holder //4
D p i l t e r Holder //la
I f l i g h t Blocking [/ Box
I
' p F i l t e r Holders | \ / /2a,b,c
[Primary Simulator
75/25 ' Beamsplitter
B&W Camera
Filter Holder //3
L
Power Supply
I. \
Transform and Display Electronics and Vector Display
Color Monitor
\
I Viewfinders
//I //2
n
Lens
-h; - c Color Camera ?/2
Filter Holder •*5
Figure 5-1. The Experimental System.
70
16 mm f/1.6 fixed aperture lens manufactured by Cosmicar. Each
camera has been modified to accept a cable connector for access to
the needed (Y, R -Y, B -Y) signals. n n
The signals themselves are in raster scanned form. As shown
in Fig. 5-2, each vertical scan is 16.6 msec, in duration. Thus,
the vertical sync pulses occur at a frequency of 59.95 Hz, while
the horizontal sync pulses have a frequency of 15.734 kHz. The
characteristics of the cameras, for a given point in the scan
sequence, will be the subject of much of the discussion of Section
5.2.
The B & W camera called for in Chapter 4 is a Panasonic model
WVllOOA vidicon camera. The lens is a Canon TV zoom lens (17 to
102 mm f/2). This camera was chosen because it can be externally
synchronized with the signals from Color Camera #1, and because its
voltage output is linear with respect to input intensity as shown in
Fig. 5-3, where the camera is set to sensitivity range "normal" and
the light control switch is set to "auto." The curve of Fig. 5-3 is
generated by adjusting the aperture of the camera to yield a maximum
output of .95 V with a nominal intensity of light input. Neutral
density filters of decreasing transmittance are then placed between
the light source and the camera to attenuate the input, while the
corresponding peak of the signal output is noted on the oscilloscope.
The television monitor shown in Fig. 5-1 is an NEC model
C13-304A 13 inch color monitor/receiver. When used here, it acts
as a monitor, displaying images from Color Camera #1. The various
controls on the front of the monitor are used to adjust the displayed
brightness and color, while controls internal to the monitor are used
to adjust the relative gains of the individual (red, green, blue)
color channels. These adjustments will be discussed in detail in Section 5.2.
The primary optical components of this system include a 55%
transmittance, 45% reflectance (55/45) beamsplitter and a (75/25)
beamsplitter (both shown in Fig. 5-1). These two beamsplitters have
71
0.1 V/div.. 2 msec/div.
(a) Vertical Scan
0.1 v/div., 10 ysec/div.
ihiiillliiiiiBii (b) Horizontal Scan
Figure 5-2. Oscilloscope Traces of Single Scan Lines.
72
spectral curves that are flat over the visible range (about 400 - 630 cry
nm) within 5.0% as shown in Fig. 5-4. Thus, it is assumed that they
do not affect chromaticity appreciably. The cylindrical lens shown
in Fig. 5-1 will be discussed in Section 5.3.
Sources A and B in Fig. 5-1 are "Chromega" color printing filter
heads. They are very useful as sources because they allow various
colors to be selected by internal combinations of subtractive filters.
These sources may be used to illuminate N.D. or colored filters
placed in Filter Holders //la and //lb in Fig. 5-1. Note that the
sources pass through the 55/45 beamsplitter so that addition of
pixels may be tested. The filters used in the external filter
holders are gelatin filters manufactured by Kodak for scientific
and technical use.
The NTSC primary simulator that appears in Fig. 5-5 is used
to calibrate the system. The three sources are 12 volt 100 watt
tungsten filament lamps which are used to illuminate red, green,
and blue filters that have been matched to the chromaticity coordinates
of the NTSC primary phosphors that are shown in Fig. 2-9.
In order to match the chromaticities of the NTSC primaries, a
measure known as "color temperature" is used. Color temperature
is the equivalent temperature in °K to which an ideal "black body"
radiator must be heated in order to give off a certain spectral
distribution of light. ' The color temperature of an incandescent
lamp (illuminant A in Fig. 2-3) is around 3000''K, and appears
yellowish when compared with illuminant C (also in Fig. 2-3), which
has a higher color temperature of about 6500''K. The color tempera
ture of the tungsten lamps (General Electric type FOR) that are in
the primary simulator is rated at SSOO 'K at 12 volts (rated voltage).
As shown in the graph of Fig. 5-6, when a tungsten lamp is operated
at 90% of rated voltage, the color temperature is at 96% of the
rated value.^^ Thus, the lamps are operated at 10.8 volts to yield
an equivalent color temperature of 3168 K. The chromaticity coordinates of the NTSC primaries may be
73
Output Signal (volts)
10
8
6
4
2
0
•
0.0 0.4 0.8 Input Light Intensity
(Normalized)
1.2
Figure 5-3. Response of the Black & White (B&W) Camera
Normalized Response
Trans. Reflect
Trans. 1 Reflect
55/45 Beamsplitter
75/25 Beamsplitter
100
80
60
40
20
0 ^ ^ - ^
_ _-
- . . - - - ^
—
_— —
* • * • • *
400 500 600 800 wavelength (nm)
Figure 5-4. Spectral Responses of the Beamsplitters.
74
Opal Glass Diffuser
Color Temp. Compensating Filters are Placed in Filter Holders with Tricolor Filters and Adjusting ( .D.) Filters.
Figure 5-5. Details of the NTSC Primary Simulator.
110
100 % of Rated Color Temperature
90 90 100 110
% of Rated Voltage
Figure 5-6. Color Temperature vs. Voltage for Tungsten Lamps
75
closely matched by the Kodak Wratten (gelatin) filters shown in
Table 5-1. Note however, that the illuminating sources must operate
at a color temperature of 6774°K. Thus, the sources must be converted
to this higher equivalent color temperature. To the observer, this
appears bluish, when compared with the lamps operating at 3168°K.
For source conversion, color temperature is expressed as a
"mired" (micro-reciprocal d,egrees) value:
Mired Value = — ^'°°°'°°° ^ (5-1) color temperature in K ^ -^
Filters that convert a source operating at one color temperature to
a new equivalent color temperature are characterized by a "mired shift
value," represented by:
(^ - ;^) X 10^ (5-2) 2 I
Here, T- represents the color temperature of the original source and
Ty is the new equivalent color temperature. Thus, to convert from
3168°K to 6774°K, a mired shift value of -168.03 is needed. This
value is approximated as -169 by a Kodak 78AA (bluish) conversion
filter (mired shift value = -196) and an 81B (yellowish) filter 67
(mired shift value = 27). These filters are placed in the filter
holders of the primary simulator (//2a, //2b, and //2c in Fig. 5-1).
With the sources converted as described above, the chromaticity
coordinates of each of the NTSC phosphors are closely matched, as
shown in Table 5-1. In the primary simulator, shown in Fig. 5-5,
these three primaries are projected onto an opal glass diffuser in
various additive combinations. Thus, the simulator can reproduce
the whole range of colors achievable within the NTSC color space
shown in Fig. 2-9. This simulator will be used to characterize
range limitations and linearity in the tests of Section 5.2.
Much of the work of constructing this experimental system
76
Table 5-1 Chromaticity Coordinate Matching for the N.T.S.C, Phosphor Primaries.
NTSC Coordinates Wratten Primary in (x,y) Filter
for Primaries to Match
Filter % Error With Coordinates Respect to at 6774 °K x,y Coordinates
X X X
Red .670 .330 //24 .668 .332 +0 .4 +0 .7
Green .210 .710 //61 .221 .705 + 5 . 3 - 0 . 6
Blue .140 .080 //47A .141 .079 +0 .8 - 1 . 3
77
involved the design and building of transform and instrumentation
electronics to interconnect, synchronize, and measure the outputs
from the system components. The electronic circuit boards, housed
with a vector display, are shown in Fig. 5-7. The simplified block
diagram of Fig. 5-8 shows the basic functions of the electronic
circuits. When the signals to be displayed are taken from the
selected camera, they are buffered, scaled, and sampled for input
to the vector display. In the display, the R -Y signal appears n
on the vertical axis. In use, it was found that the vector display
was not as accurate as displaying a single horizontal scan on the
oscilloscope.
So that the oscilloscope can display any given horizontal scan,
the synchronization pulses from the cameras are counted and compared
to an 8 bit binary code. Thus, any horizontal scan between 1 and
256 can be selected for display on the oscilloscope by setting a
bank of 8 microswitches. The Y signal is displayed on one oscillo
scope while the R -Y and B -Y color signals are low-pass filtered n n
to eliminate high-frequency noise, then displayed together on a
second (dual trace) oscilloscope. The details of these display
circuits appear in Appendix C. The remaining electronic circuits are used to scale, sample,
and transform the R -Y signal from Camera //I into a scaled sine and n
cosine function. These transform functions are multiplied by the Y signal from the B & W camera to yield the polar-to-rectangular P signal transform that was discussed in Section 4.2. As shown in
that section and in Fig. 5-8, the transform signal outputs are
rescaled, biased to the proper D.C. voltage level, and are input
back into the camera to be sent on to the monitor. Note that the
input to the final stage of the camera can be selected to be either
the normal signals passed directly from the first part of the
camera or the transformed signals described above. The details of
these electronic transform circuits also appear in Appendix C.
For all of the tests that are discussed in this chapter, the
78
Figure 5-7. Transform, Electronics, and Vector Display.
