SELECTIVE POLARIZATION IMAGER FOR CONTRAST ENHANCEMENT IN EXTENDED SCATTERING MEDIA by

SELECTIVE POLARIZATION IMAGER FOR CONTRAST ENHANCEMENT IN EXTENDED SCATTERING MEDIA by
SELECTIVE POLARIZATION IMAGER FOR CONTRAST ENHANCEMENT IN
EXTENDED SCATTERING MEDIA
by
Darren Alexis Miller
_____________________
Copywrite © Darren A. Miller 2011
A Dissertation Submitted to the Faculty of the
COLLEGE OF OPTICAL SCIENCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2011
2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
prepared by Darren A. Miller
entitled Selective Polarization Imager for Contrast Enhancement in Extended Scattering
Media.
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy
_______________________________________________________________________
Date: 7/11/2011
Eustace Dereniak
_______________________________________________________________________
Date: 7/11/2011
Tom Milster
_______________________________________________________________________
Date: 7/11/2011
Daniel Wilson
_______________________________________________________________________
Date:
_______________________________________________________________________
Date:
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
________________________________________________ Date: 7/11/2011
Dissertation Director: Eustace Dereniak
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at the University of Arizona and is deposited in the University Library
to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided
that accurate acknowledgment of source is made. Requests for permission for extended
quotation from or reproduction of this manuscript in whole or in part may be granted by
the copyright holder.
SIGNED: Darren A. Miller
4
DEDICATION
The research and time invested in the work contained within this dissertation were largely
a labor of love for, and are dedicated to, a man who has taken it upon himself to work
himself to the bone to provide for multiple generations of his family, my grandfather and
best friend, Walter Miller. It is my hope that, in some small way, my trials and successes
with respect to the pursuit of my doctorate provide validation for your own trials and
tribulations in support of the maintenance of our family’s happiness.
To my grandmother, Marie, beyond any other, I am a reflection of you. Your love and
support since my birth have shaped me into the person I am today. For this, I thank you
from the bottom of my heart.
To my father, my friends, and the rest of my family, you are all a part of me. Without all
of you, I would not have succeeded in this quest. At times I found myself faltering, but
your support allowed me to stay the course, knowing that it was my responsibility to
finish what I had started.
Lastly, to Andrea, and to the light of my life, my son, Landen, I dedicate my success and
the rest of my life to you. Here, on the page, I pledge to always be true to you and to live
my life with integrity and honor. I also pledge to you, Landen, to instill in you the same
values that were instilled in me by your grandfather, Mark, and your great grandparents,
Walt and Marie.
5
TABLE OF CONTENTS
LIST OF FIGURES ........................................................................................................... 7
LIST OF TABLES ........................................................................................................... 13
ABSTRACT ..................................................................................................................... 14
CHAPTER 1: INTRODUCTION ..................................................................................... 15
1.1 Polarimetry ........................................................................................................... 16
1.2 Cross Polarization Intensity Subtraction (CPIS) in Medical Imaging .................. 17
1.3 Dissertation Arrangement ..................................................................................... 19
CHAPTER 2: SCATTERING PHYSICS ........................................................................ 21
2.1 Mie Scattering Theory ......................................................................................... 21
2.1.1 Solutions to the Scalar and Vector Wave Equation .................................... 22
2.1.2 The Mie Problem and Application of Boundary Conditions ...................... 24
2.1.3 Multiple Scatter Analysis of Polarization Properties .................................. 29
2.2 Monte Carlo Modeling ......................................................................................... 36
2.2.1 Probability Density Function Construction and Use: Step Size ................. 38
2.2.2 PDF Construction and Use: Physical Target Scatter Angle ........................ 39
2.2.3 PDF Construction and Use: Water Vapor Particle Size .............................. 48
2.2.4 PDF Construction and Use: Spherical Particle Scattering Angle ............... 51
2.2.5 Photon Position Tracking and Counting ..................................................... 56
2.2.6 Monte Carlo Simulation Convergence and Validation ............................... 59
CHAPTER 3: LABORATORY BASED SYSTEM DESIGN ......................................... 66
3.1 System Optical Design ......................................................................................... 67
3.1.1 Optical Design: Lens Design to Field Stop (Objective) ............................. 69
3.1.2 Optical Design: Field Stop to Wollaston Prism (Collimator) ..................... 72
3.1.3 Optical Design: Complete Lens Design ...................................................... 73
3.2 Image Registration ............................................................................................... 80
3.3 Polarimetric Calibration ....................................................................................... 85
3.4 Post Registration Algorithms ............................................................................... 92
3.4.1 Post Registration Algorithms: Image Blurring Filter .................................. 95
3.4.2 Post Registration Algorithms: Iterative Polarization Image Subtraction .... 99
3.4.3 Post Registration Algorithms: DC Block Fourier Spatial Filter ............... 104
CHAPTER 4: LAB DATA COLLECTION SETUP, RESULTS, AND ANALYSIS .. 108
4.1 Laboratory Experimental Setup and Procedure ................................................. 108
4.2 Image Enhancement Metrics .............................................................................. 112
4.3 Experimental Results: Rough Metal Wrench Images ........................................ 116
6
TABLE OF CONTENTS - Continued
4.3.1 Rough Metal Wrench Images using Circularly Polarized Illumination .... 117
4.3.2 Rough Metal Wrench Images using Linearly Polarized Illumination ...... 130
4.3.3 Experimental Results: Rough Metal Wrench Image Analysis ................. 143
4.4 Experimental Results: Printed Paper Images (Depolarizing Object) ................. 148
4.4.1 Experimental Results: Printed Paper Resolution Target Images ............... 153
4.4.2 Experimental Results: Printed Paper Resolution Target Image Analysis . 161
4.5 Experimental Results: Contrast Enhancement Analysis .................................... 162
CHAPTER 5: FIELD SYSTEM DESIGN ..................................................................... 165
5.1 Wire Grid Diffraction Grating Theory ............................................................... 165
5.2 Wire Grid Polarizer Array Design and Alignment ............................................ 169
5.2.1 WGP Array Design and Alignment: Active Polarizer Area ..................... 171
5.2.2 WGP Array Design and Alignment: Detection Site Isolation .................. 174
5.2.3 WGP Array Design and Alignment: Opto-Mechanical Alignment Setup 176
5.2.4 WGP Array Design and Alignment: Coarse Angular Alignment ............. 179
5.2.5 WGP Array Design and Alignment: Precision Angular Alignment ......... 181
5.3 Bonded Wire Grid Images ................................................................................. 188
CHAPTER 6: CLOSING REMARKS ........................................................................... 196
REFERENCES ............................................................................................................... 199
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LIST OF FIGURES
1. Images of a duck heart using polarization discrimination techniques ......................... 18
2. Clinical images of a basal cell carcinoma featuring improvement using CPIS .......... 19
3. Scattering geometry for use in the derivation of Mie scattering theory ...................... 21
4. Degree of polarization ratio as a function of optical depth ......................................... 31
5. DOP as a function of optical depth in transmission and backscatter .......................... 31
6. Cross polarization discrimination as a function of optical depth ................................ 33
7. Active imaging schematic for crossed polarization intensity discrimination ...............33
8. Target contrast comparison between various polarization modes of illumination .......35
9. Histogram for step size between scattering events .......................................................39
10. BRDF measurement setup ........................................................................................... 41
11. Irradiance detection geometry ..................................................................................... 42
12. Intensity distribution for scattering angles around the specular reflection angle ........ 42
13. Corrected irradiance distribution for scattering angles around specular ..................... 43
14. Gaussian fit to average data values for the BRDF measurements of the wrench ........ 44
15. Histogram from random number trials to determine scattering angle from wrench ... 46
16. Histogram from random number trials to determine scattering angle from paper ...... 48
17. Water droplet measured counts and concentration from nebulizer ............................. 50
18. Histogram from random number trials to determine particle size of scattering event 51
19. Randomly determined Euler angle coordinate transformation .................................... 56
20. Simulation versus theory for 10,000 photons launched into anisotropic media .......... 62
21. Model flux error map (g=.500) generated by comparison with tabulated data ........... 64
22. Model flux error map (g=.750) generated by comparison with tabulated data ........... 64
23. System block diagram ................................................................................................. 66
24. Circular polarization imager ........................................................................................ 67
25. Lens layout of objective .............................................................................................. 70
26. Lens prescription of objective ..................................................................................... 70
27. MTF of stand-alone objective ..................................................................................... 71
28. Lens layout of collimator ............................................................................................. 72
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LIST OF FIGURES – Continued
29. Lens prescription for the collimator ............................................................................ 73
30. Lens layout for the re-imaging subsystem ................................................................... 74
31. Lens prescription for the re-imaging lens .................................................................... 75
32. System MTF with vertically polarized light incident .................................................. 76
33. System MTF with horizontally polarized light incident .............................................. 76
34. Effect of defocus on system MTF ............................................................................... 78
35. Raw image of a paper resolution target illuminated with unpolarized light ............... 79
36. Raw image of a rough metal wrench target illuminated polarized light ..................... 79
37. Auto-correlation of registration images ....................................................................... 82
38. Laplacian of the registration scene’s auto-correlation map ......................................... 83
39. Pre and post subtracted images in the RCP channel of the instrument ....................... 84
40. Poincare sphere representation of any polarization state ............................................ 89
41. Calibration images using linearly polarized illumination ........................................... 90
42. Calibration images using circularly polarized illumination ........................................ 91
43. Post-processed image product using the presented system ......................................... 94
44. Cross-sectional data of the post-processed image data ............................................... 94
45. 2D rect function ........................................................................................................... 96
46. Blurring filter applied to a resolution target ................................................................ 97
47. Obscured image data using only subtractive algorithms with and without blur ......... 98
48. Obscured image data using only subtractive algorithms with and without blur ......... 98
49. Post subtracted image array for an obscured cross-hatch target for Ldiff .................. 100
50. Post subtracted image array for an obscured cross-hatch target for Rdiff .................. 101
51. Post subtracted image array for an obscured resolution target for Rdiff .................... 102
52. Post subtracted image array for an obscured resolution target for Ldiff .....................103
53. Post processed images of fine resolution target using the novel Fourier filter ..........106
54. Post processed images of coarse cross-hatch target using the novel Fourier filter .... 107
55. Experimental setup using a simple radiometer, the sensor, and scattering medium .. 109
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LIST OF FIGURES – Continued
56. Targets used in the experimental data collection ...................................................... 110
57. Monte Carlo fit to simple radiometer output to correct for scatter noise .................. 111
58. Intensity image of metal wrench using circularly polarized illumination, δ = 5.11 .. 117
59. Post-processed image array of metal wrench, δ = 5.11 ............................................. 118
60. Correlation coefficient versus subtraction proportion constant for δ = 5.11 ............. 118
61. Contrast ratio versus subtraction proportion constant for δ = 5.11 ........................... 119
62. Intensity image of metal wrench using circularly polarized illumination, δ = 4.57 .. 120
63. Post-processed image array of metal wrench, δ = 4.57 ............................................. 120
64. Correlation coefficient versus subtraction proportion constant for δ = 4.57 ............. 121
65. Contrast ratio versus subtraction proportion constant for δ = 4.57 ........................... 121
66. Intensity image of metal wrench using circularly polarized illumination, δ = 4.35 .. 122
67. Post-processed image array of metal wrench, δ = 4.35 ............................................. 122
68. Correlation coefficient versus subtraction proportion constant for δ = 4.35 ............. 123
69. Contrast ratio versus subtraction proportion constant for δ = 4.35 ........................... 123
70. Intensity image of metal wrench using circularly polarized illumination, δ = 3.67 .. 124
71. Post-processed image array of metal wrench, δ = 3.67 ............................................. 124
72. Correlation coefficient versus subtraction proportion constant for δ = 3.67 ............. 125
73. Contrast ratio versus subtraction proportion constant for δ = 3.67 ........................... 125
74. Intensity image of metal wrench using circularly polarized illumination, δ = 3.17 .. 126
75. Post-processed image array of metal wrench, δ = 3.17 ............................................. 126
76. Correlation coefficient versus subtraction proportion constant for δ = 3.17 ............. 127
77. Contrast ratio versus subtraction proportion constant for δ = 3.17 ........................... 127
78. Intensity image of metal wrench using circularly polarized illumination, δ = 2.70 .. 128
79. Post-processed image array of metal wrench, δ = 2.70 ............................................. 128
80. Correlation coefficient versus subtraction proportion constant for δ = 2.70 ............. 129
81. Contrast ratio versus subtraction proportion constant for δ = 2.70 ........................... 129
82. Intensity image of metal wrench using linearly polarized illumination, δ = 4.82 ..... 131
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LIST OF FIGURES – Continued
83. Post-processed image array of metal wrench, δ = 4.82 ............................................. 131
84. Correlation coefficient versus subtraction proportion constant for δ = 4.82 ............. 132
85. Contrast ratio versus subtraction proportion constant for δ = 4.82 ........................... 132
86. Intensity image of metal wrench using linearly polarized illumination, δ = 4.57 ..... 133
87. Post-processed image array of metal wrench, δ = 4.57 ............................................. 133
88. Correlation coefficient versus subtraction proportion constant for δ = 4.57 ............. 134
89. Contrast ratio versus subtraction proportion constant for δ = 4.57 ........................... 134
90. Intensity image of metal wrench using linearly polarized illumination, δ = 4.35 ..... 135
91. Post-processed image array of metal wrench, δ = 4.35 ............................................. 135
92. Correlation coefficient versus subtraction proportion constant for δ = 4.35 ............. 136
93. Contrast ratio versus subtraction proportion constant for δ = 4.35 ........................... 136
94. Intensity image of metal wrench using linearly polarized illumination, δ = 3.67 ..... 137
95. Post-processed image array of metal wrench, δ = 3.67 ............................................. 137
96. Correlation coefficient versus subtraction proportion constant for δ = 3.67 ............. 138
97. Contrast ratio versus subtraction proportion constant for δ = 3.67 ........................... 138
98. Intensity image of metal wrench using linearly polarized illumination, δ = 3.17 ..... 139
99. Post-processed image array of metal wrench, δ = 3.17 ............................................. 139
100. Correlation coefficient versus subtraction proportion constant for δ = 3.17 ........... 140
101. Contrast ratio versus subtraction proportion constant for δ = 3.17 .......................... 140
102. Intensity image of metal wrench using linearly polarized illumination, δ = 2.70 ... 141
103. Post-processed image array of metal wrench, δ = 2.70 ........................................... 141
104. Correlation coefficient versus subtraction proportion constant for δ = 2.70 ........... 142
105. Contrast ratio versus subtraction proportion constant for δ = 2.70 .......................... 142
106. Simulated flux ratio at the focal plane for linear and circular polarized light ......... 143
107. Raw images of a rough metal wrench obscured by fog using circ and lin pol ........ 144
108. Raw irradiance image compared to modeled irradiance image, δ = 3.53 ................ 145
109. Correlation coefficient of obscured experimental image, δ = 3.53 .......................... 146
11
LIST OF FIGURES – Continued
110. Correlation coefficient of modeled obscured image, δ = 3.53 ................................. 147
111. Modeled backscattered flux for principal Stokes parameters for circ and lin pol ... 149
112. Registered irradiance images of a paper cross-hatch for circ and lin pol illum. ...... 150
113. Correlation data for a depolarizing object using circ and lin pol illumination. ....... 151
114. Contrast ratio data for a depolarizing object using circ and lin pol illumination .... 151
115. Post-processed image array of a paper cross-hatch using lin pol illumination ........ 152
116. Post-processed image array of a paper cross-hatch using circ pol illumination ...... 152
117. Intensity image of paper target using linearly polarized illumination, δ = 3.98 ...... 154
118. Post-processed image array of paper target, δ = 3.98 .............................................. 154
119. Correlation and Contrast versus subtraction proportion constant for δ = 3.98 ........ 155
120. Intensity image of paper target using linearly polarized illumination, δ = 3.53 ...... 155
121. Post-processed image array of paper target, δ = 3.53 .............................................. 156
122. Correlation and Contrast versus subtraction proportion constant for δ = 3.53 ........ 156
123. Intensity image of paper target using linearly polarized illumination, δ = 3.17 ...... 157
124. Post-processed image array of paper target, δ = 3.17 .............................................. 157
125. Correlation and Contrast versus subtraction proportion constant for δ = 3.17 ........ 158
126. Intensity image of paper target using linearly polarized illumination, δ = 2.70 ...... 158
127. Post-processed image array of paper target, δ = 2.70 .............................................. 159
128. Correlation and Contrast versus subtraction proportion constant for δ = 2.70 ........ 159
129. Intensity image of paper target using linearly polarized illumination, δ = 2.11 ...... 160
130. Post-processed image array of paper target, δ = 2.11 .............................................. 160
131. Correlation and Contrast versus subtraction proportion constant for δ = 2.11 ........ 161
132. Contrast enhancement factors for various object material and spatial frequency .... 163
133. Wire-grid polarizer interface action on incident radiation ....................................... 166
134. Slab waveguide with propagation constant, β .......................................................... 167
135. 50x magnification of the wire-grid polarizer “super pixel” ..................................... 169
136. Transmission of a WGP with a 2x2 super pixel design with 22.5˚ lin pol illum. .... 172
12
LIST OF FIGURES – Continued
137. Flux map for the 2x2 design with 22.5˚ linearly polarized illumination ................. 172
138. Transmission of a WGP with a 4x4 super pixel design with 22.5˚ lin pol illum. .... 173
139. Flux map for the 4x4 design with 22.5˚ linearly polarized illumination ................. 174
140. Transmission of a WGP with a 6x6 super pixel design with 22.5˚ lin pol illum. ..... 175
141. Flux map for the 6x6 design with 22.5˚ linearly polarized illumination ................. 176
142. Aprroximate opto-mechanical setup for active alignment of the WGP polarizer ... 177
143. Actual opto-mechanical setup for active alignment of the WGP polarizer ............. 178
144. Sample Moire pattern ............................................................................................... 180
145. Active pixel values from an actual alignment run ................................................... 182
146. Pixel projections on a rotated 6x6 super pixel design with 22.5˚ lin pol illum ....... 186
147. Difference value plot for the (1,2) pixel normalized to the (2,2) pixel .................... 187
148. Difference value plot for the (1,3) pixel normalized to the (2,2) pixel .................... 188
149. Macroscopic view of pixels with polarized, white light illumination ..................... 189
150. 22.5˚ flat field illumination images from the bonded WGP/FPA ............................ 190
151. 67.5˚ flat field illumination images from the bonded WGP/FPA ............................ 190
152. 112.5˚ flat field illumination images from the bonded WGP/FPA .......................... 191
153. 157.5˚ flat field illumination images from the bonded WGP/FPA .......................... 191
154. 40x images of a dose array illuminated with unpolarized white light ..................... 192
155. On-chip linear polarimeter images of a plastic container ........................................ 193
156. Pre-corrected linear polarization images of a plastic container ............................... 194
157. Images for comparison to contrast stretched image of a metal wrench, δ = 3.81 ......197
158. Cross sectional correlation coefficient data for the images presented in Fig. 157 ....197
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LIST OF TABLES
1. Random Number Generation of Scattering Angle, θ ................................................. 46
2. Model vs Henyey-Greenstein Function Solution for a Finite Layer, g = .500 .......... 61
3. Model vs Henyey-Greenstein Function Solution for a Finite Layer, g = .750 .......... 62
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ABSTRACT
Improved imaging and detection of objects through turbid obscurants is a vital problem
of current interest to both military and civilian entities. Image quality is severely
degraded when obscurant fields such as fog, smoke, dust, etc., lie between an object and
the light-collecting optics. Conventional intensity imaging through turbid media suffers
from rapid loss of image contrast due to light scattering from particles (e.g. in fog) or
random variations of refractive index (e.g. in medical imaging). Intensity imaging does
not differentiate between rays scattered off particles in the obscurant field and those
reflected off objects within the field. Scattering degrades image quality in all spectral
bands (UV, visible, and IR), although the amount of degradation is wavelength
dependent. This dissertation features the development of innovative system designs and
techniques that utilize scattered radiation’s deterministic polarization state evolution to
greatly enhance the image contrast of stand-off objects within obscurant fields such as
smoke, fog, or dust using active polarized illumination in the visible. The produced
sensors acquire and process image data in real time using computationally non-intensive
algorithms that differentiate between radiation that scatters or reflects from obscured
objects and the radiation from the scattering media, improving image contrast by factors
of ten or greater for dense water vapor obscurants.
15
CHAPTER 1: INTRODUCTION
Image quality is severely degraded when obscurant fields such as fog, smoke, dust, etc.,
lie between an object and the light-collecting optics. Conventional intensity imaging
through turbid media suffers from rapid loss of image contrast due to light scattering
from particles (e.g. in fog) or random variations of refractive index (e.g. in medical
imaging). Intensity imaging does not differentiate between rays scattered off particles in
the obscurant field and those reflected off objects within the field. Scattering degrades
image quality in all spectral bands (UV, visible, and IR), although the amount of
degradation is wavelength dependent.
This dissertation investigates the development of innovative system designs and
techniques that utilize scattered radiation’s deterministic polarization state evolution to
greatly enhance the image contrast of stand-off objects within obscurant fields such as
smoke, fog, or dust using active polarized illumination. These techniques differentiate
between light that scatters or reflects from obscured objects and the light from the
scattering media. This is possible because the polarization state of the light is maintained
through many scattering events allowing processing of the orthogonal polarization state
images for improving scene contrast.
Applications for research contained within this dissertation encompass any activity
that requires object identification through extended scattering media. Interested parties
would include: fire fighters attempting to identify objects within smoke, soldiers trying to
discern objects/vehicles obscured by intentionally employed smokescreens, civilians
requiring improved vision of roadways encompassed by fog or dust while driving a
vehicle.
A review introducing principles of polarimetry and cross-polarization intensity
subtraction (CPIS) in medical imaging is given in the following two subsections.
Additionally, the final subsection gives a logical explanation of the arrangement of the
dissertation.
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1.1 Polarimetry
Polarimetry is the science of quantifying an electric field’s polarization state. At the heart
of deterministic polarization state evolution of post-scattered radiation is the interaction
of the electric field with the matter on which it is impingent. In addition to polarization
intensity subtraction for image contrast enhancements, quantification of an electric field’s
evolution after interaction with a material can give an observer insight into many optical
properties of said material. These properties include, but are not limited to dielectric
properties and stress-induced birefringence.
Time-sequential measurement, division of aperture, division of amplitude, and
polarization modulation are the four broad classes of polarimeters which detect the
polarization state of incident flux through the use of polarization state analyzers and
generators (polarizers). In time-sequential measurement polarimeters, a set of flux
measurements are made sequentially in time with changes made to the independent
generator-analyzer pairs after each measurement. Division of aperture polarimeters use
multiple analyzer polarizers in parallel to detect radiation common to one aperture of the
system, essentially sub-dividing the aperture into projected analyzer super-pixels.
Division of amplitude polarimeters divide incident radiation into separate paths and
analyze each path independently. Finally, polarization modulation polarimeters utilize
dynamic polarization elements to generate and analyze independent polarization states
very rapidly, effectively modulating a single incident source as a function of the dynamic
elements’ orientation during a particular measurement.
A common representation of a polarization state is the Stokes vector representation.
In this representation a 4x1 column vector is assembled over a scene with x-y spatial
coordinates as,
⎡ S 0 ( x, y )⎤ ⎡ I 0 ( x, y ) + I 90 ( x, y ) ⎤ ⎡ I ⎤
⎢ S ( x , y )⎥ ⎢ I ( x , y ) − I ( x , y ) ⎥ ⎢ Q ⎥
K
1
90
⎥=⎢ 0
⎥=⎢ ⎥
S ( x, y ) = ⎢
⎢ S 2 ( x, y )⎥ ⎢ I 45 ( x, y ) − I 135 ( x, y )⎥ ⎢U ⎥
⎥ ⎢
⎢
⎥ ⎢ ⎥
⎣ S 3 (x, y )⎦ ⎣ I R ( x, y ) − I L ( x, y ) ⎦ ⎣V ⎦
(1)
17
where S0 describes the total energy of the beam, S1 denotes preference for linear 0º over
linear 90º, S2 for linear 45º over linear 135º, and S3 for right circular over left circular
polarization states. It is important to note that the Stokes vector representation of
polarized light can not, by nature, give any insight into the phase of a particular beam,
since it is based on irradiance measurements.
1.2 Cross Polarization Intensity Subtraction (CPIS) in Medical Imaging
When the spatial character of one of the sub-components of a Stokes vector element, i.e.
I0, I90, differs from its orthogonal partner, one can infer information about the scene upon
which the generator polarization state was incident. When imaging through nonscattering media, polarization images can provide a higher degree of contrast than
intensity images and consequently, dramatically improved object identification.
Theoretical models5 and near-field experimental data3,6 indicate that using polarization to
image through scattering media, such as fog, dust, and smoke, greatly increases contrast
vs. intensity imaging.
Michael C. Roggemann and Byron Welsh7 describe the problem of image
degradation, using conventional imaging in turbid media: “Turbid media can be
characterized by random variations in the index of refraction. As an optical field
propagates through a region of turbulence, these variations cause phase perturbations in
the wave front. Propagation causes the phase perturbations to evolve into both phase and
amplitude perturbations.” Along with these amplitude perturbations, scatter from finite
particles within the obscurant field causes image quality and contrast to be greatly
reduced.
When imaging through random media, scattering causes degradation in resolution and
contrast. The amount of scatter is directly correlated to the optical depth (number of mean
free paths) of the medium through which imaging is attempted. One scattering mean free
path (SMFP) is the average distance a photon travels in a turbid medium between
scattering events. After traveling one SMFP, the radiation has 1/e (36.8%) of its initial
amplitude not scattered. Polarization processing can greatly enhance image resolution
18
and contrast through turbid media because the degree of polarization (DOP) of radiation
is maintained through many optical depths.1,2 Therefore, if we subtract the polarization
orthogonal to the polarization we expect to receive from illuminated targets, we will see a
vast improvement in image contrast at all object ranges. This is known as CrossPolarization Image Subtraction (CPIS). Using CPIS to remove radiation from scatter, we
can effectively achieve the same image contrast seen in conventional imaging at much
greater distances.
Intensity imaging does not allow for differentiation between rays scattered off
particles in the obscurant field vs. those reflected off objects within the field. Light is
scattered directly back into the image-forming optics from the obscurant field, causing
large amounts of contrast reduction and image degradation. To ameliorate this issue,
researchers have used polarization to differentiate those rays that are scattered off the
medium from those reflected off the objects of interest. Detecting the polarization of
radiation from obscurants and objects of interest, then differentiating between the two,
allows for large contrast enhancements. This method has been used in medical imaging,
as shown in Fig. 1.8
Figure 1 – Images of a duck heart through a dense medium using (a) Intensity imaging; (b) Polarization
discrimination techniques. Dashed green lines represent common contours between (a) and (b).8
S. Jacques et al.9 developed a polarized light camera to guide surgical excision of skin
cancers. By illuminating the skin with linearly polarized light and acquiring the
difference image (CPIS), they subtract background diffusely scattered light and enhance
19
the “fabric” of the skin. As cancer destroys the complex pattern of this fabric, the
difference image reveals the cancer margin. Compared to clinical margin drawn by a
physician (unpolarized intensity image, Fig. 2a) the cancer margin is larger for the
difference image (Fig.2b). The predictions of cancer margin based on the polarized
imaging technique were confirmed by pathological analysis of the specimen.
Figure 2 – (a) clinical intensity image of basal cell carcinoma with dashed lines by doctor showing clinical
margin; (b) CPIS technique: the blue line is the cancer margin determined by observing the difference
image. The dot grid was drawn to improve the legibility of the images.9
1.3 Dissertation Arrangement
Adaptation of the CPIS technique to a remote sensing system that presents orthogonally
polarized images of the scene to separate pixels of its focal plane array (FPA) forms the
conceptual basis for developed system along with a novel Fourier filter applied to postsubtracted images. Two polarimeter class methods for implementing the technique are
presented. In the first method, a division of amplitude polarimeter utilizes a Wollaston
prism-based design to completely separate orthogonally polarized images of the scene.
The second method exploits a novel wire-grid polarizer super-pixel design and precision
alignment procedure which was bonded to a focal plane array (FPA) to generate a
division of aperture polarimeter for use in field data acquisition.
This dissertation is arranged in 4 distinct sections. The first section, encompassing
Chapter 2, addresses scatter-based design considerations, Mie scattering physics and
20
associated construction of a scattering Mueller matrix, and modeling approaches given
various design parameters.
The second and third sections examine the design, calibration, experimental results
and associated post-processing algorithms of the Wollaston prism-based and Wire-grid
polarizer-based systems respectively. The second section contains chapters 3 and 4, while
the third section holds chapter 5.
Finally, conclusions, and references shall be provided in the fourth section.
21
CHAPTER 2: SCATTERING PHYSICS
Understanding the essential physics behind the scattering processes of interest is integral
to engineering a system that harnesses the deterministic nature of the polarization
evolution of radiation after scattering events. Illumination source band, source
polarization state, optical design of receiver, and polarization analyzer state are all
consequences of the scattering processes induced by obscurant fields of various particle
size distributions.
Thus, exhaustive polarization models must be constructed with previous knowledge
of what type and size of particle one might encounter, in order to efficiently design a
system which attempts to image through scattering media. Theoretical optics problems
are all members of applied Maxwell’s equation theory. Solutions to Maxwell’s equation
for a polarized plane wave incident upon a homogeneous, spherical scattering particle of
arbitrary radius is termed Mie Theory.
