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Experimental studies on microwave detection and imaging of targets in clutter using correlation techniques

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Experimental Studies on Microwave Detection and Imaging
of Targets in Clutter Using Correlation Techniques
by
Tsz K. Chan
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
University of Washington
1998
Approved by.
Chairperson of Supervisory Committee)
Program Authorized
to Offer Degree__
Date.
£ . (iff
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UMI Number: 9836149
Copyright 199 8 by
Chan, Tsz King
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© Copyright 1998
Tsz K. Chan
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University of Washington
Abstract
Experimental Studies on Microwave Detection and Imaging of Targets in
Clutter Using Correlation Techniques
by Tsz K. Chan
Chairperson of Supervisory Committee
Professor Yasuo Kuga
Department o f Electrical Engineering
Recently, the detection and imaging o f targets in the presence of strong clutter has become
an increasingly important research area in environmental science. The major momentum
behind of the proliferation of this research discipline is the growing public awareness
of environmental issues such as the detection of abandoned land mines and imaging of
terrain and vegetation features. Since remote sensing for these environmental missions
routinely requires electromagnetic (EM) radiation probing through natural stratified media
and since most existing sensing tools (such as radars and their variants) are prone to passive
interference (such as scattering of incident EM energy) from the media themselves, the
call for an effective subsurface sensing tool for these detection and imaging applications
has become an increasingly urgent task faced by radar engineers.
This dissertation details a three-year-long research effort performed at the University
of Washington. Specifically, this research focuses on the applications of a novel corre­
lation phenomenon in various target detection and imaging problems in environmental
science. The technique examined in this dissertation makes uses of the complex angular
correlation function (ACF) and/or frequency correlation function (FCF) measurement to
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achieve enhanced visibility contrast for targets embedded in strong-clutter environment.
Experimental research is conducted at both millimeter-wave (75-110 GHz) and X-band
(7-13 GHz) frequencies. The results clearly demonstrate the relative effectiveness of this
correlation technique over conventional techniques in various detection and imaging ap­
plications. Finally, a list of research topics is appended at the end of this dissertation to
define the framework for the scope and depth o f further investigation on this topic.
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TABLE OF CONTENTS
List of Figures
iv
List of Tables
Chapter 1:
xiii
Introduction
1.1 Environmental applications of radars
1
.................................................................
3
...................................................
4
1.3 Synthetic aperture radar as an imaging t o o l .......................................................
6
1.2 Ground-penetrating radar as a detection tool
Chapter 2:
Angular Memory Effect
12
2.1
Angular memory effect in the 0 - p la n e ................................................................
12
2.2
Angular memory effect in the c j-p la n e ................................................................
16
2.3
Correlation peaks and second-order Kirchhoff approximation
.....................
23
2.4
Applications of angular correlation fu n c tio n ......................................................
26
2.5 Other forms of correlation fu n c tio n s....................................................................
27
Chapter 3:
3.1
Correlation Technique in Target Detection
Target detection at millimeter-wave freq u en cies................................................
28
29
3.1.1
Experimental s e tu p ....................................................................................
32
3.1.2
Experimental r e s u l t s .................................................................................
36
3.2 Target detection at X-band frequencies
.............................................................
37
3.2.1
Experimental s e tu p ....................................................................................
37
3.2.2
Experimental r e s u l t s .................................................................................
41
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3.3
Summary of target detection using correlation technique
Chapter 4:
4.1
43
Correlation Technique in TargetImaging
44
3-D imaging using circular S A R .........................................................................
46
4.2 Circular SAR processing a lg o r i th m ...................................................................
49
4.3 Generalized ambiguity function of circular S A R ............................................
53
4.3.1
Analytic fo rm u latio n ................................................................................
53
4.3.2
Numerical calcu latio n s.............................................................................
59
4.3.3
Experimental r e s u l t s ................................................................................
64
4.4 Experimental studies on 3-D imaging using circularS A R .............................
64
4 .4 .1 Experiment A: Confocal reconstruction of layers of spheres . . . .
4.5
4.6
4.4.2
Experiment B: Confocal reconstruction of a single s p h e r e ......
4.4.3
Experiment C: Confocal reconstruction of a model helicopter
Clutter suppression using correlation im a g in g ...........................................
69
...
73
73
4.5.1
Conventional SAR p ro c e ss in g ...............................................................
77
4.5.2
Frequency-correlation SAR p ro c e s s in g ...............................................
77
4.5.3
Angular-correlation SAR processing......................................................
78
4.5.4
Experimental comparison
......................................................................
87
Summary of target imaging using correlation technique..................................
95
Chapter 5:
5.1
67
Conclusion
100
S u m m a r y ....................................................................................................................100
5.2 Further studies
..........................................................................................................103
Bibliography
Appendix A:
105
System Calibration
109
ii
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Appendix B:
Chirp radar
114
B .l
Case A: Infinite pulse d u ratio n ................................................................................115
B.2
Case B: Finite pulse duration..................................................................................118
Appendix C:
Determination of reference slant range
123
C .l
Analytic fo rm u la tio n ...............................................................................................123
C.2
Experimental results
.........................................................................................124
C.2.1 Experiment A: Single sphere along x - a x i s ........................................... 125
C.2.2 Experiment B: Single sphere along y - a x i s ........................................... 125
Appendix D:
MATLAB codes for circular SAR processing
130
D .l
Main p ro g ram ............................................................................................................130
D.2
Other su b ro u tin e s.....................................................................................................130
iii
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LIST OF FIGURES
1.1 The simplified block diagram of a typical ground-penetrating radar. . . . .
5
1.2 The operation of an airborne SAR system. The cross-range resolution
(along the x-axis) is given by the antenna size and down-range resolution
(along the y - axis) is given by the measurement bandwidth (or pulsewidth)
of the system.............................................................................................................
1.3
7
Fourier-transform relationship between near-field and far-field variations:
broad near-field variation transforms to narrow far-field variation,
(a)
Pencil beam pattern resulting from a broad circular near-field pattern, (b)
Fan beam pattern resulting from a rectangular near-field pattern. Sidelobes
of these beams are not shown in this figure.......................................................
2.1
9
The scattering geometry of angular memory effect in 0-plane. The neces­
sary condition for strong angular correlation is governed by the generalized
Snell’s law: s in d\ —s in Oi = s in 6's —s in Qs .....................................................
2.2
13
Plot of Eq. 2.2 on the sin(0|) —sin(0^) plane. The angular memory line
and scan line intercept at the reference point (sin( 0i),s in ( 0 ,)) and are, by
definition, perpendicular to each other.
2.3
............................................................
15
Field equivalence under reciprocity condition, (a) the original scattering
situation: an a-polarized transmitter transmitting at 0T and a ^-polarized
receiver receiving at Or . (b) the reciprocal version of the original scattering
geometry: a /^-polarized transmitter transmitting at Or and an a-polarized
receiver receiving at Or...........................................................................................
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17
2.4 The scattering geometry of angular memory effect in the <p-plane. At con­
stant ^-incidence, the resulting “plane” of incidence traces out a conical
surface in 3-D space................................................................................................
2.5
19
The signature of 0 -plane ACF as a function of reference angles (o t . <ps ) .
“i” stands for reference incident angle <&, “s” stands for reference scattered
angle cps. x - and y-axes represent <z>- and o s, respectively, in degrees.
2.6
. . 22
(a) Comparison between experimental and analytical ACF signatures across
the narrow contours in Fig. 2.5 for the case (0 t,0 s) = (0°, 180°), and (b)
Comparison between experimental and analytical ACF signatures across
the wide contours in Fig. 2.5 for the case (&, 0 S) = (0°, 180°)..........................24
2.7
The three major components in the second-order Kirchhoff Approximation:
(a) first-order scattering: single bounce, (b) second-order scattering: iden­
tical doubly-bounced signals resulting from identical propagation paths, (c)
second-order scattering: identical doubly-bounced signals resulting from
time-reversed propagation paths.............................................................................
3.1
25
A far view of the millimeter-wave system used in the experiments. System
specifications of this advanced vector scattereometer are fully documented
in [23].........................................................................................................................
3.2
30
A close view of the same system shown in Fig. 3.1. In this figure, the trans­
mitting antenna (on the left hand side) is covered with a servo-mechanical
polarizer. The receiving antenna (on the right hand side) is designed to
receive both copolarized and cross-polarized scattered signals............................31
3.3
ACF magnitude of the selected natural media at millimeter-wave frequen­
cies (95-100 GHz) for (0j,0s) = (20°, —20°), along (solid line) and per­
pendicular to (dotted line) angular memory line: (a) fine sand, (b) rough
sand, (c) gravel, and (d) rock..................................................................................... 34
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3.4
Target detection with spatial and angular scans with (9i: 9S) = (20°, —20°).
Spot size of the footprint « 30A............................................................................