Color Signals from Camera //I
Sw. Color p Signals _ from Camera
Camera i i
//I ~^^ sync. ^ ^ I
Camera ' | //2
Scan L ine Sync. Se lec t
I I I I I I I
Color Signals Sync. Select to Scope To Vector
to AScope Trig / \
I
Buffer
8 Bit Switc EIZ]
Display A
Sample and Scale
Return Signals to Cam. //I
Scope
Y from P B&W Camera
Y Signals
Cam. //2
Figure 5-8. Block Diagram of Transform and Display Electronics.
79
signals are read as being positive or negative volts above or below
a baseline on the oscilloscope (A.C. input). Since the signal takes
up a relatively small area of each scan, as shown in Fig. 5-2b, the
"averaging" effect of the A.C. oscilloscope input can be neglected.
Even with the low-pass filters, the color signals have a rather
large amount of noise present, as shown in Fig. 5-9. The reading
errors will be minimized, as much as possible, by reading at the same
point on the noise "envelope" every time. This will, of course,
involve some subjectivity by the observer. Thus, an uncertainty of
about positive or negative 0.05 volts (about 10% of the peak signal
value) will be included in most of the tests and results.
5.2 Characterization, Linearity, and Range Tests
Tests were undertaken using Color Camera //I, to determine the
linearity and range characteristics achievable with the camera as
compared to the theoretical and physical characteristics predicted
in Section 3.3.
5.2.1 Initial Tests and Settings
To start, numerous gain settings on the controls of the red,
green and blue channels internal to the camera were tried, with the
inputs from the primary simulator at various levels. These preliminary
experiments showed generally that maximizing the gains of the color
channels, and thus the useful dynamic range of the color space, results
in greater nonlinearity of the signal response, so a smaller range
must be used in order to achieve good linearity. The effect of
decreased dynamic range is to reduce the signal-to-noise ratio of
the output signal, since the noise is always present at a level
comparable to that of Fig. 5-9, while the maximum achievable value
of the signal is reduced. Thus, the percent of uncertainty for a
given value of signal is greater. The dynamic range can be increased
at the expense of linearity, but large nonlinearities are unacceptable
for this experimental system, since the purpose of these experiments
80
.05 v/div, 10 ysec/div,
Figure 5-9. Noise Characteristics of a Single Horizontal Scan.
81
is to verify the linear vector space treatment that is discussed in
Chapters 2 and 3.
Initially, an attempt was made to set the gain controls of
Color Camera //I to reproduce the theoretical matrix of Eq. (2-20).
It was found, however, that the camera would not produce a linear
response over this theoretical range of values and that any constant
intensity (Y) plane in the (Y, R -Y, B -Y) color space would be n n
difficult to maintain since the three primaries are each at different
Y levels in Eq. (2-20). These difficulties arise partly because the
(Y, R -Y, B -Y) space of the camera appears as in Fig. 3-13 with the
dynamic range of the color signals dependent on Y.
Because of the difficulties associated with using different
values of Y for each of the primaries, another approach was taken.
Here, a value of .35 volts was chosen as the constant Y value. This
value was chosen because it allows a reasonably large range of color
signals to be reproduced without operating too close to the electronic
saturation point (Y :::: .45 volts). Each of the sources was adjusted
to yield a Y level of .35 volts for a nominal value of R , G , or ^ n n
B =1.0. This constant Y level was maintained for all combinations n of R , G , and B so that, for the vector equation,
n n n
R R + G G + B B = Q (5-3) n n n n n n —
the scalars R , G , and B must all sum to 1.0. The color signals n n n
were then adjusted to yield the values shown below for each of the
single primaries (all values for Y, R -Y, and B^-Y are in volts):
Red: (R =1.0): Y=.35 R -Y= .42-^ .43 B -Y=-.34^->-.35 n n n
Green: (G =1.0): Y=.35 G^-Y=-.46 H->-.47 B^-Y=-.25 - -.26 (5-4)
Blue: (B =1.0): Y=.35 B^-Y=-.ll HH--.12 B^-Y= .54^^ .55
Note that a range of signals is identified for each of the color
82
signals to correspond to the uncertainty inherent in those signals.
These values yield a color signal range that appears as in Fig. 5-10.
These settings also yield the following matrix equation for Camera //I
(all constants are in volts):
R -Y n B -Y - n -•
350
425
345
.350
- . 4 6 5
- . 2 5 5
.350
- . 1 1 5
•545
R n
G n
G - n-*
(5-5)
Note that this equation is constrained to 0.35 volts for Y, while
\* ^n' ^^ ^n ^^^S® between 0.0 and 1.0 and are unitless (recall
R„ + G + B = I.O). The matrix coefficients associated with R -Y n n n n
and B -Y are found by averaging the range of values in Eq. (5-4).
The validity of Eq. (5-5) was tested extensively by placing
N.D. filters in front of the three sources in various combinations
to yield the desired Y value of .35 volts, read for a single
horizontal scan on the oscilloscope. (Recall that the horizontal
scan appears as in Fig. 5-2b.) As shown in Table 5-2, the measured
values for Y, R -Y, and B -Y are compared with those predicted by n n ^
Eq. (5-5). The errors for R -Y and B -Y are given by that value n n °
within the measured range of signals which is closest to the predicted
value. The errors are then expressed as percent differences with
respect to the predicted value. A notation of "IR" indicates that
the predicted value fell within the range of observed signal values,
while "SM" indicates that the predicted signal value was small
(below ,025 volts) and the uncertainty is therefore a high percentage
of that signal.
These results indicate that, considering the noise and resulting
uncertainty of observed signal values, Eq. (5-5) is a reasonably
accurate model for Color Camera //I. One result that becomes apparent
is that, for smaller values of signal level the noise, and thus the
uncertainty of the signal values, is a larger percentage of the signal
value. The result is an increase in percentage error. Though the
83
R -Y (volts) n
B -Y (volts)
Figure 5-10. Color Signal Range for Initial Adjustment of
Color Camera //I.
4-1
• u
m
5 U o u u w
cn u i H o >
i n CO
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85
errors for Y are all negative, they are not critical, as the constant
Y value carries no information.
With Camera //I set as modeled by Eq. (5-5) several more tests
were run. First, a test was made to determine the color space
characteristics for various values of Y. In this test, various N.D.
filters were placed in front of each of the sources in the primary
simulator, with the other two sources blocked off so that each
primary was tested individually. The Y, R -Y, and B -Y values were n n
recorded for each filter as shown in Table 5-3. The results shown
in Table 5-3 yield the input-output relation for Y that is shown in
Fig. 5-II. (The R , G , and B notations on the graph will be n n n
explained in Section 5.2.2.) This curve was the same for each of
the primaries, showing the important property that all colors have
the same luminance characteristics in the camera. Note that the
relation shown by this graph is not linear. In order to minimize
nonlinearities, the single Y value, constant for all colors as in
Eq. (5-3) through (5-5), must be used. When the various values of the
color signals (R -Y and B -Y) are plotted for different values of Y, n n
the color space of Fig. 5-12 follows. It is this result that leads to the color space of Fig. 3-13.
The above test was repeated for each primary by setting the
aperture on the camera lens such that the Y value matched each of
those measured with the N.D. filters in Table 5-3. It was found that the R -Y and B -Y signal values were uniformly greater than
n n
those in Table 5-3, even though the Y values were the same. This
effect seemed to be due to the aperture sensing circuitry in the
camera changing the gains of the color channels inside the camera.
Thus, for all remaining tests, the aperture of the camera was kept
at the constant value used for these initial tests.
The other test that was undertaken with the camera at these
settings was a check of the addition properties of the 55/45 beam
splitter. Refer back to Fig. 5-1 for the positions of the components re ferred to in this test. Here, pixels of light were projected in
86
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87
0.4
0.3
Y Signal Output Q 2 (volts)
0.1
/ n
^ n
G n
0.0 0.4 0.8 Normalized Light Input
1.2
Figure 5-11. Luminance (Y) Signal Response for Camera //I
ieure 5-12. Color Space for Camera //I (from Table 5-3) Figu
88
various combinations, first from Chromega A, then from Chromega B
as shown in the first two columns of Table 5-4. Next, both of the
Chromegas were turned on at the same time, and a 50% transmittance
N.D. filter was placed in front of the camera (at Filter Holder #3
on Fig. 5-1). The resulting color signals for the addition of two
pixels in Table 5-4 are measured against the predicted value, given
as:
The Y value remains constant at 0.35 volts since each of the two
constant Y values from the Chromega sources are added, then attenuated
by a factor of 2. Note that the predicted values in Eq. (5-6) and
the constant Y value are the same as those values predicted in Section
3.1 when an ideal 50/50 beamsplitter is used for addition. Thus,
the method described above compensates for the 55/45 beamsplitter
used in the experimental system by setting each source to the constant
Y value detected by the camera. As described in Section 3.1 and in
Section 4.2.1, the added values would be scaled by a factor of 2
when representing added angles, so the attenuation of factor 2
would be compensated for.
These preliminary tests have shown several important character
istics that will be used throughout the experiments on this system.
The important result that a color camera can be modeled as a linear
system has been shown. The tests have also shown that the constraints
on this linear model include the facts that Y must be constant and
that the camera aperture should not be changed. Addition of pixels
with a beamsplitter has also been demonstrated, where the beam
splitter has been compensated as described.
5.2.2 Characterization of the Complete System
Camera //2 was put in place of Camera //I in Fig. 5-1 so that it
89
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90
was illuminated by the primary simulator. The camera was then
adjusted to operate at approximately the same levels as Camera #1.