2.1 Mie Scattering Theory
Central to any light scattering model is Mie scattering theory. Mie scattering theory is
based on the scattering of a plane wave by a homogeneous sphere. Briefly, the scattering
parameters are found by solving the wave equation for the electric and magnetic fields
using boundary conditions across the surface of the homogeneous, spherical scattering
sphere.
Figure 3 - Scattering geometry for use in the derivation of Mie scattering theory2.
22
2.1.1 Solutions to the Scalar and Vector Wave Equation
The author will assume the reader has a familiarity with Maxwell’s equations and the
formulation of the scalar and vector wave equations in a vacuum from them. Because of
the wave nature of light, we can assume a periodic phenomenological basis for all
quantities of interest. These quantities will have the form,
A = (α + iβ )e iωt
(2)
which are periodic functions of time. With this assumption, Maxwell’s equations assume
simpler forms and the scalar wave equation can be written as,
∇ 2ψ = − k 2 m 2ψ ,
(3)
where
k=
2π
m2 = ε −
and
(4)
λ
4πiσ
ω
.
(5)
Equations 4 and 5 describe the wave number and complex refractive index of the medium
at frequency, ω, respectively. The consequence of deriving the scalar wave equation in a
vacuum is the allowance to derive a plane wave solution traveling in the positive zdirection as,
ψ = exp[ikmz + iωt ] .
(6)
Scattering of light from small particles is a problem with spherical symmetry. As
such, it is convenient to solve the wave equation and represent following quantities in
spherical coordinates, where spatial positions are functions of r, θ, and φ, and the position
vector
in
rectangular
Cartesian
coordinates
K
r = ( x, y, z ) = (r cos φ sin θ , r sin φ sin θ , r cosθ ) .
can
be
expressed
as
The scalar wave equation is separable in spherical coordinates allowing for
independent solutions in each degree of freedom and accompanying solutions to be the
multiplication of solutions in each dimension25. This yields solutions of the form,
ψ ln =
cos lφ ⎫ l
⎬ Pn (cos θ )z n (mkr )
sin lφ ⎭
(7)
23
which consists of solutions in φ, which represent a factor of either a cos or sin; solutions
in θ being an Legendre polynomial factor, and solutions in r that can take the form of any
spherical Bessel function, defined by2
z n (ρ ) =
π
Z n +1 / 2 (ρ )
2ρ
(8)
Now that a scalar wave equation in spherical coordinates has been established, it is
required that it be related to vector quantities, E and H. Due to the periodicity mentioned
before; the vector wave equation assumes a similar form as the scalar wave equation in
Equation 3,
G
K
∇ 2 A = −k 2 m 2 A
(9)
In a homogeneous medium, the electric and magnetic fields obey the vector wave
equation presented above. Given this, one must now find vector quantities relating
directly to the scalar solution shown in Equation 7. Fortunately, such relations do exist. If
a scalar quantity, ψ, satisfies the scalar wave equation the three relations below follow2.
K
K
M ψ = ∇ × (r ψ )
K
K
mkNψ = ∇ × M ψ
(10a,b,c)
K
K
mkM ψ = ∇ × Nψ
The impact of the relations found in Equation 10 is that given a periodic scalar
solution to the wave equation, an associated periodic vector solution exists. Additionally,
the vector solution is determined exactly from its scalar counterpart. Now, a vector
solution to the wave equation for the electric and magnetic fields can be constructed by
using scalar solutions to the wave equation, u and v. With the assumption that u and v are
independent solutions to the scalar wave equation, Mu, Nu, Mv, Nv, are constructed.
Moreover, through application of Maxwell’s equations, the electric and magnetic fields
are found2,
K K
K
E = M v + iN u
K
K
K
H = m(− M u + iN v )
(11a,b)
24
2.1.2 The Mie problem and Application of Boundary Conditions
Of vast importance are the derivations of boundary conditions for tangential and normal
components of a field incident on a sharp boundary between homogeneous media. Again,
these boundary conditions are derived through the use of Maxwell’s equations. For
general boundary geometries, the boundary conditions for the tangential components are
as follows2:
K
K
nˆ × (H 2 − H 1 ) = 0
K
K
nˆ × (E 2 − E1 ) = 0
Similarly, the boundary conditions for the normal components are2,
K
K
nˆ ⋅ m22 E 2 − m12 E1 = 0
K
K
nˆ ⋅ (H 2 − H 1 ) = 0
(
)
(12a,b)
(13a,b)
where, n is a unit vector normal to the boundary surface and E and H are the electric and
magnetic fields, respectively.
As stated earlier, the Mie problem is the solution to Maxwell’s equations with a
linearly polarized plane wave incident on a homogeneous, spherical particle. In support
of a solution to the Mie problem, the origin of the system is set at the center of said
particle. Additionally, the amplitude of the incident plane wave is unity with polarization
in the x-direction. Figure 3 illustrates this setup. The fields for the incident plane wave
are written as,
K
E = xˆ exp[− ikz + iωt ]
Hˆ = yˆ exp[− ikz + iωt ]
(14a,b)
Choosing a spherical Bessel function jn derived from the Bessel function of the first
kind, the scalar solution for the incident wave outside of the scattering sphere is derived
from an infinite sum of elementary scalar plane wave solutions (eq. 7). In spherical
coordinates, the incident, scalar plane wave solutions takes the form,
∞
u = e iωt cos φ ∑ (− i )
n
n =1
v=e
i ωt
2n + 1 1
Pn (cos θ ) jn (kr )
n(n + 1)
2n + 1 1
sin φ ∑ (− i )
Pn (cos θ ) jn (kr )
n(n + 1)
n =1
∞
n
(15a,b)
25
Given this choice for the incident wave, outside the scattering sphere, it sets the form for
the scattered wave outside of the sphere. Using the associated boundary conditions and
the condition that the scattered field should have asymptotic behavior at very large
distances away from the homogeneous sphere, the scattered wave solutions take the
following form,
∞
u = e iωt cos φ ∑ − an (− i )
n
n =1
v=e
iωt
2n + 1 1
Pn (cos θ )hn( 2 ) (kr )
n(n + 1)
2n + 1 1
sin φ ∑ − bn (− i )
Pn (cos θ )hn( 2) (kr )
(
)
n n +1
n =1
∞
(16a,b)
n
The boundary conditions set at the scattering sphere’s boundary, determine the
coefficients: an and bn. The condition of asymptotic behavior at infinity supports the
choice of the spherical Bessel function factor, because it demonstrates a spherical wave
form. The Bessel function of the second kind allows derivation of hn. It has the form2,
hn(2 ) (kr ) ~
i n +1
exp[− ikr ].
kr
(17)
This is a valid choice, since it shows asymptotic behavior towards infinity. It is also not a
coincidence that it has the form of a spherical wave. Later it will be evident that postscattered waves take the form of spherical waves and behave as such.
The wave inside of the scattering sphere utilizes the fact that the fields are finite at the
origin (center of the sphere), and as such, take the following form:
∞
u = eiωt cos φ ∑ mcn (− i )
n
n =1
v=e
i ωt
2n + 1 1
Pn (cos θ ) jn (mkr )
n(n + 1)
2n + 1 1
sin φ ∑ md n (− i )
Pn (cos θ ) jn (mkr )
n(n + 1)
n =1
∞
(18)
n
The yet to be determined coefficients are found through application of boundary
conditions at the edge of the scattering sphere. Applying said boundary conditions with
the continuity of field and its first derivative at r = a (boundary of the sphere), the an and
bn coefficients are determined. They are2
26
an =
bn =
ψ n ' ( y )ψ n (x ) − mψ n ( y )ψ n ' (x )
ψ n ' ( y )ζ n (x ) − mψ n ( y )ζ n ' (x )
(19a,b)
mψ n ' ( y )ψ n ( x ) −ψ n ( y )ψ n ' ( x )
mψ n ' ( y )ζ n ( x ) −ψ n ( y )ζ n ' ( x )
where x=ka is the size parameter and y=mka. The other parameters in Equation 19
derived from the Riccati-Bessel functions27. Essentially, they are the spherical Bessel
functions multiplied by a common factor,
ψ n (z ) = zjn ( z )
(20a,b)
ζ n ( z ) = zhn( 2) (z )
Combining equations 15, 16, and 19, the scalar field solutions the incident and
scattered fields are yielded for any point outside of the homogeneous scattering sphere.
The point of interest in Mie theory is the scattered wave. It can be rewritten as,
∞
2n + 1 1
− ie − ikr +iωt
u=
cos φ ∑ an
Pn (cos θ )
kr
n(n + 1)
n =1
(21a,b)
v=
− ie
−ikr + iωt
kr
∞
sin φ ∑ bn
n =1
2n + 1 1
Pn (cos θ )
n(n + 1)
Repeating the steps taken in equations 10 and 11 to convert solutions of the scalar wave
equation to a vector-valued electric and magnetic field, one can calculate the components
of E and H:
{
}
∞
K
K
∂ Pn1 (cos θ ) ⎤
− i exp[− ikr + iωt ]
2n + 1 ⎡ a n 1
Eθ = H φ =
Pn (cos θ ) + bn
cos φ ∑
⎢
⎥
kr
∂ cos θ ⎦
n =1 n(n + 1) ⎣ sin θ
∞
K
∂{Pn1 (cos θ )}⎤
− i exp[− ikr + iωt ]
2n + 1 ⎡ bn
1
(
)
P
a
− Eφ = H θ =
+
sin φ ∑
cos
θ
⎢
⎥
n
n
kr
∂ cos θ ⎦
n =1 n(n + 1) ⎣ sin θ
(22a,b)
which can be written as,
27
K
K
− i exp[− ikr + iωt ]cos φ
Eθ = H φ =
S 2 (θ , x )
kr
K
− i exp[− ikr + iωt ]sin φ
− Eφ = H θ =
S1 (θ , x )
kr
(23,a,b)
Upon scatter by an angle θ, the azimuth angle, φ gives a projection of the scattering path
on the plane perpendicular to the incident wave’s direction. As such, when a scattering
event occurs, the polarization direction of the initial wave’s field can be decomposed
into, Es ,inc = sin φ and E p ,inc = cos φ . The scatter field’s amplitude is related to equation 23
by Es , scat = − Eφ and E p , scat = Eθ . With these substitutions, generation of a scattering Jones
matrix based solely on illumination wavelength, particle size and scattering angle is
possible:
⎛ Es ⎞
e −ikr +iωt
⎜ ⎟ =
⎜E ⎟
ikr
⎝ p ⎠ scat
0 ⎞⎛ E s ⎞
⎛ S 2 (θ , x )
⎜⎜
⎟⎜ ⎟
S1 (θ , x )⎟⎠⎜⎝ E p ⎟⎠ inc
⎝ 0
(24)
In this solution, the plane of reference is taken through the incident and scattered wave’s
direction of propagation.
Equation 24 describes the amplitude and polarization evolution of the electric field
after a single scattering event from a particle of radius a at an angle θ. This is exactly the
result for which we are looking. However, we also require knowledge of scattering and
absorption coefficients.
The scattering and extinction cross-sections Csca and Cext are related to the amount of
the energy intercepted by particle from the incident beam because of scattering and
extinction (which includes both scattering and absorption) and they are defined as:
Cext = Qext G =
Wext Wsca + Wabs
W
=
, C sca = Qsca G = sca ,
Ii
Ii
Ii
(25)
where Wsca is the energy scattering rate, Wabs is the energy absorption rate and Wext is the
energy extinction rate, Ii is the incident irradiance, G is the geometrical cross-sectional
area of the particle, Qsca and Qext are the scattering and extinction efficiency factors,
28
respectively. Both efficiency factors and the asymmetry factor g can also be defined from
coefficients an and bn of equation 19.2
2 ∞
( 2n + 1)( an an* + bn bn* )
2 ∑
x n =1
2 ∞
= 2 ∑ ( 2n + 1) Re(an + bn )
x n=1
Qsca =
Qext
g = cosθ =
4
2
x Qsca
(26)
⎡ n ( n + 2)
⎤
2n + 1
Re(an an*+1 + bn bn*+1 ) +
Re(an bn* )⎥
n( n + 1)
n =1 ⎣ n + 1
⎦
∞
∑⎢
The asymmetry parameter g is the weighted mean of cosθ with the scattering
function. The transport mean free path (TMFP) is derived from g and the scattering mean
free path, TMFP = SMFP/(1-g). The transport mean free path is the path length required
to completely depolarize polarized incident radiation. This will be a very important
parameter in later discussions on depolarization’s impact on system level design
considerations.
In order to stay in keeping with the commonly used Stokes polarization basis, a
scattering Mueller matrix must be constructed. In the Mueller calculus basis, quantities
are calculated in irradiance not field, as mentioned in Chapter 1.1. It requires a 4x4
representation that describes Ip, Is, U, and V, commonly known as modified Stokes
parameters. They describe linear polarization parallel to the plane containing the scattered
wave and the incident wave; linear polarization perpendicular to said plane; linear
polarization 45 degrees out of phase with both Ip and Is, and circular polarization
respectively. Using the determined Jones matrix in equation (24), the 4x4 Mueller matrix
is calculated as,
⎛ S2 (θ ) 2
⎜
⎛Ip ⎞
⎜ 0
⎜ ⎟
1 ⎜
⎜ Is ⎟
⎜ U ⎟ = k 2r 2 ⎜ 0
⎜
⎜ ⎟
⎜V ⎟
⎜
⎝ ⎠
⎜ 0
⎝
0
S1 (θ )
0
0
0
2
0
1
S1 (θ )S2* (θ ) + S2 (θ )S1* (θ )
2
−i
S1 (θ )S2* (θ ) − S2 (θ )S1* (θ )
2
(
(
where the modified Stokes parameters are:
⎞
⎟⎛ I ⎞
⎟⎜ 0 p ⎟
0
⎟⎜ I 0s ⎟ (27)
S1 (θ )S2* (θ ) − S2 (θ )S1* (θ ) ⎟⎜ U ⎟
⎟⎜ 0 ⎟
⎟⎜ V ⎟
S1 (θ )S2* (θ ) + S2 (θ )S1* (θ ) ⎟⎝ 0 ⎠
⎠
0
) 2i (
) 12 (
)
)
29
I p = E p E *p
I s = E s E s*
U = E p E s* + E s E *p
V = i E p E s* − E s E *p
(
(28)
)
These parameters differ slightly from those presented in Equation 1, but there is no loss
of information and are used for a simpler scattering Mueller matrix. Equation (27)
completely describes the irradiance evolution of scattered radiation after a single
scattering event from a particle of specified radius and index of refraction.
2.1.3 Multiple Scatter Analysis of Polarization Properties
This dissertation addresses concerns when imaging through dense scattering media. This
lends itself to detection of radiation that has undergone many scattering events. Modeling
or solution of these multiple scattering events must be completed in order to make
intelligent design decisions. Solving the vector radiative transfer equation (VRTE)
provides the required analytic solutions for multiple scattering, unfortunately analytic
solutions to the VRTE generally do not exist due to the fact that the integrations in the
VRTE
μ
∂
[I d ] + I d =
∂τ
2π 1
∫ ∫ S (μ ,φ , μ ' ,φ ')I (τ , μ ' ,φ ')dμ ' dφ '+ F (τ , μ ,φ )
d
0
(29)
0 −1
are over the entire solid angle and spans all possible scattering states. The parameters of
the VRTE in equation 29 are: μ is cos θ, the polar angle; φ is the azimuthal angle; Id is the
Stokes vector of radiation; S is the scattering Mueller matrix phase function; F0 is the
incident flux magnitude; τ is an optical depth. (Note the use of Id as the notation for the
Stokes vector.) As long as the scattering process remains incoherent (allows for direct
addition of scattered intensities), the VRTE may be used. In order to ameliorate the issue
of lack analytic solutions to the VRTE, various numerical techniques can be used to
circumvent the prohibitive number of calculations that would be required for an analytic
solution.
30
When these numerical techniques applied to the VRTE converge to an approximate
solution, their results fit experimental data and give physical insight into the scattering
processes themselves.
An issue of paramount concern in using the polarization discrimination techniques
presented in this dissertation is the maintenance of the degree of polarization (DOP) of
radiation reaching objects of interest and subsequently being reflected back into the
collection aperture of the optics. If the radiation is depolarized before it enters the
collection aperture, polarization discrimination techniques cannot be used to increase
image quality. The degree of polarization in the conventional Stokes vector
representation of polarized light is written as,
DOP =
S12 + S 22 + S 32
S0
Ishimaru, et al.1 use Mie scattering statistics, Fourier transform theory, and the
descrete ordinates method and the radiative transfer equation in conjunction with the S1
and S2 scattering amplitude parameters given in equation 23 to model the multiple
scattering events in a parallel-plane medium. They study the effect of the loss of DOP
and amount of cross-polarization discrimination (XPD) when pulsed linearly polarized
and circularly polarized light is transmitted and backscattered through an obscurant
medium of varying particle size and optical depth. XPD is the ratio between energy
received in the desired polarization state vs. that received in the orthogonal state.
In transmission, the DOP is plotted in Fig. 4 as a function of optical depth with
varying scattering particulate sizes, ka, which is the wave number multiplied by the
particle radius.1
31
Figure 4 – Degree of polarization ratio as a function of optical depth with varying particle sizes ka.1
To further illustrate the ability of circular polarization to maintain its degree of
polarization for large optical depths in transmission and backscatter, Ishimaru, et al.1 plot
the DOP as a function of optical depth for a fixed particle size as shown in Fig. 5.
(LP-linear polarization, CP-circular polarization)
Figure 5 – DOP in db as a function of optical depth for a fixed particle size in (a) transmission, and (b)
backscatter. Note that the scale was changed in (b).1
Figures 4 and 5 clearly demonstrate that in transmission and backscatter circular
polarization maintains its degree of polarization much longer than linear polarization
through virtually all optical depths with larger scattering particulates, such as those seen
in fog, smoke, and biological samples.
Since circularly polarized wave exhibits point symmetry, it lacks any dependence on
azimuthal angle upon scatter. A right circularly polarized (RCP) wave that undergoes an
32
azimuthal scatter of any angle maintains its exact Stokes vector, while a wave that is
partially linearly polarized that undergoes the same azimuthal scatter will have its
polarization state changed, necessarily. The azimuthal scatter angle reveals itself as a
coordinate rotation about the scatter angle, θ. In the modified Stokes vector basis, the
azimuthal angle rotation matrix is represented as,
⎡ cos 2 φ
⎢
2
⎢ sin φ
⎢− sin 2φ
⎢
⎣⎢ 0
sin 2 φ
cos 2 φ
sin 2φ
0
.5 sin 2φ
− .5 sin 2φ
cos 2φ
0
0⎤
⎥
0⎥
0⎥
⎥
1⎦⎥
(30)
This physical characteristic manifests itself when the transfer equation is decomposed
into Fourier components. As seen in equation 24, the orthogonal states are not coupled
upon scatter; therefore the equation of transfer can be determined individually for each
component. When a linearly polarized wave is normally incident on a plane parallel
scattering medium, two Fourier components are required to describe the wave’s intensity
evolution, where when a circularly polarized wave is normally incident on the same
medium, only one Fourier component is needed. Additionally, upon inspection of
Equation 30 multiplied with the modified Stokes vector seen in equation 28, the linear
states of the resultant Stokes vector are modulated by the azimuthal angle, where the
circular state remains unchanged. This, in addition to other physical interactions of light
with matter, lends itself to circularly polarized light maintaining its DOP longer than
linearly polarized light in scattering media whose mean particle size is on the order of the
wavelength or larger.
Cross-polarization discrimination, XPD=10log(Ico-pol/Ix-pol), is also of interest in
imaging through turbid media. XPD is a metric for how much polarization clutter, i.e.,
the detection of polarization orthogonal to that being sent out, Ix-pol, is introduced when
polarized radiation is detected through turbid media. Ishimaru, et al.1 plot cross
polarization discrimination (XPD) as a function of optical depth for a particle size of 2
μm.
33
Figure 6 – Cross polarization discrimination in db as a function of optical depth for 2μm particles using
0.53 μm source in (a) transmission, and (b) backscatter. Note the change of scales between (a) and (b).1
Figure 6 shows that in transmission circularly polarized light demonstrates better
performance in XPD than linear polarized light, but it actually shows a loss in XPD in the
backscattered situation. This is consistent with the change in handedness upon reflection
for circularly polarized light. Also, polarized radiation will experience a fractional loss of
DOP as a function of incident and scattering angles.18
In summary, Ishimaru, et al.1 found that circularly polarized light exhibits superior
DOP maintenance and XPD for all situations except XPD in backscatter for larger size
parameters. This is important because, depending on the depolarization character of
imaged objects, linear or circularly polarized illumination and detection schemes will
provide optimum post-processed results. Therefore, when illuminating objects through a
scattering medium which are then imaged in reflection, as seen in Figure 7,
Figure 7 – Receiver collects polarized light from scattering media and the target from different spatial
locations on FPA.6
the light must first be transmitted through the scattering medium before it reflects from
the targets of interest and travels to the collecting aperture. Essentially, the light must
transmit twice through the scattering medium.
34
One very important factor has been neglected in the discussion so far, how the objects
within the obscurant evolve the polarization state of incident illumination. Not only can
an object onto which radiation is impingent depolarize, it can drastically change the
polarization state. This is no more evident than when circularly polarized light undergoes
mirror reflection. It changes its handedness to the orthogonal state from an observer’s
perspective.
The characteristics of objects being imaged within the scattering media have as large
an impact on the choice of illumination and detection state as how the respective
polarization states evolve within the scattering media. If an object is a complete
depolarizer, then XPD in backscatter becomes a metric for which system design must
maximize. However, if an object maintains the polarization of impingent illumination,
then system design must minimize XPD in backscatter. Unfortunately, it is not as cut and
dry as the previous statement would suggest. The DOP must also be maintained as long
as possible in all situations to produce the most efficient image contrast enhancements.
Therefore, choice of the most efficient illumination and detection state will vary between
circularly polarized light and linearly polarized light for individual detection scenes and
particle distributions.
As previously stated, if the polarized illumination must also interact with the
scattering media, Figures 4, 5, and 6 suggest that circularly polarized illumination should
be used because of its ability to maintain its DOP much longer through scattering media
in transmission and in backscatter for size parameters we expect to encounter when
imaging through smoke and fog with an object that maintains incident polarization well.
These conclusions were experimentally verified in near field imaging by Lewis, et al.3
They illuminated through a scattering medium of polystyrene latex particles (0.1 μm
spherical scatterers) and detected polarized backscattered illumination from a painted
metal target. The remote sensing adaptation of their results is shown in Fig. 8 on the
following page.
35
Figure 8 – Target contrast comparison between various polarization modes of illumination and detection of
the target [no polarization (“intensity”), linear polarization and circular polarization].3 The conversion from
number of scattering mean free paths (SMFP) to range is based on 1 SMFP = 100 m, a value indicative of
moderate fog.4
The concept of the SMFP allows us to scale these results to scenarios representative
of remote sensing, i.e., 1 SMFP = 100 meters. Figure 8 shows that at a contrast of 0.01,
the target range is doubled, from 220 meters to 440 meters, when using circular
polarization vs. conventional intensity detection.
Furthermore, the target range is
approximately 30% greater when comparing circular to linear polarization. Regardless of
the type of illumination, the image contrast is eventually reduced to unusable levels at
long ranges, but using polarization processing techniques to reduce backscatter yields a
higher contrast at all target ranges than with intensity imaging alone. It should be noted
that Lewis’ data was collected by a system with a very narrow field of view, essentially
allowing only photons that directly interacted with the target (painted metal plate) and
photons that had a low number of scattering events to be detected.
Scattering models1,2,5 and near-field experimental data3,6 prove that using polarized
illumination and specific detection schemes will allow for increased contrast when
imaging through dense media. This forms the basis for this dissertation. As will be
discussed in later chapters, our system allows for remote detection of objects within
obscurant clouds using both linear and circular polarization state illumination and
36
detection to study the remote-imaging contrast enhancement of each with novel
subtractive-based Fourier filter algorithms in support.
2.2 Monte Carlo Modeling
Modeling of multiple scattering events must be completed in order to make intelligent
design decisions using available setup geometries, illumination sources, obscurant
sources, etc. As mentioned in the previous section, numerical solutions based on the
vector radiative transfer equation are used to create these models. Some of these methods
include the method of successive orders29, 30, Amburtsumian’s method29, 30, the adding
and doubling method29, 30, the Eigenvalue/Eigenvector method1, 28 approaches using the
discrete ordinates method and the Monte Carlo (MC) method16, 31.
Each method has various advantages and drawbacks relative the others. A choice was
made to use the Monte Carlo method for simplicity of programming and its ease of
allowing the implementation of complicated geometries and variable parameters. The
Monte Carlo method also maintains intermediate data products that can be studied to gain
insight on the scattering events occurring within the obscurant and how the polarization
state is evolving through the system, instead of strictly how it starts and ends, which is
what many of the aforementioned yield.
For each launched photon packet the MC method simulates how the packet
propagates through the system. The trajectories upon interaction with molecules in a
turbid medium, or objects within the medium, are determined using probability density
functions (PDFs) associated with said interactions. A probability point spread function
(PSF) is then realized given the probability the sensor receives a photon of a particular
class (backscattered, ballistic from the object, or forward scattered after interaction with
the object) within the acceptance angle of the collection optics.
Since the MC method is a statistical process, it carries with it a standard deviation that
is proportional to the square root of the number of photons received by the collection
optics. This is due to the central limit theorem which states when the probability density
is formed by n independent random variables, as we observe in the MC process, the
37
resultant distribution is a normal distribution if n is 4 or greater32. A very large number of
photon trajectories must be simulated in order to construct a solution that has physical
meaning due to the fact that only a small portion of the launched photons actually strike
the receiver, and an even smaller number have their incidence angle satisfy the NA
optical system.
The general rule to tracking a photon packet’s polarization/position goes as follows:
1. Create photon’s initial coordinates, direction, and polarization. The initial
coordinates are set by a random number game in association with the size of the
collimating optics exit pupil size. The initial direction is set by the divergence of
the illumination source. If the source is collimated, the initial angles are set to 0
for all points on the collimating optics’ exit pupil. The initial polarization is either
circular or linear depending on the experiment.
2. Determine next collision site and type (position and particle size)
3. Change direction and polarization according to type of collision
4. Score and go back to step 2...
5. The path terminates when the photon reaches the system boundary, then it is
tracked with its last known directions to the entrance pupil of the collection
optics. If it satisfies angular and positional requirements to be received, the
photon is counted as part of the system flux.
Monte Carlo methods have at their core a random number generation scheme that
determines all probabilistic parameters to be used. In our case these parameters are step
size between subsequent scattering events (R); size of particle with which to be interacted
(a); angles of scatter (θ,φ) from particles in an obscurant medium, where θ is the
scattering angle and φ is the azimuthal angle specifying the new direction after the
scattering event, and finally, the scattering angle after the photon packet hits physical
targets within the scattering media (rough metal, paper, etc...) i.e. (bi-directional
reflectance distribution function (BRDF)). Each of these parameters is determined at each
required step by using a random number generator between 0 and 1, and finding the value
that corresponds to that particular probabilistic marker. Linear interpolation is used to
38
increase the accuracy of values determined by PDFs that are not analytic functions of
their respective variable parameters.
2.2.1 Probability Density Function Construction and Use: Step Size
The first PDF to be constructed is the step size between scattering events. In a vacuum,
the step size is infinite. However, when particles are introduced, the distance between
subsequent scattering events is derived using Beer’s law,
I = I 0 exp[− τ ]
(31)
τ = d / SMFP
(32)
where,
is the optical depth, d is the length of the medium, and SMFP is the scattering mean free
path. The SMFP is also a deterministic value given the extinction cross section Cext and
the volume density of the scattering particles, N:
SMFP =
1
C ext N
(33)
Due to the fact that Beer’s law has a simple exponential form; integral inversion may
be used to construct the desired function. We wish to calculate the step size, R, as a
function of a random number between 0 and 1. This is accomplished by finding the PDF
for the step size, which is simply a normalized form of Beer’s law when integrating over
the length of the scattering medium31, d.
R (ξ )
∫μ
s
exp[− μ s R']dR' = 1 − exp[− μ s R ] = ξ
(34)
0
where μs is the inverse of the SMFP, and ξ is a random number between 0 and 1. When
we solve for R in Equation 34, we find the step size between scattering events as a
function of a random number31:
R(ξ ) = −
ln (1 − ξ )
μs
(35)
39
As will be evident in following chapters, the tank in which the scattering medium is
contained is 33 cm deep. For an optical depth, τ = 5, the step size R is calculated 100,000
times to illustrate the statistics associated with the result shown in Equation 35.
Figure 9 – Histogram of Equation 35 with d=33, and τ = 5.
The mean of the distribution approaches the length of the container divided by the optical
depth, as expected. That leads to the conclusion, that, on average, a photon should travel
a length, d on occasion of the τth scattering event and that the probability that a photon
traverses a path equal to d within the obscurant without being scattered is exp[-τ].
2.2.2 Probability Density Function Construction and Use: Physical Target
Scatter Angle
Perfectly smooth surfaces will reflect incident rays only in the specular direction, where
rough surfaces will scatter incident rays in an array of directions. The spread of these rays
is determined by the depth and slope of the micro-structure of the scattering object. As
the micro-structure becomes deeper and more severe in slope, the more pronounced the
scattering angle becomes.