3.5
35
(a) ACF magnitude as a function of footprint: (9i,9s) — (2 0 °,—20°)
and {O'^Q's) = (10°. —10°). (b) RCS as a function of footprint: (O'^d’s) =
(6i, 9S) = (20°, —20°). It is clear from this comparison that ACF technique
results in higher target visibility contrast than RCS technique............................ 36
3.6
The top view of the X-band bistatic target detection system. In the course
of the experiments, the system scans over the composite target-soil medium
contained in the sand box.......................................................................................
3.7
39
Frequency dependence of the dielectric parameters o f the soil medium
used in the experiment: (a) attenuation constant, (b) relative permittivity
constant, (c) ACF magnitude along (solid line) and perpendicular to (dotted
line) the angular memory line for {9U9S) = (2 0 °,-2 0 °) and (d) ACF
magnitude along (solid line) and perpendicular to (dotted line) the angular
memory line for (9i, 6s) = (20°. —4 0 ° )................................................................... 40
3.8
Object buried at 6 cm below the surface with (9Z,9S) = (30°,—30°): (a)
radar cross section as a function of footprint. (0-,0^) = (30°,—30°), (b)
ACF magnitude as a function of footprint. {9\,9's) = (50°,—50°). Object
placed above the surface with (9i,9s)
=
(30°,—30°); (c) radar cross section
as a function of footprint. (9'i ,9's) = (30°,-30°), (d) ACF magnitude as a
function of footprint. (9^,9^) = (50°,—50°).............................................................. 42
4.1
(a) The geometry of spotlight-mode linear SAR. (b) Altitude ambiguity
caused by propagation paths of equal lengths (dx = d?)........................................ 45
4.2
(a) The geometry of spotlight-mode circular SAR. (b) Altitude ambiguity
resolved by propagation paths of different lengths(dx^ d2).................................. 47
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4.3
Different window functions are used at different data-acquisition positions
in order to capture the “clean” responses from target located at the focal
point............................................................................................................................
4.4
Geometry of a circular SAR system.
Focal point r Q being located at
(x0, ~ h 0)....................................................................................................................
4.5
51
55
Magnitude variations of the generalized ambiguity function x (T ,r0) pro­
jected along the x - , y -, and 2-axes. Bandwidth: 7-13 GHz.
Sweep
time = 4 s. Depth h 0 of the focal plane = 1 m below the flight track.
Depression angle 6dp = 45°. r c = (0, —h0).........................................................
4.6
61
Pixel resolution (in terms of wavelength) along the x —, y—, and 2 -axes as
a function of depression angles 6dp for circular SAR system. Bandwidth:
7-13 GHz. Sweep time = 4 s. rQ= (0, —h), where h — a x tan(0dp). . .
4.7
62
Pixel resolution (in terms of wavelength) along the x - , y- , and 2-axes as
a function of depression angles Qdp for linear SAR system. Bandwidth:
7-13 GHz. Sweep time = 4 s. r 0 = (0, —h ), where h = a x ta n (9dP). ■ ■ 63
4.8
2-D circular SAR image of a conducting sphere in free space. The sphere
was located at a distance of 15 cm (or 5A at a center frequency of 10 GHz). 65
4.9
1-D extraction of data points along the x-axis that contains the bright
image in Fig. 4.8.
The 3dB peak width of the generalized ambiguity
function of circular SAR system is approximately 0.25A.................................
66
4.10 Schematic for 3-D confocal imaging of layers of metallic spheres using
circular SAR. Bandwidth: 7-13 GHz..................................................................
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68
4.11 Experimental result for the imaging experiment shown in Fig. 4.10: (a)
no sphere, corresponding to “layer (a)” in Fig. 4.10, (b) one sphere, cor­
responding to “layer (b)” in Fig. 4.10, (c) three spheres, corresponding to
“ layer (c)” in Fig. 4.10 and (d) no sphere, corresponding to “layer (d)” in
Fig. 4.10......................................................................................................................
70
4 .12 Schematic for 3-D confocal imaging of a metal sphere suspended in free
space using circular SAR. Bandwidth: 7 -1 3 ........ G H z......................................
71
4.13 Experimental result for the imaging experiment shown in Fig. 4.12. The
image is displayed as a stack of uniformly spaced (vertically) contours.
The apparent dark line that connects the top and bottom contour layers is
produced by the PLOT3 routine in MATLAB and has nothing to do with
the actual image.........................................................................................................
72
4.14 Schematic for 3-D confocal imaging o f a palm-sized model helicopter
suspended in free space using circular SAR. Bandwidth: 7-13 GHz. . . .
74
4.15 Experimental result for the imaging experiment shown in Fig. 4.14. The
image is displayed as distributed clusters o f dots for enhanced visibility
about its details. Note that the comer structures at the helicopter tail result
in the corresponding strong reflection in the image. On the other hand,
the longitudinal dimension of the image agrees with the actual length of
the h elico p ter............................................................................................................
75
4.16 Schematic of bandwidth partitioning in the frequency-correlation SAR for
(a) 4 partitions and (b) 8 partitions........................................................................
79
4.17 Schematic of angle partitioning in the angular-correlation SAR for (a) 3°,
(b) 30°, (c) 45°, (d) 90°, (e) 120° and (f) 150°. Angles are not drawn to
scale.............................................................................................................................
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81
4.18 Conventional SAR image of a 63 mm conducting sphere on top of absorber
material.......................................................................................................................
83
4.19 Angular-correlation SAR image of a 63 mm conducting sphere on top of
absorber material. The image was processed with a decorrelation angle of
2°. The bright annular ring results from the highly-correlated terms in the
summation mechanism in Eq. 4.41.......................................................................
84
4.20 Angular-correlation SAR image of a 63 mm conducting sphere on top of
absorber material. The image was processed with a decorrelation angle of
20°. Note that the previous bright annular image in Fig. 4.19 has dimmed
significantly as a result of using large decorrelation angle (i.e. 20 °) in this
case..............................................................................................................................
85
4.21 Angular-correlation SAR image of a 63 mm conducting sphere on top of
absorber material. The image was processed with multiple decorrelation
angles of 20°, 40° and 60°.
Note that the previous annular image in
Fig. 4.20 has disappeared almost completely. This image demonstrates
the importance of using large number of large decorrelation angles in
angular-correlation SAR processing in low-clutter environment......................... 86
4.22 Experiment A:
a
schematic for microwave imaging in medium-clutter en­
vironment using different kinds of SAR processing methods. The size of
spheres (diameter = 25 mm) is much larger than the size (mean diameter
ss 3 mm) of the gravel particles............................................................................
88
4.23 Experiment A: conventional SAR processing continues to produce faithful
image of the scene in medium-clutter environment. The two spheres are
indicated unambiguously by the bright dots in the Figure.....................................89
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4.24 Experiment A: frequency-correlation SAR processing results in an im­
age strikingly similar to that produced by conventional SAR processing.
This mage demonstrates that the benefit of applying frequency correlation
technique is not obvious in not only low-clutter environment, but also
medium-clutter environment...................................................................................
91
4.25 Experiment A: angular-correlation SAR processing results in an image
strikingly similar to that produced by conventional SAR processing. This
image demonstrates that the benefit of applying angular correlation tech­
nique is not obvious in not only low-clutter environment, but also mediumclutter environment..................................................................................................
92
4.26 Experiment B: a schematic for microwave imaging in heavy-clutter envi­
ronment using different kinds of SAR processing methods. The size of
spheres (diameter = 25 mm) is about the same as that (mean diameter ~
30 mm) of the gravel particles..............................................................................
93
4.27
Experiment B: a top view of Fig. 4.26..............................................................
94
4.28
Experiment B: conventional SAR
processing failstodisplay thecorrect
image of spheres in heavy-clutter environment.Sphere
size = 25
mm,
mean gravel particle size s; 30 mm, bandwidth = 7-13 GHz............................. 96
4.29 Experiment B: frequency-correlation SAR processing results in an im­
age strikingly similar to that produced by conventional SAR processing.
Again, frequency-correlation SAR processing fails to display the correct
image of spheres in heavy-clutter environment. Sphere size = 25 mm,
mean gravel particle size « 30 mm, bandwidth = 7-13 GHz. This im­
age demonstrates that in heavy-clutter environment, frequency-correlation
SAR processing may not be an effective means for clutter suppression. . .
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97
4.30 Experiment B: angular-correlation SAR processing results in an image that
correctly accounts for the presence of the spheres. In fact, it is the only
means among other SAR processing schemes examined in this investiga­
tion that brings clutter level down to a reasonably low level, leading to
clear visibility of spheres in the presence of strong clutter. Sphere size =
25 mm, mean gravel particle size ~ 30 mm, bandwidth = 7-13 GHz. This
images demonstrates that in the presence of strong clutter, angular corre­
lation SAR processing is an effective tool of achieving a higher degree
of clutter suppression compared with the conventional SAR technique,
resulting in a higher target-to-clutter ratio...........................................................