The constant Y level was again set at 0.35 volts. The matrix
equation for Camera //2 now appears as shown (the units on Y, R -Y, n
and B -Y are in volts): n
R -Y n
'— n -' 2
350 .350 .350
455 -.485 -.095
.345 -.235 .535
R n
G n
B ~ n
(5-7)
As in Eq. (5-5) the constraint that R + G + B = I ( Y = 0.35 volts) n n n
must hold. As before, various values for R , G , and B were input n n n ^
to test the validity of Eq. (5-7). The results, shown in Table 5-5,
verify that the above matrix equation is valid as a model for Camera #2.
Once the response of Camera #2 was verified, the camera was
positioned as shown in Fig. 5-1 to face the television monitor that
is driven by Camera //I. The entire system is shown in Fig. 5-13,
but the cylindrical lens in front of Camera //2 is omitted for the
following tests. (Note that Fig. 5-13a corresponds to Fig. 5-1.)
N.D. filters were placed in front of Camera #2 so that the output of
the monitor would drive the camera at approximately the 0.35 volt
level for Y_. The phosphor simulator was again used as a light
source for Camera #1, and the cameras were spatially aligned so that
the image detected by Camera //I, when displayed on the monitor,
appears the same size and at the same position for Camera //2.
When the system was operated in this mode, it was found that
the blue or red colored pixels, when displayed on the monitor, had
a higher luminance value than for a green pixel. When the phosphor
drives on the monitor were readjusted to yield equal luminance levels
for the three stimuli, the color balance was not correct because
there was too much green. These results indicate that the monitor
phosphors operate at nominal luminance levels that are higher for
91
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Figure 5-13. Photographs of the Experimental Svst em
93
green or red than for b lue , as predicted by the theore t ica l matrix in Eq. (2-20):
R = I : n
G = I : n
B = 1 : n
Y = .299
Y = .587
Y = .114
(5-8)
Because constant Y values are needed for all colors displayed
on the monitor, the R , G , and B gains in Camera //I were readiusted n n n •*
such that each primary was of an equal Y value on Camera #2, with
Camera #2 aimed at the monitor. Now, the Y values from Camera //I for
each primary at its maximum value of 1.0 appear as shown:
R = I . O : n G = 1 .0 :
n B = 1 .0 :
n
^2
^2
^2
= .350 v o l t s
= .305 v o l t s
= .245 v o l t s
(5-9)
These values can be converted to normalized linear tristimulus
values for Y by using the graph of Fig. 5-11. In the figure, the
three primary stimuli are marked at their appropriate points on
the curve. The normalized Y values for each of the primaries can
then be found along the x (horizontal) coordinate of the curve. These
Y values for Camera //I are:
R = 1 .0 : n
G = I . O : n
B = 1 .0 : n
Y = 0 .75
Y = 1.00
Y = 0 .50
(5-10)
Since these values only cover a limited part of the curve in Fig. 5-11
they can be approximated by a linear relationship as shown by the
dotted line. Thus, even though a constant Y value is not maintained
as was assumed in Section 5.1, it can be expected that nonlinearities
from this range of Y values will not be a major problem.
94
The matrix-form equations for the output signals from Camera //I
are shown below:
R -Y n B -Y "n -
750
460
270
1.000
-.365
-.215
.500
-.100
.440
R n G n B ^ n-i
(5-11)
The R^, G^, and B^ inputs from the primary simulator are unitless
values between 0.0 and 1.0 as is the normalized Y value. R -Y, B -Y n n
and their corresponding matrix coefficients are given in volts.
For Camera //2, the Y value remains constant at .375 volts. When
this value is normalized to a value of I.O for Y , the corresponding
matrix form for the camera response follows, where Camera #2 is aimed
at the monitor that is driven by Camera #1:
Y
R -Y n B -Y - n -J 2
=
1.000
.195
^.320
1.000
-.325
-.155
1.000
-.085
.50^
R n
n B — n-
(5-12)
As held for Eqs. (5-5) and (5-7), the constraint that R + G + B = 1 n n n
again holds for both Eqs. (5-II) and (5-12).
These equations were tested as system models by using the inputs
from the primary simulator in various combinations. The results are
shown in Table 5-6, where the measured values for Y, R -Y, and B -Y n n
are compared to the values predicted by Eq. (5-11) for Camera //I and
by Eq. (5-12) for Camera #2. Each point in Table 5-6, with a measure
of its associated errors, was plotted in a form similar to that of
Fig. 5-14. Note that Fig. 5-14a, plotted for Camera //I, represents
the projection view of a diagonal slice on the Y plane as it would
appear in the inset in the figure, while Fig. 5-14b, plotted for
Camera #2, is a constant Y slice (Y = .375 volts) as appears in Fig.
3-13. The plots of Fig. 5-14 are useful both as a mapping of the
range of signals that may be reproduced and as a means by which to
95
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97
R n
B -Y INSET n
(3-D Projection)
-0.4
R -Y (volts) n
-0.4
B -Y (volts)
(a) Color Signal Range for Camera //I.
R -Y (volts) n
B -Y (volts) n
(b) Color Signal Range for Camera f/2
Figure 5-14. Color Signal Range for the Cameras in Svstem Form
98
study patterns in the errors which occur.
The errors in Table 5-6 are given in volts, rather than percent.
In general, they are reasonable, when compared with the magnitudes of
the predicted values. They are large however, when considering the
accuracy needed for number processing.
The errors in Camera //I seem to be due to the greater nonlineari
ties that arise as a result of the use of a non-constant Y mapping
with color. While the errors seem to have some patterns (the errors
seem smaller for negative values of B -Y), they are mostly due to
noise in the measured values. Note that the measured Y values are
uniformly high, with respect to the values predicted by Eq. (5-11).
The errors in Camera //2 show a slight negative offset for values of
R -Y while desaturation of the phosphors seems to be the primary
cause of errors by B -Y. These results demonstrate that television n
phosphors can be difficult to work with. Like vidicon tubes, phosphors
tend to show some hysteresis in that they "remember" previous signal
inputs. As they warm up, over a period of several hours, they tend
to desaturate slightly and reduce their brightness, which would
account for the "high" readings for Y and for some of the desatura
tion effects in Camera //2.
While errors are present, they are not too severe. Thus, Eqs.
(5-11) and (5-12) give a reasonably accurate linear response model
for the entire system of Camera //I and the combination of the monitor
and Camera #2. These matrices model the system response as shown in
Fig. 5-15, where [A] is the matrix of Eq. (5-11) which characterizes
the response of Camera #1 to the (R^, G^, B^) inputs while [C] , the
matrix of Eq. (5-12), characterizes the overall response of the
system from those inputs to the outputs of Camera //2. The system
response from the signals of Camera #1, (Y^, (\-Y)^* ^ n' - l ^°
the system output (Y^, (V^^2' ^V^^2^ ^^ matrix [B] . The system
matrices are related as:
99
Optical Path
Electrical Path
Camera //2 Monitor
A
/
Camera //I Primary Simulator
a
R -Y n B -Y n
= [A]
R n
n B n
R -Y n B -Y - n -
[c]
R n
n B n
R -Y n B -Y n
[B] R -Y n B -Y n
Figure 5-15 Matrix Models for the Complete Experimental Svstem.
Y
R -Y n
B -Y i - n -1 2
= [C]
R n
G n
B ' - n
It follows that
[B][A]
R n
n B •- n-"
= [B]
100
R -Y n B -Y
(5-13)
[C] = [B][A] (5-14)
,-1, Multiplying both sides by the inverse of [A] ([A] ) yields the m
following:
[B] = = [C][A]""^ =
1.351
- . 0 9 3
.035
.466
.605
- . 1 1 7
.843
.050
1.062
(5-15)
These characterization tests have shown the important results
that, by using constraints on Y, it is possible to model this
experimental system as a linear system. While errors do arise from
the effects of phosphor memory, etc., they are not too severe. A
more complete model could be derived from Eqs. (5-11) - (5-15) by
making extensive error measurements and including error compensations
in the equations. That is not necessary however, for the simple
linear models developed here. Also, as will be shown in Section 5.3,
using only a part of the available color signal range results in
negligible errors due to system effects.
5.3 Complex Number Mathematics
Because the experimental system was successfully modeled as
linear, the operations of polar-form complex multiplication, polar-
to-rectangular conversion, and rectangular-form complex addition, as
described in chapters 3 and 4, may be reproduced.
101
5.3.1 Multiplication
Polar-form multiplication follows directly from the tests of
Section 5.1. The angles can be represented and added by the (R -Y) n 1
color signal by using the maximum range possible. Shown in Fig. 5-I4a, this yields the following scaling for 9, cj) and m from Eq.
(3-1) :
0 = (|) = (R^-Y)^ X 7r(rad) ^ = (\-Y)^ x 2TT(rad)
.35(volts) .35(volts) ^ " ^
where (R -Y), is in volts. Note that (R -Y)^, rather than (B -Y), u ± n 1 n 1
yields maximum range, as shown in the figure.
Complex number amplitude multiplication was demonstrated by
setting the aperture on the B & W camera such that, at the amplified output Y , a maximum value of 10 volts is obtained. Then, back-to-
P
back N.D. filters were used, in combination with the linear response
of Fig. 5-5, to yield a physical representation for the magnitudes
(or amplitudes) of Eq. (3-1):
C = AB = T,T. = Y /lOv (5-17) 1 2 p
Here, T^ and T« are the intensity transmittances of the N.D. filters
and are scaled directly, as in Eq. (3-10).