The scattering from surface is often quantized in a metric called the bidirectional
reflectance distribution function (BRDF), in which the irradiance distribution for
scattering angles around the specular reflection angle are tabulated to gain insight into
40
exactly how the surface will scatter incident radiation. A rough surface can also induce an
amount of depolarization for the incident polarization state of the illumination source33.
A Mueller matrix can be constructed that describes how scattering from a surface
will polarize unpolarized light or depolarize polarized light. In the conventional Stokes
vector representation, it is written as33,
~
M BRDF
Px
⎡1
⎢P 1 − 2D
x
ν
= R⎢
⎢0
0
⎢
0
⎣⎢ 0
0
0
Py
− Pz
0⎤
0 ⎥⎥
Pz ⎥
⎥
Py ⎦⎥
(36)
where R represents the BRDF for scatter, Px is the parameter that turns unpolarized light
into partially polarized linearly polarized light, Pz is the parameter that turns linearly
polarized light into partially circularly polarized light (elliptically polarized), Py is the
parameter that maintains the polarization of incident light, and Dv is a depolarization
parameter. Therefore, for a normalized input, the degree of polarization of an interaction
with the scattering object can be written as33,
P = Px2 + Py2 + Pz2
(37)
Making the assumption that the azimuthal component of scatter about the specular
reflection angle is equal for all azimuthal angles, measurements were made to
characterize the BRDF and depolarization induced by scatter for two objects of interest: a
rough steel wrench and a piece of copier paper. These were the two objects that were
later placed within the tank and obscured with water vapor from an ultra sonic nebulizer,
which will be explained in subsequent chapters detailing the experimental setup and
experimental results.
The experimental setup is illustrated in Figure 10, where a photo-detector was
translated, via optical rail, parallel to the scattering object to measure the flux at various
angles away from the specular reflection angle. By design, the incident angle was not
normal to the scattering surface. This allows for the BRDF to be measured on both sides
of the specular reflection angle, θinc.
41
Of note, is the fact that the vector normal to the photo-detector’s viewing window
was parallel to the vector normal to the scattering surface. This maintains a constant
geometry, reducing angular errors caused by rotation of the detector to the desired
viewing angle. This introduces a loss in flux proportional to cos(θoff-specular), which is
calculated out upon data analysis and PDF construction.
Not shown in Figure 10 is that the photo detector and laser source (632.8 nm) were
equipped with polarization optics to illuminate and detect in desired polarization states
consistent with determination of the parameters introduced in Equation 36. They were
oriented in such a manner as to not obscure the ray angle to be measured at a specific rail
position.
Figure 10 – BRDF measurement setup.
The vertical position on the rail, relative to the target area illuminated by the laser
source, of the photodetector determines the angle of detection. Unfolding the path,
because light always travels from left to right (Optical Sciences joke!), the detection of a
single angle, θd, is illustrated in Figure 11.
42
Figure 11 – Irradiance detection geometry34
Multiple experimental trials were conducted measuring the irradiance for various
polarization states on the photodetector at one degree intervals around the specular
reflection angle determined by the incident angle of the source on the scattering surface.
The results for a rough metal target are given in Figure 12,
Normalized Irradiance vs Scatter Angle (Rough Metal)
1
0.9
0.8
0.7
Trail 1
0.6
Trial 2
0.5
Trial 3
0.4
Trial 4
Trial 5
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Angle About Specular Reflection (degrees)
Figure 12 – Intensity distribution for scattering angles around the specular reflection angle.
43
Another important factor must be considered to produce a physically valid model for
the BRDF from a scattering surface given the detection geometry given in Figure 11.
Upon inspection of Figure 11, it is evident that as the detector is translated away from the
specular reflection angle (maximum irradiance angle), not only does the angle incident on
the increase, but the distance away from the source point on the scattering increases. A
cos3(θ) factor on the detected flux is manifested as a result of the detection geometry used
in the experimental BRDF determination.
Averaging the irradiance values at each angle over the sets of data determined in each
trial, and correcting for the cos3(θ) factor, produces a BRDF that increasingly approaches
the valid BRDF as the number of trials approaches infinity. For the number of trials that
were completed in support of this dissertation, an irradiance distribution over the
scattering angles about the specular reflection was determined by applying the inverse
cos3(θ) factor to the experimental data, then averaging the values at each angle for which
a measurement was taken. It is displayed in the following figure:
Corrected Average Irradiance vs Scatter Angle (Rough Metal)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Angle about Specular Reflection (degrees)
Figure 13 – Average irradiance distribution for scattering angles around the specular reflection angle, and
corrected for geometric losses in flux.
44
An analytic function is fit to the average data set seen in Figure 13 in order to allow
for the construction of a probability density function associated with the scattering from
the rough metal object to be used as a target within the scattering medium.
The form of the averaged data set in Figure 13 is observed to be Gaussian. A
Gaussian function has the following form,
⎡ − ( x − b )2 ⎤
f ( x )Gaussian = A exp ⎢
⎥
2
⎣ σ
⎦
(38)
The parameters, A, b, and σ2 were determined using the Solver function in Excel where
the deviations from the data in Figure 13 from a best guess Gaussian estimate are
calculated. The square root of the sum of the squared deviations is calculated as the error
metric for which the solver will minimize upon changing the parameters in Equation 38.
The b parameter was intentionally set to zero, because the specular reflection shall always
exhibit the maximum reflected radiance, and the angular origin of the system for which
the Gaussian function is being defined lies at the specular reflection angle.
Gaussian BRDF Fit: 1.02exp[-x2/24.42]
1.2
1
0.8
Inorm
Average
0.6
Gaussian Fit
0.4
0.2
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Angle about Specular Reflection (degrees)
Figure 14 – Gaussian fit to average data values for the BRDF measurements made to construct the PDF
required in the scatter model for rough metal.
45
The solver determined that the best Gaussian fit to the average data was the function
given in Equation 39,
⎡ − x2 ⎤
f ( x )Gaussian = 1.02 exp ⎢
⎥
⎣ 24.42 ⎦
(39)
Since an analytic solution has been calculated from the data set, the PDF can now be
constructed by integrating the Gaussian function over all space and dividing the integrand
by the result. This results is a function that, when integrated over all possible values, will
produce a value of 1. That is,
PDFBRDF =
f (x )
∞
∫ f (x )dx
=
⎡ − x2 ⎤
1
exp ⎢
⎥
8.759
⎣ 24.42 ⎦
(40)
−∞
This function will produce the probability of scattering at any particular range of angles,
x to x+dx.
As was done in Section 2.2.1, we now must determine, from the PDF, the angle of
scatter from the object as a function of a random number between 0 and 1. Unfortunately,
the integral inversion technique is not applicable in this circumstance, because the
definite integral of a Gaussian function involves the error function for which analytic
inversion is not possible. In this case, a technique is employed in which the possible
range of integration is divided in to N sub-ranges. The integral of the PDF is calculated
for each sub-range and its value is added to the sum of the integrals of the sub-ranges
complete before it. When this process integrates from 0 to the maximum value of 1, a set
of N elements is defined, each of which corresponding to the probability of scattering in a
particular direction. For the case of scattering angle, the probability of scattering in a subrange is a member of the vector valued density function, pi.
pi (scat [θ i −1 → θ i ]) =
θi
∫ PDF (θ )dθ
(41)
θ i −1
A probability marker vector is formed via cumulative sum with N members, who
correspond to a specific scattering angle:
46
i −1
K
Pi = pi + ∑ p n
(42)
n =1
The probability marker vector, Pi, allows for the determination of θ (ξ ) , the angle of
scatter as a function of a random number between 0 and 1. The process of finding θ using
a random number is detailed in Table 1:
Table 1 - Random Number Generation of Scattering Angle, θ
Task
1
Description
A random number generator produces a value between 0 and 1, ξ.
2
Find the two values in the vector, P i , that form a boundary about ξ. These
correspond to a specific member of the sub-range angles used in the
determination of P i .
3
Determine the exact proximity of ξ to the bounding members of P i to use
linear interpolation to find an approximate scattering angle that more closely
approximates the exact angle of scatter.
The form of the PDF determines the form of the statistical spread of data when
accessing the parameter associated with that particular PDF. As such, we should expect
to observe a Gaussian envelope with a mean approaching zero for a histogram built by
repeating the process detailed above for finding a scattering angle from the rough metal
surface. Such a histogram is built in Figure 15,
Histogram of 10,000 Random Number Trials for Scattering Angle from Rough Metal
350
Mean = -0.0060
Stdev = 3.2296
300
250
#
200
150
100
50
0
-10
-5
0
5
10
Scattering Angle about Specular Reflection Angle (degrees)
Figure 15 – Histogram built from repeated random number trials to determine scattering angle from a
rough metal surface. The data has a mean of -0.0060 and standard deviation of 3.2296.
47
The above figure validates the process by demonstrating a mean that is very close to zero,
and a Gaussian-like form.
For each of the experimental trials, the Stokes vector was also measured at the angles
of interest to quantify the depolarization induced by the scattering process. Over the
range of angles of interest, the depolarization was approximately equal for both linear and
circular polarization states and was very small <3%. Therefore, Mueller matrix upon
scatter from the rough metal surface with incident illumination at approximately 15
degrees from normal is approximated as,
~
M dep , scat
0
0
0 ⎤
⎡1
⎢0 0.97
0
0 ⎥⎥
⎢
= BRDF *
⎢0
0
0 ⎥
− 0.97
⎥
⎢
0
0
− 0.97⎦
⎣0
(43)
The range of angles for which measurements were take only constituted a small
portion (20˚) of the possible range of angles. This choice was made, because the detecting
instrument was not sensitive enough to resolve meaningful data past this point.
Depolarization and polarizing behavior indicative of scattering at larger angles33 could
not be observed due to the lack of sensitivity. For the purposes of the system presented in
this dissertation, the approximate depolarization model shown in Equation 43 suffices,
because the NA of the instrument will not accept large angles. Because of obscurant
scattering, photons that did strike the rough metal at large angles of incidence could
potentially have their global reference scattering angle reduced to detectable orientations,
but the probability of this occurring is very low, and therefore not statistically significant
in relation to the probability of other photon trajectories being accepted by the designed
receiver.
The same process was repeated for copier paper as the scattering surface. The copier
paper approximately acted as a Lambertian reflector. The average experimental data
demonstrated a very flat intensity distribution with scattering angle about the specular
reflection angle. Additionally, it acted as a very strong depolarizer due to the fact that its
approximate Lambertian nature lends itself to a large amount of multiple scattering
48
events within the micro-structure of the paper, effectively randomizing detected
polarization from the paper.
Using the exact process for determination of the histogram found in Figure 15, a
histogram was built using the random number determination for scattering angle for the
copier paper:
Histogram of 100,000 Random Number Trials for Scattering Angle from Copier Paper
1200
1000
# in Bin
800
600
400
200
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Scattering Angle about Specular Reflection Angle (degrees)
Figure 16 – Histogram built from repeated random number trials to determine scattering angle from a
copier paper’s surface.
2.2.3 Probability Density Function Construction and Use: Water Vapor
Particle Size
The size distribution of particles within obscurant clouds ultimately dictates the scattering
behavior of the cloud as a whole. As evidenced by equations 26 and 27, the induced
polarization or depolarization at various angles of incidence and scatter, the fractional
losses due to absorption, and all other physical processes related to interaction with
incident polarized radiation are governed by the particle size distribution.
Accurate modeling of the scattering process must have, at its core, an efficient way to
determine particle sizes of those bodies that interact and scatter impingent illumination.
49
For our purposes, a specific type of ultrasonic nebulizer was employed to produce water
vapor clouds to that were contained within a transparent enclosure.
Nebulizers convert low viscosity liquid into fine particles. Liquids enter a reservoir at
the base of which is a high frequency vibrating element. The vibrating element forces
waves to move through the liquid, the liquid then begins to be pushed upward, breaking
the surface tension of the liquid, creating gaseous cloud on the surface of the liquid. An
air-flow mechanism then ejects the cloud through an exit aperture of the nebulizer to a
desired location. The mean particle size of the gaseous cloud formed in this process is
calculated by Equation 44.
d part
⎛ T
= 0.73⎜⎜ 2
⎝ ρf
⎞
⎟⎟
⎠
1/ 3
(44)
where T is the surface tension of the liquid, ρ is the density of the liquid, and f is the
frequency of the vibrating element in the reservoir.
These nebulizers provide a very inexpensive source for the creation of particle clouds.
However, characterizing a well defined size distribution is required for accurate modeling
when studying scatter processes within the produced obscurant clouds.
There is an array of techniques to quantify the particle size distribution of a cloud of
particles. One of these techniques was presented by Arnot, et al.35 in which the optical
depth of a cloud is measured over a broad-band infrared source using a Fourier transform
spectrometer. Ultimately, inversion of these measurements yields the size distribution of
the cloud being viewed by the system35.
This technique was not used in support of this dissertation, however, their results
were. To obtain an inexpensive particle cloud source whose size distribution was well
characterized, the exact same model of the nebulizer used in Arnot’s paper35 was
acquired. They measured the following optical depth and associated droplet diameter of
the specific ultrasonic nebulizer:
50
Figure 17 – Water droplet measured counts (solid binned curve) and concentration retrieved by inversion
of the greater and lesser optical depths of (a), (thin solid and dotted curves, respectively).35
Following similar processes to fill the chamber with the obscurant cloud that were
used by Arnot, et al., it is reasonable to conclude that a particle cloud with the same size
distribution would be produced with the same nebulizer. This assumption forms the basis
for the formulation of the particle size PDF construction.
By direct inspection of Figure 17(b), two vectors were created containing the direct
count measurement and the water droplet diameter stepping in 0.5 micron increments.
These vectors are SC and SP, respectively. The probability, pi, that a photon will interact
with a particle of a radius consistent with a corresponding value in the vector SP is the
density function member,
pi =
SC i
(45)
N
∑ SC
i =1
i
A cumulative sum of the elements of the vector formed by pi will produce the probability
marker vector, Pi (eq. 42) for interaction particle size. Linear interpolation is not used in
this case, because during the MC method many photon packets are launched and,
generally, multiple scattering events occur for each photon packet. At each interaction, a
scattering Mueller matrix must be calculated (eq. 27). There are many computations that
are involved in the determination of these Mueller matrices, so exact computation of the
scattering Mueller matrix for each event is computationally prohibitive. Therefore, only
51
Mueller matrices for the set values of particle size are tabulated before the MC runs, as
will be explained in the next section.
Instead of linear interpolation after establishing a random number in proximity to two
values in the probability marker vector, the member Pi that is closest to a random number
between 0 and 1 is used to determine which value of particle size shall be used in this
particular instance. This process is repeated 100,000 times to demonstrate its feasibility to
produce a distribution similar to that seen in the solid bin curve in Figure 17(b) in the
following histogram:
Histogram of 100,000 Random Number Trials for Particle Size
12000
10000
# in Bin
8000
6000
4000
2000
0
1
2
3
4
5
6
7
8
9
Particle Diameter (microns)
Figure 18 – Histogram built from repeated random number trials to determine particle size with which to
be interacted in a scattering event.
Figure 18 indicates that the process by which we statistically sample the constructed
probability marker vector for interaction particle size is valid, in that its form matches an
experimentally verified curve35.
2.2.4 Probability Density Function Construction and Use: Spherical Particle
Scattering Angle (θ, φ)
With the size of the scattering particle determined, the Mueller matrix for any given set of
incident and scattering angles is yielded through application of Equation 27 in concert
with the calculation of its background parameters. It is through the use of this Mueller
52
matrix that the scattering statistics associated with the deflection angles θ, and φ is
derived.
First, the irradiance after a scattering event from the spherical particle must be
quantified. After a scattering event, the incident photon’s change in direction is described
by two angles: the scattering angle, θ, which describes the magnitude of the angular
deviation, and the azimuth angle, φ, which describes the direction of scatter about the
incident photon’s Poynting vector direction. With incident Stokes vector, S0, the scattered
Stokes vector is found as,
S sca = M (θ , φ )S 0 = M (θ )R (φ )S 0
(46)
where
⎡ cos 2 φ
⎢
sin 2 φ
R (φ ) = ⎢
⎢− sin 2φ
⎢
⎢⎣ 0
sin 2 φ
cos 2 φ
sin 2φ
.5 sin 2φ
− .5 sin 2φ
0
0
cos 2φ
0⎤
⎥
0⎥
0⎥
⎥
1⎥⎦
(47)
and M(θ) is the 4x4 scattering Mueller matrix found in Equation 27. The total irradiance
scattered in the (θ,φ) direction is given by the sum of the first two components of vector
S sca in the modified basis in which Equation 27 was derived.
U sin 2φ ⎞
U sin 2φ ⎞
2⎛
2⎛
2
2
isca = S 2 (θ ) ⎜ I p cos2 φ + I s sin2 φ +
⎟ + S1 (θ ) ⎜ I s cos φ + I p sin φ −
⎟
2 ⎠
2 ⎠
⎝
⎝
(48)
where S1 (θ ) and S 2 (θ ) are complex amplitudes defined in equations 22 and 23.
Using the result found in Equation 48, a phase function can be constructed by
dividing isca by a factor consistent with the square of the wave number multiplied by the
scattering cross section. With division of k2Csca and an incident Stokes vector of unit
irradiance, integration over all possible angles on the unit sphere (all solid angles)
produces a value of one. The integrand of Equation 49 constructs the PDF for scattering
from a homogeneous sphere of a given radius.
2π π
∫∫ k
0 0
i sca
sin θdθdφ = 1
C sca
2
(49)
53
As with the previous two sections, integral inversion is impossible to construct an
analytic relation for the scattering angles as a function of random numbers. Another
complication in finding these angles is that the construction of the scattering Mueller
matrices is very time intensive, lending itself to choosing discrete angle values for which
the scattering Mueller matrices will be defined and forming a library to be accessed by
the MC simulation. The largest complication in determining the deflection angles is that
they are coupled. The PDF is a coupled function of θ and φ, because they are both
required to form isca.
Amelioration of this issue requires a valid technique to either decouple the angles or
find them in conjunction with each other. A rejection technique in which the inversion of
probability function P (θ , ϕ ) can be performed with the simplest rejection technique finds
the angles in conjunction with each other. A random number generator determines three
numbers over their possible full range, they are θ ∈ [0,π ],
[
ϕ ∈ [0,2π ] and
]
max
f ∈ 0, isca
(θ ,ϕ ) . If the value of f is less than isca (θ ,ϕ ) , the doublet (θ,φ) is accepted,
otherwise it is rejected. The probability p ( f < i sca (θ , ϕ )) is always less than 0.5 (the case
of small particles), because the scattered intensity is peaked toward the forward and the
backward regions and becomes increasingly smaller as the diameter of the sphere
increases.16 This technique is very inefficient, since a majority of the random draw angle
doublets are always rejected. As the particle size increases, more and more doublets are
rejected, slowing down the associated simulations by a factor equal to the probability that
a doublet will be accepted.
A decoupling procedure is preferred for its ability to not incur a loss in simulation
speed if it maintains accurate values. Such an algorithm is detailed by Kaplan, et al.16 in
which a modified probability density function is used to find the deflection angles
independent of one another.
The decoupling method formed by Kaplan first studies the first three elements of the
Stokes vector incident upon the scattering location, I, Q, and U, and formulates the
following relations16:
54
Q = IW cos 2φ 0
(50)
U = IW sin 2φ 0
where W is the degree of linear polarization of the incident Stokes vector and φ 0 is the
orientation of the major axis of polarization. He also defines16,
π
T1 = π ∫ S1 (θ ) sin θdθ
2
0
π
(51)
T2 = π ∫ S 2 (θ ) sin θdθ
2
0
A random number between 0 and 1, u1, is then generated. If u1 = T1/(T1+T2), then let β=1,
else let β=2. Through the composition method16, the probabilities for the two deflection
angles are then decoupled and the probability for given deflection angle in azimuth is
given by16
sin 2 (φ − φ0 )
1
+W
, if i0 = 1;
π
2π
1
cos 2 (φ − φ0 )
P(φ ) = (1 − W )
+W
, if i0 = 2.
π
2π
P(φ ) = (1 − W )
(52)
At this point another random number between 0 and 1 is generated, u2. If u 2 ≤ 1 − W ,
then the PDF for the azimuth deflection is considered uniform, and as such the
probability for all possible angles is equal to 1/2π. In this case, the deflection for azimuth
is φ = 2πξ , where ξ is a random number between 0 and 1. For the case where
u 2 > 1 − W , the probability of a particular scatter in azimuth is given by the following16:
P(φ ) =
sin 2 (φ − φ 0 )
, if i0 = 1;
π
2
cos (φ − φ 0 )
P(φ ) =
, if i0 = 2.
π
(53)
Again in a process similar to that given in the previous sections, a probabilistic marker
vector Pi is formed and a random number determines, through linear interpolation, an
approximate azimuth deflection angle that more accurately represents the azimuth
deflection without interpolation.
55
With the deflection angles now decoupled, the scattering angle, θ, can be determined.
Similar to what was done in section 2.2.2, the possible range of integration, [0, π], is
divided in to N sub-ranges. At each sub-range the following integral is performed,
m0 i (θ i −1 → θ i ) =
∫θ (I
θi
)
S 2 (θ , x ) + I 0 y S1 (θ , x ) dθ
2
0x
2
(54)
i −1
which leads to the probability to be scattered in the ith sub-range,
p0 i =
m0 i
.
N
∑m
i =1
(55)
0i
The individual members of mi were yielded by studying the total irradiance output over
the sub-range, [θi-1, θi], at zero azimuth angle of a scattering event with incident modified
Stokes parameters, I0x and I0y on a particle with a single size parameter, x.
Due to the complicated mathematical formulation for the S parameters in Equation
54; a special numerical integration technique known as Legendre-Gaussian quadrature
was required to determine mi. Briefly, Gaussian quadrature evaluates the integrand at
specific values within the range of integration and takes a weighted sum of the result of
these evaluated sums. This technique allows for numerical integration of complicated
integrands using very few evaluation points.
The values of p0i again form a vector valued density function. A probability marker
vector is formed via cumulative sum with N members, who correspond to a specific
scattering angle:
i −1
K
P0 i = p 0 i + ∑ p0 n
(56)
n =1
The probability marker vector, P0i, allows for the determination of θ (ξ ) , the angle of
scatter as a function of a random number between 0 and 1 using the exact process that
was given in Table 1.
A probability marker vector is defined for every discrete particle size of interest and
incident linear polarization state so that angles of scatter can be defined in the MC
method for each available member of SP, the particle size vector defined in Section 2.2.3.
56
2.2.5 Photon Position Tracking and Counting
Now that the processes to generate all of the required variable parameters have been
treated, what remains is the ability to track a photon’s position in a global coordinate
frame to determine its relation to the boundary of the obscurant medium, the objects
within the medium, and the collection optics of the receiver.
A photon’s trajectory is set by the Euler angles presented in the previous section, θ
and φ. These angles, in conjunction with the step size parameter set forth in Section 2.2.1
allow for the tracking of a photon’s exact location in the global coordinate frame as well
as any intermediate reference frame. First, the azimuth angle rotates incident coordinate
frame about the direction z-axis (axis parallel to the photon’s direction of propagation),
then the scattering angle, θ, rotates the post-azimuth rotated coordinate frame in the new
x-z plane. The resultant coordinate transfer is the new scattered photon’s coordinate
frame. The coordinate frame transformation is illustrated in the following figure.
Figure 19 – Randomly determined Euler angle coordinate transformation23.
Mathematically, the transformation of the original coordinate frame from the azimuth
angle of rotation is represented as,
e xinc = cos(ϕ )e x0 + sin(ϕ )e 0y
0
0
e inc
y = − sin(ϕ )e x + cos(ϕ )e y .
e
inc
z
=e
(57)
0
z
The subsequent scattering angle rotation then transforms erinc in the following fashion,
57
e xsca = cos(θ )e xinc − sin(θ )e zinc
e ysca = e inc
y
.
(58)
e zsca = sin(θ )e xinc + cos(θ )e zinc
Using the Euler angle formulation and associated coordinate transformations, the
global coordinate frame, and all intermediate scattering frames, can be tracked. The
coordinate positions within these frames are also tracked as a result.
The position of a photon within the global coordinate frame is tracked using a simple
relation:
⎛ x⎞
⎛ x⎞
⎜ ⎟
⎜ ⎟
⎜ y⎟ = ⎜ y⎟ + R *ez
⎜ ⎟
⎜z⎟
⎝ ⎠ n +1 ⎝ z ⎠ n
(59)
where R is the step size. As more and more scattering events, which correspond to more
and more coordinate transformations, ez evolves as a function of the azimuth and
scattering angles randomly generated for each scattering event.
If a photon is found to impinge on an object set within the obscurant medium, it is
backtracked to the point at which the intersection would have took place, and the
scattering statistics associated with that object’s surface are applied.
Additionally, if a photon is found to impinge on a wall of the containment cell
consistent with a probability of detection from the collection optics, Snell’s and Fresnel’s
laws36 are respected and a new position and Stokes vector are calculated based on the
material and thickness of the containment wall. The Fresnel coefficients determine the
reflectance and transmittance associated with the wall of the container, as such, when a
photon reaches the wall, they are applied in the form of another random number game. If
a random number between 0 and 1 is less than the reflectance coefficient, the associated
photon is reflected, otherwise it is transmitted (absorption was neglected in the MC
simulations for this dissertation). If said photon is found to transmit, it is tracked to the
plane of the entrance pupil of the collection optics. If the photon’s position and angular
trajectory satisfies the acceptance conditions of the receiver, then the photon is traced to a
detector position and is counted as a flux constituent for that detector.
58
Due to the fact that a detected photon’s polarization state is represented in the
coordinate frame determined by its last scattering event; it must be transformed back into
the global coordinate frame to gain an accurate representation of its state from the
observer’s perspective.
The scattered photon’s coordinate frame is compared to the global coordinate frame
to calculate a singular transformation on the global coordinate frame that will yield the
scattered photon’s coordinate frame. This transformation will give values for θ and φ that
are entered into1
~
Rscat ⇒ global
⎡ cos 2 θ cos 2 φ
⎢
cos 2 θ sin 2 φ
=⎢
⎢− cos 2 θ sin 2φ
⎢
0
⎢⎣
sin 2 φ
0.5 sin 2φ cos θ
cos 2 φ
sin 2φ
− 0.5 sin 2φ cos θ
cos 2φ cos θ
0
0
0 ⎤
⎥
0 ⎥
0 ⎥
⎥
cos θ ⎥⎦
(60)
to give
⎛Ix ⎞
⎛Ix ⎞
⎜ ⎟
⎜ ⎟
~⎜ I y ⎟
⎜Iy ⎟
= R⎜ ⎟ .
⎜U ⎟
U
⎜ ⎟
⎜ ⎟
⎜V ⎟
⎜V ⎟
⎝ ⎠ global
⎝ ⎠ scat
(61)
The result in Equation 61 is then used to determine how the photon is counted as far as
the system is concerned. Since a photon is a quantized energy packet, it can not be
divided into fractions. Therefore, a random number game is used to determine how the
system detects the photon based on the values of the modified Stokes vector in the global
coordinate frame. The possible orthogonal analyzer polarization states of the detector are
right and left circular; or horizontal and vertical linear, as will be discussed in following
chapters.
Given the detection state of the system, an accepted photon will be detected in one of
the two orthogonal states. The probability of detection in each state is,
59
⎧ p x = I x / (I x + I y )
If detection = linear ⎨
⎩ p y = I y / (I x + I y )
1 + V / (I x + I y )
⎧
=
p
⎪⎪ R
2
If detection = circular ⎨
1 − V / (I x + I y )
⎪ pL =
⎪⎩
2
For a linear polarization detection scheme, the normalized versions of Ix and Iy give the
respective probability of detection for each. For a circular polarization detection scheme,
the probability of detection is given by subtraction and addition of the normalized version
of V from 1, then dividing by 2 for pL and pR respectively.
The random number generated for detection state is then compared to these
probabilities to determine in which state the photon will be detected. A photon’s energy
is Plank’s constant multiplied by the frequency oscillation of the electric field. The
counting procedure normalizes by this value, such that the energy of a photon is unity.
2.2.6 Monte Carlo Simulation Convergence and Validation
The Monte Carlo simulation to be used in support of this dissertation completes the
following steps:
1. Create photon’s initial coordinates, direction, and polarization. The initial
coordinates are set by a random number game in association with the size of the
collimating optics exit pupil size. The initial direction is set by the divergence of
the illumination source. The source in question is a continuous wave, collimated
HeNe laser source with initial angles set to zero for all points for all points on the
collimating optics’ exit pupil. The initial polarization is either circular or linear
depending on the experiment.
2. A random number generation scheme determines the distance the photon travels
before it interacts with a particle of radius, a, which is also determined by a
random number generations scheme (Section 2.2.1 and 2.2.3).