98
A. 1 ACF signature measurement of a large tilted metal flat plate. The way
the antennas are moved describes an angular memory line for (Oi,d3) =
(30°. —30°) on the s i n ^ - s i n ^ plane. As expected, the correlation level
approaches to 1 over a wide range of variable incident angles 6\
B .l
110
Frequency-time plot of transmitted pulse, received pulse, and the de­
chirped pulse. The net effect of this chirping/de-chirping process is to
compression of a T-long pulse into a 1/B -long pulse, allowing fine spa­
tial resolution................................................................................................................ 116
B.2 Delay-time characteristic o f a de-chirping filter matched to the transmitted
chirp pulse, assuming linear FM modulation......................................................... 117
B.3 Plot of Eq. B.22 with sweeping time T = 4 s and chirp bandwidth B = 6
GHz. The 3dB width of the de-chirped signal is about 1.057I B ...................... 122
C .l
Experimental determination of reference slant range. Ideally, the center of
rotation should be the same as the center of illumination (except for the
obvious altitude difference)....................................................................................... 127
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C.2
Experimental setup for determination of reference slant range in which a 63
mm conducting sphere, located at an altitude o f h below the circular SAR
flight track, was placed at a distance of 15 cm along the ar-axis relative
to the center of rotation. Three independent reflection measurements were
made to solve for variables p, (f> and h ................................................................ 128
C.3
The relative locations of the calculated center of illumination and the
ideal center of illumination. The negligible spatial discrepancy verifies
the superb construction accuracy of the circular SAR system used in this
experimental research................................................................................................. 129
D. 1
The structural components of CORRSAR.M....................................................... 131
D.2
MATLAB code of CORRSAR.M............................................................................ 132
D.3
MATLAB code of CORRSAR.M, continued from Fig. D.2
.......................... 133
D.4
MATLAB code of CORRSAR.M, continued from Fig. D.3
.......................... 134
D.5
MATLAB code of CORRSAR.M, continued from Fig. D.4
.......................... 135
D .6
MATLAB code of GETDATA.M
D.7
MATLAB code of V O L S IZ E .M ............................................................................ 137
D .8
MATLAB code of F C U S G R ID .M .........................................................................138
D.9
MATLAB code of L E N S G R ID .M .........................................................................139
.........................................................................136
D. 10 MATLAB code of R A D E X .M ............................................................................... 140
D . l l MATLAB code of S C R E E N IN F O .M .................................................................. 141
D.12 MATLAB code of K A ISE R .M ............................................................................... 142
D.13 MATLAB code of GATING.M ............................................................................... 143
D. 14 MATLAB code of C H O P B A N D .M ...................................................................144
D. 15 MATLAB code of F R E Q _A C F.M ...................................................................... 145
D.16 MATLAB code of T IM E _ A C F .M .........................................................................146
D.17 MATLAB code of N O R M A L .M ............................................................................147
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LIST OF TABLES
3 .1
Dimension and absorption characteristics of the four natural media used
in the millimeter-wave experiments
4.1
.................................................................
32
Specification comparison between linear SAR and circular S A R .................
48
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ACKNOWLEDGMENTS
To come to the pleasure you have not you must go by a way in which you
enjoy not. - St. John o f the Cross (1542-1591)
Before I express my wholehearted appreciation to a number of individuals in this
section, I would like to take up some space here to share with you, my dear readers,
something invaluable I learned about experimental research succinctly summarized by St.
John of the Cross’s counsel on one’s pilgrimage centuries ago.
Experimental research such as the one outlined in this dissertation, by its very timeconsuming nature, is an endeavor no less demanding than any other research areas. In the
course of my 5-year training in the Ph.D. program at the University o f Washington, I have
the privilege of sailing through a number o f “dark nights” (both physically in laboratory
and psychologically in mind) during my sojourn over “research plateau” (a term coined
by Prof. Ishimaru) where I hardly foresaw further breakthroughs ahead. The reason why
I consider it a privilege (as opposed to agony!) to experience dark nights is that although
the dark nights are both unpleasant and inevitable along any non-trivial research path,
they contain the necessary nutritious ingredients for a research novice to grow from being
dependent to being independent in capability, from being naive to being experienced in
skills, and from being arbitrary to being thoughtful in conclusion making. O f tremendous
benefits to him during dark nights, such novice therefore should ask and continue to ask
even though there seems to be no answer, seek and continue to seek even though there
seems to be no solution, and knock and continue to knock even though the door seems
to remain shut. Very often at a certain point along this long journey of work in patience
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and steadfastness, however, a kind o f elegant inspirtation suddenly sparks, lighting up all
previous dark nights in much the same way as the sun’s radiance breaks into the dark at
daybreak. After all, perseverance is the lead to an once hidden answer, the map for an
once overwhelming labyrinth and the key to an once shut door.
I wish to express my deepest appreciation to all members, present and past, of the
Electromagnetics and Remote Sensing Laboratory (ERSL) at the University o f Washington.
In particular, special thanks must first go to Professor Kuga for his guidance and financial
support. Throughout my years in the program, he has been the single most influential
mentor to me who constantly enriched and expanded my research interests with his new
insights and unparallelel experimental expertise. On many occasions, in particular, his
heuristic approach to technical problems and broad readership of technical literature have
been such a great help to me - much like a lighthouse to the ships wandering in the
sea at night, saving me from stumbling over many potential “research potholes” invisible
to a naive researcher like me. These potholes are diverse in nature and include from
re-inventing the wheel, searching randomly in the dark, to majoring in minor ideas while
missing the whole picture altogether. In short, he has been a timely “rescue” to me on
a number of occasions that could have infectiously eclipsed my research impetus and
progress in the future.
In physical chemistry, a catalyst is a substance that speeds up a chemical reaction by
providing an alternative reaction path of lower activation (start-up) energy. By the same
token, if Professor Kuga’s guidance and my work are considered as reaction components,
Professor Ishimaru’s insightful counsel is definitely the catalyst responsible for many
exothermic “research reactions” over the years in the past. While his legendary counsel is
inseparable from his being an authoritative figure in the field, his smile and kind advice are
also inseparable from his being a great nurturing advisor to young researchers - feeding
them with encouragement, implanting them with sound research philosophy and finally
xv
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letting them grow as mature and independent minds of various personal styles. His artistic
way of doing research and bird’s eye view of entangling technical issues have constantly
surprised me by joy. To me, his office has always been synonymous with “powerhouse” a place where one could regain personal and technical momentum to walk through curtains
of uncertainties with calmness and confidence.
Furthermore, I want to acknowledge the contributions by many undergraduate students
who unreservedly dedicated their time and talent to my research. They not only helped me
identify and solve a wide range of technical issues “behind the scene”, but also accepted
and carried on uncomplainingly many of my demanding requests. I truly owed them a lot.
Over the years, there had been many helpful students who facilitated my work in one way
or another. Of particular significance to my research among these talented individuals are
(1) Canh Kha, (2) Kenneth Pinyan and (3) Hatim Saleem. Mr. Kha is a professional
whom I enjoyed chatting and working with him very much. He was always a good source
of humor which I benefited most, especially in times of research obstacles. On the other
hand, Mr. Pinyan is an all-rounder specializing in a wide spectrum of skills, ranging from
heavy-duty work like metal machining to Iight-duty work like computer programming. In
fact, this research will remain largely incomplete without the circular SAR system that
he constructed with intense dedication over months of hard work. Last but not least, Mr.
Saleem is another gifted individual whom I owed a lot. His uncomplaining work attitude
and inquisitive personality have enabled greatly my research, making my work to proceed
in the smoothest possible extent.
Finally, I would like to express my gratitude to the following fellow classmates, both
former and present, for their friendship and support. Their presence is truly essential to
the my research life at ERSL: Jun-Ho Cha, Chi-Te Chen, Todd Elson, Leibing Huang,
Bertin Koala, Charlie Le, Seung-Woo Lee, Qin Li, Chien-Min Lin, Jun Liu, Garfield
Mellema, Katsuhiro Ono, Kyung Pak, Christopher Penwell, Phillip Phu, John Rockway,
xvi
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Lynn Sengers, Geoffrey Wang, Ji-hae Yea, Cynthia Young, Guifii Zhang, and Hui Zhao.
This alphabetically sorted list is by no means exhaustive and I deeply apologize to those
whom I miss. Before I close this section, I must reserve a unique space for our ERSL’s
program coordinator: Ms. Noel Henry. From the date I joined this group till now, she
has been a great helper to many of us in the group. Her office is a truly wonderful
place for at least 3 good reasons. First, it is the place where students can obtain accurate
information such as submission deadlines, reimbursement policy, and travel arrangements.
The way she organizes information enables her to handle all these repetitive requests
with grace and meekness, even at the peak of busy periods. Secondly, it is the place
where faculty can rely on for accurate information when it comes to the time for financial
and research issues - anything from proposal deadlines, contact information, to budget
handling. Finally, perhaps the most important reason of all, it is the place where both
students and faculty can seek emotional support and replenishment - her advice, help,
smile, humor and encouragement all add up to make her office a uniquely helpful place
for us to stop by. I am truly thankful for her unconditional dedication to everyone of us.
xvii
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DEDICATION
This dissertation is unreservedly dedicated to my parents, Lap Chan and Miu-King
Lam-Chan, for their unconditional love and unselfish sacrifice to me since my childhood.