5.3.2 PoIar-to-RectanguIar Conversion
A 2-quadrant electronic polar-to-rectangular converter was con
structed as shown in the block diagram of Fig. 5-16. The 8-quadrant
(+2TT to -2-n) transform specified in Chapter 4 was not constructed due
due to dynamic range limitations, but a design for an 8-quadrant
transformer appears in Appendix C, with the circuit diagrams and
details of the existing 2-quadrant transform. For 2 quadrants, 4*
appears as:
102
^ = ^ V ^ ^ 1 . TT/2 (rad) (5-18) .35(volts)
C appears as in Eq. (5-17), while a single scan of (R -Y) appears
in Fig. 5-17.
The (R -Y) input is sampled such that every 4 scans, the value
to be input into the sine and cosine transform circuitry is updated.
Such sampling is necessary so that the slower time response of that
circuitry will not cause problems. A sampled-form scan is shown in
Fig. 5-17. The outputs of the sine and cosine circuits are multiplied
with Y , and are biased to the proper d.c. level. The signals are P
rescaled by experimental adjustment such that the real and imaginary
parts of the complex number product, given by
Re = C cos ^ 0 < C < 1.0
Im = C sin- -Tr/2 < ^ < 7T/2 (5-19)
appear on the linear range of the color signals as shown:
Re' = -(B -Y)' volts 0 < Re'< 1.0 n 1 .215 volts 0 > (B -Y): > -.215
n 1
Im' = (R -Y)' volts -I.O < Im'< 1.0 n 1 .400 volts -.40 < (R -Y)_ < .40
n 1
(5-20)
The limits on Eq. (5-20) result from the limits on Eq. (5-19). The
negative sign in front of (B -Y)^ is a result of that color signal
being inverted in the camera at the point where the transformed signal
returns.
The range of values for the transformed color signals appears in
Fig. 5-18, where a complex number vector representation is shown. The
length or magnitude of the vector determines the saturation of the
observed color, while the angle determines the hue.
103 H .Svnc . V Sync.
-^
"n R -Y n _
m
Amplify & Offset
Sample/ Hold Circuit
Absolute Value
Y in P — from B&W Camera
Scale & Offset
10 V max.
Sine & Reinvert 4-*
Delayed Sample
Multiply
Cosine
"I -X-
Delayed Sample
Rescale & Bias
Multiply Rescale & Bias
(R -Y) into -• n
Camera //I
(B -Y) into ->• n Camera //1
Figure 5-16. Polar-to-Rectangular Transform Circuitry
2 v/div., 10 ysec/div.
Figure 5-17. Sampled Form and Input Form of the R^-Y Signal
104
-0.4 -0.22
Complex Vector Representation
(R -Y)' (volts) n
(B -Y)' (volts) n
-0.4
Figure 5-18 Range of Values for Transformed Color Signals
105
Eq. (5-19) was tested against Eq. (5-20) for various values,
with results shown in Table 5-7. The test was performed by setting
the input such that a given (R -Y) value (a given 9 value) was
maintained, at a Y value of about .375 volts. The aperture of the
B & W camera was then adjusted to give different magnitudes for Y P
(C, in complex number notation). Note that, for the limited range
used, the Y^ value did not vary greatly, and errors were very small.
One reason for the small errors is the lack of noise in the transformed
color signals, as shown with the input Y signal in Fig. 5-19. Note
that these signals have very little noise, compared to Fig. 5-2b.
On Color Camera //2, the complex number components were found by
observation of the (R -Y). and (B -Y)„ values given in the first n z n z
th ree rows of Table 5-7 . They a re modeled a s :
Re^ = -(\-^^2 " •22((R^-Y)2 + -105) U((R^-Y)2 + .105)
.200 7200 X/ ( 5 - 2 1 )
Im^ = ((R -Y)_ + .105)/ .195 z n z
where U(x) is a unit step function at (R -Y)^ = -.105 (since C/J_ =0.0
corresponds to (R -Y)_ = -.105(v) and (B -Y) = O.O(v)). n z n ^
The fact that Re^ depends on both (B^-Y)2 and (R^-Y)^ follows
from the plots of Fig. 5-14, where the coordinates at Camera //I are
distorted at the output of Camera //2. This interaction of coordinates
is also predicted by Eq. (5-15) where:
(R _Y)^ = -.093 Y + .605(R^-Y) + .050(B -Y) ^ n ^2 1 n 1 n 1 (5_22) (B^-Y)2 = .035 Y^ +-.I17(R^-Y)^ + I.062(B^-Y)^
Note that both color signals are almost independent of Y^, and that
(B -Y)2 is more dependent on (R^-Y)^ than (R^-Y)2 is on (B^-Y)^ (a
factor of 0.117 vs. a factor of .050). Ideally, (B^-Y)2 depends
only on (B^-Y)^, while (R^-Y)2 depends only on (R^-Y)^
Table%-7 also compares the validity of Eq. (5-21) against the
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107
2 v/div., 10 ysec/div.
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IHR (a) The Y Product Signal.
0.1 v/div., 10 ysec/div.
(b) The (R -Y)' and (B -Y)' Transform Signals n 1
Figure 5-19 Single Horizontal Scans for the Product and Transform Signals.
108
predictions of Eq. (5-19). The results were again very good, with
relatively small errors. The errors were generally larger than those
compared for Eq. (5-20) due to a reduced signal-to-noise ratio on
Camera #2.
The results discussed for Table 5-7 are perhaps the most
important results of this entire chapter, because they show that a
relatively accurate polar-to-rectangular complex number representation
can be achieved in an actual physical system. A range of linear values
was identified and used to represent and transform complex numbers,
using color signal representations.
5.3.3 Complex Number Addition
In the final test, the method of using a lens to add complex
numbers (described in Sections 3.2 and 4.2.4) was demonstrated. Here,
two pixels of different hue and equal luminance were displayed in a
vertical distribution as shown in Fig. 5-20. When the cylindrical
lens of Fig. 5-1 (also shown in Fig. 5-13) was put in place, the
double pixel was vertically integrated and spread as in Fig. 5-21,
where the viewfinder output of Camera //2 is shown. Using this
arrangement, the results shown in Table 5-8 were obtained, where
the measured values were taken for each pixel. The lens was then
put into place to obtain a summation pixel. The prediction for
the summation, given in rectangular form as:
Re . (B^-Y)^ = (B^-Y)^ + (B„-Y)^
Im . (R^-Y)^ = (R„-Y)^ + (R^-Y)^
was tested for errors. Shown in Table 5-8, these errors are due to
fluctuations in luminance and size between the pixels. Relative pixel
size and the alignment of the lens is especially critical and care is
required to insure that both pixels are equally weighted. If one
pixel has a larger area, or if the lens collects more light from one
of the pixels, that pixel will have a greater effect in the addition
109
Pixel b (e.g. red)
Pixel a (e.g. green)
Figure 5-20 Double-Pixel Display for Addition Testing (Pixels Appear as Different Colors).
110
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Ill
of Eq. (5-23). When measuring the additive sum, it is important
to use a point that is spatially centered between the two added
pixels so that each is weighted equally.
Addition using a lens was demonstrated in a simplified I-D form
as shown here. While the test was only for 2 pixels, it did show
that addition can be done for the color signals of a camera on a
constant Y slice of the color space. It is interesting to note
that the diverging incoherent light waves from the monitor form a
virtual image at Camera //2 when passed through the lens.^^ Thus,
the waveform is spread at the focal point of the lens as shown in
Fig. 5-21, rather than focused to a small, intense point as
predicted in Sections 3.2 and 4.2.4. As a result, different normal
izing scale factors than those given in Eqs. (4-23) and (4-24) may be
needed.
5.3.4 Summary
The complex number operations of multiplication, polar-to-
rectangular transformation, and addition have been realized, using
the experimental system as described. While these operations have
been achieved using only a small number of pixels with limited
accuracy, the principles behind the operations have been shown.
The results indicate that, in order to do the large number of opera
tions required for a good approximation of most space-variant optical
systems, linearity, accuracy, and dynamic range must be improved.
The results from this chapter specifically indicate that
tristimulus-based processing systems should use primaries that
operate at a single constant intensity level, so that a maximum
linear range can be achieved. The results also emphasize the fact
that physical tristimulus systems have many limitations with respect
to the theoretical 3-D vector spaces that model them. This was
especially apparent in this experimental system, which showed that
changes in luminance can not be completely separated from changes
in color, and that only a small portion of the total tristimulus
112
(a) Viewf inder of Camera //2 Without Lens i n P l ace
(b) Viewf inder of Camera //2 With Lens in P lace
Camera //2
(focal length)
(c) Placement of Lens.
Figure 5-21 Using a Lens to Perform Pixel Addition
113
vector space model will yield a linear result. Again, the limiting
factors seem to be dependent on the dynamic range and linearity
properties of the color cameras, monitors, and sources that are used.
Among these limitations, the most serious problems occur due to the
vidicon tubes and color CRT phosphors used in the components of the
system. Possible alternatives to these devices will be discussed in
the final chapter.
CHAPTER 6
CONCLUSION
This thesis has investigated the basic principles and methods
of tristimulus-based incoherent optical processing. In summary,
the basics of tristimulus theory have been presented and the linear
vector space properties of theoretical tristimulus color space
representations have been successfully demonstrated. These properties
are used, as shown in Chapter 3, to represent, multiply, and add
complex numbers. By the use of a sampled-form approximation, these
operations may be used to represent general space-variant systems as
described by the complex superposition integral. A system archi
tecture, which makes use of color television equipment and the NTSC
(television) tristimulus color spaces has been designed and an
experimental demonstration system has been built. The results of
experiments done with this system have shown that the basic operations
required for a tristimulus-based processor are physically realizable.