60
3. Change direction and polarization according to type of collision based on the
scattering statistics for the surface or particle size determined in step 2. (Section
2.2.2 and 2.2.4)
4. Score and go back to step 2...
5. The path terminates when the photon reaches the system boundary. Physical laws
of refraction and reflection are obeyed to change the polarization and direction of
the photon, and then it is tracked with its new directions to the entrance pupil of
the collection optics. If it satisfies angular and positional requirements to be
received, the photon is counted as part of the system flux. (Section 2.2.5)
6. The detection analyzers of the experiment determine in what state the photon is
detected through yet another random number game. (Section 2.2.5)
As the previous list attests, MC process is statistical in nature. Therefore, a statistical
metric is required to conclude that the simulation has converged to a solution with desired
accuracy. The standard deviation of global reference frame normalized Stokes vector
components for all of the qualifying photons is calculated for the Stokes components
consistent with the orthogonal detection states of the receiver (horizontal and vertical
linear, or right and left circular). When the standard deviation for these photons reaches
0.015, application of the Central Limit Theorem yields the conclusion that the results are
within 3% of the mean value for the Stokes components in question at a 95% confidence
level16. When this occurred, the solution was considered to have adequately converged
and the simulation ceased the firing of photons. This convergence metric was used due to
its successful implementation demonstrated by Kaplan, et al.16.
By way of validating the Monte Carlo simulation presented in this chapter, a
comparison is made between published tabulated values and simulation results for the
first three Stokes parameters when incident flux is reflected and transmitted through a
plane-parallel medium. Results are compared for anisotropic scattering media, where the
asymmetry factor is defined to be greater than zero (g>0).
61
To validate the simulation’s ability to efficiently track the overall transmitted and
reflected flux from a non-absorbing anisotropic medium, tabulated data yielded from the
Henyey-Greenstein Function solution for finite layers29 with an incident collimated flux
of 10,000 photons was used as a theoretical standard by which to compare.
The magnitude of anisotropy is represented by the asymmetry factor, g, defined in
Equation 26. It represents the cosine of the average scattering angle for a medium. As g
diverges from a value of zero, the medium for which the factor is being defined is
considered increasingly anisotropic. Tables 2 and 3 present the model’s data in
comparison with the Henyey-Greenstein Function solution for an asymmetry factor of
0.500 and 0.750 respectively.
In total, ten trials tracking the trajectories of 10,000 photons launched normal to the
face of a finite plane-parallel scattering layer with a specified optical depth were carried
out by the Monte Carlo simulation. The “Reflected Photons (Model)” list in Table 2 and
Table 3 represents the average value over the ten trials for the appropriate optical
thickness. Here optical thickness and optical depth are used interchangeably. The
standard deviation for the model-produced data is also given in the following two tables.
Table 2 - Model vs Henyey-Greenstein Function Solution for a Finite Layer, g=0.500
Optical Thickness
0.03125
0.0625
0.125
0.25
0.5
1
2
4
8
16
32
Reflected Photons (Theo)
54
108
219
445
898
1761
3203
5090
6890
8210
9032
Reflected Photons (Model)
62.3
120.0
240.3
480.8
944.3
1821.5
3196.8
5071.9
6861.3
8179.1
9017.4
Standard Deviation (Model)
9.2502
7.6158
12.437
28.3188
27.849
30.5441
56.7838
44.5557
27.2399
53.5754
25.083
62
Table 3 - Model vs Henyey-Greenstein Function Solution for a Finite Layer, g=0.750
Optical Thickness
0.03125
0.0625
0.125
0.25
0.5
1
2
4
8
16
32
Reflected Photons (Theo)
21
43
87
179
373
787
1632
3116
5064
6880
8204
Reflected Photons (Model)
16
31.3
65.8
146
316
721.9
1565.9
3112.9
5086.9
6919.6
8212.1
STD (Model)
3.5901
7.6746
6.6633
17.075
16.5059
30.5812
40.3938
32.1505
70.2827
34.3712
23.048
Figure 20 gives a graphical representation of the data listed in Tables 2 and 3 with error
bars representative of the standard deviation of the model data.
Figure 20 – Simulation of 10,000 photons launched into anisotropic media and its comparison to theory.
The simulation shows very good performance for the completed trials tracking the net
flux transmitted through and reflected from a non-absorbing media. The simulation
demonstrates a better fit to the theoretical curves for optical depths greater than one and
for less anisotropy. Fortunately, the model fits to within fractions of a percent of the
theoretical values when the optical depths reach values that would truly obscure objects
within the scattering medium (τ > 1.5). At optical depths less than this, contrast is not
appreciably affected, but the MC simulation demonstrates degraded performance as is
evident by inspection of the counts at lower optical depths in Figure 20.
63
With the presented MC simulation demonstrating quality performance for tracking
the total reflected and transmitted flux from an anisotropic medium, the overall tracking
of a photon within a scattered medium is validated. However, the model still must be
validated with respect to net reflected flux at various incident angles. Once validation has
been completed for these trials, the polarization output as a function of incident and
scattered angles external to the scattering media of interest are yielded by simple
polarization state tracking defined in Equation 46. The polarization tracking is validated
through this procedure, because the probability of scattering at specific angles is a
function of the polarization state defined in the local scattering reference frame at the
scattering location within the medium as demonstrated in Equation 54 and Equation 55.
Agreement with net flux measurements at variable incident angles demonstrates
polarization dependant tracking.
Using the developed Monte Carlo simulations, reflected flux as a function of incident
angle on a plane parallel medium was calculated and compared to tabulated values29 for a
particle size distribution consistent with asymmetry factors of 0.500 and 0.750.
In total, ten trials tracking the trajectories of 10,000 photons launched at a specified
angle of incidence to the face of a finite plane-parallel scattering layer with a determined
optical depth were carried out by the Monte Carlo simulation. The respective error maps
for simulations at the two asymmetry factors represent the error of simulated data with
respect to the tabulated data. Simulated data points at each asymmetry factor were found
by the average value over the ten trials for the appropriate optical thickness. Here optical
thickness and optical depth are used interchangeably.
The error maps produced by comparison of simulated data with Van de Hulst’s
tabulated data29 are given in Figures 21 and 22 on the following page,
64
Error Map for g = 0.50
0.15
cos(incident angle)
1
.9
0.1
.7
0.05
.5
.3
0
.1
.5
1
2
4
8
-0.05
Optical Depth (# Mean Free Paths)
Figure 21 – Reflected flux error map produced by calculating the deviation of model data from theoretical
tabulated values29 for a particle distribution yielding an asymmetry factor of 0.500. The color bar represents
the percent error from the tabulated data when multiplied by 100.
Error Map for g = 0.75
0.15
cos(incident angle)
1
.9
0.1
.7
0.05
.5
.3
0
.1
.5
1
2
4
8
-0.05
Optical Depth (# Mean Free Paths)
Figure 22 – Reflected flux error map produced by calculating the deviation of model data from theoretical
tabulated values29 for a particle distribution yielding an asymmetry factor of 0.750. The color bar represents
the percent error from the tabulated data when multiplied by 100.
65
Agreeing with the trend shown in Table 2 and Table 3, the largest errors in the MC
simulation data correspond to when the optical depth parameter in the simulation was set
at lower values and at lower angles of incidence in the reflection case.
These validation simulations were considered sufficient because the magnitude of
error in optical depth regions of interest never broached 3%. The regions of interest in the
optical depth regime encompass optical depths greater than 2. In these regions we start to
observe obscurant levels that can hinder image contrast appreciably.
66
CHAPTER 3: LABORATORY BASED SYSTEM DESIGN
An instrument capable of experimentally verifying contrast enhancements of
backscattered light when illuminating with circularly and linearly polarized light was
designed, aligned, and constructed in support of this dissertation. It allowed for
experimental data collection to verify the many models and results presented in Section
2.1.3. The instrument was designed to have the capability of switching easily between a
circularly polarized mode and a linearly polarized mode to study the contrast
enhancement differences between the two. The instrument also uses a unique design to
present orthogonally polarized images on the same FPA, allowing for snapshot capability
and single path detection. A schematic of the instrument is given in Figure 23.
Figure 23 – Circularly polarized image of object is collimated through a quarter wave plate and Wollaston
prism, then re-imaged to split the image into right and left circular components. (NOT TO SCALE)
In addition to the optical design of the instrument, novel algorithms were created in
support of this dissertation. The algorithms vastly improve the image contrast
enhancements past what conventional orthogonal image subtraction can achieved. These
algorithms use CPIS as their base, but extend into the study of the spatial frequency
spectrum of post-subtracted images to attempt to enhance object-based spatial
frequencies relative to spatial frequencies associated with the obscurant.
67
A computationally non-intensive image registration algorithm was developed to
determine the appropriate pixel shifts to align the orthogonally polarized images for
allowance of direct pixel to pixel image subtraction.
3.1 System Optical Design
Quality optical design of a system that can present orthogonally polarized images of the
same scene on the same FPA was an objective of the research detailed in this dissertation.
The following design allows for ease of image registration in a common path instrument
that can process data in a real-time environment.
A schematic of the polarization imager that we propose to build is shown in Figure
24.
Figure 24 – A circularly polarized image of an object is collimated through a quarter wave plate and
Wollaston prism, then re-imaged to split the image into right and left circular components.
For this receiver, we assume a circularly polarized field is incident on the sensor. The
fore-optics, which include the objective lens, the field stop, and a collimating lens,
provide an intermediate image plane, and limit the field of view (FOV) that will be
imaged by the instrument, preventing overlap between the adjacent images on the FPA.
Next, the quarter wave plate converts the circularly polarized field into linearly polarized
light, which is decomposed by the Wollaston prism into two orthogonally polarized
constituents. At the interface of the prism, a small angular divergence is produced
68
between these orthogonally-polarized components, and the two ray bundles continue to
diverge until they are incident on the re-imaging lens, which focuses the rays to produce
two polarization images on the FPA as shown in Figure 24.
The sensor may also be easily converted into a linear polarization imager by rotation
or removal of the quarter wave plate (see Figure 24). Then the two split images would be
linear polarized in two orthogonal directions. For both circular and linear polarization
configurations, cross-polarization image subtraction and other algorithms may be
achieved using image processing software to be presented in following sections of this
chapter.
It is clear that the optical component driving the design and performance of the
system schematically presented in Figure 24 is the Wollaston prism. A Wollaston prism
is formed when two birefringent wedge prisms are cemented together with their
respective ordinary and extraordinary axes orthogonal to one another. An angular
divergence as a function of incident radiation’s polarization state is manifested because of
the index mismatch between the extraordinary and ordinary indices at the sloped interface
of the cemented crystals.
The Wollaston prism available to support the design of the receiver had a rectangular
aperture 12.1 mm on its short side with birefringence, Δn = ne − no = −0.1699 for each
negative uniaxial crystal. The prism produced a total angular deviation of about 1.84
degrees at 632.8 nm for orthogonally polarized, normally incident rays. These parameters
drove the design of the sensor. Additionally, this system utilized all common, off-theshelf lenses from the Thorlabs catalogue.
Using a general rule in optical design to avoid very sharp ray bending whenever
possible, coupled with the aperture size of the Wollaston prism and available lens
elements, the approximate lens diameters were chosen to be either 25.4 mm (1 inch) or
12.7 mm (1/2 inch). The system was designed in four steps:
1. Lens design to image at the field stop location. This is the system objective.
2. Design collimation optics with the Wollaston prism set at the intermediate pupil
location defined by the image of the entrance pupil.
69
3. Design re-imaging system that minifies the field stop to a magnitude that allows
for the orthogonally polarized images to lie side by side on the image plane and
not be clipped by the aperture of the FPA. Care was taken here to allow for a large
enough field of view to gain useful image products.
4. Optimize the design with each subsystem in conjunction.
At each step, the lens design was accomplished by introducing lenses with variable
thicknesses, radii of curvature, and glass types. When an optimized solution was
produced in CodeV, a lens within the design was removed in favor of its closest
approximation available in the Thorlabs catalogue of lenses. The Thorlabs lens
approximation often consisted of two sphero-planar lenses placed with their planar sides
in contact. The solution was then optimized again with the Thorlabs lens specifications
frozen. If the newly optimized solution exhibited quality performance, the next Thorlabs
lens was inserted in favor of its optimized partner. This process was repeated until all
optical elements in the sub-system were off-the-shelf components. However, if it was
found that the replaced lens degraded performance appreciably, another lens was added
to attempt to correct for the aberrations induced from the Thorlabs lens substitution. This
iterative process was used in each of the three steps outlined above.
3.1.1 Optical Design: Lens design to Field Stop (Objective)
The design of a quality objective is imperative for end-system imaging performance. A
large entrance pupil was desired to minimize the f/# of the system and maximize its light
collection capabilities. However, small f/#’s induce aberrations and therefore degrade
performance. Thus, the system stop was chosen to be located at the first lens in the
objective. The clear aperture of this lens was 25.4 mm.
The field of view of the system was set by inspection of available polarizer apertures
to be set in the path of the illumination source and the distance at which the objects
would be placed away from the objective. A full field of view (FFOV) of 1 degree was
chosen in support of the likely detection geometries. Remember that the field stop will
70
determine the true field of view of the system, but the system was optimized at a FFOV
of 1 degree.
The available lens elements were all spherical singlet lenses. Given these choices the
objective lens was designed with the use of three plano-convex lenses and two planoconcave lenses seen in the following figure:
12.50
System Objective
Scale:
2.00
MM
01-Mar-11
Figure 25 – System objective shown imaging to an intermediate plane where the field stop is located.
The size of the image at the field stop is 1.828 mm. The lens prescription for the
objective shown in Figure 25 was determined to be:
Figure 26 – Objective lens prescription.
71
The surface names for each element in the lens prescription above are the appropriate part
numbers in the Thorlabs catalogue.
The modulation transfer function (MTF) is often used as a metric by which imaging
performance is measured. It describes resultant contrast at spatial frequencies being
imaged the system for which the MTF is defined, where the contrast at a specific spatial
frequency, η, can be defined by,
C=
I max (η ) − I min (η )
I max (η ) + I min (η )
(62)
The contrast will always have a maximum of one, when a spatial frequency can be
perfectly imaged. Diffraction limits the ability of a system to resolve spatial frequencies
to a cutoff spatial frequency defined by the inverse of the product of the illumination
wavelength and the f/#. Precisely, the MTF is the auto-correlation of the pupil error
function. When a spherical wavefront is perturbed from its nominal shape at optical
interfaces and is not corrected by following surfaces, the auto-correlation of the pupil
function will stray from the diffraction limit. The MTF for the designed objective is given
in Figure 27.
DIFFRACTION LIMIT
AXIS
MTF for Objective
DIFFRACTION MTF
01-Mar-11
T
R
1.0 FIELD ( 0.50 O )
T
R
-1.0FIELD ( -0.50 )
WAVELENGTH
632.8 NM
WEIGHT
1
O
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.2
0.1
30
60
90
120
150
180
210
240
270
SPATIAL FREQUENCY (CYCLES/MM)
Figure 27 – Modulation transfer function for the objective.
300
72
For the field of view defined in the design process, the objective performs very well.
Figure 27 includes the bounding diffraction limited performance for this system. With
observation of the on and off axis performance, this objective design was determined to
provide sufficient performance in support of overall system design. There exist many
metrics by which to study imaging performance, but the MTF provides the most intuitive
understanding of the overall system performance in the author’s opinion. As such, it is
used in further discussions about image performance.
3.1.2 Optical Design: Field Stop to Wollaston Prism (Collimator)
Following the design of an objective that performed well over the desired field of view,
the next subsystem was designed to present collimated rays to the Wollaston prism.
Perfect collimation is not required, but as the tangential rays from a common field point
become more divergent, the angular spread of the rays induced by the Wollaston prism
will not be uniform. This non-uniformity can be corrected by intelligent optical design,
but it adds an additional degree of complexity that would otherwise not exist.
To design the collimator sub-system, the optical design software was placed in an
afocal mode, and the process set forth in Section 3.1 was followed. The resultant design
consisted of three sphero-planar optical elements, of which one was plano-concave and
two were plano-convex.
11.36
Field Stop to Wollaston (Collimator)
Scale:
2.20
MM
01-Mar-11
Figure 28 – System collimator shown from the field stop location to a distance away from the final element
where the Wollaston prism is to be placed.
73
This design demonstrated an angular divergence from the chief ray to the meridonal
ray of 0.0067˚ for the on-axis field, and 0.0269˚ for the off-axis fields. The endpoint of
the collimator design in Figure 28 approximates to the location of an intermediate pupil
plane formed by imaging the stop through the system. At this location, the angular
deviations are present, but the positional deviations are at a minimum between the
individual field rays. The lens prescription from the intermediate image location (field
stop location) is given as:
Figure 29 – Lens prescription for the collimator sub-system
Due to the minimum amount of spatial variation between separate field points at
intermediate pupil planes, they provide prime locations for the placement of zero-power
optical elements. Phenomena with positional dependence on said optical elements will
then affect the fields equally over the face of the zero-power element in question. In our
case, our zero-power element is the Wollaston prism.
3.1.3 Optical Design: Complete Lens Design
What remained in the lens design was to take the collimated rays after they were deviated
from the Wollaston prism and re-image them onto a focal plane. Care was exercised to
maintain a small enough image spread on the focal plane to ensure no clipping occurred.
At the same time, a large enough image spread was required to separate the orthogonally
74
polarized images on the focal plane such that the full field of view of each was
uninterrupted by the other.
The design of the re-imaging lens presented a far more challenging design problem
than did the first two sub-systems. The angular deviation incurred at the Wollaston prism,
coupled with the intermediate collimated radiation to be imaged produced a wavefront
that was far removed from a spherical wavefront. Using custom lenses would allow for a
far simpler solution with less required elements, but custom fabricated lenses are
expensive and not within the budget of research done in support of this dissertation. As
such, eight spherical lenses were required to provide quality re-imaging of field
distribution at the field stop.
Without the benefit of custom lens fabrication, the performance still fell short of
desired. As a result, the re-imaging lens subsystem was inserted and the system as a
whole was redesigned with specific constraints on the optimization routine to maintain
intermediate image and final pupil locations. Maintaining these locations allowed for the
conservation of the relative positions of the field stop, the Wollaston prism, and the FPA
with respect to adjacent lens elements. The optimization routine perturbed only air spaces
between lenses to converge to a solution. The resultant re-imaging subsystem is
demonstrated in the following figure:
13.89
Re-Imaging System (Wollaston to FPA)
Scale:
1.80
Figure 30 – Re-imaging subsystem.
MM
01-Mar-11
75
The re-imaging subsystem performed well after system level optimization. The lens
prescription of the re-imaging lens was,
Figure 31 – Lens prescription for the re-imaging lens
After the entire system was optimize in support of image quality at the final image plane,
the degree of collimation at the Wollaston prism, and the image quality at the field stop
were degraded due to the variation of air spaces between virtually every lens
combination. However, a quality overall system MTF and sufficient separation on the
FPA are the metrics for success in this optical design. If intermediate success metrics
need suffer in favor of end-system success metrics, it is acceptable.
With inclusion of the Wollaston prism in the design, polarization ray tracing was
required to obtain the MTF for the entire system. The MTF in the Figure 32 represents
the system’s performance when vertically polarized light was impingent on the first
element of the objective; similarly the MTF in Figure 30 represents the system’s
performance when horizontally polarized light was incident on the first element of the
system. In the optical design, the Wollaston prism had its first crystal’s ordinary axis in
the horizontal direction and its second crystal’s extraordinary axis in the horizontal
direction.
76
DIFFRACTION LIMIT
MTF for Vertical Pol
arized Light
Y
X
0.0 FIELD ( 0.00 O )
DIFFRACTION MTF
Y
X
1.0 FIELD ( 0.50 O )
Y
X
-1.0FIELD ( -0.50 O )
01-Mar-11
WAVELENGTH
632.8 NM
WEIGHT
1
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.2
0.1
36
72
108
144
180
216
252
288
324
SPATIAL FREQUENCY (CYCLES/MM)
Figure 32 – System MTF with vertically polarized light incident.
DIFFRACTION LIMIT
MTF for Horizontally
Polarized Light
Y
X
0.0 FIELD ( 0.00 O )
DIFFRACTION MTF
Y
X
1.0 FIELD ( 0.50 O )
Y
X
-1.0FIELD ( -0.50 O )
01-Mar-11
WAVELENGTH
632.8 NM
WEIGHT
1
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.2
0.1
36
72
108
144
180
216
252
288
324
SPATIAL FREQUENCY (CYCLES/MM)
Figure 33 – System MTF with horizontally polarized light incident.
360
X
Y
360
X
Y
77
Comparing figures 32 and 33 shows the modulation transfer functions for incident
vertically and horizontally polarized light to be very similar. Upon close inspection,
however, they do differ most noticeably in the on-axis field for the tangential rays (red,
solid curves). The MTF for the vertically polarized light demonstrates better performance
in this regard. Also, it is evident that the sagital rays (dashed curves) perform superior to
the tangential rays at all field locations with the exception of the on-axis field for the
tangential rays. Lens symmetry disappears for off-axis fields; therefore the MTF is not
rotationally symmetric for off-axis fields. In addition, the Wollaston prism induces a
system level asymmetry that is reflected in the differences in performance between
equivalent field angles for different incident polarization states. Note that the MTF
performance
MTF performance comparison between the various incident polarization states
allowed for a design decision to be made for the polarization state of the illumination
system. The illumination consistent with the channel that was to be used as the object
channel was chosen to be vertically polarized if the Wollaston prism was oriented with
the first ordinary axis horizontal. Conversely, if the Wollaston prism was oriented with its
first ordinary axis in the vertical direction, illumination would be delivered in the
orthogonal state.
The optical design thus far has been treated as if linearly polarized light was incident
and detected, but Figure 21 is entitled Circular Polarization Imager. This is due to the fact
that the imager can switch easily between detecting in orthogonal circularly polarized
states or orthogonally polarized linear states by inclusion or exclusion of a quarter wave
plate oriented in the appropriate direction. The eigenstates of a Wollaston prism are any
two orthogonally polarized linear states as are the eigenstates of a quarter-wave retarder.
If the quarter-wave retarder’s fast axis is oriented 45 degrees from an eigenstate of the
Wollaston prism, incident circularly polarized light will be forced into one of the
eigenstates of the Wollaston prism depending on if it is right or left circularly polarized
effectively yielding system eigenstates that are right and left circular.
78
A tolerance analysis of the final design was completed that used the Thorlabs
spherical singlet tolerances on each of their lens elements, with appropriate compensators
set on individual and group lens element translations, the design was considered
realizable. The set of compensators that produced the best statistical performance defined
number of lens tubes and their respective lengths required to mount and align the system
properly. These lens tubes and retaining rings were also procured from Thorlabs.
After the system constructed and aligned, the resultant images displayed an
appreciable amount of polarization astigmatism. What is meant by polarization
astigmatism is that the images rendered on the FPA of orthogonal polarization had
differing positions where they would be considered in focus. The Wollaston prism was
found to be the culprit in this astigmatism, as such; it couldn’t be corrected from a lens
design prospective unless other expensive optical elements were introduced into the
system design. From an image registration perspective (treated in the following section),
it is very important that the orthogonally polarized images are as close to exact copies of
each other as can be achieved. The polarization astigmatism coupled with image
registration requirements demanded the addition of defocus at the focal plane to match
the orthogonally polarized images at the final image plane.
Unfortunately, the intentionally applied defocus generated degraded MTF
performance for the entire system. A simulation reported the effect on the MTF when
0.2mm of defocus was added to the optimized system design,
DIFFRACTION LIMIT
Vertically Polarized
.2mm defocus
Y
X
0.0 FIELD ( 0.00 O )
DIFFRACTION MTF
Y
X
1.0 FIELD ( 0.50 O )
Y
X
-1.0FIELD ( -0.50 O )
05-Mar-11
WAVELENGTH
632.8 NM
WEIGHT
1
DEFOCUSING 0.00000
1.0
0.9
0.8
0.7
M
O 0.6
D
U
L 0.5
A
T
I
O 0.4
N
0.3
0.2
0.1
36
72
108
144
180
216
252
288
324
360
SPATIAL FREQUENCY (CYCLES/MM)
Figure 34 – Effect of defocus on system MTF
X
Y
79
The physically realized system required a defocus larger than 0.2 mm to accurately
match the orthogonally polarized images; therefore the performance of the system was
degraded even further than that shown in Figure 31, but it maintained an MTF value over
0.2 for the spatial frequencies of the resolution target. This allowed for quality
performance as an imager to generate functional images for use in the image subtraction
post processing algorithms. A subset of these images is presented in Figures 35 and 36.
Figure 35 – Paper resolution target illuminated with unpolarized light.
Figure 35 demonstrates the system’s ability to render equivalent images to the focal
plane when unpolarized light is incident on the objective. The Wollaston prism should
split the irradiance equally into the two channels, because unpolarized light can always be
decomposed into an equal superposition of orthogonally polarized constituents. As Figure
35 attests, the Wollaston prism does indeed split the incident irradiance into equal parts.
Additionally, the lens design ensures that the respective images be split apart enough at
the image plane as to not interfere with each other.
Figure 36 – Rough metal wrench target illuminated with circularly polarized light.
80
Figure 36 exhibits the system’s capability to display the image of a scene in only one
channel when incident radiation is of a polarization consistent with an eigenstate of the
system. In Figure 36, the system was set in circular polarization detection mode and a
rough metal object was placed in the system’s field of view with right circularly polarized
illumination. Upon close inspection of the previous figure, a small amount of flux is able
to be perceived in the orthogonal channel. This is attributed to the fact that the rough
metal wrench in the field of view induced some depolarization (Chapter 2.2.2).
3.2 Image Registration
Regardless of the mode in which the optical system is detecting (circular or linear), the
orthogonally polarized images presented to the focal plane ultimately need to be
subtracted from one another. This requires that the individual images on separate pixels
on the FPA be registered to each other such that spatial structure in one image correlates
exactly to the spatial structure seen in its orthogonal partner when unpolarized light is
incident on the sensor.
What was not modeled in the treatment of the optical design in the previous section
was that in practice, the Wollaston prism we used induced polarization images that were
astigmatic with respect to one another.
This manifested itself as blurring of the
laboratory setup’s post processed images, seen in experimental images in following
chapters. We attempted to reduce this effect by defocusing the sharp polarization subimage towards the focus of the orthogonal polarization image, to a point where the
blurring was uniform in both images. This situation is preferred over the alternative of
using a sharp image and a blurred one, because better performance is achieved using
image subtraction of like images, thus avoiding image misregistration.
The paper resolution target seen in Figure 35 was used to find the appropriate pixel
shift values using an autocorrelation technique. The Convolution Theorem states that if
we have two functions f(x) and s(x), whose Fourier transforms are F(p) and S(p),
81
respectively, then the inverse Fourier transform of the product of their Fourier transforms
can be written as
ℑ−1 [F ( p ) S ( p )] =
1
=
2π
1
2π
∞
∫ F ( p ) S ( p )e
ipx
dp
−∞
⎛
⎞
−ipγ
ipx
∫−∞⎜⎜⎝ −∫∞ f (γ )e dγ ⎟⎟⎠S ( p)e dp
∞
∞
(63)
interchanging the order of integration
= ∫ f (γ )
1
2π
(∫ S ( p)e
−ip ( x −γ )
)
dp dγ
(64)
∞
=
∫ f (γ )s( x − γ )dγ =
f ( x) * s ( x)
−∞
This is the convolution of f with s. For every value of x, we compute the integral,
eventually obtaining the complete convolution. The final integral in equation (64)
contains the term, s(x-γ). This means that in order to do the convolution, one flips the
original function, s(x), and integrates the product for all values of x. There is a special
brand of convolutions called auto-convolutions. Auto-convolutions are when f(x) = s(x).
The Correlation Theorem follows the exact same line of logic, except we use S*(p),
instead of S(p). By changing this one detail we jump to the final step to find
∞
∫s
*
(γ ) f ( x + γ )dγ = s * ( x) ⊕ f ( x)
(65)
−∞
For every value of x, like before, we compute the integral. This differs from the
convolution in one respect, however. In the correlation, we do not flip the function. We
keep its original orientation. Then we range through all values of x, computing the
integral at every value. Since we do not flip the original function, the correlation gives a
measure of how much alike the two functions are. If the functions are well matched, the
correlation will be very sharp, but if the functions are not alike, we will obtain a broad
correlation.
The two-dimensional correlation is very important in pattern recognition. If the
correlation is very sharp, then what you are comparing is very much alike. To efficiently
82
register the orthogonally polarized images to one another for future processing, the autocorrelation of a registration scene (a scene with varied spatial structure across the field of
view) is computed and displayed in Figure 37. The registration scene was chosen to be a
resolution target, because of its varied spatial structure in multiple dimensions. This
effectively compares the registration scene with itself.
Auto-correlation of Alignment Image
100
200
Y Pixel Location
300
400
500
600
700
800
900
200
400
600
800
1000
1200
X Pixel Location
Figure 37 – Auto-correlation of registration images.
The maximum in the very center of the auto-correlation is indicative of all auto
correlations. The regions of interest in this particular auto-correlation image are the
secondary maxima flanking the primary maximum in the geometric center of the autocorrelation image. These areas correspond to the correlation values associated with the
orthogonally polarized images comparing to each other.
In optical pattern recognition, finding the maximum values associated with the autocorrelation signature is a poor metric for recognition. There are various other metrics that
yield better recognition results37, one of which is to study the sharpness of the change of
correlation values. This is mathematically equivalent to taking the Laplacian of the
correlation map:
83
[
]
(
)
(
∂ 2 f * ( x, y ) ⊕ f ( x, y ) ∂ 2 f * ( x, y ) ⊕ f ( x, y )
F ( x, y ) = ∇ f ( x , y ) ⊕ f ( x , y ) =
+
∂y 2
∂x 2
2
*
)
(66)
The location of the maximum values of the Laplacian of the correlation in regions of
interest defined by the correlation corresponds to likely pattern recognition.