Without their emotional support and care over years, this dissertation would not have been
a reality. Even though they do not understand even one single equation or plot in this work
at their education level, they understand and make me understand the transforming power
of enduring love. The following verses (from I Corinthians 13) are reverently presented
to honor their love for me:
Love is patient, love is kind.
It does not envy, it does not boast, it is not proud.
It is not rude, it is not self-seeking,
it is not easily angered, it keeps no record of wrongs.
Love does not delight in evil but rejoices with the truth.
It always protects, always trusts, always hopes, always perseveres.
Love never fails.
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Chapter 1
INTRODUCTION
Since its introduction in early 40’s, radar, an acronym name for “fladio Detection And
hanging” , has been in extensive use in a wide range of applications. Simply stated, a radar
system consists of a directive antenna which serves as an interfacing device between the
region under surveillance and the internal transmitter/receiver electronic circuitry system.
A radar operates by radiating a sequence of bursts of electromagnetic energy into the region
illuminated by the antenna beam. If there is a reflective target within the illuminated
volume, it will absorb a fraction of the incident energy and scatter the rest of the energy
in other directions. Most radars deployed for remote-sensing applications operate at the
frequency range of 0.1-10 GHz. At these frequencies, electromagnetic radiation exhibits
long penetration depth into most natural media such as clouds, light rain, vegetation
and snow. In the case of a monostatic radar, only the portion of the scattered energy
observed in backscattering direction is received. Assumming point scatterer and freespace propagation, the relationship between the received power PR and the transmitted
power Pt is given by the well-known radar equation [21, 25]
Pr
PT
C iG a A2
{ A -K fR \p f
....
K }
where G\ is antenna gain of the transmitter (a measure of how spatially directive the
transmitting antenna emits power, Go the antenna gain of the receiver (a measure o f how
spatially directive the receiving antenna receives power), A the wavelength in free space
at the operating frequency, R i the range from the transmitting antenna to the scatter, Ro
the range from the scatter to the receiving antenna, and finally, a the radar cross section
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2
(RCS) of the illuminated target.
From Eq. 1.1, it is important to note that the received power PR decreases rapidly as
ranges (R \ and Ro) increase (i.e. received power oc 1/distance4). As a result, for longrange surveillance applications it is necessary to have a powerful transmitting antenna
that is capable of sending high-power (on the order of 106 watts) microwave radiation
into free space. Note that Eq. 1.1 expresses the return power from a point target while
neglecting propagation, polarization and other system losses. Modifications for Eq. 1.1
become necessary in practical systems to account for scattering by distributed targets such
as clouds, rough surfaces and canopy layers. Note also that in the case of monostatic radar
systems, it is the part of the energy backscattered to the radar antenna that constitutes the
observed echo, from which one can deduce the following parameters useful for target
identification and characterization [3, 25]:
• range,
• radial velocity,
• angular direction,
• size, and
• shape.
The history of radar development spanned over half of a century, with applications ex­
tending from military surveillance to environmental sensing. A short list of this eventful
evolution process is presented chronologically below [21 ]:
• Early radars
1. 1922 - Continuous-wave (CW) radar
2. 1934 - Pulse radar
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3
3. 1940 - Airborne radar
4. 1946 - Imaging radar
• Imaging radars
1. 1952 - Side-Looking Airborne Radar (SLAR)
2 . 1966 - Extensive mapping by SLAR
3. 1952 - “Dopper Beam Sharpening” system
4. 1958 - Synthetic Aperture Radar (SAR)
• Space radars
1. 1973 - Skylab scattereometer (non-imaging)
2 . 1978 - Seasat SAR and scattereometer
3. 1981 - Shuttle Imaging Radar (SIR-A)
4. 1983 - Shuttle Space Lab (European) SAR
5. 1984 - Shuttle Imaging Radar (SER-B)
6 . 1990 - Shuttle Imaging Radar (SIR-C)
7. 1993 - Earth Observing System (EOS) SAR
I.I
Environmental applications o f radars
During the past 50 years of research and development, radar technology has witnessed
significant growth in capability, variety, and functionality. Presently its applications are
diverse and penetrate deep into many aspects of our daily life, ranging from military
surveillance and weapon control to civilian navigation for air/sea traffic safety, law en­
forcement, weather forecasting [2], and remote sensing of geological media [12, 28, 30].
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4
Although radar continues to be an indispensable remote-sensing tool for modem mili­
tary and civilian applications, it is the evolution from these applications to environmental
applications, which calls for superb subsurface sensing capability, that has been injecting
momentum to much of the radar research in recent years [22]. Examples of these envi­
ronmental applications include detection of trenches, landfill debris, grave sites, chemical
spills, nuclear waste, underground utility lines, abandoned active land mines, as well as
imaging of terrain and vegetation regions. As a result of the significant impact of these
critical issues to our environment, the call for an effective subsurface sensing tool for
these detection and imaging applications has become an increasingly urgent task faced by
radar engineers.
Depending upon the functions it performs, a remote-sensing radar could be categorized
either as a (1) detection radar and (2) imaging radar. From an application point of view,
detection radars usually deal with those issues where spatial resolutions for revealing
details of targets are of secondary priority. However, imaging radars deal with those
issues where fine spatial resolutions are necessary for accurate target identification. In
subsequent sections of this chapter, working principles of ( 1) ground-penetrating radar
(GPR) as a detection tool and (2) synthetic aperture radar (SAR) as an imaging tool will
be addressed.
1.2
Ground-penetrating radar as a detection tool
GPR was invented in the early 70’s for detecting engineering and environmental targets
in the upper 10 m ground layer of the earth [22]. When used in practice, GPR’s operate
in close vicinity of the soil-air interface. Essentially, a GPR is similar to a time-domain
reflectometer which transmits high-power transient pulses into the ground and receives
the reflected transient echoes from the region under surveillance, as shown in Fig. 1.1.
With reference to Fig. 1.1, a GPR system is usually towed continuously by a vehicle
over the ground at uniformly spaced intervals. The distance between consecutive intervals
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5
transient
source
transient
receiver
signal
processing
an tennas
ground surface
\
\
i
i
*
i
i
i
/
i
buried target
Figure 1.1: The sim plified block diagram o f a typical ground-penetrating radar.
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6
is called the trace spacing and is usually less than 0.3 m. Each time the antennas traveled
over one of these intervals, the GPR performs the following sequence of operations [22]:
1) the transient source emits a pulse into the ground, 2) the transient receiver is then turned
on to wait for the radar echo from the ground, and 3) after a brief period of time (usually
less than 1 p s) the receiver is turned off. The time-domain information that is recorded
while the receiver is turned on is called a trace. Such a trace can be used to determine
the a) size, b) shape, and c) depths of targets buried under the illumination spot on the
ground surface. By mapping these recorded time-domain traces for each interval into their
space-domain equivalent via a knowledge of certain medium parameters (e.g. dielectric
constant), a two-dimensional RCS plot of the area under ground can be constructed.
Finally, it is important to realize that simple as the concepts of GPR’s are, however,
reliable detection and clear image construction are often undermined by practical imple­
mentation challenges, which include careful antenna design to avoid mutual coupling and
rejection to interference due to scattering by random rough surfaces.
1.3
Synthetic aperture radar as an imaging tool
In contrast to the small-scale illumination area achievable by GPR’s, SAR’s are used for
surveillance over a large-scale illumination area [11, 18, 31], as shown in Fig. 1.2. In
practical environmental imaging applications, SAR technology makes use of airborne or
spacebome radars for terrain mapping. To understand the operation of SAR, one may
begin with an examination at how a radar of given shape directs its power in space [21],
From antenna theory, it is a well-known fact that near-field variation and far-field
variation bear a Fourier-transform relationship. This succinct Fourier relationship is a
direct result of applying far-field approximation in the Green’s function formulation for
antenna radiation [13]. In view of this result, a radar of wide aperture (i.e. broad nearfield antenna pattern) produces a narrow far-field antenna pattern, and vice versa. In
Fig. 1.3 two common types of radar are illustrated. In Fig. 1.3(a), the near-field antenna
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7
v # -
&
►y
(down-range)
/
/ pixel
footprint
x (cross-range)
Figure 1.2: The operation o f an airborne SAR system. The cross-range resolution (along the x -a x is) is given
by the antenna size and down-range resolution (along the y -a x is) is given by the measurement bandwidth
(or pulsewidth) o f the system.
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8
pattern exhibits circular symmetry over a broad circular aperture. As a result, the radar
produces a sharp far-field antenna beam that bears a circular symmetry similar to that
of the antenna aperture. Because of its obvious shape, the beam so produced is known
as a pencil beam. In Fig. 1.3(b), on the other hand, a radar with a rectangular aperture
produces a far-field antenna beam that exhibits an elongated elliptical shape, with the
narrow near-field variation along the shorter dimension o f the rectangle being transformed
to a broad far-field variation, and the broad near-field variation along the longer dimension
of the rectangle being transformed to a narrow far-field variation. The resulting beam is
commonly known as a fa n beam.