The limited dynamic range and the nonlinearities inherent in
color television equipment remain the major problems with practical
tristimulus-based optical processing. These problems require further
investigation and the application of new devices that are specifically
designed or adapted for linear optical computing. The experimental
system, presently used for demonstration purposes, is severely
limited as a practical information processor. While "fine tuning"
the system and adding electronic processing to reduce noise could
result in some improvements in performance, the changes would
probably not produce many benefits.
In order to build a processor that is more practical, detection
and display devices are needed that will provide substantial improve
ments in dynamic range and linearity. One possibility is to replace
the existing vidicon tube cameras with charge coupled device (CCD)
cameras. Unlike vidicon tubes, CCD imaging arrays are solid state,
have a linear response to input light, and have a spectral response
114
115
that extends into the infrared. This wide spectral response would
make new tristimulus spaces possible that extend beyond the visible
range, providing a greater range of physically realizable tristimulus
values. The dynamic range of a CCD camera is potentially very large
(1000:1), so processing accuracy may be greatly increased over that
achievable with a vidicon camera, as discussed in Appendix D.
CCD imaging arrays are already in use in a filmless 35 mm color
camera system (the "Mavica") that will soon be introduced by the Sony
corporation. This type of system, which makes use of tristimulus
methods for color reproduction, may be directly applicable to the
kind of work that has been described here. A preliminary discussion
of CCD arrays and their potential application to tristimulus-based
optical processing appears in Appendix D.
The CRT monitor may be the weakest component of the system
described in this work, because of the phosphors, which introduce
nonlinearities and hysteresis in objectionable amounts. Solid-state
replacements for the color CRT have not yet been developed, though
alternatives are currently being studied. For number processing,
color CRTs could possibly be replaced by raster scanned and modulated
laser display devices or LED arrays (where the coherence of the
laser is somehow reduced or eliminated). These, and other types of
display devices should be investigated as potential candidates for
tristimulus-based processing schemes. Again, displays should be
studied for the possibility of extending tristimulus methods beyond
the range of the visible spectrum.
Subsequent investigations into the use of tristimulus methods
may also be extended to include various types of hybrid optical-
electronic or optical-optical systems with architectures that may
differ from the one presented in this work. For example, the elec
tronic polar-to-rectangular transform could possibly be modified or
eliminated by the use of various incoherent and/or coherent optical
processing methods, as discussed in Appendix C.
In conclusion, this work has successfully shown the basic
116
principles of using tristimulus-based methods for incoherent optical
processing of complex number information. While the results indicate
that these methods can be used in linear processing arrangements,
there are still many practical problems to be overcome. Thus, further
research on the topic area of this thesis is warranted and should be
undertaken. Additional investigations should include non-standard
and extended tristimulus systems, the use of sensing and display
devices that have improved linearity and dynamic range, various
hybrid and optical processing architectures, and new information
interface and input/output techniques.
APPENDIX A
EQUIVALENT 1-D EQUATIONS FROM SECTION 4.2
117
118
The equivalent I-D representations for each of the applicable
equations in Section 4.2 follow. These equations are numbered to
correspond directly to those in Section 4.2, where the 2-D forms of
the equations appear.
h(i,m) = A(i;m)/(l)(i;m)
g(m) = B(m)/9(m) (4-13A)
(R -Y)^ (i;m) = 0.5(|)(i;m) n i.
TT
(R -Y). (m) = 0.5(|)(m) n z
TT
(4-I4A)
Y^ = 0.5
(R -Y).(i;m) n 1
llB^-Y)^ = OJ
^2=0.5
(R -Y) (m) n i.
(B^-Y) = 0 n z _
R^^(i,m)
G _(i,m) nl
^B^^(i,m)
G;^(m)
L'n2 ") .
(4-15A)
R^^(i;m) = 2R e '° ^ ^ ^%(^'°^^ G^^(i;m) = ilG 0(i;m) ^ hG^^(.U^)
B^^(i;m) = B^e^^'"^^ ^ "^^^^'^J^
(4-16A)
\ = 1 + '-^2 (\-Y)^(i;m) = i (R -Y)i(i;in) + hi^^-Y) ^^^'^^^
( B % ) ^ = (B^-Y)i + (B^-Y)2 = 0
(4-I7A)
(R -Y) (i;in) = {0.5(t>(i;ni) + 0.59(m)} n H' ^ TT
+ n.5((j)(i:m) + e(m)) = 0.5y(i;m) 2-n 27T
(4-18A)
119
T^(m) = B(m)
T2( i ;m) = A(i ;m)
1 ^ (B -Y)**(i)<=cRe{^ E C( i ;m) / ' t ' ( i ;m)}
m—1
M ^ (R - Y ) * * ( i ) = - ^ Im{ E h ( i ; m ) g ( m ) }
n M
(B _Y)**r i ) = % ^ Im{ E h ( i ; m ) g ( m ) } n ' ' M ,
m=I
(4-19A)
C( i ;m) = A(i;m)B(m) (4-20A)
Y = 0 . 5 ( c o n s t a n t ) ou t
(R^-Y)*( i ;m) = Y ( i ; m ) s i n [ 2 - ^ (R-Y)^ ( i ;m) ]
= I m { C ( i ; m ) / T ( i ; m ) } (4-21A)
(B - Y ) * ( i ; m ) = Y ( i ; m ) c o s [ 2 ; r ^ (R-Y)^ ( i ;m) ] n p u.D T
Re{C( i ;m) /Y( i ;m)}
I ^ (R -Y)**( i )«Im{:^ E C( i ;m) /H ' ( i ;m)} n M T
m=I (4-22A)
m=l
M (4-23A)
i ( i ) = 7 ^ ( ( B - Y ) * * ( i ) + j ( R - Y ) * * ( i ; m ) ) 0 . 5 n "
M M ^ (4-24A)
= E R e [ h ( i ; m ) g ( m ) ] + j E Im[h( i ;m)g(m)]
m=l °i=l
APPENDIX B 4
A REAL-TIME SPATIAL LIGHT MODULATOR
120
121
The pixel-by-pixel multiplication described in Section 4.2.2
requires back-to-back real-time spatial light modulators. As
described in Chapters 3 and 4, these are operated in an electronically
addressed transmittance mode. This mode was chosen, however, merely
as a convenient means of illustrating the principle of complex number
modulus multiplication. In practice, many different types of archi
tectures and modes may be used.
Most spatial light modulators (SLM's) that are currently in use
are optically addressed in that a "write" light image pattern (usually
incoherent illumination) is projected onto the modulator, which
controls the reflection or transmission of a "read" light (usually
coherent). The primary devices now in use include the £Ockels
_readout optical modulator (PROM), the l iquid £rystal Mght valve 72-74.
(LCLV), and the micro-channel spatial 2.i§ t modulator (MSLM).
Many other devices exist, in various stages of development, but only
the devices mentioned above and specifically, the LCLV will be dis
cussed here. The LCLV is of particular interest, because it has been
characterized for incoherent tristimulus systems.
The operation of the LCLV for pixel-by-pixel multiplication is
shown in Fig. B-1, where the appropriate pixel representations for
A(i,j;m,k) appear on CRT #1 as various luminance patterns, while the
patterns for B(m,k) are "written" onto a nematic LCLV by the output
of CRT //2. When no light comes from CRT #2, the polarizer and
analyzer form a crossed polarization pair which will not allow light
to pass through to the camera. The pattern from B(m,k), when written
onto the LCLV from CRT //2, will change the polarization of the light
reflected from the LCLV, which will in turn, allow light to pass
through the analyzer in a pattern that is proportional to the product
of the patterns of CRT #1 and #2.' ^ Thus, the final pattern at the
camera is proportional to A(i,j;m,k) x B(m,k).
Note that, unlike Fig. 3-8, the input pattern for A(i,j;m,k) is
provided directly by the CRT, thus eliminating one of the expensive
SLMs shown in Fig. 3-8.
122
CRT //I.
A(i,j;m,k)
3 Polarizer
B(m,k)
Polarized Analyzer
Bias Voltage
Figure B-1. Use of an Optically Addressed Liquid Crystal Light Valve (LCLV) for Pixel-by-Pixel Multiplication.
123
The performance parameters of the Hughes Corporation LCLV are
shown in Table B-1. Note that the contrast ratio, which determines
the usable dynamic range of the device, is limited to about two orders
of magnitude, and the response time is slow. Because of the relatively
slow response time, the LCLV is used to represent B(m,k), in Fig. B-1,
since B(m,k) does not change until all values of i,j have been evalu
ated, as described in Section 4.2.
The other devices mentioned here can possibly be used in place
of the LCLV, in arrangements similar to that of Fig. B-1. The
specific device, architecture, and transfer characteristics can be
designed according to system and user requirements. For applications
and specifications, consult the literature and the manufacturer's
data on various SLMs.
As a low-cost alternative to the LCLV, a CRT could be con
structed with internal back-to-back grids that modulate an electron
beam. If A(i,j;m,k) controls the first grid and B(m,k) controls the
second, the product of the two variables appears on the electron
beam at the face plate. If the signals are "preprocessed" such
that the light output from the CRT is linear with respect to the
input signals, the product A(i,j;m,k) x B(m,k) may be detected at 78
the B & W camera.