The natural logarithm of the Laplacian of the auto-correlation of the registration
image is shown in Figure 38 to more easily demonstrate the spatial structure of the
Laplacian.
Natural Logarithm of Laplacian of Auto-Correlation of Registration Scene
100
Y Pixel Location
200
300
400
500
600
700
800
900
200
400
600
800
1000
1200
X Pixel Location
Figure 38 – Laplacian of the registration scene’s auto-correlation map.
The location of the maximum values of the Laplacian of the auto-correlation will yield
the desired pixel shifts to register the orthogonally polarized images to one another. One
can envision have two exact copies of the registration scene displayed in Figure 35 lying
perfectly on top of one another. The bottom image stays still, while the top image is
moved by the X and Y pixel shift values. Upon movement of the calculated Y and Y
pixel shifts, orthogonally polarized images are laying upon one another as good as can be
achieved without intermediate linear interpolation such that a spatial location in one
84
image will correspond to the same spatial location of its orthogonally polarized partner.
This registration procedure can only align to half of a pixel in the X and Y directions.
After the registration procedure was complete, the registration scene was moved in
the system’s field of view and another image was taken with unpolarized light incident on
the scene. Due to the required defocus treated in the previous section, the pre-subtracted
image appears blurry, but the post subtracted image is the image of import in this figure.
Figure 39 – Pre and post subtracted images of a resolution target in the RCP channel of the instrument on
the same intensity scale.
Figure 39 demonstrates the feasibility of the registration procedure. Like Figure 35,
this scene was illuminated with unpolarized light, and a system level defocus was
induced to allow image matching for the RCP and LCP channel of the system with
unpolarized light incident on the receiver. This leads to the assumption that a well
registered system should have a dark post-subtracted image, which is evident in Figure
39. Upon inspection of the post-subtracted image, one can notice a small amount of
leakage on the periphery of the field stop. The leakage was attributed to the fact that the
astigmatism present in the instrument was not perfectly balanced by simple defocus. The
leakage was considered tolerable for the purposes of this dissertation.
85
3.3 Polarimetric Calibration
For the purposes of the instrument presented in this dissertation, absolute radiometric
calibration was not required or attempted. However, since the instrument and its
associated algorithms depend heavily on accurate polarization readings, polarimetric
calibration was accomplished through the use of a data reduction matrix technique.
In said technique the goal is to determine a system measurement matrix, W. The
measurement matrix rows are composed of the first row of the Mueller matrix of each
analyzer configuration measured by the system during the calibration procedure. With W
constructed in such a manner a series of flux measurements will be yielded for each
analyzer configuration and incident Stokes vector39,
Pn ,m = Wn ,mS n ,m
(67)
The system’s data reduction matrix is then generated by pseudoinverse of W. The
pseudoinverse, W-1, is used in the following fashion to produce the output Stokes vector
for a set of intensity measurements output by the detector:
S n ,m = Wn−,1m Pn ,m
(68)
In the conventional Stokes vector basis (Equation 1), a Mueller matrix consists of a
square matrix with 16 elements. It is generally represented as,
⎛ m00
⎜
⎜m
M = ⎜ 10
m
⎜ 20
⎜m
⎝ 30
m01
m02
m11
m12
m21
m31
m22
m32
m03 ⎞
⎟
m13 ⎟
m23 ⎟
⎟
m33 ⎟⎠
(69)
Each element in the system path that can have its polarization action on incident radiation
characterized by a linear diattenuator and a retarder Mueller matrix. In general, the
Mueller matrix describing diattenuation of an optical element oriented at an angle, θ,
relative to the x-axis is written as39,
⎛ (Tx + Ty )
⎜
⎜ (Tx − Ty )
1
M D (Tx , Ty , θ ) = R(− θ )⎜
0
2
⎜
⎜ 0
⎝
(T
(T
x
x
− Ty )
+ Ty )
0
0
0
2 TxTy
0
0
⎞
⎟
0 ⎟
R(θ )
0 ⎟
⎟
2 TxTy ⎟⎠
0
(70)
86
where Tx and Ty are the irradiance transmittances in the x and y directions, and R(θ) is the
Mueller rotation matrix about the optical axis in the conventional Stokes basis,
0
0
⎡1
⎢0 cos(2θ ) sin (2θ )
R(θ ) = ⎢
⎢0 − sin (2θ ) cos(2θ )
⎢
0
0
⎣0
0⎤
0⎥⎥
0⎥
⎥
1⎦
(71)
The diattenuator Mueller matrix describes the transmission ratio of irradiance polarized
parallel and perpendicular to θ and the total irradiance.
Often, the diattenuation Mueller matrix is normalized to the first element of
MD(Tx,Ty,0) which corresponds to normalizing the Stokes vector produced after
interaction with the diattenuator to the total irradiance Stokes component, S0.
The retarder Mueller matrix describes a relative phase shift, δ, between orthogonally
polarized vectors incident on the element for which the retarder Mueller matrix is being
defined. It is generally written as39,
⎛1
⎜
⎜0
M ret (δ , θ ) = R(− θ )⎜
0
⎜
⎜0
⎝
0
0
1
0
0 cos δ
0 sin δ
0 ⎞
⎟
0 ⎟
R(θ )
− sin δ ⎟
⎟
cos δ ⎟⎠
(72)
The relative phase shift parameter in the retarder Mueller matrix is defined by the
thickness of the retarder and the relative difference in refractive index between rays
traveling parallel to the ordinary and extraordinary axes of the element perceive upon
traversing the optical element.
δ=
2πd
λ
(no − ne )
(73)
where d is the physical thickness of the element, no is the ordinary refractive index, ne is
the extraordinary refractive index, and λ is the wavelength of the light interacting with the
element.
When the appropriate Mueller matrices are formed for each element in a system, a
total system Mueller matrix is formed for each analyzer configuration. An analyzer
configuration exists for each state of polarization that can be presented to a detector
87
array. An ideal system measurement matrix is formed by taking the top row of each
analyzer configuration’s Mueller matrix and constructing a matrix of these rows,
Wideal
⎛ A1 ⎞ ⎛ m00,1
⎜ ⎟ ⎜
⎜ A ⎟ ⎜ m00, 2
=⎜ 2⎟=⎜
#
#
⎜ ⎟ ⎜
⎜ A ⎟ ⎜m
⎝ K ⎠ ⎝ 00, N
m01,1
m02,1
m01, 2
m02, 2
#
#
m01, N
m02, N
m03,1 ⎞
⎟
m03, 2 ⎟
# ⎟
⎟
m03, N ⎟⎠
(74)
The top row of each analyzer configuration’s Mueller matrix was used because it
corresponds to the total irradiance passed through the final polarization element which is
what conventional focal plane arrays detect. If a detector array induced retardance or
diattenuation, it would have to be considered part of the analyzer Mueller matrix and be
modeled appropriately.
In an actual calibration procedure, a series of known polarization states (generator
states) are sent into the system with a polarization generator for each possible analyzer
configuration and the output flux is detected. The incident polarization states are written
in a matrix, whose columns are formed by their individual Stokes vectors,
K
S inc = (S1
K
S2
⎛ s0,1
⎜
K
⎜ s1,1
" SQ ) = ⎜
s
⎜ 2,1
⎜s
⎝ 3,1
s0 , 2 " s 0 ,Q ⎞
⎟
s1, 2 " s1,Q ⎟
s 2 , 2 " s 2 ,Q ⎟
⎟
s3, 2 " s3,Q ⎟⎠
(75)
where there are a total of Q incident, known generator states in the calibration procedure.
Similarly, the detected flux for each incident state at every analyzer configuration is
written in a similar matrix form,
K
P = (P1
K
P2
⎛ p1,1
⎜
K
⎜ p 2,1
" PQ ) = ⎜
#
⎜
⎜p
⎝ K ,1
p1, 2
"
p 2, 2
"
#
%
"
p K ,2
p1,Q ⎞
⎟
p 2 ,Q ⎟
# ⎟
⎟
p K ,Q ⎟⎠
(76)
There are a total of K analyzer configurations and a total of Q known polarization states
incident on the system in the calibration procedure.
The ideal system measurement matrix written in Equation 67 is just that, ideal. It is
formed with perfect polarization analyzer vectors and does not represent a physically
88
realizable system. As such a calibration coefficient matrix is introduced to compensate
and describe the actual measurement matrix by multiplication with Wideal. The calibration
procedure is written in matrix form:
Pk ,q = C calib Wk ,ideal S q ,inc
(77)
The real system measurement matrix is CcalibWn,ideal and is begotten through
pseudoinverse of Sq,inc. The calibration matrix is similarly formed by an additional
pseudoinverse operation that is detailed in the following equation:
C calib = Pk ,q S −q1,inc Wk−,1ideal
(78)
With a well defined calibration matrix established, the calibrated output for each pixel
on the detector array is determined by yet another pseudoinverse operation. The postcalibrated Stokes vector at each pixel is determined as,
S n ,m = (Ccalib Wn ,m ) Pn ,m
−1
(79)
Ultimately, the accuracy of the calibration procedure is dependant on the ability of the
polarization generator to output well characterized states. As such, the generator must be
characterized independent of the optical system. The HeNe laser source used as the first
element in the polarization generator output random polarization. The laser was
collimated using a high NA microscope objective in conjunction with a spatial filter and
an extended positive lens. Collimation was tested using a shear plate interferometer.
Following the collimating lens was a dichroic film linear polarizer with an extinction
ratio at 633nm of 20,000:1. Finally, a quarter-wave linear retarder was placed in the
collimated laser path. The combination of the linear polarizer and quarter-wave linear
retarder allowed for sufficient polarization state coverage for generator characterization.
A second dichroic film linear polarizer was set in front of a photodetector and rotated
through full revolutions for linear input states from the generator. The data from these
trials fit to Malus’s Law to such a degree that the generator linear states were considered
perfectly polarized in relation to the resolution of the CCD used in the system.
Malus’s Law states that when a perfect polarizer is placed in front of a perfectly
polarized beam, the output irradiance is proportional to cos2(θ), where θ is the angle
between the polarizer axis and the electric field orientation of the incident beam.
89
This procedure was repeated for when the quarter-wave linear retarder was oriented
45 and 135 degrees from the first linear polarizer’s orientation. In this case, virtually no
variation was observed, for both cases, when the second linear polarizer was rotated in
front of the photodetector. This was the theoretically expected behavior. Again, with
respect to the CCD’s radiometric resolution, the generated circularly polarized states
were considered to be perfectly polarized. Intermediate elliptically polarized states were
also considered perfect for purposes of this dissertation due to the extreme cases fitting so
well to expectation values.
With the generator source characterized to perfect, with respect to the CCD’s
resolution, the calibration procedure was completed for both linear and circularly
polarized detection orientations of the system each time any element in the optical train
was adjusted. The calibration was completed sampling many states on the Poincaré
sphere. The Poincaré gives a visual representation of all possible polarization states,
Figure 40 – Poincaré sphere representation of any polarization state.
The three axes shown in the previous figure respectively represent the last three
Stokes parameters normalized by the first: S1/S0, S2/S0, and S3/S0. As one goes around the
equator of the sphere, every linearly polarized state is represented. As one travels on in
90
latitude from the equator to the pole of the sphere, the polarization becomes increasingly
elliptical, to the point of completely circular at the poles. The ellipticity angle is defined
by the linear polarization state defined at the equatorial position closest to the elliptically
polarized state. Depolarization is also represented in the form of sub-spheres defined
within the outer Poincaré sphere to the limit when the radii of a sub-sphere approaches
zero, at which point light is completely depolarized.
Flux from the generator was aimed directly into the objective of the system with
appropriate divergence to cover the field of view for each of the generator polarization
states for each detection scheme of the system. Flux for each detector was registered and
processed by way of the calibration procedure detailed above. An automated procedure
with rotating generator elements would yield a more accurate calibration coefficient
matrix quickly, but such a technique was not used due to the manual calibration
procedure giving quality results as can be observed in the following two figures.
Unaltered Linear Polarization Detection
Y Pixel Position
250
200
100
150
200
100
300
50
400
100
200
300
400
500
X Pixel Position
600
700
Calibrated Linear Polarization Detection
Y Pixel Position
250
200
100
150
200
100
300
50
400
100
200
300
400
500
X Pixel Position
600
700
Figure 41 – Pre and post calibrated images with linear polarized light consistent with an eigenstate of the
linear polarization detection system incident on the objective.
91
Unaltered Circular Polarization Detection
Y Pixel Position
250
200
100
150
200
100
300
50
400
100
200
300
400
500
X Pixel Position
600
700
Caibrated Circular Polarization Detection
Y Pixel Position
250
200
100
150
200
100
300
50
400
100
200
300
400
500
X Pixel Position
600
700
Figure 42 – Pre and post calibrated images of circularly polarized light consistent with an eigenstate of the
circular polarization detection system incident on the objective.
Figures 41 and 42 illustrate the calibration’s effectiveness as well as the system-level
orthogonal polarization leakage present in the pre-calibrated images for both detection
schemes. It should be noted that the leakage is exaggerated because most of the entrance
pupil of the system is illuminated with high enough flux to saturate the detector. In
practice, the flux at the detectors is appreciably lower. The images seen in the precalibrated members of the figures were produced by illuminating the system objective
directly with vertically polarized light in the linear polarization detection scheme and
with right circularly polarized light in the circular polarization detection scheme. The
seemingly perfect calibration performance in the previous two figures is attributed to the
fact that the system was illuminated by an exact detection eigenstate of the system.
Detection of polarizations that stray from these eigenstates shall exhibit lesser postcalibration performance; however the post-calibration performance always exceeds the
pre-calibration performance.
92
The leakage is attributed to a combination of three factors. First, there was a
possibility that the Wollaston prism was not being perfectly aligned in the system relative
to the incident polarization states. A second possibility was that the extraordinary and
ordinary axes of the individual crystal elements of the Wollaston prism were slightly
misaligned. Lastly, the retarder required in the circular polarization scheme could have
had some misalignment as well. Each of these factors could individually produce leakage
in the orthogonal channel. The exact source of the leakage was not investigated, because
it induced very little error in the post-calibrated images and comprised a fraction of the
flux seen in the orthogonal channel. Even without calibration, the subtractive algorithms
used in the system minimize the leakage’s effect relative to actual signal present the
channel where leakage would exist.
Figure 42 also features slightly higher leakage values than those seen in Figure 41.
The inclusion of an additional optical element was determined to be the culprit for this.
The retarder not only introduces possibility for angular mismatches, but adds an element
without AR coating. Any radiation reflecting off of the retarder toward the detector in the
circular polarization scheme will have its polarization forced into the orthogonal state.
The reflected radiation is then detected in the orthogonal polarization channel, whereas
reflection in the linear polarization detection setup will be registered in the same state as
transmitted light, if it is horizontally or vertically polarized. Any component of S2 in the
Stokes vector will be flipped into the negative state upon reflection.
3.4 Post Registration Algorithms
Following calibration and image registration the orthogonal images presented to the focal
plane are processed relative to their interaction with one another when objects of various
depolarization character are viewed within an obscurant medium.
The image subtraction approach finds its basis in the fact that, depending on the
depolarization induced by scatter from an object, detection in polarization states parallel
and perpendicular to the active illumination’s state will yield one detection state with a
higher amount of object structure than its orthogonal partner. Subtraction of these
93
orthogonal states will effectively reduce the amount of obscurant in the channel that
houses the largest amount of object structure.
Pure image subtraction approaches give modest improvements in contrast. Large
contrast enhancements are found in a multi-tiered algorithm approach. First, the image
subtraction is done in iterative steps, at each step increasingly greater fractions of one
polarization image is subtracted from its orthogonally polarized counterpart. However,
the channel used as the subtractive element is blurred using a correlation-based
smoothing filter to attempt to obscure object structure, but to maintain the rough
obscurant structure and amplitude for efficient subtraction. Following the blurred
subtraction at each step, the DC Fourier component is appreciably reduced in the
resultant spatial frequency spectrum. Obscuring media is largely manifested as an equal
increase in flux over the entire field of view at the final image plane. Drastically reducing
the DC Fourier component of the spatial frequency spectrum effectively eliminates this
equal increase in flux. Finally, the spatial frequency spectra of the pre-subtracted image
and that of a specified subtraction iteration step are compared. A Fourier spatial filter is
formed by study of those spatial frequencies that demonstrate large changes upon
subtraction the blurred, orthogonal image subtraction.
Experimental data was taken covering available permutations of optical depth, object
depolarization character, object spatial structure, illumination source polarization state,
and ambient illumination. Figure 43 gives a sample data product using our system when
illuminating a metal wrench with the letters “D AL” in the field of view with a circularly
polarized HeNe laser using the briefly described algorithmic process above.
Summarily, the experimental setup for the images featured in Figure 43 consisted of
the rough metal wrench oriented vertically at the rear of an enclosure into which water
vapor was injected. The wrench stood roughly 1.305 meters from the system’s objective
and the enclosure. The enclosure featured a depth of 30.5 cm.
94
Figure 43 – (a) Unobscured image of a section of a metal wrench with the letters “D AL” clearly
visible.(b) Same field of view obscured through an optical depth of 3.5 of water vapor (fog). (c) Wrench
image through obscurant (fog) with only orthogonal polarization image subtraction used. (d) Wrench image
through obscurant using the novel Fourier filter. (Blue, red and green lines represent the cross-sections seen
in Figure xx7 for the respective images)
Figure 44 illustrates the pixel values corresponding to the cross sections highlighted
in Figure 43.
Figure 44 – Demonstration of novel Fourier filter applied to obscured image data seen in Figure 43. Notice
how the green (filtered) data fits the blue (unobscured) data much better than the red (obscured).
Figures 43 and 44 give a sample of how the algorithms written in support of imaging
though obscurant media at remote distances produce very good results with the 1-D
95
correlation coefficient, r, is used as a metric for cross sectional data matching. An
exhaustive treatment of their performance with respect to various experimental conditions
will be given in following chapters.
3.4.1 Post Registration Algorithms: Image Blurring Filter
To reiterate, the image subtraction approach finds its basis in the fact that, depending on
the depolarization induced by scatter from an object, detection in polarization states
parallel and perpendicular to the active illumination’s state will yield one detection state
with a higher amount of object structure than its orthogonal partner.
When an object maintains the incident active illumination’s degree of polarization
after scatter occurs at the object’s surface, the optimum detection situation occurs. The
channel that detects the orthogonal polarization of the channel corresponding to the
object detection channel will contain only obscurant spatial structure, where the object
detection channel will contain both the object and the obscurant. Subtraction of these two
channels produces no loss of object information due to no common object information
existing in both channels.
Unfortunately, depolarization is inherent in virtually all scattering events consistent
with light impingent on macroscopic object surfaces. As a result, common object
information will exist in orthogonally polarized images. If orthogonal polarization image
subtraction is to be used to its best effect, the object commonality between channels must
be reduced.
Mitigation of the existence of common object information shared between channels is
found in the form of a blurring/defocusing filter applied to the channel consistent with the
least resolved object information. This channel varies as a function of the degree of
depolarization induced by the object’s surface. Following chapters shall discuss this
further.
The blurring filter aims to homogenize the subtraction channel’s image enough to
obscure fine object structure, but to maintain the obscurant’s course spatial structure. The
96
blurring filter takes the form of a two dimensional rectangle function. In one dimension,
the rectangle function is defined as40,
⎧
x − x0
⎪ 0, for b > 0.5
⎪
x − x0
⎛ x − x0 ⎞ ⎪
rect ⎜
= 0.5
⎟ = ⎨0.5, for
b
⎝ b ⎠ ⎪
x − x0
⎪
⎪ 1, for b < 0.5
⎩
(80)
The two dimensional rectangle function is simply the multiplication of rect functions in
each dimension40,
⎛ x − x0 y − y 0 ⎞
⎛ x − x0 ⎞
⎛ y − y0 ⎞
rect ⎜
,
⎟ = rect ⎜
⎟rect ⎜
⎟
d ⎠
⎝ d ⎠
⎝ b
⎝ b ⎠
(81)
Two-Dimensional Rectangle Function
-3
x 10
2
1.5
1
0.5
0
100
90
100
80
90
70
80
70
60
60
50
50
40
40
30
30
20
20
10
Y-axis
10
0
0
X-axis
Figure 45 – 2D rectangle function with b = d = 25.
The blurring filter in the form of a 2D rectangle function is applied to the subtractive
information channel using a two dimensional convolution with the function shown in
Equation 81 divided by the product of the rectangle widths in each dimension. This
maintains the average pixel value over the entire image.
97
At each point in the convolution process, the value of the convolution is the average
value of the pixels around it in a rectangle defined by the individual sizes of the rect
functions used to construct the blurring filter. All blurring filters used in support of this
dissertation were square. The operation for blurring an image is demonstrated,
Imageblur = f subtract (x, y )* *
1
⎛ x⎞
⎛ y⎞
rect ⎜ ⎟rect ⎜ ⎟
2
b
⎝b⎠
⎝b⎠
(82)
Figure 46– Blurring filter applied to a resolution target, and imaged by the polarization imager. (a) Raw
image rendered by the sensor. (b) Image of the target seen in (a) with the blurring filter applied.
A resolution target was imaged with the polarization imager using ambient lighting
conditions in the laboratory to demonstrate the action of the blurring filter in Figure 46.
This image was chosen as a representation of the blurring filter’s action because it
contains some fine structure (group numbers) and some course structure (bar targets and
blocks). As was desired, we maintain a semblance of the rough structure, but the fine
structure has all but disappeared. Of note, in Figure 46, is that the outer ring defined by
the field stop is also blurred. In subtraction images, this will manifest as a slightly
brighter ring on the periphery of the post-subtraction images, because the correlation
process at the periphery used the black area outside of the field of view for those pixels
close to the outer ring.
Figures 47 and 48 comprise four separate images of objects within an obscurant cloud
of water vapor. Each sub-image illustrates a phase in the subtraction algorithm. The
contrast enhancement is evident between the intensity image and the subtracted images.
Further, upon comparison of the subtracted image with blur and that without, a very small
improvement is made between the two in the image with the course spatial structure
98
(Figure 48), but an appreciable contrast enhancement is seen in the image with the fine
spatial structure relative to the one with coarse spatial structure (Figure 47).
Figure 47 – Paper resolution target obscured by thick water vapor imaged with the polarization imager
with only conventional polarization image subtraction and polarization image subtraction with blur.
Figure 48 – Paper black and white cross-hatch target obscured by thick water vapor imaged with the
polarization imager with only conventional polarization image subtraction and polarization image
subtraction with blur.
99
As expected, the blurring filter enhanced the contrast of fine spatial structure within
the field of view more than that of the image with coarse spatial structure with only
conventional image subtraction used as the enhancement tool. It is used at the first step in
the contrast enhancement algorithms featured in this dissertation.
3.4.2 Post Registration Algorithms: Iterative Polarization Image Subtraction
Following the correlation-based overlap algorithm and the application of the blurring
filter, a simple subtractive algorithm is used. This algorithm differs from a full image-toimage subtraction. Because different objects will depolarize incident radiation at varying
degrees, the algorithm takes factors of one image and subtracts it from its orthogonally
polarized partner. This process is repeated a specified number of times, each time
yielding a slightly different image product. In a scene with objects of the same material
and/or depolarization characteristic, there will be a subtraction factor that yields the best
image contrast for that scene.
Equation 83 demonstrates the algorithm used to determine Ldiffi and Rdiffi, the
resultant difference images for proportional subtraction of one image with the blurred
version of the other, using the detection states consistent with the circular polarization
detection state of the polarization imager.
Ldiff i ( x, y ) = CPL( x, y ) − X i ⋅ (CPR ( x, y )* *B( x, y ))
Rdiff i ( x, y ) = CPR ( x, y ) − X i ⋅ (CPL( x, y )* *B( x, y ))
(83)
where B(x,y) is the blurring filter defined in Equation 82, Xi is a multiplicative factor
between 0 and 1, CPL(x,y) is the pre-processed left circularly polarized image, and
CPR(x,y) is the pre-processed right circularly polarized image. The algorithm for image
subtraction in the linear polarization detection state of the imager has the exact form as
Equation 83, but with vertical and horizontal detection states instead of right and left.
As the value of Xi ranges from 0 to 1 larger proportions of the irradiance of the
blurred image is subtracted from the orthogonally polarized object image channel until
the entire blurred image is subtracted off.
100
As mentioned before, there will be a sweet spot for the value of Xi where the largest
image contrast yielded from the subtraction process is optimized. This value for Xi varies
depending on the depolarization characteristic of the objects being imaged. The
depolarization characteristics of an object being imaged also dictates which detection
channel will yield the best imaging contrast. Channel determination is much more
pronounced for objects that maintain the polarization state of incident radiation upon
scatter.
The following figure demonstrates the iterative subtraction process for a depolarizing
object, within a thick obscurant medium, consisting of a black and white cross hatch
printed on copier paper, when illuminating with right circularly polarized light, for Ldiffi
(Equation 83). When Xi equals zero, no subtraction is being employed.
Figure 49 – Post subtracted images for Ldiff with X Є [0, 0.95] for a cross-hatch target obscured by dense
water vapor.
101
It is readily apparent that the optimum proportion of the blurred imaged to be
subtracted off of the orthogonally polarized image lies between X = 0.2 and X = 0.3.
Before these values the obscurant dominates the pixel values causing a large loss in
image contrast. After these proportion constant values, the object information is washed
out in the subtraction process.
A companion figure to the one given above gives Rdiffi for the same object and
optical thickness. It is given in Figure 50. Comparisons between Figures 49 and 50 show
that the optimum proportion factor for subtraction occurs sooner for Figure 49, which is
due to immediate backscattered light to be discussed later.
Figure 50 – Post subtracted images for Rdiff with X Є [0, 0.95] for a cross-hatch target obscured by dense
water vapor.
.
Figures 49 and 50 demonstrate the simple iterative subtraction algorithm that
comprises a step in the contrast enhancement algorithms. They lead to the conclusion that
full image subtraction is far from the optimum subtraction solution. Only a proportion of
102
a blurred orthogonally polarized image need be used as the subtraction element. Also,
that the optimum proportion factor will vary based on detection parameters and object
depolarization character. It is difficult to observe the differences in contrast ratio between
the optimum proportional subtraction images by study of Figures 49 and 50. The only
behavior that is readily apparent is that the optimum proportional factor is biased towards
lower factors for Ldiff and towards higher factors for Rdiff. To illustrate a variation in
contrast ratio performance, subtraction images for a depolarizing object with fine spatial
structure for Ldiff and Rdiff are presented in Figures 51 and 52.
Figure 51 – Rdiff images for a resolution target obscured by thick water vapor. X values range in the same
manner as Figure 49.
103
Figure 52 – Ldiff images for a resolution target obscured by thick water vapor. X values range in the same
manner as Figure 49.
When an object with fine spatial structure is imaged in each channel via the
subtraction process given in Equation 83, differences in optimum contrast ratio can be
readily observed between the image sets in Figure 51 and 52. The Rdiff images
demonstrate higher contrast than the Ldiff images do. Briefly, this is attributed to
immediate backscattered light from the obscurant existing primarily in the left circularly
polarized image channel for the polarization helicity consistent with the source. The
immediate backscattered light also explains why each set of images has its optimum
subtraction proportion constant, Xi, biased towards larger or smaller values. Due to autoscaling completed to show the spatial structure in the two previous figures it is difficult to
ascertain the relative pixel values in the image between the Rdiff and Ldiff images. The
Rdiff images have pixel values less than those seen in Ldiff. This is also a consequence of
immediate backscatter into the system objective. Because the left circularly polarized
channel contains a higher proportion of obscurant scattered light, it will have a reduced
imaging capacity and produce less contrast enhancement when the algorithms are
completed using it as the object channel. This observation is supported by the fact that the
104
images of Rdiff exhibit higher contrast than those of Ldiff. A similar behavior is exhibited
by linearly polarized difference images. In either case, it is readily apparent that
orthogonally polarized image subtraction does buy enhancements in image contrast.
3.4.3 Post Registration Algorithms: DC Block Fourier Spatial Filter
The blurring algorithm coupled with the iterative orthogonally polarized image
subtraction yield contrast enhancements when imaging through obscurant media relative
to normal intensity imaging. The bulk of the contrast enhancements presented in this
dissertation are facilitated by a novel DC block Fourier Spatial filter.
The first step in the formation of the filter consists of taking the post subtracted image
products, like those seen in the previous section, and drastically reducing the DC
component of the subtracted image’s Fourier spectrum. A comparison is then made
between the pre-subtracted polarization image consistent with the channel being used as
the object channel and subsequent post-subtracted polarization images in spatial
frequency space. Difference inspection then chooses specific spatial frequencies that
most likely correspond to object spatial structure in the scene and enhances them relative
to the unselected spatial frequencies. The filter is then applied by direct multiplication in
spatial frequency space to the image’s spatial frequency spectrum. The novel Fourier
filter is applied in three distinct steps:
1. The DC component of the spatial frequency spectrum of the post-subtracted
image is drastically reduced or eliminated.
imageCPLi ( x, y ) = Ldiff i ( x, y ) − ℑ[Ldiff i ( x, y )] ξ ,η =0
(84)
2. Define filter through spatial frequency difference inspection relative to a threshold
value. Filter values are function of the difference’s deviation from threshold.
Qi (ξ ,η ) = ℑ[imageCPLi ]
Fdiff ,i (ξ ,η ) = f (abs(Qi − Q1 ) ≥ thresh )
(85)
3. Apply filter to image’s spatial frequency spectrum and transform back into
position space.