SAR can produce high-resolution images based on the generation of an effective long
antenna (hence producing sharp a far-field antenna beam) by signal (Doppler) processing
means rather than by the actual size of a long physical antenna. With reference to Fig. 1.2,
the cross-range and down-range resolutions of an image pixel are determined by the
boresight beamwidth (and hence the antenna aperture size L) and transmitted pulsewidth,
respectively. In a theoretical case, it can be shown that the cross-range resolution (Aar)
and down-range resolution (A y) can be expressed as [5]
A* = !
(1.2)
± y = 02 B
R sin
c. 9n
(1-3)
where B represents the measurement bandwidth o f the SAR system, c the speed of light
in free space, and 9 is the depression angle measured from the horizon.
It is important to note that the high quality of SAR imaging, however, does not happen
by accident or without cost. In practice, a successful SAR system requires demanding
system specifications and regular verification/monitoring of performance over time. Flight
path irregularity, phase errors due to oscillator instability, requirement of high-speed data
recording
media for data
surface scattering, are
acquisition, and strong speckles due torandom volume and
but a few practical and costly issues that have to be dealt with in
SAR applications [25].
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(b)
Figure 1.3: Fourier-transform relationship between near-field and far-field variations: broad near-field
variation transforms to narrow far-field variation, (a) Pencil beam pattern resulting from a broad circular
near-field pattern, (b) Fan beam pattern resulting from a rectangular near-field pattern. Sidelobes o f these
beams are not shown in this figure.
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10
From the discussion above it is apparent that the usefulness of GPR’s can be severely
limited by interference due to random scattering from ground surfaces. Likewise, the
quality of SAR images can be adversely impaired by speckles as a result of random
scattering from terrain and vegetation media.
In this regard, optimum detection and
imaging using G PR’s or SAR’s therefore demand suppression of clutter return produced
by random multiple scattering.
In subsequent sections of Chapter 2, a heuristic account of achieving clutter suppression
using a correlation phenomenon will be presented. Constructed in light of a scattering
phenomenon known as memory effect, this technique makes uses of the complex angular
correlation function (ACF) and/or frequency correlation function (FCF) measurement to
achieve enhanced visibility contrast for targets embedded in clutter. Among numerous
environmental applications of these correlation techniques are, for instance, detection of
abandoned active land mines and topographical imaging of natural terrains.
To make the presentation of this work as logically smooth as possible, this dissertation
is organized in a chronological sequence for those critical research studies that surfaced
in this research effort.
Chapter 2 presents the concept of memory effect of scattering by random media.
Counter-intuitive as this correlation phenomenon seems to be, it is emphasized that the
only necessary condition for this effect is random scattering, may it be single or multiple
scattering. In addition, the rationale behind applications of this phenomenon in detection
and imaging problems is discussed, and a way of quantifying the memory strength using
ACF is introduced.
In chapter 3, experimental studies on subsurface target detection using ACF measure­
ment are presented. To demonstrate the effectiveness of this correlation technique over the
traditional technique based on RCS acquisition, a comparison between the detection results
using ACF and RCS measurement is made. It is shown that in detection situations where
the main clutter source comes from random scattering, ACF technique generally works
better than traditional RCS technique, resulting in higher target visibility in strong-clutter
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II
environment.
As an extension of applications of this ACF technique in other areas, the subject of
SAR imaging is explored in Chapter 4. The chapter presents a chronological account
on the development of a novel three-dimensional (3—D) SAR confocal imaging system.
Specifically, the concept of circular SAR is presented, together with analytic derivation and
calculations of the system’s generalized ambiguity function [25]. In addition, experimental
studies on the 3-D confocal imaging capability of circular SAR are presented. In parallel
to the organization of Chapter 3, a comparison between imaging results using correlation
and traditional techniques is made. It is shown that in imaging situations where the main
clutter source comes from random scattering, correlation-SAR processing generally works
better than traditional SAR processing, resulting in higher target visibility in strong-clutter
environment.
Finally, concluding remarks, together with a list of topics proposed for further research
in this area are given in Chapter 5.
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Chapter 2
ANGULAR MEMORY EFFECT
When a wave is incident upon a two-dimensional (2-D ) rough surface, it undergoes
variable degrees of absorption, scattering, and depolarization. Depending upon surface
roughness and dielectric parameters, the resulting co-polarized and cross-polarized scat­
tered waves are in general in all directions and exhibit completely random phase fluc­
tuations. At first glance, it is intuitive to assume that these scattered waves contain no
statistically coherent information (such as the incidence direction) of the original incident
waves.
2.1
Angular memory effect in the 6-plane
In a series of theoretical and experimental studies on wave transmission in diffusive me­
dia [7, 8] during the late 80’s, however, condensed-matter physicists found that such
intuition is only a partial truth. The new findings, long overlooked by classical studies,
show that the direction of incident waves can be deduced from diffused field measurement
by virtue of a phenomenon known as the angular memory effect. Basically, this effect
describes how the changes in the direction of the incident wave are “remembered” , or cor­
related, by the diffused scattered waves when a certain relationship between incident and
scattering angles is satisfied. Simply stated, this relationship, as shown in Fig. 2.1, depicts
the existence of strong correlation between scattered waves when the difference between
the transverse incident wave vectors is equal to the difference between the transverse
scattered wave vectors [16, 19].
Mathematically, angular memory effect can be characterized by ACF. Denoting the
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13
2(0.)
Es(0s;0i)
n
random rough surface
Figure 2.1: The scattering geometry o f angular memory effect in 0-plane. The necessary condition for
strong angular correlation is governed by the generalized Snell’s law: s in # ’ - sin # , = s in # ’ - s in # s
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14
unprimed E s(9i, 0S) the reference scattered wave observed at 9S due to an incidence at 0*
and the primed E a{9'i.9's) the variable scattered wave observed at 9's due to an incidence
at 0-, the ACF T(9'i: 9's-, 9i, 9S) is defined as
Q's, 9ii9S) =< E s(0i;0s)E s'(0;,0;) >
(2.1)
where the angle brackets denote an ensemble average measurement over a large number
of independent realizations, signifying the statistical nature of scattering involved. The
asterisk mark
appearing within the angle brackets refers to complex conjugation. All
the experimental results included in this dissertation assume TM -, or p-polarization. The
reason for this choice of polarization is that EM energy can be most efficiently coupled
into a given scattering medium with minimal scattering at Brewster-angle incidence, which
exists only for TM polarization. With reference to Fig. 2 .1, the condition for strong angular
correlation is reiterated here as
sin 0, —sin 0* = sin 9S —sin 0S
(2.2)
where (0 i,0 s) and (0-.0^) refer to the reference and variable antenna angles, respectively,
in which ACF is to be measured. On the sin(0’) — sin(0’) plane in Fig. 2.2, Eq. 2.2
represents a straight line passing through the point (sin(0;),sin(0s)) with a slope of +1.
Previous theoretical work suggests that along this line there exists a high, non-uniform
level of correlation but rapid bilateral decorrelation away from the line [16]. Because of
its association with high angular correlation, this particular line is appropriately known as
the angular memory line corresponding to antenna angles (0*. 9S. On the other hand, the
line that passes through the same reference point (sin(0j), sin(0s)) but with a slope o f -1
(i.e. perpendicular to the angular memory line), is called the scan line.
In general, there exist along the angular memory line two points, A ( s i n ^ ) , s in ^ ^ ) )
and B (sin(0^2).sin(0j2)), where the strongest correlation occurs.
At point A where
(sin(0'1), sin( 03!)) = (sin(0t), sin(0s)), the variable scattered wave coincides with the ref­
erence scattered wave. Therefore, Eq. 2.1 reduces to an autocorrelation operation and thus
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15
sin(6s)-axis
sin(0i)-axis
/
/
/
/
/
&^
(sin(-es),sin(-0i))
-O
<y*
S
y
S
/
#<$>
y
0.
y
y
/
/
/
/
/
/
(sin(0i),sin(es))
'- • • V
<s>
Figure 2.2: Plot o f Eq. 2.2 on the sin (0t ) - s i n ( 0 3) plane. T he angular memory line and scan line intercept
at the reference point (sin (# i), sin (0 s )) and are, by definition, perpendicular to each other.
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16
attains a maximum. Because of this self-correlation operation, the ACF measurement is
actually identical to RCS (intensity) measurement. At point B where (sin(#-2) ,s in ( ^ 2))
= (sin(—0S), s in (—0 J ), however, the variable and reference scattered waves are related
by reciprocity relationship [28], which in turn establishes (ignoring polarization depen­
dence for the time being) the equivalence between the variable scattered wave observed at
0's2 = —Oi due to an incidence at 0'i2 = —0S and the reference scattered wave observed at
0S due to an incidence at 0,. Consequently, in this case Eq. 2.1 again reduces to another
autocorrelation operation and thus attains a maximum.