Table B-1 Specifications and Typical Performance Levels 124 of a Liquid Crystal Light Valve (lp= line pairs).
Voltage bias range 5-15 V rms
2 5 AC frequency range 10 -10 Hz
Maximum clear aperture 46 mm diameter
2 Imaging light power (at 525 nm) 150 yw/cm
Spatial frequency at which (<I% interharmonic distortion) >15 Ip/mm
modulation is 50% (unspecified linearity) >30 Ip/mm
Limiting resolution (Air Force Resolution Chart) >40 Ip/mm
Response time (Rise - 90%) 40 msec
(To full contrast) (Decay 100 - 10%) 30 msec
Contrast ratio (at one wavelength) >100:1
Signal/noise ratio in Fourier plane >35 dB
(for spatial frequencies >5 Ip/mm)
Optical flatness (peak-to-peak distortion) >3X/4
2 Excitation energy to full contrast 60 ergs/cm
APPENDIX C
INSTRUMENTATION AND TRANSFORM ELECTRONICS
125
126
In order to display results and to do the polar-to-rectangular
transform, electronic circuits were constructed. These circuits,
with their respective connections between the various cameras in
the system, were constructed as shown in block diagram form in
Fig. C-1. In the figure, dotted lines denote digital synchronization
and switching signal paths.
As shown, the Y, R -Y, and B -Y signal paths in Camera //I are n n
broken at the first stage of the camera, while the horizontal (H)
and vertical (V) synchronization pulse trains are tapped. The
(R^-Y)^ signal is used as the scaled angle input into the polar-to-
rectangular transformer, while Y , from the B & W camera, is scaled P
for use as the magnitude input. Note that the B & W camera is synchronized from Camera //I. The output signals from the transformer can be input into the B -Y and R -Y signal channels at the final
n n ^ stage of Camera //I for transformed operation, or the original B -Y
n
and R -Y signals can be input into the final stage to allow the
camera to function in a normal mode. The two modes are selectable
at Switch Point //I (S.P. #1 in Fig. C-1). The resulting NTSC
composite signal drives the monitor. The (Y, R -Y, B -Y) signals from the first stage of Camera //I,
n n those tapped from Camera #2, the ((R -Y)', (B -Y)') signals from the ^ '^ n n transform circuit, and the Y signal from the B & W camera are
P selected in various combinations at S.P. //2 for input into the
display electronics. All signals entering the display section are
buffered, while the selected color signals are low-pass filtered.
The signals are then sent to the inputs of the oscilloscopes, while
the color signals are additionally tapped, sampled, and scaled for
input to the x-y vector display, shown in Fig. 5-7.
For the oscilloscopes to operate properly, any signal being
displayed must be properly synchronized with its source. Thus, the
synchronization pulse trains from Camera //I cr //2 may be selected at
S.P. #3. The selected sync pulses are counted and compared with an
input 8 bit binary code to allow the selection of any given horizontal
127
/-c B&W Camera
fViV I I I I
Sync. Driver
Optical Sources ^
Color Camera //I
F i r s t Stage
Y^, (R„-Y)j , (B^-Y)^
n 1.
NTSC Monitor
Color Camera #2
NTSC Composite Signal
Color Camera //I
^Final Stage
Polar-to-Rectangular Transform Circuit
{(R^-Y)', (B^-Y)'
Display Inputs
Scan ^^^^wu^HQ g^^ Electronics r- „
iz Sw.
Y,, (R„-Y),, (B„-Y)
fj
Display Buffers
R -Y n
>
R -Y, B -Y n ' n
Low-Pass Filters :>
Trigger
To Oscilloscope
Inputs
V B -Y n
Vector Display Circuits
R -Y, B -Y _I1 !—S To Vector
Display Optical Signal Path Analog Electronic Signal Digital Electronic Signal
Figure C-1. System Block Diagram.
128
scan line to be displayed on the oscilloscopes. Alternately, the
vertical sync pulse train may be used to synchronize the oscillo
scopes for display of an entire vertical scan as shown in Fig. 5-2a.
The circuits for the synchronization of the B & W camera
appear as in Fig. C-2a. These circuits buffer and lengthen the H
and V pulses from Camera //I so that they are able to drive the H and
V synchronization inputs to the B & W camera. The timing diagrams
for these pulses appear in Fig. C-2b and C-2c.
The display electronics appear as in Fig. C-3a, where the
selected input signals are buffered, the color signals are low-pass
filtered to reduce noise, and the signals are displayed on oscillo
scopes. The color signals are tapped, the 6.2 volt d.c. offset is
taken out, and the signals are switched on and off during each
vertical scan as shown in the timing diagram of Fig. C-3b. This
switching prevents the vector display from showing unwanted informa
tion by sampling at the point where the signal occurs. The sampled
signals are then scaled and input into the vector display, where they
appear as shown in Fig. 5-7.
In order to allow the oscilloscopes to display any of the 252
horizontal scan lines, the horizontal sync pulses are counted by
two 4-bit counters, linked such that they count from 0 to a maximum
of 256 and are reset by every vertical pulse. As shown in Fig. C-4,
the 8 bit code from the counters is compared to an 8 bit input code
that is selected by a combination of microswitches. When the count
code matches that selected by the switches, the Sync Out line is
pulsed. The scopes are triggered from this selected pulse, once for
each vertical scan. Thus, any single horizontal scan can be displayed
The block diagram of the 2-quadrant polar-to-rectangular trans
form circuit is shown in Fig. C-5. Again, the dotted lines represent
digital synchronization and switching signals. The circuit is syn
chronized such that the input is sampled once for every four
horizontal scan lines so that the slow responses of the sine and
circuits do not cause problems. This design has several cosine
I V Sync^ from I Camera | //I I
1 I
H Sync.I from ^ Camera •
129
Buffer
Multivibrator
//I I I Multivibrators
Buffer
(a) Block Diagram
V Sync. to B&W Camera
H Sync . to B&W Camera
H Sync 63 ys
Multivibrator //I
Multivibrator //2
(b) Horizontal Signal Timing Diagram
Multivibrator Output
16.7ms
V Sync
(c) Vertical Signal Timing Diagram
Figure C-2. Synchronization Electronics for the B&W Camera
130
in
Unity Gain Buffer
Y Signal to Input on Oscilloscope
n m
Buffer
Buffer
n m
Low-Pass Filter
R -Y n to Input on Oscilloscope
Low-Pass Filter
Gain & Offset Adjust Switch
V Sync ,
to Input on Oscilloscope
-Y to Vector
„ . Display Gain ^ ^ Adjust
-Y to Vector Display
Analog Signal
I I Timing I i Digital Signal
(a) Block Diagram
V Sync. 1
Timing Signal to Switch
V e r t i c a l
1 1
llliiii>iii....H.fliHnnuiiHi«idllllllllllllllIllllIlimuii.uiiiMltl^ Scan
(b) Timing Diagram f*~ on ~H
Figure C-3. Display Electronics.
V in H in
Count Reset
4 Bit Counter
B
D
Count Reset
4 Bit Counter
H
4 Bit Comparator
Link
4 Bit Comparator
Sync Select Output
131
A'
B'
C 8 Bit Switch
tD' Bank
E*
tF'
G'
rH'
Figure C-4. Scan Line Selection Electronics.
cu &0 CO 4J C/3 / - ^
/ - ^ r-\ >* r H CO
1 O CO (U
c u c Pi Pi "H ^w'
» / — V
>! 1
c PQ
> N
00 CO 1 3 2 CO C 4-1 -H CO 60
CO O rH B •U CO M
C ^
(U rH CO CO CO O -H CO P Q 0)
D d <4J
^ 3
r
L
I 1
^ u p (L> O
- j ^ 1 c re
- H ' . _
6 CO U CJO CO
CJ
o PQ
C O U 4-1
a OJ
iH
w 6 M O
<4H CO C CO }H
H
CO
rH 3 00 C CO 4-1
a 0)
£3(5 I )H CO
o OH
i n I
u
>H 3 00
P3 >
133
disadvantages in that only one new value of angle information can
be displayed for every four scan lines. Also, only two of the
eight quadrants needed for full representation of complex number
products can actually be used. This circuit was used successfully,
however, in the experiments described in Chapter 5.
The two toggled D flip-flops, shown in Fig. C-5 are triggered
by positive-going pulses as shown in the timing diagram of Fig. C-6.
The first flip-flop drives the second, which triggers a monostable
multivibrator as shown by Pulse Lines #1 through //4 in Fig. C-6.
The input signal from (R -Y)^, shown in Line #5 of Fig. C-6, is
adjusted such that its d.c. offset is zero, while its gain is raised
to yield peak signal values of close to ±10 volts. This signal is
then sampled and held by the sample/hold (S/H) circuit that is
controlled by Pulse Line //4. The output of the circuit is then
adjusted and the absolute value is taken such that the output appears
as in Pulse Line //6 of Fig. C-6.
The absolute value circuit is necessary because the sine and
cosine resolver circuits are only operable for single quadrant
(non negative) signals. Shown in detail in Fig. C-7, it uses a
comparator, strobed from Pulse Line //4, to compare the signal to
reference ground. If the signal is greater than zero, the positive
output of the comparator is high, while its complementary output is
low. If the signal is less than zero, the opposite occurs. These
two digital outputs control analog switches so that, if the signal
is greater than zero it is inverted. If the signal is less than
zero it is not inverted. These two outputs are fed into a summing,
inverting amplifier. Since one of the switches is always off, the
output of the summing amplifier displays the absolute value of the
sampled signal, scaled to a maximum peak at 10 volts.