105
[
]
imagei ( x, y ) = ℑ−1 Qi (ξ ,η )Fdiff (ξ ,η )
(86)
The mathematical process detailed above is for a post-subtraction image with the object
channel corresponding to the left circularly polarized detection area of the focal plane
array. The function, f, seen in Eq. 85, simply doubles the Fourier space value of spatial
frequencies above thresh, and does not change those below thresh. The threshold value
was defined at a quarter of the maximum Fourier space value of Qi. The choice of f and
thresh were made through an iterative process of performance maximization. This
process was not involved, and is an area where it is anticipated that future research could
provide quite large performance increases.
Unfortunately, the flux due to the scattering media is present in the spatial frequencies
corresponding to object structure. Even after the image subtraction process detailed in the
previous section, objects that depolarize incident radiation will exist in both detection
channels, therefore difference inspection will not perfectly select only object spatial
frequencies.
Scattered flux from the obscurant medium is manifested in its greatest capacity in the
exact lowest spatial frequency, which is the DC component of the spectrum. Greatly
reducing the DC component vastly increases image contrast in and of itself. Enhancing
the spatial frequencies corresponding to object spatial structure adds more contrast with
the added effect of smoothing out very fine spatial structure corresponding to noise in the
detector, speckle if one is using a coherent source, and very fine compressions and
rarefactions of the obscurant media in the field of view. This is attributed to the fact that
the majority of objects that define the extent of the spatial filter in frequency space will
create a modified low-pass spatial filter.
To demonstrate the performance of the DC block Fourier spatial filter, the images in
Figure 51 are compared, at chosen subtraction iterations, to the same images, but with the
filter applied. The “Filter off” columns correspond exactly to the same X value in Figure
51, but with a different color map. The “Filter on” columns correspond to the exact X
value image in Figure 51, but with the filter applied.
106
Figure 53 – Post processed images of a resolution target with application of the DC block Fourier spatial
filter on and off for X = [0, 0.90] in 0.10 steps. The second and fourth columns correspond to images with
the filter applied, while the first and third columns correspond to images without the filter applied.
The filter off, X = 0, sub-image equates to the pre-subtracted, resolution target
obscured by water vapor, where the filter on, X = 0, sub-image is the pre-subtracted
image with only its DC value reduced. The real power of DC reduction is illustrated in
comparison of the two X = 0 sub-images.
Another feature of the DC block Fourier filter is that coarse spatial features that the
subtraction process might wash out, due to higher probabilities of their existence in both
orthogonally polarized detection channels, are differentiated from the obscurant flux by
virtue of the vast reduction in the DC spatial frequency component in the resultant
Fourier spectrum. Inspection of Figure 50 relative to Figure 51 shows how the coarse
spatial structure loses some magnitude of contrast enhancement versus finer structure.
The following figure illustrates the novel filter’s ability to provide quality contrast
enhancement for coarse spatial structure as well.
107
Figure 54 – Post processed images of a depolarizing cross-hatch target with application of the DC block
Fourier spatial filter on and off for X = [0, 0.90] in 0.10 steps. The second and fourth columns correspond
to images with the filter applied, while the first and third columns correspond to images without the filter
applied.
The “Filter off” columns in Figure 54 are analogous to the images corresponding to the
same X values in Figure 50.
With the exception of the object used in Figure 43, all imaged objects shown in
Chapter 3.4 were printed on copier paper that exhibited a large amount of depolarization
(Chapter 2.2.2). Chapter 3.4 served to give a conceptual basis for the algorithms used to
increase contrast when imaging through remote scattering media.
108
CHAPTER 4: LAB DATA COLLECTION SETUP, RESULTS AND ANALYSIS
The ultimate goal of the Wollaston prism-based system and its associated algorithms was
to provide a proof-of-concept prototype system capable of imaging actively illuminated
objects through optically thick obscurant media and studying the contrast enhancements
yielded after post-processing.
In support of this goal, a lab-scale experimental setup that provided the means to
measure the optical depth of water vapor injected into an extended containment unit in
real time. It also provided the means to actively image targets placed at a defined object
plane at various optical depth levels.
After raw image data was acquired, the raw data was processed using the algorithms
presented in Chapter 3, and their results were analyzed using simulation data provided by
the Monte Carlo simulation steps detailed in Chapter 2.2.
4.1 Laboratory Experimental Setup and Procedure
A lab-scale, stand-off distance, experimental setup was generated to provide proof-ofconcept data using the presented technique.
Included in the setup is an additional
collimated source and radiometer on the back side of the obscurant container. These
additions allowed for optical depth tracking while images were being registered by the
system. The radiometer was placed as far away from the enclosure as possible in an
attempt to minimize flux at the detector caused from scattering events. Since the
radiometer’s detector had a physical extent, flux exterior to the radiometer source from
scattering events inside of the detection cone of the photodetector necessarily contributed
to the radiometer’s reading. This “noise” manifested as an underestimation of the optical
depth of the medium.
The enclosure had a physical width of 1 foot, and the imaging system stood
approximately 1 meter from the front face of the enclosure. The experimental setup is
shown in Figure 55.
109
All system components were designed for performance in the visible region of the
electro-magnetic spectrum. The source that illuminated the object plane was a collimated
HeNe laser (0.6328 μm). The scattering medium used in the experiments was water
vapor (fog) produced by a commercially available ultrasonic nebulizer. The nebulizer
generated water droplets by vibrating a reservoir of standing water at ultrasonic
frequencies.
The manufacturers of the specific nebulizer we used provided an
approximate particle size distribution. The particle sizes ranged from 1 μm to 10 μm.
However, as treated in Chapter 2.2.3, the specific particle size distribution for the
nebulizer model in question was determined. The particle distributions equated to a size
parameter, (2π/λ*particle radius) distribution that ranged between 10 and 100.
Figure 55 – Experimental setup to provide preliminary data using the presented technique. Not shown is an
angular offset of the enclosure relative to the source. This offset reduces reflected flux from the enclosure’s
front face detected by the system. (Not to scale)
In order to gain an insight on how depolarization of incident radiation induced by
object scatter affects image contrast enhancement, objects of vastly different
depolarization character were used as targets in the obscurant cloud: a metal wrench for
an object that maintains the degree of polarization of incident radiation, and a resolution
target printed on common copier paper. In addition, a black and white cross hatch printed
110
on common copier paper was used as a target to study effects of course spatial structure
on image contrast enhancements when using the algorithms presented previously.
Of importance when studying the contrast enhancements yielded from the techniques
presented in previous chapters is to have a base image to which post-processed images
can be compared. The base images were formed simply by acquiring an image of targets
at the object plane before any obscurant was injected into the enclosure.
Figure 56 – Base image set to which post processed images are compared. (a) Section of wrench being
eventually imaged through the fog in our enclosure. (b) Image taken with our instrument, zoomed in on the
“D AL” section of the wrench. (c) Black/white cross-hatch printed on copier paper for use as a target. (d)
Image taken with the instrument, zoomed in on the central intersection cross-hatch. (e) Resolution target
printed on copier paper for use as a target. (f) Image taken with the instrument, zoomed in on the high
resolution elements of the “0” group. Each of the images, [(b), (d), (f)], used collimated HeNe illumination.
The procedure through which raw images were acquired first involved getting a base
reading of the radiometer with no scattering media present in the enclosure. The nebulizer
then injected the obscurant into the enclosure until the radiometer reading reached a
111
steady state low value. At this point, the amount of water vapor escaping the enclosure
equaled the amount of water vapor entering the enclosure. Following establishment of the
steady state value, the nebulizer was turned off and raw image data was acquired in five
second intervals. Radiometer values were registered each time a raw image taken. When
the nebulizer was turned off, the water vapor slowly dissipated out of the enclosure.
Taking data in five second intervals sampled optical depths with sufficient resolution to
form an efficient data set for post processing and analysis.
This procedure was repeated for each target in Figure 56 for circularly polarized and
linearly polarized illumination. The true optical depth from the radiometer readings were
found by using running the Monte Carlo simulation with the radiometer detection
geometry. The extended radiometer readings were fit to the simulation’s output for true
optical depth as a function of the apparent optical depth in the radiometer detection
scheme. Only ballistic photons contribute to the true optical depth of the medium.
Ballistic photons are those photons that traverse the medium without being scattered.
True Optical Depth from Photodetector Reading 7
y = 1.6385x ‐ 0.7182
True Optical Depth 6
2
R = 0.9954
5
4
True Optical Depth
3
Linear (True Optical
Depth)
2
1
0
1
2
3
4
5
Measured Optical Depth
Figure 57 – Results from the Monte Carlo simulation ran for the radiometer detection geometry.
The results from the Monte Carlo simulation produced a function into which the optical
depth measured by the extended radiometer could be entered to find the true optical depth
112
of the medium that corresponds to ballistic photons that reach the detector. The function
had a strong linear fit coefficient (R2 = 0.9954). It was found to be,
δ TRUE = 1.6385δ measured − 0.7182
This linear relationship between measured and true optical depths for the radiometer fails
when the measured optical depth is a very low value. As the measured optical depth
approaches very low values, δ TRUE ≈ δ measured for the extended radiometer.
4.2 Image Enhancement Metrics
Once the raw image data has been registered and the post processing algorithms have
been applied, the resultant data products must have established metrics by which to
measure their performance relative to the raw image data. Also, it is convenient to have a
metric available to compare the post processed images with the unobscured image.
Obviously, these metrics require prior knowledge of the objects being imaged to establish
their image’s spatial distribution.
Two imaging metrics were used to measure the performance of the post processed
image products. The first metric measured the image contrast in regions of interest
defined specifically for each target. These regions of interest are defined in areas of the
target where a single spatial frequency dominates. This produces contrast as a function of
spatial frequency by treating the shadowed “dark area” of an image as a location that
should have minimum reflected radiance and the area immediately adjacent with high
reflected radiance as a location that should have maximum reflected radiance. The
contrast is calculated by subtracting the average pixel value in this dark area from the
average pixel value in the bright area divided by the sum of the average pixel value in
these respective areas.
C (ξ ,η ) =
I max,(ξ ,η ) (Δxbright , Δybright ) − I min,(ξ ,η ) (Δxdark , Δydark )
I max,(ξ ,η ) (Δxbright , Δybright ) + I min,(ξ ,η ) (Δxdark , Δydark )
(87)
113
where ξ and η are spatial frequencies in the x and y directions where the regions of
interest are defined, Δx and Δy are extended areas in bright and dark zones within the
regions of interest over which averages are taken to create Imax and Imin.
Averaging is required over the regions of interest, because a coherent source was used
as the illumination. Laser speckle manifested in the image plane and was consequently
imaged by the system. The speckle, evident in Figure 56(b,d), generated a non-uniform
spatial distribution for all image spatial frequencies in addition to polluting accurate
contrast measurements for intensity images at high optical depths.
Contrast is an apt performance metric when studying obscured images, because the
obscurant flux does not differentiate between object spatial frequencies. For multiple
scattering processes, much of the obscurant flux at the image plane is due to
backscattered flux that didn’t interact with the object itself. That obscurant flux at the
image plane that did interact with the object has had its object-space positional and
angular orientations altered in such a fashion that there is virtually no correlation between
object-space points and image-space points for flux that has undergone an additional
scattering event after interaction with the object. The lack of correlation lends itself to an
approximately uniform flux increase across the image plane that is attributed to the
scattered flux. A uniform flux increase will relate exactly to a loss in contrast. Therefore,
measuring the change in image contrast for known objects within obscurant media is very
much like measuring the effect of the obscurant itself.
In addition to the flux increase across the imaged field of view, the scattering process
yields less ballistic photons from the object itself, because as the obscurant medium
becomes more dense, less ballistic photons will exist due to a higher probability of
scattering. For small fields of view, where the transit distance to the receiver is
approximately the same for all object points, the object flux will be uniformly affected
across the field of view. When transit distances start to deviate appreciably, exterior
portions of the field of view will appear darker, because the probability of scattering
increases due to the effective optical depth increasing by a 1/cosθ factor, where θ is the
114
angle measured from the optical axis of the receiver to an object point of interest. This
can be conceptualized as an equivalent increase in optical depth for off-axis field points,
⎛ δ ⎞
exp(− δ ) = exp⎜ − o ⎟
⎝ cos θ ⎠
(88)
The contrast is unaffected by this phenomena, however, because it reduces “bright”
and “dark” areas by the same factor at adjacent field points. Therefore, contrast still
remains a quality metric by which to measure contrast losses or enhancements in the
presence of dense scattering media.
The second metric by which to measure the magnitude of improvements made by the
presented system relative to normal intensity imaging finds the degree to which an
obscured imaged (post or pre-processed) compares to the unobscured image. Again,
using a two-dimensional correlation technique delivers such a value quickly. A two
dimensional Pearson Product Moment Correlation is calculated between the image of the
object with no obscurant present (Figure 56(b,d,f)) and the obscured image of interest, be
it processed by the presented algorithms or raw. For a two-dimensional data set, the
Pearson correlation is found by,
r=
∑∑ (A
mn
m
)(
− A Bmn − B
)
n
(
)
(
)
(89)
2⎛
2 ⎞⎞
⎛
⎜⎜ ∑∑ Amn − A ⎜ ∑∑ Bmn − B ⎟ ⎟⎟
⎝ m n
⎠⎠
⎝ m n
Pearson’s correlation coefficient measures the linear relationship between two variables,
and ranges between -1 and 1. A value of -1 depicts a perfect negative correlation between
two variables; likewise a value of 1 depicts a perfect positive correlation between two
variables. A value of 0 represents no relationship between two variables. In statistical
terms, the Pearson’s correlation coefficient is the covariance of Amn and Bmn divided by
B
the product of their standard deviations.
For the purposes of this dissertation, Amn is the unobscured image pixel values, and
Bmn is the obscured image pixel values being correlated to A. If the normalization factor is
B
ignored, each value in the Pearson correlation coefficient double sum addresses a pixel
value in the arrays being compared. The pixel values in each array are linearly related to
115
their respective averages and the product of these differences are logged as the value to
be used in the double sum.
The presence of the scattering medium reduces the average pixel value across the
field of view as a function of the optical depth to a point where the scattering medium is
the only discernable object of interest within the field of view for very large optical
depths. Therefore, to find an accurate representation of the comparison of the obscured
image array and the image array of the object without the presence of scattering media,
the respective images are normalized to where their average values are equal to one
another.
∑∑ (A
rnorm =
mn , norm
m
)(
− C Bmn ,norm − C
)
n
⎛
⎞⎞
⎛
⎜⎜ ∑∑ ( Amn,norm − C)2 ⎜ ∑∑ (Bmn ,norm − C)2 ⎟ ⎟⎟
⎠⎠
⎝ m n
⎝ m n
(90)
where C is the common average of the pixel values for the newly normalized pixel
arrays, Amn,norm and Bmn,norm. With this normalization completed, each value of the double
B
sum in the numerator of Equation 90 represents the product of the deviations of each
pixel from a common array average. Normalization of each array average to a common
value produces a more realistic array-wide comparison between images that accounts for
flux reductions induced by the presence of the scattering medium.
It should be noted that due to the normalization a completely obscured image will
yield a correlation coefficient of 0.500. Therefore, that is used as the base value for which
no correlation exists between the pre-obscured object and the post-processed images. If
correlation values start to dip beneath 0.500, there begins to exist a degree of negative
correlation for parts of the compared images.
Pearson’s correlation coefficient gives a scalar-valued representation of how alike
two complete datasets are. Whereas localized image contrast measurements take a portion
of a single image array and measures a value consistent with Equation 87. These metrics
were deemed sufficient to measure the performance of the system presented in this
dissertation.
116
The contrast measurements give a value for how much contrast enhancement can be
achieved relative to an obscured version of the image. Using the contrast measurement
alone as an image enhancement metric is insufficient, because you gain no information
on exactly how the algorithms recover the spatial structure of the unobscured object. The
Pearson’s correlation coefficient addresses the issue left unresolved by use of contrast
measurements alone, and compares the original target image to the obscured target
image.
4.3 Experimental Results: Rough Metal Wrench Images (Polarization
Maintaining Object)
When imaging the metal wrench, two regions of interest were defined to produce a
contrast measurement. Referring to Figure 56b, the first region of interest resides left of
the lettering and contains the extended dark and bright regions to be used for minimum
and maximum values in the contrast measurement. The second region of interest at which
contrast is measured was found by treating the shadowed “dark area” of the wrench
image lettering as a location that should have minimum reflected radiance and the area
between the “D” and “A” as a location that should have maximum reflected radiance.
The contrast is calculated by subtracting the average pixel value in this dark area from the
average pixel value in the bright area divided by the sum of the average pixel value in
these respective areas as shown in Equation 87.
The value of the contrast ratio for post-subtracted images can be a bit misleading in
this application, however. For raw images that feature very low contrast to begin with, it
is possible for the contrast value to be very close to unity, but have little in the way of
discernable object structure in the image. This is due to the “dark area” of the image
being subtracted to zero, and the “bright area” of the image being largely zero, with the
exception of a few bright pixels that will give very large contrasts, but little in the way of
image quality. This is why the contrast ratio must be leveraged against the Pearson
correlation coefficient to give a quantitative value of the contrast ratio increase for an
image that fits the original object well.
117
4.3.1 Experimental Results: Rough Metal Wrench Images using Circularly
Polarized Illumination
This section will give a repetitive presentation of the experimental data associated with
illuminating the rough metal wrench with collimated circularly polarized 632.8nm laser
illumination through a subset of optical depths consistent with sufficient image
obscuration to warrant processing the image to gain enhancements in contrast. Refer to
sections 4.3.3 and 4.5 (pages 123 and 142) for the summary/analysis of the results.
The image to which obscured and post processed images are compared is given in
Figure 56b. The contrast for the low spatial frequency structure in the Figure 56b was
calculated to be Crob = 0.8931 and corresponded to approximately 50 pixels on the focal
plane, where the high spatial frequency structure’s contrast spanned approximately 11
pixels and featured a pre-obscured contrast of 0.2623. The low spatial frequency feature
corresponded to 0.464 lp/mm, and the high spatial frequency feature corresponded to
2.110 lp/mm in object space by relating the system’s magnification to the spatial
frequencies on the focal plane. By comparison to Figure 56b, the following four figures
give data products for an optical depth of 5.11.
Figure 58 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 5.11 with collimated circularly polarized illumination.
118
Figure 59 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 5.11.
Correlation Coefficient vs Subtraction Proportion Constant, δ = 5.11
1
0.95
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 60 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 59
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
58.
119
(A)
(B)
1.2
1
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
Contrast Ratio
0.8
0.6
0.4
0.2
0.15
0.2
0.1
0
0.05
-0.2
0
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 61 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
5.11 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 60.
Note the scale change between (A) and (B).
The obscured intensity image, the post-processed image products for various values
of the subtraction proportion constant, the correlation coefficient comparing postprocessed data and the intensity image to the pre-obscured object, and the values of
contrast for the defined region were given in the previous four figures. These data
products will be given presently for selected optical depths for raw images of the rough
metal wrench. After the data products for the selected optical depths are repetitively
presented in the subsequent pages for both linearly polarized and circularly polarized
illumination, a discussion of their results will follow.
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 4.57 using a circularly polarized source are given in
the following four figures:
120
Figure 62 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 4.57 with collimated circularly polarized illumination.
Figure 63 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 4.57.
121
Correlation Coefficient vs Subtraction Proportion Constant, δ = 4.57
1
0.95
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 64 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 63
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
62.
(A)
(B)
1.2
1
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
Contrast Ratio
0.8
0.6
0.4
0.2
0.15
0.2
0.1
0
0.05
-0.2
0
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 65 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
4.57 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 64.
Note the scale change between (A) and (B).
122
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 4.35 using a circularly polarized source are given in
the following four figures:
Figure 66 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 4.35 with collimated circularly polarized illumination.
Figure 67 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 4.35.
123
Correlation Coefficient vs Subtraction Proportion Constant, δ = 4.35
1
0.95
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 68 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 67
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
66.
(A)
(B)
1.2
1
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
Contrast Ratio
0.8
0.6
0.4
0.2
0.15
0.2
0.1
0
0.05
-0.2
0
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 69 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
4.35 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 68.
Note the scale change between (A) and (B).
124
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 3.67 using a circularly polarized source are given in
the following four figures:
Figure 70 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.67 with collimated circularly polarized illumination.
Figure 71 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.67.
125
Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.67
1
Correlation Coefficient
0.95
Filter On
Sub Only
Unobscured Object
Intensity
0.9
0.85
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 72 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 71
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
70.
(A)
(B)
1
0.9
0.8
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
0.7
Contrast Ratio
Filter On
Sub Only
Unobscured Object
Intensity
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.2
0.05
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 73 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
3.67 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 72.
Note the scale change between (A) and (B).
126
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 3.17 using a circularly polarized source are given in
the following four figures:
Figure 74 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.17 with collimated circularly polarized illumination.
Figure 75 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.17.
127
Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.17
1
Correlation Coefficient
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
Filter On
Sub Only
Unobscured Object
Intensity
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 76 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 75
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
74.
(A)
(B)
1
0.9
0.8
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
0.7
Contrast Ratio
Filter On
Sub Only
Unobscured Object
Intensity
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.2
0.05
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 77 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
3.17 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 76.
Note the scale change between (A) and (B).
128
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 2.70 using a circularly polarized source are given in
the following four figures:
Figure 78 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 2.70 with collimated circularly polarized illumination.
Figure 79 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.70.
129
Correlation Coefficient vs Subtraction Proportion Constant, δ = 2.70
1
Correlation Coefficient
0.95
0.9
0.85
0.8
0.75
0.7
0
Filter On
Sub Only
Unobscured Object
Intensity
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 80 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 79
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
78.
(B)
(A)
1
0.3
0.9
0.25
0.7
Contrast Ratio
Contrast Ratio
0.8
0.6
0.5
0.4
0.3
0.1
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.15
0.1
Filter On
Sub Only
Unobscured Object
Intensity
0.2
Filter On
Sub Only
Unobscured Object
Intensity
1
0.05
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 81 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
2.70 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 80.
Note the scale change between (A) and (B).
130
Much more raw image data of the metal wrench obscured by water vapor at varying
optical thicknesses was taken, but optical depths lower than those given in this subsection
were deemed unnecessary to present due to the ability of the system to render excellent
image contrast and correlation to the pre-obscured object for higher optical depths.
4.3.2 Experimental Results: Rough Metal Wrench Images using Linearly
Polarized Illumination
Like the previous section, this section gives repetitive presentation of the experimental
data associated with illuminating the rough metal wrench with collimated linearly
polarized 632.8nm laser illumination through a subset of optical depths consistent with
sufficient image obscuration to warrant processing the image to gain enhancements in
contrast.
The same object was used without any movement of the target area or of the
enclosure. Therefore, the image contrast for the “high” and “low” spatial frequency
structures in the scene of the rough metal wrench are the same when illuminating with the
linear state.
To reiterate, the contrast for the low spatial frequency structure in the Figure 56b was
calculated to be Crob = 0.8931 and corresponded to approximately 50 pixels on the focal
plane, where the high spatial frequency structure’s contrast spanned approximately 11
pixels and featured a pre-obscured contrast of 0.2623. The low spatial frequency feature
corresponded to 0.464 lp/mm, and the high spatial frequency feature corresponded to
2.110 lp/mm in object space by relating the optical system’s magnification to the spatial
frequencies on the focal plane.
Like the previous subsection detailing the experimental results with circularly
polarized illumination on a rough metal wrench, this subsection will give the same
repetitive presentation of the intensity image, a grid of post-processed images, the
correlation between the pre-obscured object and post processed images, and the contrast
ratios for linearly polarized illumination. Again, a discussion of the results of this and the
previous subsection will be given immediately following the linearly polarized data.
131
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 4.82, using a linearly polarized source, are given in the
subsequent four figures:
Figure 82 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 4.82 with collimated linearly polarized illumination.
Figure 83 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 4.82.
132
Correlation Coefficient vs Subtraction Proportion Constant, δ = 4.82
1
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.8
0.7
0.6
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 84 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 83
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
82.
(A)
(B)
0.9
0.8
0.7
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
0.25
Contrast Ratio
0.6
Contrast Ratio
Filter On
Sub Only
Unobscured Object
Intensity
0.5
0.4
0.3
0.2
0.2
0.15
0.1
0.1
0.05
0
-0.1
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 85 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
4.82 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 84.
Note the scale change between (A) and (B).
133
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 4.57, using a linearly polarized source, are given in the
subsequent four figures:
Figure 86 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 4.57 with collimated linearly polarized illumination.
Figure 87 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 4.57.
134
Correlation Coefficient vs Subtraction Proportion Constant, δ = 4.57
1
0.95
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 88 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 87
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
86.
(A)
(B)
0.9
0.8
0.7
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
Filter On
Sub Only
Unobscured Object
Intensity
Contrast Ratio
Contrast Ratio
0.25
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.2
0.05
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 89 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
4.57 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 88.
Note the scale change between (A) and (B).
135
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 4.35, using a linearly polarized source, are given in the
subsequent four figures:
Figure 90 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 4.35 with collimated linearly polarized illumination.
Figure 91 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 4.35.
136
Correlation Coefficient vs Subtraction Proportion Constant, δ = 4.35
1
0.95
Correlation Coefficient
0.9
Filter On
Sub Only
Unobscured Object
Intensity
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 92 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 91
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
90.
(A)
(B)
0.9
0.8
0.7
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
Filter On
Sub Only
Unobscured Object
Intensity
Contrast Ratio
Contrast Ratio
0.25
0.6
0.5
0.4
0.3
0.2
0.15
0.1
0.2
0.05
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 93 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
4.35 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 92.
Note the scale change between (A) and (B).
137
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 3.67, using a linearly polarized source, are given in the
subsequent four figures:
Figure 94 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.67 with collimated linearly polarized illumination.
Figure 95 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.67.
138
Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.67
1
Correlation Coefficient
0.95
Filter On
Sub Only
Unobscured Object
Intensity
0.9
0.85
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 96 – Correlation coefficient as a function of X relating the post-processed images seen in Figure 95
to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X
are also given for the images with only image subtraction used and for the intensity image seen in Figure
94.
(A)
(B)
0.9
0.8
0.35
Filter On
Sub Only
Unobscured Object
Intensity
0.3
Filter On
Sub Only
Unobscured Object
Intensity
0.7
Contrast Ratio
Contrast Ratio
0.25
0.6
0.5
0.4
0.15
0.1
0.3
0.05
0.2
0.1
0
0.2
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0
0
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 97 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
3.67 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure 96.
Note the scale change between (A) and (B).
139
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 3.17, using a linearly polarized source, are given in the
subsequent four figures:
Figure 98 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.17 with collimated linearly polarized illumination.
Figure 99 – Post processed images of a metal wrench with application of the DC block Fourier spatial filter
enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.17.
140
Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.17
1
Correlation Coefficient
0.95
Filter On
Sub Only
Unobscured Object
Intensity
0.9
0.85
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 100 – Correlation coefficient as a function of X relating the post-processed images seen in Figure
99 to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of
X are also given for the images with only image subtraction used and for the intensity image seen in Figure
98.
(B)
(A)
0.9
0.3
0.8
0.25
Contrast Ratio
Contrast Ratio
0.7
0.6
0.5
0.4
0.2
0.15
0.3
0.2
0.1
0
0.1
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.05
0
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 101 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
3.17 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure
100. Note the scale change between (A) and (B).
141
The results for the image of the rough metal wrench obscured by a water vapor
obscurant with an optical depth of 2.70, using a linearly polarized source, are given in the
subsequent four figures:
Figure 102 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 2.70 with collimated linearly polarized illumination.
Figure 103 – Post processed images of a metal wrench with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.70.
142
Correlation Coefficient vs Subtraction Proportion Constant, δ = 2.70
1
Correlation Coefficient
0.95
Filter On
Sub Only
Unobscured Object
Intensity
0.9
0.85
0.8
0.75
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 104 – Correlation coefficient as a function of X relating the post-processed images seen in Figure
103 to the object with no obscurant present seen in Figure 56b. The correlation coefficients as a function of
X are also given for the images with only image subtraction used and for the intensity image seen in Figure
102.
(A)
(B)
1
0.3
0.9
0.25
Contrast Ratio
Contrast Ratio
0.8
0.7
0.6
0.5
0.2
0.15
0.4
0.3
0.2
0
0.1
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.05
0
Filter On
Sub Only
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
1
Subtraction Proportion Constant
Figure 105 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
2.70 for: (A) the low spatial frequency feature (0.464 lp/mm) (B) the high spatial frequency feature (2.11
lp/mm). The data members of this figure are the same as those described in the caption given for Figure
104. Note the scale change between (A) and (B).
143
4.3.3 Experimental Results: Rough Metal Wrench Image Analysis
Perhaps the most striking feature, besides the large increases in contrast and correlation
relative to intensity imaging and pure image subtraction, is that for every optical depth
circularly polarized illumination and detection performs superior to linearly polarized
illumination and detection for the rough metal wrench, with the exception of contrast
ratio for the high spatial frequency structure at the lowest presented optical depth.
A linear increase in performance for correlation and contrast ratio is observed for the
linear polarization case. In the circular polarization case, there exists a maximum value
for the correlation coefficient that does not correspond to full valued image subtraction
(X=1). Also, the contrast ratio performances for the circular polarization case generally
demonstrate a non-linear increase.