Before proceeding to the next section, a few remarks concerning the second type of
field equivalence (the one established under reciprocity condition) are given here. First,
reciprocity condition requires not only reciprocal antenna positions, but also reciprocal
polarizations [28].
This scenario is illustrated in Fig. 2.3.
Second, it is difficult in
practice to obtain exact time-reversed propagation paths Pi and P>, where Pi is defined
to be incidence-at- 07'-and-scattering-at- 0 R and P2 to be incidence-at-0fl-and-scattering-at9t , as shown in Fig. 2.3(a) and (b). Therefore, the resulting level of correlation will
be lower compared with that resulting from the first kind of field equivalence (the one
established from autocorrelaton condition).
2.2
A ngular m em ory effect in the o-plane
In previous section, it is emphasized that angular memory effect is an inherent property of
random scattering and becomes apparent as long as a certain phase matching condition (the
generalized Snell’s law given in Eq. 2.2) is fulfilled. Although developed for scattering
in the plane of incidence (the 2-D 0-plane), satisfaction of this phase matching condition
can be extended to 0—plane. In fact, This independence of angular memory effect on
coordinate planes makes surveillance platforms based on this correlation technique both
flexible and practical to implement.
First-order Kirchhoff approximation (KA1) is used in the following analysis to demon-
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17
(a)
a-polarized transmitter
transmitting at &r
P-polarized receiver
receiving at 0R
original
(b)
a-polarized receiver
receiving at 0t
P-polarized transmitter
transmitting at 0R
reciprocal version
Figure 2.3: Field equivalence under reciprocity condition, (a) the original scattering situation: an a polarized transmitter transmitting at Or and a /3-polarized receiver receiving at O r . (b) the reciprocal
version o f the original scattering geometry: a ^-polarized transmitter transmitting at Or and an a-polarized
receiver receiving at 6 t -
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18
strate angular memory effect in the o-plane. It is assumed that the single-bounce scattering
mechanism takes place only in the upper infinite hemisphere, as shown in Fig. 2.4
Denote the umprimed reference incident and scattering wave vectors by the reference
transverse wave numbers Ki and K s and the primed variable incident and scattering wave
vectors by the variable transverse wave numbers K x and K s , one has
sin 6i cos (bii
Ki
=
k
sin Q{ sin d>iy
—cos OiZ
Ki — k cos OiZ
(2.3)
sin 93 cos 0 sx
Ks
— k
sin 0Ssin d>sy
co s9sz
ks
+ k cos 6sz
(2.4)
sin 6'i cos o ’iX
k
sin 6'i sin 0 xy
— cos 9\z
k x
— k cos OiZ
(2.5)
sin 9S cos o sx
k
sin 6S sin 0 sy
cos 6 'sz
ks
+ k cos 9sz
( 2 .6 )
where Ki = k(sm0iCos<piX -f- sinfl, sinewy), ks — A:(sin0s cosd>si: + s in 0 s sind»sy), k / =
A:(sin 0- cos
+ sin 0- sin d^y), and
= A;(sin 6's cos <p’sx -F sin 0’s sin <p’sy ) . In addition,
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19
z
X
Figure 2.4: The scattering geometry of angular memory effect in the <jhplane. At constant ^-incidence, the
resulting “plane” o f incidence traces out a conical surface in 3 -D space.
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20
it can be shown that the correlation between the reference scattered field represented by
K s and the variable scattered field represented by K s is proportional to the exp(—
where
ud = w c = I17-17!"
(2-7^
where V —
TcJ— «7 and v — kJ — k~'. Obviously, maximum angular correlation results
when U2 =
0.This requires
|(«7 - k ~) - (a; - « " ') |
=0
(2.8)
Note that Eq. 2.8 necessarily reduces to Eq. 2.2 in the plane of incidence. The details of
this reduction are presented in the following analysis. With
k~
— k (sin 9i cos O ii 4 sin 9{ s in &{/)
TcJ
= k(sin 9S cos psx 4- sin 9S sin o sy)
Ki
= k(sin9i cosO ii 4 s i n ^ sin (pty)
k^ ’
— k(sin 9S cos p'sx 4 sin 9S sin o sy)
obtained previously, Eq. 2.8 can be expanded as
0
—
^i)
(^5
)J”
=
|[(A;sin9S cos o s — Arsing cosOj) — (A:sin9S co so s — Arsing co s0 J]£ ’
4
[(A: sin 9S sin o s — k sin 0; sin ©;) — (A: sin 9's sin o s — k sin 9\ sin 4>i)\y[
=
\A x 4 B y \2
=
A2 + B 2
(2.9)
where A = ( k s m 9 s cos<ps — Arsing cosd>i) — (ksin9'acos(p's — k s in 9[ cos (pj and B =
(A: sin 9S sin <ps — k sin 0* sin (pi) — (k sin 9's sin <p's —k sin 9\ sin 0 ’). The condition that A 2 4
B 2 = 0 requires both A = 0 and 5 = 0. That is,
(k sin 9s cos (ps — k sin 9i cos (pi) — (k sin 9S cos (ps — k sin 0* cos ©’)
=
0 (2.10)
(A:sin0s sin<?)s — A rsin^sinipi) - (Arsin^sintpg — A :s in ^ s in 0 |)
=
0 (2.11)
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21
In the plane of incidence where tpi = <t>\ — 0° and ®s - <ps = 180°, Eq. 2.10 and Eq. 2.11
result in
sin 9{ — sin 0£ = sin 9S —sin 6S
(2.12)
which is exactly Eq. 2.2, the condition for strong angular correlation in the emplane.
Next, another special case of significance to practical implementation is considered.
In this case, the surveillance platform moves along a horizontal circular orbit at a fixed
angle 6 from the 2-axis. Therefore, 9{ = 9\ = 6S = 6's = 9. As a result, Eq. 2.9 becomes
[(cos <ps —cos (pi) — (cos©^ —co so £)]2 + [(sin<f»s —sintpi) —(sin o s —sin©’)]2 - 0. (2.13)
Given a fixed pair of (o£. ©s), Eq. 2.13 is a function of two variables: o\ and o s. Assigning
angular dependence to Eq. 2.13 as
r(6-, <PS : 6 i , o s )
—
[(cosos —coso,) — (cosol —cos©’)]2
+
[(sinos —sinOj) — (sin o l —sinO j)]2.
(2.14)
a plot of r(o [. ol; ®i, 6 S) for various combinations of (®i: o s) is shown in Fig. 2.5.
As evident in the figure, the signature of r(o [, o s; o *. o s), depending on the choice of
reference angles {®i,®s), can vary from straight lines to elongated contours. In the case
of (oi-Os) = (0°.0°), the transmitting and receiving antennas are positioned in specular
configuration. In general, maximum angular correlation results as long as the two antennas
lie in the same plane of incidence, independent of the orientation o f the plane relative to
x - or y-axis. This independence of orientation is manifested by the straight line strips in
the figure. In the case of (o,, o s) = (0°, 180°) (or (0°, —180°)), however, the transmitting
and receiving antennas sire positioned in backscattering (monostatic) configuration. For
maximum correlation it is necessary that the variable antenna configuration coincides with
the reference antenna configuration. As this coincidence takes place only at a point on the
(
plane, the resulting maximum correlation occurs at a point where (<p£, ®s) = (0[, <ps).
This unique ACF signature
is qualitatively demonstrated experimentally (using
gravel
particles asthe random medium with particle dimension rj A) and analytically (assuming
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-200
0
200
Figure 2.5: The signature o f ©-plane ACF as a function o f reference angles (4>i.0s )• “i” stands for
reference incident angle <f>i, “s" stands for reference scattered angle <bs . x - and y —axes represent o[ and
(t)s , respectively, in degrees.
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23
first-order random surface scattering) using KA1. The results are shown in Fig. 2.6.
Note that the apparent underestimation by K A l (demonstrated by the relatively slow
decorrelation from the correlation maximum) implies that the actual scattering mechanism
could well be beyond first-order in nature. The insufficiency of KAl formulation can be
remedied by including second-order scattering effects, as explained in the next section.
2.3
Correlation peaks and second-order Kirchhoff approximation
Analytically, the correlation peaks discussed in Chapter 2.1 can be described by different
components of second-order scattering. Recent studies on scattering by high-slope 2 -D
random rough surfaces based on the second-order Kirchhoff approximation (KA2) [1]
indicate that for the case of reference antenna positions located in the backward direction,
which is the case deployed in this research and in most practical applications, the firstorder scattering component in KA2, as shown in Fig. 2.7, gives rise to a broad response
along the angular memory line when single scattering dominates but two peaks when
multiple scattering dominates as a result of the second-order ladder and cyclic scattering
components in KA2, as shown in Fig. 2.7. Specifically, the ladder term gives rise to the
correlation peak at point A in Fig. 2.2 where the autocorrelation condition is satisfied,
whereas the cyclic term gives rise to the correlation peak at point B in the same figure
where the reciprocity condition is satisfied [16]. Furthermore, it can be shown that the
lateral width of ACF is on the order of 1ID , where D is the illumination spot size
expressed in wavelengths, and is independent of surface roughness [1]. This theoretical
prediction is found to be in good agreement with experimental observations on millimeterwave scattering from random rough surfaces [4]. In summary, given a reference pair of
antenna angles (0*, 03) there exists high correlation along the angular memory line defined
by Eq. refsnell, with correlation peaks located at (sin (^),sin (0 j)) = (sin(0i), sin(0s)) and
(sin(0'),sin(0j)) = (sin(—0S), sin (—0*)), but low correlation elsewhere on the sin(0j) —
sin(0[,) plane.