The single-quadrant sine and cosine resolver circuits, which
also appear in detail in Fig. C-7, form the heart of the polar-to-
rectangular transform circuit. They use a multifunction integrated
circuit (Model 4302, manufactured by Burr-Brown Co.) to do a
1
CO
> o
J J d 134
q CJN 00 NO i n ro CN
i n I
CD
00
o
CO O
o 4-1
TJ c o a CO (U >H
M O
a CO u cu .o
CD U
• H c o >H
4-1
a
rH
B u o
14H CO
c CO
H V-i
o
B CO
00 CO
00 G
•H e
•H C-H
vO I
o
Ei
135
truncated power series approximation of the sine and cosine functions
and are scaled such that 0 to 10 volts input corresponds to an angle
of 0 to 90°, while 0 to 10 volts peak output corresponds to output
number values of Q.0 to 1.0:
„ 2.827 ^ ^ = ^out = ^° ^^ (%> = 1-57IE^ - 1.592 - ^
Cosine: E^^^ = 10 cos (9E^) = 10 + 0.365E^ - 0.428 E^^'^^^
Since the cosine function is positive for the two quadrants
shown, the output does not need to be changed. The sine function,
however, must be inverted if the input signal is less than 0. Thus,
the positive/negative logic signals from the absolute value circuit
are used, as shown in Fig. C-7 to reinvert the output of the sine
resolver, if needed. The operation of the circuit closely corresponds
to that of the absolute value circuit.
Because the response times of the sine and cosine circuits are
slow, as shown by Lines //7 and #8 of Fig. C-6, they must be sampled
at a delayed time. Thus, as shown in Fig. C-5, two more S/H circuits
are used. These circuits are controlled such that they sample at the
fourth scan line after the sine and cosine are updated with a new
input value, as shown in Line #9 of Fig. C-6.
The outputs of the S/H circuits are multiplied with the Y input
from the B & W camera, scaled as shown in Fig. C-6. The resulting
signals are rescaled and biased to their original levels as discussed
in Section 5.3.2.
The 2-quadrant polar-to-rectangular transform circuit of Fig. C-5
could be extended to the needed eight quadrants by the use of range
switching, but the problems with the slow-response circuits and
dynamic range limitations indicate that a different approach should
be tried. In one approach, shown in Fig. C-8, an acousto-optic (A-0)
cell can be used to displace a light beam across a sine or cosine
amplitude mask, in proportion to the input (Rj -Y)-| signal from the
136
^ > H ^ H
•3- I =fe
<u Pi • CO 00 C E C CO
O C CO " H 4-1 CO 4J O C_3 pi4 CO
xi CO 00
(U 3
CO >
4J 3
rH O CO <
/ 1 \ 4-1 U (U > c M
00 O
r-A CO C <
O 1 4J CO U CO Ou S o a
A
5 o •-J
H
CO rH •H CO
4-1 (U
Q
•H 3 u u
• H CJ
CO
a •H c o 4-1 O (U
rH
w B u o
U H CO cs CO u H
I
CJ
0) »H
C rH <U o =tfc
c H >H . CO 1 g O O Ceo
r-i CO c
•H
cu on CO 4J
CJ 4J PQ CJ fe CO
137
r-{ CO C 00
•H CO
00
c •H CO CO cu o o M
PL,
CO
o •H
c o M +J U CU
y-{
a
o 1 <:
1
u cu >
•H M
Q
I f
v-i o u u
CO
<u
o I
<
o I o 4-1 CO 3 o CJ
< : 00
c • H CO
3 B }H O
IM CO c CO u H ^ CO
rH 3 00 CO 4-1
o (U
P:; I o 4-1 I
u CO
rH
o P-. c CO
-a CO 3 a-
I 00
00 I
CJ
OJ
u 3 00
138
television camera. The output signal at the detector is then multi
plied with the input Y signal from the B & W camera to yield the
real or imaginary parts of the rectangular-form result.
ILL
APPENDIX D
CHARGE-COUPLED DEVICE IMAGING CAMERAS
139
140
As discussed in Chapter 6, vidicon tube cameras are noisy, have
a limited dynamic range, are nonlinear in response between input light
and output signal, and show hysteresis (memory) effects. Thus, they
are not very suitable for accurate optical processing. Charge-coupled
device (CCD) cameras, however, are solid state imaging devices that
have a wide dynamic range, a very linear response between input
light and output signal, and a wide spectral bandwidth. With suitable
adaptation, they may be used in tristimulus processing systems.
CCD imaging arrays operate by using the semiconductor photo
electric effect in which incident photons dislodge electrons in the
material and create charge "packets" on an array of storage elements.
By appropriate clocking and synchronization, these charge packets are
created by periodic exposures, then passed on to a storage register,
where they are transported from one storage element to the next in a
charge-coupled shift register pattern. These repeated exposures
result in framed video output signals that are proportional to input
light patterns, as they generate patterns of charge on the array.
The imaging array may either be in a line-scanned form which senses
one line of information at a time, or in an x-y area form which
delivers an entire field of video information from each exposure 1 80 cycle.
The CCD camera has a very large dynamic range or maximum signal
to noise ratio (SNR), compared to the vidicon camera. For the
Fairchild Company CCD cameras, with a response shown in Fig. D-1,
the SNR is greater than 200:1, when the maximum signal level is 81
compared to the peak-to-peak noise value. This dynamic range can
be increased, however, as the limiting SNR of the CCD imaging array
itself is as high as 1000:1 (5000:1 when compared to rms system
noise). In contrast, the best SNR achievable with a vidicon tube
camera is often less than 100:1, when compared merely with rms
• 82 system noise.
The operation of a CCD imaging array allows for a linear
response between input luminance and output signal. Thus, y
141
(discussed in Section 2.3) is approximately 1.0, as shown in Fig.
D-1. This linearity is, of course, very desirable for tristimulus-
based optical processing.
The wide spectral response of the CCD imaging array is shown
in Fig. D-2, for a Texas Instruments model TClOl image sensor. "
The array has a reasonably flat response over the visible spectral
range (400-600 nm). With compensation, the CCD array can be made
flat into the infrared region (about 900 nm), allowing extended
tristimulus values, and infrared optical processing. Optical
resolution is reduced, however, when the infrared region is used,
because long-wavelength photons tend to imbed themselves deep in
the substrate of the CCD array, causing cross-talk between storage 84
elements. Thus, a blocking filter is often used, as shown in Fig.
D-2.
The characteristics discussed here make the CCD imaging array
very desirable for use in tristimulus-based processing schemes. If
three synchronized CCD arrays are used in place of the imaging tubes
of Fig. 2-14, a color camera with excellent linearity and range
will result. Thus, the wide spectral bandwidth, linear response, and
dynamic range associated with CCD arrays make possible new and
expanded tristimulus architectures with improved performance and
flexibility.
AGC Off
Analog Video Output (mv peak)
1000
100
10
142
Saturation
Figure D-1.
1.0
.001 .01 0.1 1.0 10
Relative Scene Highlight Brightness
Input-Output Response of a CCD Imaging Camera.
Peak-to-Peak Noise
ithout Filter 10
Sensitivity V
yJ/cm^ 1.0
0.4
0.1 400 600 800 1000
Incident Wavelength- nm 1200
Figure D-2. Spectral Response of a CCD Imagimg Array,
FOOTNOTES
1. M.A. Monahan, K. Bromley, R.P. Bocker, "Incoherent Optical Correlators," Proceedings of the IEEE, 65, 121 (1977).
2. S. Lowenthal, P. Chavel, "Noise Problems in Optical Image Processing," Proceedings of the ICO Jerusalem Conference on Applications of Holography and Optical Data Processing, E. Wiener and J. Shamir, Eds., 45-55 (Plenum Press, New York, 1977).
3. A.W. Lohmann, "Incoherent Optical Processing of Complex Data," Applied Optics, 16 , 261 (1977).
4. I. Glaser, "Representing Bipolar and Complex Imagery in Noncoherent Optics Image Processing Systems: Comparison of Approaches," Optical Engineering, 20, 568 (1981).
5. D.S. Tavenner, "Space-Variant Incoherent Optical Processing Using Color," M.S. Thesis, Texas Tech University, 4-5 (Lubbock, Texas, 1981).
6. I. Glaser, op. cit., 568.
7. J.W. Goodman, Introduction to Fourier Optics, 107 (McGraw Hill, New York, 1977).
8. I. Glaser, o£. cit., 568.
9. D.S. Tavenner, o£. cit., 4-5.
10. J.F. Walkup, "Space-Variant Coherent Optical Processing," Optical Engineering, 19, 339 (1980).
II R J. Marks II, J.F. Walkup, M.O. Hagler, "Sampling Theorems for Linear Shift-Variant Systems," IEEE Transactions on Circuits and Systems, CAS-25, 228-233 (1978).
12. G. Wyszecki and W.S. Stiles, Color Science, 239 (Wiley & Sons,
New York, 1967).
13. W.K. Pratt, Digital Image Processing, 61 (Wiley & Sons, New
York, 1978).