These phenomena can be explained by studying the polarization state of the flux from
the obscurant itself for circularly and linearly polarized light detected by the receiver. The
Monte Carlo simulation was ran to determine the flux at the focal plane due to the
radiation scattered from the obscurant for each illumination state.
Ratio of Fog Scatter in Detection Channels (Reflective Object)
180
S1ob/S1diff
S3ob/S3diff
160
140
Ratio
120
100
80
60
40
20
0
2.5
3
3.5
4
4.5
5
5.5
6
Optical Depth (# SMFP)
Figure 106 – Simulated flux ratio at the focal plane for circularly polarized illumination (Blue) and linearly
polarized illumination (pink).
144
Figure 106 demonstrates an important behavior when imaging in reflection using
circularly polarized illumination or linearly polarized illumination. When imaging in
reflection using circularly polarized illumination, the ratio of the scattered flux between
the object detection channel and the channel used as the subtraction channel will have a
value near one. When the ratio is unity, the scattered flux is split evenly between the
channels, and image subtraction algorithms will yield favorable results. Conversely,
when this ratio has very large values, as those seen in the linear detection case, most of
the flux from scatter exists in the object detection channel. When such large proportions
of the scattered flux exist in the object detection channel for an object that does not
depolarize, image subtraction alone will not appreciably increase image contrast, or
algorithmic values that have a dependence on image subtraction proportions (X) will be
flat across the proportional factors.
The large ratios featured in linearly polarized illumination are physically attributed to
reflected vertically or horizontally polarized radiation not having its state flipped, as is
observed in reflected circularly polarized illumination. Figure 107 illustrates the
experimental agreement with the simulation results found in Figure 106.
Figure 107 – Raw images of a rough metal wrench obscured by water vapor of an optical depth of 4.35 for:
(a) linearly polarized illumination and (b) circularly polarized illumination.
145
The absolute values of the pixels in Figure 107 are secondary to the qualitative
agreement with the simulation’s results. Observation of Figure 107a shows a marked
reduction in the flux in the subtraction channel relative to the object channel for linearly
polarized illumination. As was predicted by the model, circularly polarized illumination
yielded flux values in the subtraction channel that are on the order of those in the object
detection channel (Figure 107b), giving a much stronger variation of the post-processed
images with the subtractions factor, X.
Even though the Monte Carlo simulation was written to given quantitative values for
the polarization state of the flux through the receiver for the targets in question, the
speckle pattern associated with the source and small spatial variation of illumination flux
across the targets are difficult to quantify to produce an exact theoretical solution. Still,
an approximate theoretical solution for an optical depth of 3.53 for the image of the rough
metal wrench illuminated by circularly polarized light was produced.
Figure 108 provides the experimental and theoretical intensity images of the wrench
target for an optical depth of 3.53. The theoretical intensity image was generated by
running the Monte Carlo simulation using an estimate of the spatially varying reflectance
of the wrench and the numerical aperture of the physical instrument to determine
acceptance and rejection of photons impingent on the objective of the system.
Figure 108 – Experimentally produced intensity image of the rough metal wrench obscured by an optical
depth of 3.53 (Left). The theoretically produced intensity image of the rough metal wrench obscured by the
same optical depth is given on the right. Both were illuminated with circularly polarized light.
146
As expected the speckle pattern in the experimentally produced image gives it a much
more grainy appearance than that of the theoretically produced image. The image
enhancement algorithms were ran on both of these “raw” images. The correlation
coefficient is given for the base intensity images in comparison to the wrench object
given in Figure 56b. It is also given on the post processed images as a function of the
subtraction factor, X.
Experimental Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.53
1
Correlation Coefficient
0.95
0.9
0.85
0.8
0.75
Filter On
Unobscured Object
Intensity
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 109 – Correlation coefficient as a function of X relating the post-processed images to the image of
the wrench with no obscurant present seen in Figure 56b. The correlation coefficients as a function of X are
also given for the experimental intensity image seen in Figure 108.
147
Theoretical Correlation Coefficient vs Subtraction Proportion Constant, δ = 3.53
1
Correlation Coefficient
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subtraction Proportion Constant
Figure 110 – Correlation coefficient as a function of X relating the post-processed images to the
approximate object reflectance of the wrench with no obscurant present seen in Figure 56b. The correlation
coefficients as a function of X are also given for the theoretical intensity image (aqua line) seen in Figure
108.
Comparing Figures 109 and 110, one finds that the values of correlation for no image
subtraction (X = 0) are in agreement for the experimental and theoretical curves. The
experimental correlation coefficient for the intensity image is -1.6% off of the theoretical
value. At X = 0, the experimental correlation coefficient for the post-processed image
product is -0.5% off of the theoretical value. For higher proportions of image subtraction
are used, the post processed curves deviate to the point where at X = 1, the experimental
value of the correlation coefficient is 5.49% off of the theoretical value. The theoretical
values of correlation are wholly dependant the approximated reflectance values, so the
error percentages are also approximations. Still, there exists quality agreement with what
is experimentally observed and what is theoretically calculated.
The reduction of correlation at higher values of X for the experimentally produced
data could be attributed to a non-uniformity of illumination across the target produced by
speckle or non-uniform illumination on the collimating optics. For the theoretically
produced images, no speckle was present, and the target was illuminated by perfectly
collimated, uniform illumination. When using image subtraction, non-uniformity of the
148
source across the field of view will induce localized areas where more or less subtraction
is being conducted, producing a steeper slope towards lower correlation values for higher
values of X. This is observed in Figure 109 in comparison to Figure 110.
The system and its associated algorithms were generated as a proof-of-concept
prototype. As such, much larger errors that those produced by comparison of Figures 109
and 110 could be tolerated, as long as the data shows a large improvement in image
quality relative to the intensity image, which was the goal of this research. In fact, since
no radiometric calibration was attempted on the lab-based instrument, it is possible that
larger errors than those seen here would be found. The important observation from the
previous two figures is that the technique by which the orthogonally polarized images
were rendered and the post-processing produced results are in keeping with theoretical
predictions.
4.4 Experimental Results: Printed Paper Images (Depolarizing Object)
The post-processed images of the rough metal wrench showed that, for an object does not
depolarize incident radiation, a large enhancement of contrast and correlation to objects
within the field can be yielded from the presented technique. Unfortunately, many objects
do not maintain a high degree of polarization when impingent polarized radiation is
scattered by said objects. It is important for the designed system to also produce large
image enhancements for depolarizing objects as well.
Figure 6b1 predicts that linearly polarized light will produce higher cross polarization
discrimination than circular. The Monte Carlo simulation was ran to determine the value
of the S3 component of the backscattered Stokes vector for circularly polarized
illumination, and for the value of the S1 component of the backscattered Stokes vector for
linearly polarized illumination. The values found in Figure 6b could not be used, because
the detection geometry of the polarization imager used in this dissertation is not taken
into account. Therefore, the MC simulation will give values pertinent to the experimental
setup.
149
Principal Backscattered Stokes Parameter
0.96
S1 Backscattered
S3 Backscattered
Normalized Irradiance
0.94
0.92
0.9
0.88
0.86
0.84
2.5
3
3.5
4
4.5
5
5.5
6
Optical Depth (# SMFP)
Figure 111 – Normalized backscattered Stokes parameters. S3 value is given for circularly polarized
illumination, where the S1 value is given for linearly polarized illumination.
Figure 111 shows qualitative agreement with Figure 6b, in that the backscattered
linearly polarized light exhibits higher values for all optical depths that directly correlates
to higher cross-polarization intensity discrimination (XPD).
The consequence of higher XPD for linearly polarized light is that less scattered flux
will pollute the object detection channel for linearly polarized illumination than it will for
circularly polarized illumination. As such, we expect to observe better performance for
post-processed images when using linearly polarized light when the target depolarizes
incident radiation. This effect exists when the target does not depolarize incident
radiation, but backscattered flux is swamped by the polarization of forward scattered flux
in that case. The degree of polarization of the forward scattered flux for depolarizing
objects is essentially zero, so the polarization activity of the backscattered flux is
responsible for variations in orthogonally polarized images, highlighting the XPD effect.
Experimental verification of the previous argument is given in the following figures
obscured by an optical depth of 2.86. The first of these figures illustrates the companion
150
images for the intensity using circularly polarized and linearly polarized illumination
respectively.
Figure 112 – Registered intensity images for when circularly polarized (left) and linearly polarized (right)
illumination was used.
The second and third figures highlighting linear polarized illumination’s ability to
yield better contrast enhancements, given on the next page, demonstrate linear
polarization’s ability to give higher contrast ratios for depolarizing object over the entire
subtraction proportion spectrum X = [0,1], as well as its ability to recover the object’s
original spatial information by appreciably higher correlation coefficients for lower X
values.
On the page subsequent to the page containing Figures 113 and 114, the
accompanying post-processed images are given to illustrate the image products for a
depolarizing object obscured by an optical depth of 2.86 illuminated by linear and
circularly polarized light.
151
Figure 113 – Correlation coefficient as a function of X relating the post-processed images to the image of
the cross-hatch with no obscurant present seen in Figure 56d. Correlation curves are defined for: post
processed images using circular and linearly polarized illumination (blue and pink); intensity images for
each illumination state (yellow and aqua); and for the pre-obscured image.
Figure 114 – Detected contrast ratio versus the proportional subtraction constant, X, for an optical depth of
2.86 for the low spatial frequency cross hatch printed on copier paper. Data members are common to the
ones given in Figure 113.
152
Figure 115 – Post processed images of a cross hatch printed on copier paper with application of the DC
block Fourier spatial filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.86 using
linearly polarized illumination.
Figure 116 – Post processed images of a cross hatch printed on copier paper with application of the DC
block Fourier spatial filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.86 using
circularly polarized illumination.
153
Figures 113 through 116 give experimental support to the theoretically-backed and
modeled conclusion that using linearly polarized illumination yields higher contrasts and
correlation coefficients than using circularly polarized illumination. Again, this is
attributed to the backscattered flux detected by the receiver having higher XPD for
linearly polarized illumination than for circularly polarized illumination of depolarizing
objects.
In keeping with higher performance existing for linearly polarized illumination and
detection, further treatment of data products consistent with depolarizing targets will only
consider linearly polarized illumination. Refer to sections 4.4.2 and 4.5 (pages 141 and
142) for the summary/analysis of the results.
4.4.1 Experimental Results: Printed Paper Resolution Target (Depolarizing
Object)
When imaging the resolution target printed on copier paper, three regions of interest were
defined to produce contrast measurements. Referring to Figure 56e, they were vertical
elements 4, 5, and 6 of the resolution group within the field of view. The spatial
frequencies of those elements were determined to be approximately 2.11, 2.58, and 3.31
lp/mm in object space, and were imaged at a distance of 1.3 meters. Again, the contrast is
calculated by subtracting the average pixel value in this dark area from the average pixel
value in the bright area divided by the sum of the average pixel value in these respective
areas as shown in Equation 87.
The paper resolution target gives a good measure of how the system can enhance
contrast and correlation relative to an intensity image. It allows for approximate
quantification of contrast as a function of spatial frequencies.
The following presentation shall mirror what was given in Sections 4.3.1 and 4.3.2. A
repetitive treatment of the data products for various optical depths through which the
resolution target was imaged is given. The first figure shall illustrate the intensity image
registered by the instrument. Following the intensity image, a 3x3 display giving postprocessed image products is exhibited. Finally, graphs detailing the correlation and
154
contrast ratios as a function of the subtraction proportion constant are offered to give
quantitative values for each.
The results for the image of the paper resolution target obscured by a water vapor
obscurant with an optical depth of 3.98, using a linearly polarized source, are given in the
subsequent three figures:
Figure 117 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.98 with collimated linearly polarized illumination.
Figure 118 – Post processed images of a paper resolution with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.98.
(a) Correlation Coefficient, δ = 3.98
(b) Element 4 Contrast Ratio
1
0.8
Contrast Ratio
Correlation Coefficient
155
0.8
0.6
0.4
0.2
0
Filter On
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.6
0.4
0.2
0
-0.2
0
1
(c) Element 5 Contrast Ratio
0.6
Contrast Ratio
Contrast Ratio
1
(d) Element 6 Contrast Ratio
0.6
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.4
0.2
0
-0.2
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
Figure 119 – (a) Correlation coefficient as a function of X relating post-processed images of a paper
resolution target imaged through an optical depth of 3.98. The contrast ratios are also given in (b), (c), and
(d) for spatial frequency features of 2.11, 2.58, and 3.31 lp/mm in object space respectively. The object to
which the data are correlated is seen in Figure 56f.
The results for the image of the paper resolution target obscured by a water vapor
obscurant with an optical depth of 3.53, using a linearly polarized source, are given in the
subsequent three figures:
Figure 120 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.53 with collimated linearly polarized illumination.
156
(a) Correlation Coefficient, δ = 3.53
(b) Element 4 Contrast Ratio
1
1
Contrast Ratio
Correlation Coefficient
Figure 121 – Post processed images of a paper resolution with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.53.
0.8
0.6
0.4
0.2
0
Filter On
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.5
0
-0.5
0
1
(c) Element 5 Contrast Ratio
0.8
Contrast Ratio
Contrast Ratio
1
(d) Element 6 Contrast Ratio
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
Figure 122 – (a) Correlation coefficient as a function of X relating post-processed images of a paper
resolution target imaged through an optical depth of 3.53. The contrast ratios are also given in (b), (c), and
(d) for spatial frequency features of 2.11, 2.58, and 3.31 lp/mm in object space respectively. The object to
which the data are correlated is seen in Figure 56f.
157
The results for the image of the paper resolution target obscured by a water vapor
obscurant with an optical depth of 3.17, using a linearly polarized source, are given in the
subsequent three figures:
Figure 123 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 3.17 with collimated linearly polarized illumination.
Figure 124 – Post processed images of a paper resolution with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 3.17.
(a) Correlation Coefficient, δ = 3.17
(b) Element 4 Contrast Ratio
1
1
Contrast Ratio
Correlation Coefficient
158
0.8
0.6
0.4
0.2
0
Filter On
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.5
0
-0.5
0
1
(c) Element 5 Contrast Ratio
0.8
Contrast Ratio
Contrast Ratio
1
(d) Element 6 Contrast Ratio
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
Figure 125 – (a) Correlation coefficient as a function of X relating post-processed images of a paper
resolution target imaged through an optical depth of 3.17. The contrast ratios are also given in (b), (c), and
(d) for spatial frequency features of 2.11, 2.58, and 3.31 lp/mm in object space respectively. The object to
which the data are correlated is seen in Figure 56f.
The results for the image of the paper resolution target obscured by a water vapor
obscurant with an optical depth of 2.70, using a linearly polarized source, are given in the
subsequent three figures:
Figure 126 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 2.70 with collimated linearly polarized illumination.
159
(a) Correlation Coefficient, δ = 2.70
(b) Element 4 Contrast Ratio
1
1
Contrast Ratio
Correlation Coefficient
Figure 127 – Post processed images of a paper resolution with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.70.
0.8
0.6
0.4
0.2
0
Filter On
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.8
0.6
0.4
0.2
0
0
1
(c) Element 5 Contrast Ratio
0.5
Contrast Ratio
Contrast Ratio
1
(d) Element 6 Contrast Ratio
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
Figure 128 – (a) Correlation coefficient as a function of X relating post-processed images of a paper
resolution target imaged through an optical depth of 2.70. The contrast ratios are also given in (b), (c), and
(d) for spatial frequency features of 2.11, 2.58, and 3.31 lp/mm in object space respectively. The object to
which the data are correlated is seen in Figure 56f.
160
The results for the image of the paper resolution target obscured by a water vapor
obscurant with an optical depth of 2.11, using a linearly polarized source, are given in the
subsequent three figures:
Figure 129 – Intensity image of the rough metal wrench obscured by water vapor totaling an optical depth
of 2.11 with collimated linearly polarized illumination.
Figure 130 – Post processed images of a paper resolution with application of the DC block Fourier spatial
filter enacted for X = [0.10, 0.90] in 0.10 steps for an optical depth of 2.11.
(a) Correlation Coefficient, δ = 2.11
(b) Element 4 Contrast Ratio
1
1
Contrast Ratio
Correlation Coefficient
161
0.8
0.6
0.4
0.2
0
Filter On
Unobscured Object
Intensity
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.8
0.6
0.4
0.2
0
0
1
(c) Element 5 Contrast Ratio
0.5
Contrast Ratio
Contrast Ratio
1
(d) Element 6 Contrast Ratio
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
Subtraction Proportion Constant
1
Figure 131 – (a) Correlation coefficient as a function of X relating post-processed images of a paper
resolution target imaged through an optical depth of 3.98. The contrast ratios are also given in (b), (c), and
(d) for spatial frequency features of 2.11, 2.58, and 3.31 lp/mm in object space respectively. The object to
which the data are correlated is seen in Figure 56f.
4.4.2 Experimental Results: Paper Resolution Target Image Analysis
Observation of the correlation coefficients for each presented optical depth for the paper
resolution target images yields the conclusion that the overall structure of the postprocessed images are very similar for each, and that the emergence of enhanced contrast
might be balanced out by the widening of the subtraction channel image, due to the
blurring filter, when calculating the correlation coefficient. A slight variation in
correlation is observed, but not appreciably so.
The most notable feature of the data given in the previous section is the large contrast
enhancements found by using the novel DC block Fourier filter algorithms and associated
blur-based subtractions. For seemingly completely obscured intensity images, gains in
contrast allow for recognition of the underlying resolution target.
The contrast improvement is observed for each spatial frequency in the target scene.
The increase in contrast obeys similar functional dependence on X for each optical depth.
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Study of the contrast ratio graphs for the depolarizing object lends itself to the
conclusion that the decrease in contrast, due to obscurant scatter, acts independently of
the spatial frequency content of targets within the scene. Radiation that has been scattered
after interaction with a depolarizing object within the scene will have its angular and
positional orientations altered such that the detector onto which a post-scattered photon is
incident has no correlation to the detector site upon which a photon that has not been
scattered would be impingent. This is why fog and other obscurants have a locally
uniform irradiance distribution where the optical thicknesses are equal.
There are locations in the contrast graphs where the contrast of the post-processed
images actually exceeds the contrast of the object being obscured. In a real situation,
where objects within the obscurant are unknown, there is no way for the algorithms to
know the exact spatial structure of said objects. Therefore, it is likely that the exact
contrasts of the obscured objects will not be recovered, but that the overall shape of the
objects will be recovered by use of the presented algorithms. This is also true of the metal
wrench images shown in previous sections.
4.5 Experimental Results: Contrast Enhancement Analysis
Through evaluation of the ratio of the post-processed contrast to the pre-processed
contrast for the optimum illumination and detection polarization states, as well as the
optimum subtractive proportion constant, a quantitative measure for the performance of
the sensor and its associated algorithms is given in Figure 132.
It is important to note that the optimum contrast enhancements, when imaging the
metal wrench, were received when illuminating and detecting with circularly polarized
states. Conversely, when imaging the depolarizing copier paper resolution target, all
optimum contrasts were yielded when illuminating and detecting with linearly polarized
states.
The physical justifications for such results were given in Sections 4.3.3 and 4.4.
Namely, linearly polarized detection and illumination are ideal for imaging depolarizing
objects through obscurant media using orthogonally polarized image subtraction as the
algorithmic basis for post-processing, because the strength of the principal backscattered
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component dominates the image discrimination. The result from Monte Carlo simulation
written in support of this dissertation (Figure 111), as well as previously published
numerical simulations (Figure 6b), agree that linearly polarized radiation will exhibit a
stronger principal backscattered component.
3
Contrast Enhancement Factor (Cpost/Cpre)
10
Metal Wrench (0.464 lp/mm)
Metal Wrench (2.11 lp/mm)
Copier Paper (2.11 lp/mm)
Copier Paper (2.58 lp/mm)
Copier Paper (3.31 lp/mm)
2
10
1
10
0
10
2
2.5
3
3.5
4
4.5
Optical Depth, δ, (# of SMFP's)
5
5.5
Figure 132 – Contrast enhancement factors, for varying object spatial frequencies, relative to normal
intensity imaging when viewing a metal wrench (polarization maintaining object) and a printed resolution
target on copier paper (depolarizing object) through an increasingly thick cloud of water vapor (fog).
When the obscured object does not strongly depolarize incident radiation, optimum
obscurant subtraction occurs when there is a comparable amount detected flux from the
scattering media itself in both orthogonally polarized images. Allowing for postprocessed images to highlight object structure and minimize obscurant structure.
Circularly polarized illumination and detection performs much better in this regard
relative to linearly polarized illumination and detection (Figure 106).
164
In short, the model and experimental data suggests that, when actively illuminating
and imaging remote objects through dense scattering media, one should use linearly
polarized illumination and detection for depolarizing objects and circularly polarized
illumination and detection for object that maintain incident polarization.
For the objects used in this section, we observed contrast enhancement factors that
approach and exceed a factor of 100 for larger optical depths, if one extrapolates the data
from the copier paper to higher optical depths. Due to the diffuse scattering from the
copier paper and the limited irradiance from the source, actual experimental data for
larger optical depths was difficult to obtain for the copier paper.
It is evident from Figure 132, however, that the presented Wollaston prism-based
sensor and its associated algorithms provide large contrast enhancements, relative to
intensity imaging, at all obscuring optical depths in real-time.
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CHAPTER 5: FIELD SYSTEM DESIGN
Results in the previous chapter give quantitative backing for the techniques presented in
this dissertation. The results in Chapter 4 gave proof of concept data for imaging through
obscurant media at a stand-off distance, but the distance for which the laboratory data
was taken was approximately 1.3 meters.
In order to do take field data at greater extended distances, the instrument must be
modified to the point where it can be transported, reassembled, and protected from the
environment without appreciable loss of system performance. Opto-mechanical mounting
structures could be used to accomplish this, but a more elegant and compact solution was
chosen in support of this task.
A novel wire-grid polarizer array structure and alignment technique was derived to
bond a wire-grid polarizer (WGP) array directly to a camera’s focal plane array (FPA)
using common, manually operated, opto-mechanical structure. The WGP and associated
elements were designed for use in the visible electromagnetic spectrum. With intelligent
design and alignment of the wire-grid polarizer array to the FPA, the system’s optical
design became trivial. Only a common C-mount lens need be used, instead of the
intermediate image planes, field stops, and Wollaston prisms used in the laboratory-based
system. Additionally, the image registration used in the Wollaston-based system is not
necessary. Adjacent pixel values are used as subtractive partners, because the WGP array
effectively makes a polarization analyzer of each pixel on the FPA.
5.1 Wire-Grid Diffraction Grating Theory
Wire-grid diffraction gratings use specific spatial structure to support specific guided
modes within the wire-grid to transmit electromagnetic radiation of a specified electric
field orientation determined by the wire-grid structure. Using modern fabrication
techniques, micro-polarizer arrays can be created where individual polarizer structure has
periodicity smaller than the wavelengths of radiation in the visible spectrum41.
These wire-grid diffraction gratings are used as polarizing beam splitters and
polarizers. When S-polarized light is incident on the wire-grid nano-structure, the electric
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field of the incident radiation is parallel to the grooves of the grating and very high
reflectivity is observed for said polarization. However, when P-polarized light is incident
on the structure, the electric field incident on the grating is perpendicular to the wires,
and very high transmission is observed for P-polarized light. An example of a wire-grid
polarizer is given in the following figure41:
Figure 133 – A wire-grid polarizer is displayed with its action on incident unpolarized light featured. The
S-polarized component of the incident light is reflected, where the P-polarized component is transmitted41.
With the nano-grid structure causing such polarization dependant reflection and
transmission, a diffraction grating orientation consistent with a period sufficient to reflect
and transmit visible radiation can be used as a polarizing element in the visible
electromagnetic spectrum.
Commonly, a physical interpretation of the action of the WGP is given by studying
surface plasmon resonances and electron mobility within the wires of the structure.
However, when the periodicity of the structure is less than half of the wavelength of light
within the dielectric, the condition for excitation of surface plasmons does not hold1.
Therefore, an alternative explanation must be given.
167
A metallic waveguide theory given by Xu, et al.41, delivers the required explanation
by studying the propagation constants for guided waves and their relationship to the
refractive index between wires and their separation.
The treatment begins with study of a single groove and the wires immediately
surround it. This establishes three regions where two interfaces with differing refractive
indices are infinitely extended in the down the z axis. Figure 134 demonstrates the
treatments setup41,
Figure 134 – Slab waveguide infinitely extended in the z-axis direction, and independent of y-axis
behavior.
If the system’s origin is set in the exact center of the slab waveguide, the index
distribution is as follows:
⎧n0 , − w / 2 < x < w / 2
n=⎨
x ≥ w/2
⎩n1 ,
(91)
The field within the waveguide is described by a linear superposition of transverse
electric (TE) and transverse magnetic (TM) modes. The TE modes of the waveguide have
their electric fields parallel to the y-axis; conversely, the TM modes have their magnetic
fields parallel to the y-axis. A propagation constant, β, is defined for waveguide modes
whose field is proportional to exp(± iβz ) due to guided waves traversing the waveguide
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in the z-direction. When the wires in region I and III are perfect conductors, the
propagation constants for TE and TM modes are determined41,
β m ,TE
m 2π 2
= k n −
, m = 1, 2, 3, ...M for TE modes,
w2
β m ,TM =
2
0
2
0
(m − 1)2 π 2 , m = 1, 2, 3, ...M for TM modes.
k 2n2 −
0
0
(92ab)
w2
where k0 is the wave number of the incident radiation in a vacuum, and m is the mode
number for the respective TE and TM modes.
The existence of varying behavior for the guided modes between TE and TM modes
is attributed to the (m-1) factor in Equation 92b, in favor of the m factor in Equation 92a.
When w < λ (2n0 ) , every TE mode is purely imaginary, however, due to the (m-1)
−1
factor, there always exists a TM mode for which the real part of the propagation constant
is non-zero.
As mentioned before, the guided modes are proportional to exp(± iβz ) , therefore
purely imaginary propagation constants lend themselves to exponential decay, while realvalued propagation constants lend themselves to sinusoidal variation with z.
Wire-grid polarizers are designed with w < λ (2n0 )
−1
to guarantee guided modes
exhibit expected polarization transmitting and reflecting behavior. Incident electric fields
polarized perpendicular to the grooves of the WGP will excite only TE modes in the
waveguide. If the design parameter w < λ (2n0 )
−1
holds true, these TE modes undergo
exponential decay due to their imaginary propagation constant. As a result, very little
transmitted light is polarized perpendicular to the grooves of the WGP. On the other
hand, an incident electric field polarized parallel to the wire-grid grooves will excite TM
modes for which a real propagation constant exists and dominates the transmission
process, since it is always the primary guided mode (m=1).
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5.2 Wire-Grid Polarizer Array Design and Alignment
Fabrication and alignment of wire-grid polarizers to be bonded to a focal plane array for
use in the visible spectrum is not a trivial process. Aligning two independent structures to
sub-micron accuracy is required to produce a bonded array with efficient polarization
discrimination capabilities.
The wire-grid array structure was chosen by working with tradeoffs between spatial
resolution, polarizer contrast ratio, and throughput. Because of the reflective nature of the
wire-grid structure for specific polarizations, a structure was chosen that would minimize
the chance that radiation of one polarization would bleed into adjacent pixels that should
only be accepting polarization of a different state. The geometric cross-talk and FPAbased cross-talk was minimized, and the throughput was maximized for a 4 polarization
state (22.5˚, 67.5˚, 112.5˚, 157.5˚) wire-grid polarizer array by choosing a 6x6 pixel
“super pixel” structure seen in Figure 135.
Additionally, the 6x6 “super pixel” design drastically simplified the alignment
process as it pertained to coarse angular alignments and transverse alignments in the
plane of the detector array.
Figure 135 – (a) 50x magnification image of a section of a wire-grid polarizer array with unpolarized light
illumination. (b) 50x magnification image of a wire-grid polarizer array with 112.5o linearly polarized
illumination in transmission.
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The WGP array shown in Figure 135 demonstrates the 6x6 “super pixel” design. The
super pixel allows for detection of four linearly polarized states. Each detection state is
45 degrees from its neighboring states. Each detection eigenstate in the super pixel
utilizes a 2x2 pixel area, with a full pixel-sized opaque buffer zone between any adjacent
active areas. These opaque bars reduce the spatial resolution and the throughput of the
system, but drastically increase the contrast ratio performance of the polarizer array,
because the full pixel opaque area between differing polarizer states in the super pixel
assists in reducing all sources of pixel cross-talk, geometric and otherwise.
The term “super pixel” is used, because, ignoring aliasing effects, a normal intensity
imaging camera would like to have a single point in object space be sampled by a single
pixel in image space. However, in using an on-chip polarimeter, a single point in object
space is required to be sampled over every polarization state in the design. In our case,
we want the point spread function to be spread over the 6x6 super pixel shown in Figure
135(a).
Also of note in the end design of the WGP to eventually be bonded onto a camera’s
focal plane is the fact that the linear polarization eigenstates are rotated 22.5˚ from those
in the conventional Stokes vector. The process through which the WGP was fabricated
used an Electron beam lithographer that utilizes a fracturing algorithm in the writing
process which splits exposure areas into very small sections which are exposed
independent of neighboring areas. The resolution of the step size between neighboring
exposure sites is limited, therefore sharp angled structure relative to the basis directions
that define how the element can moved relative to the E-beam machine have a higher
probability to contain errors. The 22.5˚, 67.5˚, 112.5˚, 157.5˚ eigenstate super pixel
design minimizes errors inherent in the writing algorithms of the E-beam exposure
process by keeping each wire-grid angle to be written an equal angular distance away
from the nearest movement basis state of the electron beam’s translation stage. The
conventional linear Stokes parameters can be calculated from the detection of these
states, so there is no loss in information by detecting in these states relative to detecting
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in the conventional linear polarization states used in the determination of the Stokes
vector.