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24
(a) ACF signature across the narrower contours for phi.= 0 °, phig = 180°
KAl
Lfo
Experiment
02
-20
-15
-10
•5
0
5
transmitter angle phi. (primed) in degrees
10
15
20
15
20
(b) ACF signature across the wider contours for phi.= 0 °, phig = 180°
KAl
••=
0.6
u_
o
Experiment
•20
-15
-10
•5
0
5
transmitter angle phi. (primed) in degrees
10
Figure 2.6: (a) Comparison between experimental and analytical ACF signatures across the narrow contours
in Fig. 2.5 for the case (<f>i,<t>s ) = ( 0 ° ,1 8 0 ° ), and (b) Comparison between experimental and analytical
ACF signatures across the wide contours in Fig. 2.5 for the case (0 t, <j>3) = (0 ° ,1 8 0 ° ).
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25
E,
first-order Kirchhoff scattering: single bounce
second-order Kirchhoff scattering: double bounce
(ladder term)
second-order Kirchhoff scattering: double bounce
(cyclic term)
Figure 2.7: The three major com ponents in the second-order Kirchhoff Approximation: (a) first-order
scattering: single bounce, (b) second-order scattering: identical doubly-bounced signals resulting from
identical propagation paths, (c) second-order scattering: identical doubly-bounced signals resulting from
time-reversed propagation paths.
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26
2.4
Applications o f angular correlation function
Recall from discussion in previous sections that while strong angular correlation exists
along the angular memory line, rapid decorrelation occurs across the angular memory line
(i.e. along the scan line). It is important to realize that the achievement of this low level
of ACF (clutter) does not come from signal processing means or novel hardware design.
Rather, it is merely a direct result when random scattering is the dominant scattering
mechanism, which is commonplace in many practical remote-sensing problems [12].
In contrast to random scattering produced by clutter, scattering by most discrete manmade targets (such as land mines), however, is usually characterized by single or double
bounces from well-defined (and hence deterministic) scattering centers with slowly varying
angular memory signature over a wide range o f angles on the sin(0j) —sin(6^) plane. As
a result, the rapid decorrelation off the angular memory line due to clutter scattering does
not find its counterpart in the case of scattering by these man-made targets. Thus, if
ACF measurement is performed at a point far away from the angular memory line, low
correlation will result if the illuminated region covers a region where there is no target,
but high correlation otherwise.
Numerically, the following argument may provide useful insights to ACF’s relative
effectiveness over the traditional RCS technique: in obtaining ensemble average over in­
dependent measurement samples of scattering by clutter, addition of complex ACF (com­
plex numbers with magnitude and phase) could result in a low correlation level for the
ACF detection technique, addition of scalar RCS (non-negative numbers with magnitude
only) does not necessarily result in a low speckle level for RCS detection technique. In
principle, the target visibility using ACF contrast should therefore be higher than that
using RCS contrast.
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27
2.5
Other forms o f correlation functions
Given the definition o f ACF as a correlation function that measures the similarity be­
tween two variables in angular domain, it is possible to devise other forms of correlation
functions defined in domains other than angular domain. Depending upon the nature of
applications, one form of correlation may be more effective than another. For instance,
in considering the polarization similarity between two scattered fields, one could form
polarization correlation function that makes use of the Poincard’s parameters commonly
used in polarization characterization [15]. The resulting polarization-correlation technique
may be more suitable than angle-correlation technique in detecting and imaging polarized
targets such as submarine’s periscope surfacing out of the sea. On the other hand, fre­
quency correlation function [1], which considers similarity of frequency content between
two scattered fields, could be more effective than other forms of correlation functions
in detection and imaging applications where clutter exhibits high frequency-dependence,
which is a rather common characteristic for naturally occurring random volume media. In
the next chapter, the rationale for using ACF technique in clutter-suppressed detection ap­
plications will be exemplified. In Chapter 4. on the other hand, the development of various
correlation techniques in clutter-suppressed imaging applications will be presented.
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Chapter 3
CORRELATION TECHNIQUE IN TARGET DETECTION
While related studies were conducted in the past to demonstrate ACF signature and
its use in detection of a target embedded in numerical clutter [4, 29], it is important
to evaluate the effectiveness of ACF technique in actual remote-sensing settings. For
this reason, the application of this detection technique in realistic clutter environment is
presented in this chapter. To achieve this, realistic clutter is simulated by using natural
media such as 1) fine sand, 2) rough sand, 3) gravel, and 4) rock, in which the combined
scattering of both surface and volume media behaves as strong interfering source to the
detection radar system.
In practical remote sensing applications, sensing radars usually operate at low frequen­
cies (e.g. 0.1-1 GHz) as pointed out in Chapter 1. This requirement is necessary in order
for deeper energy penetration into natural media such as canopy layer or soil medium with
high moisture content. If one were to simulate this frequency requirement in laboratory,
he or she would have to construct forbiddenly large-scale experimental setup consisting
of, for example, heavy antennas, long supporting booms, large observation bench and so
forth.
As a result of this inherent construction difficulty in low-frequency scattering studies,
a compromise must be made in laboratory studies in which the operating frequencies are
‘reasonably’ scaled-up (or equivalently, wavelengths being scaled-down), thus reducing
the dimension of experimental setup necessary for observing wave phenomena.
Adopting this scale-up strategy, application of ACF detection technique at millimeterwave frequencies (75-110 GHz, or wavelength A = 3 mm at a center frequency of 100
GHz) is considered. In this regard, experimental equipment can be maintained at a man­
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29
ageable size in laboratory settings so that the existence of angular memory effect for
different media can be confirmed quickly with our existing facilities documented in [23]
and [24],
Next, the application of ACF technique in detection at X-band frequencies (7-13 GHz,
or wavelength A = 3 cm at a center frequency of 10 GHz) is investigated. At a scale-down
factor of 10 from millimeter-wave frequencies, the results at X-band frequencies should
demonstrate accurately enough the effectiveness of this correlation technique at ever lower
frequencies (e.g. UHF to L-band) in actual remote sensing applications.
3.1
Target detection at millimeter-wave frequencies
Millimeter-wave (75-110 GHz) experiments are performed to study the ACF signature
of a number of natural scattering media selected in this investigation. The experiments
involve the use of a previously constructed bistatic radar system [23], which is shown in
Fig. 3.1 and Fig. 3.2. The system is calibrated using a large flat conducting plate as the
known target. Such system calibration is necessary to ensure that both the magnitude and
frequency responses are corrected and compensated for measurement accessories before
any measurement is made. The relevant details of this crucial procedure can be found
in [4] and is briefly repeated here in Appendix A for readers’ convenience.
In the course of the experiments, the selected media under examination are kept dry in
order to maintain minimal absorption loss. The detailed dimension and absorption char­
acteristics of these four media at millimeter-wave frequencies are tabulated in Table 3.1.
Note that the average size, which is usually expressed in terms of wavelength A (= 3
mm at a center frequency of 100 GHz), of media particles ranges from submillimeter
to tens of millimeters. Besides, it is worthwhile to observe from this table that certain
relationship exists between attenuation and particle dimension. On one extreme, for small
entities such as fine and rough sand particles, attenuation due to multiple scattering (as
characterized by scattering cross section os in transport theory [12]) is fairly small. On
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30
Figure 3.1: A far view o f the millimeter-wave system used in the experiments. System specifications o f
this advanced vector scattereometer are fully documented in [23].
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31
Figure 3.2: A close view o f the same system shown in Fig. 3.1. In this figure, the transmitting antenna (on
the left hand side) is covered with a servo-mechanical polarizer. The receiving antenna (on the right hand
side) is designed to receive both copolarized and cross-polarized scattered signals.
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32
the other extreme, for large entities such as rock particles, the major scattering mechanism
is mainly first-order in nature. The resulting single-bounce scattering does not contribute
strongly to the overall scattering loss, and thus exhibit low attenuation. At Mie resonance
region where the particle size is comparable with A, however, attenuation due to multi­
ple scattering becomes exceedingly high, as shown in the case of gravel particles whose
dimension is on the same order of A. Both surface and volume scattering are significant
in this particular case. As a result, gravel medium exhibits the highest attenuation among
the selection.