14. G. Wyszecki and W.S. Stiles, op. cd£., 233.
15. W.K. Pratt, o£. cit_., 64.
143
144
16. R.W.G. Hunt, The Reproduction of Colour, Third Edition, 107 (Wiley & Sons, New York, 1975).
17. G. Wyszecki, "Colorimetry," Handbook of Optics, 9-4 (Optical Society of America; McGraw-Hill, New York, 1978).
18. C.J. Bartleson, "Colorimetry," Optical Radiation Measurements, 43 (Academic Press, New York, 1980).
19. G. Wyszecki and W.S. Stiles, 0£. cit., 235.
20. C.T. Chen, Introduction to Linear Systems Theory, 16 (Holt, Rinehart & Winston, New York, 1970).
21. C.J. Bartleson, o£. cit., 45.
22. C.T. Chen, o£. cit., 20.
23. C.J. Bartleson, 0£. cit., 45.
24. W.K. Pratt, o£. cit., 74.
25. G. Wyszecki, o£. cit., 9-4.
26. Ibid., 9-24.
27. R.W.G. Hunt, o£. ci^., 107.
28. G. Wyszecki, op. it., 9-4 - 9-5.
29, M.D. Buchanan and R. Pendergrass, "Digital Image Processing, EOSD, 14, no. 5, 29 (March, 1980).
M
30. G. Wyszecki, 0£. ci±., 9-2.
31. R.H. Wallis, "Film Recording of Digital Color Images," 66 (imaging Processing Institute, University of Southern California,
1975).
32. W.K. Pratt, o£. cit_., 85.
33. Ibid., 75.
34. Ibid., 737.
35. H. Ennes, Television Broadcasting, second edition, 68-69 (H.W. Sams, Indianapolis, Ind., 1979).
145
36. D.S. Tavenner, o£. cit., 273
37. W.K. Pratt, op. cit., 737.
38, R.W.G. Hunt, o£. cit., 458,
39. Ibid., 459.
40. R.H. Wallis, o£. £it., 70.
41. H. Ennes, 0£. cit., 70.
42. R.W.G. Hunt, o£. ci£., 400.
43. R.H. Wallis, o£. £it., 70.
44. H. Ennes, o£. cit., 112.
45. Ibid., 113.
46. R.H. Wallis, o£. cit., 63-65
47. D.S. Tavenner, 0£. cit., 282.
48. J.F. Walkup, o£. cit., 342.
49. D.S. Tavenner, 0£. cit. 19.
50. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 243 (McGraw-Hill, New York, 1965).
51. Ibid., 242.
52. H. Anton, Elementary Linear Algebra, second edition, 162 (Wiley & Sons, New York, 1977).
53. J.W. Goodman, 0£. cit., 18-19.
54. R.J. Marks II, J.F. Walkup, M.O. Hagler, "A Sampling Theorem for Space-Variant Systems," J. Optical Society of America, 66, 918 (1976).
55. R.J. Marks II, "Sampling Theory for Linear Integral Transforms," Optics Letters, 6, 7 (1981).
56. B.H. Soffer, et. al., "Real-Time Implementation of Nonlinear Processing Functions," Technical Report F49620-77-C-0080 (Air Force Office of Scientific Research, Washington, D.C, September 1980).
146
57. T. Shimomura, "Color Gamut of a Nematic Liquid Crystal Display under Ambient Illumination," Applied Optics. 20, 4156-4160 (1981)
58. J.B. Flannery, Jr., "Light-Controlled Light Valves," IEEE Transactions on Electron Devices, ED-20, 941-953 (1973).
59. C. Ward, et. al., "MicroChannel Spatial Light Modulator," Optics Letters, 2> 196-198 (1978).
60. W.P. Bleha, L.T. Tipton, et. al., "Application of the Liquid Crystal Light Valve to Real-Time Optical Data Processing," Optical Engineering, 17, 371-384 (1978).
61. Nippon Electric Co., Ltd., "NEC Portable Color Service Manual, Model VCI-2I~E," ser. no. 4026, 2.
62. Rolyn Optics Co., "Optics for Industry," Catalog 879, 44 (1979).
63. Eastman Kodak Co., "Kodak Filters for Scientific and Technical Uses," Book No. 0-87985-029-9 (Eastman Kodak Co., 1980).
64. C. Rainwater, Light and Color, 14-17 (Golden Press, New York, 1971).
65. G. Wyszecki and W.S. Stiles, ££. cit., 43-53.
66. Oriel Corporation, "Complete Catalog of Optical Systems & Components," DI4 (Oriel Corporation, Stamford, Conn., 1979).
67. Eastman Kodak Co., o£. cit., 16-25, 30.
/ .68. - E. Hecht, "Theory and Problems of Optics," Schaum's Outline ^^ Series, 64 (McGraw-Hill, New York, 1975).
69. Fairchild Co., "CCD the Solid State Imaging Technology" (Fairchild Co., I98I).
70. N. Mokhoff, "Video," IEEE SPECTRUM, 19 , no. 1, 71 (IEEE, 1982).
71. A.R. Tebo, "Cockpit Displays - Works of Ingenuity and Splendor," EOSD, 13, no. 7, 41 (July, 1981).
72. Itek Corp., "PROM - Pockels Readout Optical Modulator," Technical Data Sheet (circa 1980).
73. W.P. Bleha and P.F. Robusto, "Optical to Optical Image Conversion with the Liquid Crystal Light Valve," Proceedings of the SPIE, 317, paper //32 (1981).
147
74. C. Ward, op. cit., 196.
75. T. Shimomura, o£. cit., 4156.
76. W.P. Bleha, op. cit., (Optical Engineering), 373.
77. W.P. Bleha, E. Wiener-Avnear, J. Grinberg, "Optical Data Processing Liquid Crystal Light Valve," Proceedings of the SPIE, 201, 123 (1979).
78. M.O. Hagler, personal conversation. May 27, 1982.
79. Burr-Brown Co., "General Catalog," 4-69 (Burr-Brown Co., Tucson, Az., 1978).
80. Fairchild Co., o£. cit., 123-124.
81. Ibid., 83.
82. J.A. Hall, "Problems in Photo-Electronic Imaging," Optical Engineering, 13, no. 2, G57 (1974).
83. Texas Instruments, Incorp., "Product Notes: Image Sensors," 9 (Texas Instruments, Inc. Semiconductor Group, circa 1981).
84. Fairchild Co., o£. £it_., 82.
SELECTED BIBLIOGRAPHY
Anton, H. Elementary Linear Algebra. New York: Wiley & Sons, 1977.
Bartleson, C. J. "Colorimetry." Optical Radiation Measurements. New York: Academic Press, 19^7
Bleha, W. P., L. T. Tipton, et. al. "Applications of the Liquid Crystal Light Valve to Real-Time Optical Data Processing." Optical Engineering. 17 (July/August, 1978): 371-384.
Bleha, W. P. and P. F. Robusto. "Optical to Optical Image Conversion with the Liquid Crystal Light Valve." Proceedings of the SPIE, 317 (1981): paper #32.
Bleha, W. P., E. Wiener-Avneau, and J. Grinberg. "Optical Data Processing Liquid Crystal Light Valve." Proceedings of the SPIE, 201 (1977): 122-124.
Buchanan, M. D. and R. Pendergrass. "Digital Image Processing." EOSD, 14 (March, 1981): 29-36.
Chen, C. T. Introduction to Linear Systems Theory. New York: Holt, Rinehart & Winston, 1970.
Ennes, H. Television Broadcasting. 2nd Ed. Indianapolis, Ind.: H. W. Sams Co., 1979.
Flannery, J. B., Jr. "Light-Controlled Light Valves." IEEE Transactions on Electron Devices, ED-20 (November, 1973): 941-953.
Glaser, I. "Representing Bipolar and Complex Imagery in Noncoherent Optics Image Processing Systems: Comparison of Approaches." Optical Engineering, 20 (July/August, 1981): 568-572.
Goodman, J. W. Introduction to Fourier Optics. New York: McGraw Hill, 1977.
Hall, J. A. "Problems in Photo-Electronic Imaging." Optical Engineering, 13 (March-April, 1974): G57-G58.
Hecht, E. "Theory and Problems of Optics." Schaum's Outline Series. New York: McGraw Hill, 1975.
Hunt, R. W. G. The Reproduction of Colour, 3rd Ed. New York: Wiley & Sons, 1975.
148
149
Lohmann, A. W. "Incoherent Optical Processing of Complex Data." Applied Optics, 16 (February, 1977): 261-263.
Lowenthal, S. and P. Chavel. "Noise Problems in Optical Image Processing." Proceedings of the ICO Jerusalem Conference on Applications of Holography and Optical Data Processing, eds. E. Wiener and J. Shamir. New York: Plenum Press, 1977.
Marks, R. J. II. "Sampling Thoery for Linear Integral Transforms." Optics Letters, (January, 1981): 7-9.
Marks, R. J. II, J. F. Walkup, M. 0. Hagler. "A Sampling Theorem for Space-Variant Systems." Journal of the Optical Society of America, 66 (1978): 981.
Marks, J. R. II, J. F. Walkup, M. 0. Hagler. "Sampling Theorems for Linear Shift-Variant Systems." IEEE Transactions on Circuits and Systems, CAS-25 (1978): 228-233.
Mokhoff, N. "Video." IEEE Spectrum, 1^ (January, 1982): 71.
Monahan, M. A., K. Bromley, R. P. Bocker. "Incoherent Optical Correlators." Proceedings of the IEEE, 65 (January, 1977): 121.
Papoulis, A. Probability, Random Variables, and Stochastic Processes. New York: McGraw Hill, 1965.
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