5.2.1 Wire-Grid Polarizer Array Design and Alignment: Active Polarizer Area
The first design parameter to be considered in the creation of the wire-grid polarizer array
was how large to make each polarizer in the super pixel relative to the pixel size of the
FPA onto which the WGP would be bonded.
In the absence of cross talk and alignment errors, the polarizer size for each state in
the super pixel design would one pixel with no opaque region about the polarizers,
creating a 2x2 super pixel. However, cross-talk was found to be the largest factor
contributing to losses in contrast ratio of the bonded arrays. Also, perfect alignment is
impossible. There always exists a small amount of misalignment regardless of the
alignment technique.
In each design step, the probability of detecting accurate pixel values was weighed
against spatial resolution, potential cross-talk, and ease of alignment. To further illustrate
why a 2x2 super pixel design would present problems in alignment, and maximize cross
talk, a pixel value map was created. Each value in the following map corresponds to the
flux an appropriately sized detector pixel would detect as a 7 pixel x 7 pixel section of a
wire-grid polarizer is translated in x and y with collimated 22.5˚ linear polarized
illumination. The pixel value map is given Figure 5, and steps in 1/100th pixel increments.
Figure 136, on the following page, shows the transmission of the 7x7 section of the 2x2
super pixel wire grid polarizer under examination.
172
Figure 136 – Transmission of a 7 pixel by 7 pixel region of a repeated WGP with a 2x2 super pixel design
with collimated 22.5˚ linearly polarized illumination incident on the polarizer array. Each polarizer
element within the 2x2 super pixel exhibits the exact geometric extent of a pixel on a detector to which it
would be aligned.
Figure 137 shows that, for the WGP design shown in Figure 136, only when a pixel is
perfectly aligned with the WGP does it give the appropriate value of the flux. Any shift
will cause the value to be reduced as a function of neighboring pixels. This map also
gives a feel for the opportunity of cross-talk between pixels. Every value in the map is a
weighted function of the transmission of the four polarizers in its immediate vicinity. The
dependency of the flux through the pixel on every neighboring polarizer maximizes the
effect of cross-talk from any source.
Figure 137 – Flux detected by a single pixel as the polarizer array shown in Figure 136 is sifted over it as a
function of the shift values in x and y with collimated 22.5˚ linearly polarized illumination incident on the
polarizer array. The pixel geometry is exactly matched to a single polarizing element in the 2x2 super pixel
design.
173
The design of the super pixel then evolved through the desire to maximize the
probability that the correct flux at a focal plane pixel would be detected for any given
translational shift error in the x and y directions. In order to feature this state of detection,
the flux map of the newly generated super pixel design would have to exhibit maximum
flux for a full pixel sized shift in both x and y. Such a map exists for when a 4x4 super
pixel design is used where each detection state uses a 2x2 pixel area. The transmission for
a 7x7 area of such a WGP design is given in Figure 138.
Figure 138 - Transmission of a 7 pixel by 7 pixel region of a repeated WGP with a 4x4 super pixel design
with collimated 22.5˚ linearly polarized illumination incident on the polarizer array. The dashed lines
represent the projection of a focal plane array’s pixel sizes onto the 4x4 WGP design.
Common to this design and the 2x2 design is that there still exists no opaque region
between adjacent polarizers in the super pixel. There is a sharp transition from detection
in one polarization state to detection in another polarization state. The associated pixel
flux map is presented on the next page.
174
Figure 139 – Flux detected by a single pixel as the polarizer array shown in Figure 138 is sifted over it as a
function of the shift values in x and y with collimated 22.5˚ linearly polarized illumination incident on the
polarizer array.
As was desired, the flux map for the 4x4 super pixel design exhibits a full pixel extent
in both x and y for which the flux is maximized. This is observed by studying the extent
of the bright squares present in Figure 139. They appear as rectangles, because of the
aspect ratio of the figure. The extent of the maximum value squares yields a situation
where one full pixel will always exist directly under each analyzer in the 4x4 design.
Neglecting cross talk effects, this produces a design that will always generate the correct
flux values by addressing the pixel directly under each analyzer state. This design is
completely insensitive to translational alignment errors due to this feature.
5.2.2 Wire-Grid Polarizer Array Design and Alignment: Detection Site
Isolation
Unfortunately, bleeding between adjacent pixels is still readily apparent in the flux map
for the 4x4 design. In the presence of cross talk, the design is still insensitive to
translational alignment errors, but it will exhibit large errors when cross talk is
appreciable.
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The final step in the design of the WGP super pixel minimized the potential for cross
talk and maintained the translational insensitivity featured in the 4x4 design. The 6x6
design shown in Figure 135(a,b) was generated through the addition of an opaque Al
stripe between each detection state. The experimental transmission of a fabricated WGP
using the 6x6 super pixel was given in Figure 135b for detection of a polarization
eigenstate of the structure before the dicing and alignment process. The theoretical
transmission for incident polarized light in a polarization eigenstate of the 6x6 super pixel
design is given in Figure 140.
The 6x6 super pixel design features a 2x2 pixel area for each polarization state
detected within the super pixel. It differs from the 4x4 design, because it utilizes a full
pixel opaque stripe between adjacent active areas. In order to maintain translational
alignment insensitivity, it was shown that a 2x2 active area was required for each
polarization state. To maintain the integer pixel periodicity of the super pixel and use a
2x2 pixel active area for each polarizer, the addition of an opaque stripe in the super pixel
was required to be a full pixel in size at the smallest. Larger opaque strips could have
been added, but at the cost of a large loss in spatial resolution when imaging with a
bonded focal plane array.
Figure 140 – Transmission of a 7 pixel by 7 pixel region of a repeated WGP with a 6x6 super pixel design
with collimated 22.5˚ linearly polarized illumination incident on the polarizer array. The dashed lines
represent the projection of a focal plane array’s pixel sizes onto the 6x6 WGP design.
176
This design was deemed superior to the previous two designs because it featured
protection against pixel cross talk, which is the largest factor for degraded super pixel
performance. It also shared the translational alignment insensitivity of the 4x4 super pixel
design. The associated pixel flux map is given in Figure 141.
Figure 141 – Flux detected by a single pixel as the polarizer array shown in Figure 140 is sifted over it as a
function of the shift values in x and y with collimated 22.5˚ linearly polarized illumination incident on the
polarizer array.
The pixel flux map for the 6x6 design demonstrates that the use of such a design will
feature no mixture of flux from different polarizations at a single pixel. The pixel-wide
opaque stripe prevents this mixture. It also drastically reduces other sources of cross talk;
because each polarization detection site is isolated from it’s the other detection sites by a
full pixel.
5.2.3 Wire-Grid Polarizer Array Design and Alignment: Opto-Mechanical
Alignment Setup
Some alignment considerations were accounted for during the design process. The
alignment consideration for which the super pixel design process took most account was
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producing a design with insensitivity to translational alignment errors. Unfortunately, the
opto-mechanical system built to carry out the real-time alignment was made using
available components. This project did not have the resources to produce a mechanical
system that had completely independent translation and rotation degrees of freedom.
Coupled rotation and translation creates many issues in precision alignment to elements
that are on the order of microns. Translation in any direction has small to large
components of translation in undesired directions. Similarly, rotation about any axis will
induce rotations about orthogonal axes in addition to translation if the axis of rotation
doesn’t pierce the elements to be rotated at the correct location and angle.
To account for coupled motion, quantification of the amount of coupling between
each rotational and translational degree of freedom can be carried out. This is not a trivial
process and for the mounting equipment available, not deterministic. Simply touching
some of the actuators caused movement and rotation of the WGP relative to the FPA. As
such, coupled movement quantification was not carried out, and the actual design of the
WGP eliminated the need for high resolution translational alignment.
What remained was the creation of an opto-mechanical structure that would allow for
free rotation and translation of the WGP to the FPA of a camera while it was active
detection mode.
Figure 142 – Approximate opto-mechanical setup for active polarizer alignment to the CCD.42
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An opto-mechanical setup that allowed for real-time alignment of the polarizer array
to the focal plane using an alignment algorithm developed in LabView was generated.
The alignment algorithm shall be treated in following sections. The setup included two
XYZ translation stages, with one including a rotation stage for rotational variation of the
polarizer array with respect to the CCD. The stage holding the polarizer incorporated a
vacuum holder generally used to pick up delicate electrical components. It also allowed
for small tilts of the stage about two orthogonal axes. A Fiber-Lite Fiber Optic
Illuminator and variable polarizer are used to produce the required polarized illumination
for the system in conjunction with a high-quality spectral filter.
The opto-mechanical structure built in support of the alignment process was based
heavily on the structure presented in Figure 142 and is demonstrated in Figure 143.
Figure 143 – Actual alignment setup used to align the WGP to the FPA.
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5.2.4 Wire-Grid Polarizer Array Design and Alignment: Coarse Angular
Alignment
The actual alignment of the WGP to the FPA is done in two separate steps. In the first
step, coarse angular and translational alignment is carried out by viewing the active FPA
pixel values in a macroscopic sense. Translational alignment is completed by simply
aligning a side of the WGP to the appropriate side of the FPA. As the WGP is lowered
closer to the FPA, angular misalignment is evident by the observation of the pixel values
varying at an angle. Normally, completing the angular alignment by eye in such a fashion
still leaves the angular alignment of the WGP appreciably away from the desired
alignment.
The 6x6 super pixel design allows for very fine angular alignments to be made simply
by eye. The opaque aluminum stripes induce a Moiré pattern with the reflection off of the
micro structure of the FPA across the active image.
A Moiré pattern is formed between the transmission distributions of two periodic
structures when they are viewed in parallel. What is very useful about their existence is
that a macroscopic irradiance modulation can be observed and related to structure that is
very much smaller than the extent of the modulation. They are observed in many places
in nature. When one stares out of a screen door or window screen at meshed patio
furniture, it appears as if lines are formed on the patio furniture. This is a manifestation of
a Moiré pattern. Let the transmission distribution of two periodic structures being viewed
in parallel as similar to the coherent addition of the following temporally invariant
electric fields in an interferometric process.
K K
K
E1 (r ) = E0 exp ik1 ⋅ r
K K
K
E 2 (r ) = E0 exp ik 2 ⋅ r
(
(
)
)
(93)
where r is the positional vector, [x,y,z], and k1 and k2 are the periodicity of the structures
projected on each axis. Simply adding these fields, then multiplying the addition by its
complex conjugate yields the irradiance pattern.
K K K
I = 2 + 2 cos k1 − k 2 ⋅ r
[(
) ]
(94)
where E02 = 1, and the wavelength of the resultant Moiré fringe pattern is found to be,
180
2π
dm = K K
k1 − k 2
(95)
An example of a Moire pattern by forcing an angular misalignment between similar
periodic structures with a wavelength of d is given in Figure 144.
Figure 144 – Sample Moiré pattern.
With the presence of the formed Moiré pattern coarse angular alignment of the WGP
to the FPA can be achieved to within a degree by rotating the element to the point where
the Moiré pattern disappears. The 6x6 super pixel design’s opaque stripes were periodic
to three pixel sizes, where the FPA was periodic to one pixel size. The camera FPA used
in the alignment process had a pixel size of 6.7 microns, therefore the resultant
wavelength of the Moire pattern equated to the following
2
2
⎡⎛ cos α
1 ⎞ ⎛ sin α ⎞ ⎤
⎟⎟ + ⎜⎜
⎟⎟ ⎥
d m = 2π ⎢⎜⎜
−
⎢⎣⎝ 20.1μm 6.7 μm ⎠ ⎝ 20.1μm ⎠ ⎥⎦
1/ 2
(96)
where α is the angle of misalignment between the wire grid polarizer array and the FPA.
When the angle of misalignment becomes very small it becomes difficult to resolve the
pattern, because the periodicity of the pattern covers very few pixels, therefore it can only
be used for coarse angular alignments.
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5.2.5 Wire-Grid Polarizer Array Design and Alignment: Precision Angular
Alignment
When aligning micro-polarizer arrays to a focal plane arrays on the pixel level,
commonly a single region of interest is studied and the extinction ratio in said region is
used as the metric for which alignment is optimized43. There are drawbacks to this
technique, however. The most obvious is the polarization angle of the source illuminating
the array during alignment. When actively illuminating the FPA during alignment with
and studying only a single region of interest on the focal plane, alignment is being
optimized to the angle of polarization of the source and not to individual pixels on the
focal plane. Any angular deviation from perfectly horizontally or vertically polarized
illumination with respect to the FPA axes will yield the exact angular error in the
alignment process. In addition, when studying the pixel values in a single region of
interest, the alignment can seem locally optimized, when in actuality there exists a small
angular deviation that is manifested in a global variation in super pixel row locations.
That is to say that the super pixel locations will drift from one row at the locally
optimized spot to a separate row downstream on the FPA.
For focal plane arrays with fewer pixels, local optimization can be used, because the
pixel drift will be negligible and flux variations due to angular misalignments at opposing
pixel locations may not be able to be differentiated from the noise in the detector.
For extended arrays with many pixels a novel alignment process was generated to
account for pixel drift induced by local optimization and alignment error due to
misalignment of the source polarization with respect to the FPA axes. In the novel
alignment process, a camera actively registers two equally sized regions of interest on
opposing ends of the focal plane. The regions are of the same size and are defined on the
same rows of the FPA and the columns of the regions are defined such that the first
column of the regions of interest is an integer number of super pixels away. This ensures
that for a perfectly aligned array, the regions would be exact copies of one another.
These regions of interest are then correlated in the same manner shown in the image
registration procedure given in Chapter 3.2, Equation 66, except that only the unshifted
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value of the Laplacian is used to gain understanding of the similarity between the two
defined regions. The figure of merit used in the alignment process for measuring the
likeness of the two regions is given as the Laplacian of the two-dimensional correlation
between the two regions of interest evaluated at x and y equaling zero.
(
)
(
⎛ ∂ 2 ROI 1* ( x, y ) ⊕ ROI 2 ( x, y ) ∂ 2 ROI 1* ( x, y ) ⊕ ROI 2 ( x, y )
A(0,0 ) = ⎜⎜
+
2
x
∂
∂y 2
⎝
)⎞⎟
⎟
⎠ x , y =0
(97)
where ROI1 and ROI2 are the pixel values of the specified regions of interest. A(0,0) will
be maximized only when the angular alignment of the FPA with the WGP exhibits the
least angular error that the opto-mechanical structure used in the alignment procedure can
provide.
Using a specific opto-mechanical structure; a correlation gradient algorithm, and UV
curing adhesive, a wire-grid polarizer with the 6x6 super pixel design was bonded to a
1360x1024 6.7μm pixel CMOS chip. Below is a section of the LabVIEW program
written to interface with the DATAray laser profiling camera to complete the alignments
using the ROI correlation technique.
Figure 145 – Active alignment pixel values with polarized light incident on the CMOS chip during
alignment.
Collimated illumination of a polarization state consistent with a polarization eigenstate of
the super pixel was used during the alignment process. The array being aligned in Figure
13 was not yet perfectly aligned, which explains the listed pixel values not exhibiting a
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high contrast ratio. However, the contrast ratio of the super pixel, wire-grid polarizers
after the dicing process was not extremely high over a broad band.
The 6x6 super pixel design WGP was placed under a 40x microscope and illuminated
illuminated with white light to determine the polarizer’s broad band contrast ratio after
the entire fabrication and dicing processes were complete. The following sets of data are
for 22.5˚, 67.5˚, 112.5˚, and 157.5˚ linear polarized illumination respectively. The large
variation about the plateau averages for each set of data is attributed to the Fresnel
diffraction pattern caused by the nano-structure of the wire-grids defined for the design.
V Profile Analyzing 22.5 and 157.5 Lin. Pol States
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V Profile Analyzing 157.5 and 22.5 Lin Pol States
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(D)
To reiterate, the data given on the previous two pages was found by viewing the
transmittance of the 6x6 wire grid polarizer under a 40x microscope using: (A) 22.5˚,
(B)67.5˚, (C) 112.5˚, and (D) 157.5˚ linearly polarized illumination. The extinction ratio
for each polarization detection site is inferred by taking the ratio of the maximum value
and the minimum value of the transmittance for each illumination state. The pre-aligned
broadband extinction ratio for the WGP in question is shown to be approximately 12:1 to
15:1.
After alignment of the WGP to the FPA was conducted, the angular error in
alignment was calculated by inspection of the change in pixel values for a common pixel
member in super pixels across the array when a polarization eigenstate of the super pixel
is incident on the bonded array. The value of the flux at any pixel in the array is
proportional to the inverse of the ratio defined by the area of the pixel and the area over
the pixel that allows for flux to propagate normally through the WGP.
The geometry in question is demonstrated in the following Figure where the dashed
lines represent the FPA pixels under a 6x6 super pixel design.
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Figure 146 – Pixel projections on a rotated 6x6 super pixel design with 22.5˚ linearly polarized
illumination.
As stated before, there always exists at least one full FPA pixel under the 6x6 design.
The other pixels under a detection state in the super pixel will have their flux reduced by
the proportion discussed above relative to the full pixel value. Under the very small
angles of misalignment found in the alignment process, an integral relationship for the
difference in the amount of uncovered area a common pixel in the super pixel design
features at two separate locations on the same row can be written,
R1, 2
Area 2 − Area1 =
R1,1
∫ (θ * (R + x ' ') + y ' ')dR − ∫ (θ * (R + x ') + y ')dR
0
R0 , 2
0
0
0
(98)
R0 ,1
where θ is the angle of misalignment of the WGP relative to the FPA, R is the positional
coordinate on the row where the addressed pixels reside, and x0 and y0 are the positional
misalignments induced by the angular misalignment at a R distance away from a defined
origin,
x0 ' = R' cos θ , x0 ' ' = R' ' cos θ
y0 ' = R' sin θ , y0 ' ' = R' ' sin θ
(99)
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Completing the integrals, subtracting, and using the small angle approximation for the
relations in Equation 99, one finds,
ΔArea ≈ 2rθ * (R' '− R')
(100)
where r is the x width of the pixels being addressed. When each pixel value in its super
pixel is normalized by the maximum pixel value in its super pixel, every pixel value in
the super pixel is then related to the FPA pixel value for which none of its area is blocked
by an opaque stripe. Through this normalization, the change in pixel values for a common
pixel location within the super pixel across the array on the same row will give
information on the angular misalignment.
ΔPVnorm ≈ θ * (R' '− R')
(101)
From the previous approximation, the linear slope of the set of data detailing the
difference in common pixel values for the same pixel location in each super pixel on the
same row will yield the angular value of the misalignment of the WGP relative to the
FPA.
The true differences in pixel values have a quadratic dependence, because multiple
terms were stricken from Equation 98 by use of the small angle approximation that would
give quadratic dependence.
Figure 147 – Difference value plot for the (1,2) pixel when normalized to the (2,2) pixel.
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Figure 148 – Difference value plot for the (1,3) pixel when normalized to the (2,2) pixel.
A second degree polynomial is fit to the data in Figures 147 and 148. As
hypothesized, there exists a quadratic dependence on the difference values, but the
coefficient on the quadratic term is approximately three orders of magnitude less than the
coefficient on the linear term. Through the process detailed in Equations 98 through 101,
we found that the coefficient on the linear term directly corresponds to the angle of
misalignment between the WGP and FPA. From second degree polynomial fits above,
the higher value on the linear term was used to calculate an angle of misalignment of
0.0038 degrees.
Such a small value of misalignment validates the novel alignment process and its
associated algorithms.
5.3 Bonded Wire-Grid Polarizer Images
Ultimately, the bonded WGP/FPA combination is required to be used as an imaging
polarimeter that can detect the linearly polarized elements of the Stokes vector. After the
WGP was bonded to the focal plane array of the camera in question, multiple images
were taken to get a feel for the pre-calibrated performance of the camera. Due to
geometric and focal plane pixel cross talk effects, the 12:1 – 15:1 extinction ratios seen in
the cross sectional data in the previous section was reduced for the bonded array.
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For 22.5˚ linear polarized illumination, the following image over a section of the
focal plane was registered,
4
x 10
5
100
4.5
Y Pixel Location
200
4
3.5
300
3
400
2.5
500
2
1.5
600
1
700
0.5
100
200
300
400
500
600
700
800
900
1000
X Pixel Location
Figure 149 – White light illumination of 22.5˚ linearly polarization shone directly on the bonded
WGP/FPA.
The coarse periodic nature of Figure 149 is an artifact of the program used to view the
pixel values at a low zoom. This modulation did not exist at the pixel level. Of note in
Figure 149 was the fine periodic nature induced by the well aligned wire grid polarizer.
At each super pixel, the pixel that demonstrated behavior consistent with being
completely uncovered defined the pixels that were used to register the value for each
detection polarization state within the super pixel. The registered images for 22.5˚, 67.5˚,
112.5˚, and 157.5˚ linearly polarized white light illumination on the focal plane are given
in Figures 150, 151, 152, and 153.
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Figure 150 - 22.5˚, 67.5˚, 112.5˚, and 157.5˚ images for 22.5˚ linearly polarized illumination on the FPA.
Figure 151 - 22.5˚, 67.5˚, 112.5˚, and 157.5˚ images for 67.5˚ linearly polarized illumination on the FPA.
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Figure 152 - 22.5˚, 67.5˚, 112.5˚, and 157.5˚ images for 112.5˚ linearly polarized illumination on the FPA.
Figure 153 - 22.5˚, 67.5˚, 112.5˚, and 157.5˚ images for 157.5˚ linearly polarized illumination on the FPA.
The extinction ratios for each set of images were found by taking the average value
over the entire field for orthogonal states. They were calculated to be 8.4887, 8.6048,
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8.4627, and 8.4267 for 22.5˚, 67.5˚, 112.5˚, and 157.5˚ linearly polarized white light
illumination respectively. Again, this loss in contrast ratio was attributed to pixel cross
talk between polarization states.
Using a dose array fabricated during polarizer development, the bonded chip with a
40x objective, and a C-mount tube attached to the camera, magnified the dose array’s
wire-grid structure. Realize that the dose array is used to determine what development
parameters yield the best results, so the extinction ratios of the following imaged object is
not very high. Still, studying the images in Figure 154, we see that the on-chip imaging
polarimeter exhibits quality performance.
Figure 154 – On-chip linear imaging polarimeter 40x images of a wire-grid polarizer dose array.
In a well aligned array, we should see areas of bright diagonal to areas of dark in each
image; also the dark areas in each image should correspond to bright areas in the
orthogonal image state.
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Finally, to demonstrate the on-chip polarimeter’s ability to resolve objects with varied
spatial structure, another c-mount lens was mounted onto the camera and a small plastic
container illuminated by an ambient laboratory lamp was imaged by the on-chip system.
The images of the small plastic container for each detection state of the on-chip linear
polarimeter are given in Figure 155.
Figure 155 – On-chip linear imaging polarimeter images plastic container with the word “RELEASE”
evident on the upper left part of the rendered image. Ambient laboratory illumination was used as the light
source.
The word “RELEASE” is plainly evident in the unpolarized image of the plastic
container. The 6x6 super pixel design will cut down the spatial resolution of the camera
by 1/6th, but it still can deliver images with sufficient spatial resolution to be used as an
imager.
Calibration of the on-chip polarimeter used the same process detailed in Chapter 3.3.
A polarization data reduction matrix was generated using the cross-sectional data in the
previous section.
Additionally, since pixel projections into object space within a common super pixel
do not overlap, there exists a lack of perspective that requires treatment to reduce edge
194
effects in actual polarization images. In a the snapshot, linear polarimeter produced in the
WGP bonding, the possible polarization images are found using,
⎡ S 0,r ( x, y )⎤ ⎡ I 22.5 ( x, y ) + I 112.5 ( x, y )⎤
K
⎢
⎥
S r ( x, y ) = ⎢ S1,r ( x, y )⎥ = ⎢⎢ I 22.5 ( x, y ) − I 112.5 ( x, y )⎥⎥
⎢⎣ S 2,r ( x, y )⎥⎦ ⎢⎣ I 67.5 ( x, y ) − I 157.5 ( x, y )⎥⎦
(102)
Equation 102 is simply a reduced and rotated version of Equation 1. Figure 156
demonstrates these edge effects due to the parallax induced by the super pixel
architecture.
Figure 156 – Pre-corrected linear polarization images of a plastic container using the 6x6 super pixel
WGP/FPA design.
Again, the word “RELEASE” is plainly evident in the unpolarized image, S0,r, of the
plastic container. As expected, the bulk of the S1,r and S2,r images have a value of zero,
given that the scene is unpolarized. The super-pixel design of the wire-grid polarizer
exhibits issues in image registration for Stokes vector determination. When the
polarization state of a scene is being quantified, the spatial irradiance distribution of a
particular polarization state of a scene must be directly registered to its orthogonal partner
in the Stokes vector basis.
Detecting the polarization irradiance from the same scene with spatially separated
pixels induces a slight change in perspective for each detection site in the super pixel.
That is, each polarization image will view the scene from a different perspective, creating
image registration issues. When attempting precision Stokes polarimetry with super pixel
detection, errors in the detected Stokes parameters at the edges of objects shall be
observed do to this lack of common perspective. In the pre-calibrated scene presented
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above, an average degree of polarization of 1.74% was for the S1,r image and 1.95% for
the S2,r image. The edge effects from the lack of perspective as well as slight transmission
differences between orthogonal detection states are the culprits for the mean degree of
polarization.
Though the field-portable system was generated to take data in external and obscured
conditions, no extended range images have been acquired by the field system at this time.
Mother Nature can be uncooperative at times.
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CHAPTER 6: CLOSING REMARKS
From the work presented in this dissertation, a proof-of-concept system was generated
and tested that exhibited the ability to register orthogonally polarized images on the same
focal plane using polarized active illumination. The acquired images were subsequently
processed by novel algorithms to provide large contrast enhancements and correlation
coefficients relative to intensity imaging for both depolarizing objects and objects that do
not appreciable depolarize incident radiation. Orthogonally polarized image collection on
the same focal plane and computationally non-intensive algorithms lend themselves to an
agile system that can acquire and process images in real time that provide drastic
increases in image contrast at all spatial frequencies normally detectable by the optical
system.
A novel wire-grid polarizer array structure and alignment technique was derived to
bond a wire-grid polarizer (WGP) array directly to a camera’s focal plane array (FPA) to
produce a field system that mirrored the lab-based system’s ability to acquire and process
data in real time. The field system eliminated the forward optical elements that drove the
design of the lab-based system by acquiring orthogonally polarized images on a pixel by
pixel basis using a 6x6 super pixel design. Unfortunately, field data has not yet been
taken using the bonded chip linear polarimeter.
In addition to the sensor and algorithm developments presented in this dissertation,
experimental image data of remote objects within scattering media, yielded important
conclusions about the role of object depolarization in the choice of active illumination
and detection polarization states. For illumination in the visible through water vapor
(fog), active circularly polarized illumination and detection produces higher contrast
images than linear for objects that maintain incident degree of polarization. Conversely,
active linearly polarized illumination and detection produces higher contrast images for
depolarizing objects.
It is also important to present a quick comparison to the use of contrast stretching
commonly used in image processing. Figures 157 and 158 demonstrate that even though
contrast stretching will improve the contrast of the intensity image, it actually produces
197
lower correlation to the actual intensity image and the resultant contrasts, through the use
of common contrast stretching, are not as good as the ones using the presented
techniques.
Figure 157 – (a) Unobscured image of the metal wrench given in Figure 56(b). (b) Obscured image of
metal wrench through an optical depth of 3.81. (c) Post-processed image of the image seen in (b). (d) Image
seen in (b) with a contrast stretch applied.
Figure 158 – Correlation coefficients with respect to the unobscured image of the metal wrench for the
cross sections defined in the images presented in Figure 157.
The experimental process through which the raw images were acquired and the
optical depth was tracked was not digitally synced. As such, the optical depth
198
measurements for each presented image include error that was not able to be accurately
quantified. The extended radiometer also included error in the determination of the
optical depth. Non-ballistic photons that needed to be rejected added to the photodetector flux of the radiometer, yielding an underestimation of the optical depth. The
Monte Carlo simulation was used to attempt to correct for this effect, but geometric
measurements required to sufficiently correct for the underestimation were rough, at best.
Even with loose quantification of the optical depths through which imaging was
conducted; the presented system provides the means to image at remote distances through
thick obscurant media in real time. This leads to the conclusion that the techniques
presented here provide a system that can see farther in the same optical thickness or
through the same distance at thicker optical depths relative to intensity imaging. The
increase in imaging range through an obscurant of the same optical depth relative to
intensity imaging was difficult to quantify experimentally, because the enclosure used to
trap the water vapor obscurant saturated at a specific optical depth where a steady-state
situation existed where the vapor diffusing through the container equaled the vapor
entering the container. At this optical depth, image contrast sufficient to still detect the
presence of an object and some of its features was produced through use of the presented
algorithms. Still, by inspection of the experimental results shown in Chapter 4, it is
evident that sufficient contrast exists well into optical depths of 5 or more, lending the
results to an increase in imaging range through obscurant media by a factor of 2 or more.
I
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