Table 3 .1: Dimension and absorption characteristics o f the four natural media used in the millimeter-wave
experiments
Medium
Attenuation
Skin Depth
Shape
M ajor Axis
M inor Axis
fine sand
3.14 dB/cm
1.38 cm
m sphere
0.09 mm
0.09 mm
rough sand
3.70 dB/cm
1.17 cm
~ sphere
0.99 mm
0.99 mm
gravel
6.21 dB/cm
0.70 cm
« ellipsoid
8.34 mm
4.59 mm
rock
3.74 dB/cm
1.16 cm
ss ellipsoid
26.82 mm
14.52 mm
3.1.1
Experimental setup
At a Fixed reference transmitter angle o f 0l = 20° and a fixed reference receiver angle of
9S = 40°, the TM co-polarized ACF signature along and perpendicular to the angular
memory line for each of the selected media is measured and the result is shown in
Fig. 3.3.
The shape of ACF for the media demonstrates a pattern strikingly similar
to that for two-dimensional conducting random rough surfaces [4]: along the angular
memory line the correlation peaks occur at the autocorrelation and reciprocity points, with
rapid decorrelation occurring along the scan line. It is important to note that within the
angular resolution of measurement, the angular width of angular memory line is also the
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33
same for all four selected media and does not depend on the roughness characteristics
of the medium of interest, as consistent with theoretical predictions based on KA2 [16].
Thus, it is appropriate to infer that angular memory effect is a quite “universal” scattering
phenomenon in that its occurrence does not depend on the mechanisms that lead to random
scattering, but rather on random scattering itself.
The above observation of “universality” is important for practical reasons since it
suggests that the existence of angular memory effect in a wide range of actual remote
sensing applications wherever random scattering exists. This existence of angular memory
effect makes ACF technique applicable for improved measurement in a wide range of
detection issues.
To study the applicability of ACF measurement in detection o f a target buried in a
natural scattering medium, a long conducting cylinder of diameter a = 3 mm is buried
under a rough sand medium at a depth of d = 6 mm. With (0{,0S) = (20°, —20°) and
= (3 0 °,—30°) (note that the point {0\,9's) = (3 0 °,—30°) is far away from the
angular memory line corresponding to (9i: 9S) — (20°. —20°), as a requirement from the
discussion in Section 2.4), the correlation measurement then proceeds with two different
types of scan, namely, angular scan and spatial scan, as shown in Fig. 3.4. While angular
scan involves moving the antennas in the plane of incidence (i.e. the 0-plane) from 0°
to 40°, spatial scan allows the antenna footprint to scan continuously over the sand-air
interface (i.e. along a straight line). The footprint (spot size « 90 mm) is moved over
a distance of 200 mm, with 100 mm from either side of the cylinder, in increments of 2
mm during the experiment. In other words, at each of these 101 footprint locations, ACF
is measured as a function of frequency. Because of the stationary nature of the problem,
only frequency samples (but not spatial samples) are available for ensemble averaging.
With a measurement bandwidth of 35 GHz and a decorrelation bandwidth of 1 GHz,
about 35 independent frequency samples are available to each single look of correlation
measurement.
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34
X 10'4
(a) fine sand
X 10“*
4
4
<D3
3
(b) rough sand
E
U.
Lt_
<,
c.
o
o
o
0
0
10
20
30
40
50
0
60
10
x 10 -3
(c) gravel
x10-3
2
2
031.5
1.5
c» 1
cn .
cn I
30
40
50
60
50
60
(d) rock
ea 1
E
E
LL.
L i_
O
<c.
05
o
<c,
'
0
0
20
incident angle in degrees
incident angle in degrees
10
20
30
40
incident angle in degrees
50
60
0.5
0
0
10
20
30
40
incident angle in degrees
Figure 3.3: ACF magnitude o f the selected natural m edia at millimeter-wave frequencies (9 5 -1 0 0 GHz) for
(0 i , 9 s ) = (20°, —2 0 °), along (solid line) and perpendicular to (dotted line) angular memory line: (a) fine
sand, (b) rough sand, (c) gravel, and (d) rock.
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35
antennae
angular scan
spatial scan
footprint
rough sand
cylinder
Figure 3.4: Target detection with spatial and angular scans with (6 i , 6 s ) = (2 0 ° , - 2 0 ° ) . Spot size o f the
footprint « 30A.
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36
x lO ’3
(a) Detecton using A C F
x 10*3
1a
1.8
1.6
1 .6
(b) Detection u sin g R C S
1.2
I
0.6
0.6
0.4
0.4
02
0.2
-100
Figure 3.5:
0.8
•so
SO
10O
• 10O
sam pling distance in mm
(a) ACF magnitude as a function o f footprint:
50
•SO
sam pling d ista n c e in mm
sampling
(0 i , 8 3) =
100
( 2 0 ° ,- 2 0 ° ) and (0 [ , 0 S) =
( 1 0 ° ,- 1 0 ° ) . (b) RCS as a function o f footprint: (0t , 0 s ) = (6t , 0 s ) = (2 0 °, - 2 0 ° ) . It is clear from
this comparison that ACF technique results in higher target visibility contrast than RCS technique.
3.1.2
Experimental results
In Fig. 3.5(a), the target visibility in terms of ACF magnitude is plotted as a function of
footprint locations x expressed in millimeters, with the object buried at approximately x
= 0 mm. It is clear from the Figure that the magnitude of ACF attains a reasonably low
level at regions where the buried object is absent. This low level o f magnitude signifies
the fact that angular correlation due to incoherent scattering by natural media, when
measured across the angular memory line, is quite insignificant. On the other hand, as the
footprint approaches the region where the object is buried directly underneath, however,
the magnitude of ACF rises rapidly. This increase in correlation level can be explained
by observing that in coherent scattering by those deterministic objects with well-defined
scattering centers, the scattered fields exhibit high degree of coherent information and
therefore, the correlation between the fields maintains at a high level.
As target visibility depends only on the target-to-clutter ratio, an unambiguous decision
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37
can be made about the absence/presence of the buried target based on the above rapid
Iow-to-high transition of correlation level. When compared with ACF, the traditional RCS
approach produces a weaker visibility contrast with the results shown in Fig. 3.5(b). As
evident from the figure, RCS plot shows only a slowly varying profile of intensity, making
it difficult to discriminate the target from the noisy surrounding medium.
To summarize, it is demonstrated that in target detection applications where the clutter
sources (including surface and volume scattering) primarily exhibit random scattering, the
proposed ACF technique is superior to the traditional RCS technique, resulting in higher
target visibility contrast.
3.2
Target detection at X-band frequencies
As illustrated in the previous section on millimeter-wave scattering, the proposed ACF
technique is shown to be more effective than the traditional RCS technique in terms of
target-to-clutter ratio.
In practical subsurface remote sensing applications, millimeter-
wave radiation, however, has limited usefulness because of its significant attenuation in
common geophysical media [25]. For ground penetration applications, UHF (0.3-1 GHz)
to L-band (1-2 GHz) is a more realistic frequency band o f choice since radiation in these
frequency bands can penetrate deep into geophysical media such as moist soil/vegetation
layers or canopies.
In this section, the results of a scaled-down (from the UHF- and
L-bands) experimental investigation are presented. The details of this investigation are
described in the following sections.
3.2.1
Experimental setup
A bistatic far-field scattereometer operating at X-band frequencies is constructed to inves­
tigate the angular memory effect of scattering by a natural medium. The whole assembly is
shown in Fig. 3.6. It consists of two rigid booms, with each of them supporting an antenna
radiating at TM polarization. The reason for using this particular choice of polarization is
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38
that EM energy can be most efficiently coupled (with minimal scattering) into a scattering
medium at Brewster-angle incidence, which exists only for TM polarization. To allow
for bistatic scanning capability, the two booms are controlled by independent stepping
motors and can move freely in the plane of incidence (i.e. the 0-plane). Since the whole
bistatic radar assembly sits on a horizontal translation stage driven by a stepping motor,
the resulting system can perform not only angular-scan measurement, but also spatial-scan
measurement, similar to the millimeter-wave system described in Section 3.1.1.
The natural medium under examination is typical garden soil housed in a container
with dimensions of approximately 1 m x 1 m x 1 m. The soil is kept dry and exhibits
a slightly rough soil-air interface. The attenuation constant a and relative permittivity er
of the medium are determined using transmission measurement. The dependence of a
and er on frequency is shown in Figs. 3.7(a) and (b). From the measurement the relative
permittivity and attenuation constant of the medium are found to have mean values of
3.13 and 0.50 dB/cm, respectively, over frequencies.
To verify the existence of angular memory effect for this particular scattering medium,
ACF is measured along and perpendicular to the angular memory line for two antenna
configurations:
(0i,6s) = (2 0 °,—20°) and (0i,0s) = (2 0 °,—40°).
As mentioned in
Chapter 2.1, correlation peaks occur at angle pairs where autocorrelation condition or
reciprocity condition is satisfied. For the case of (0*, 0S) = (20°. —20°), the autocorrelation
and reciprocity peaks coalesce together into a single peak, as shown in Fig. 3.7(c). For
the case of (0t,0s) = (20°,—40°), on the other hand, the autocorrelation peak occurs
at (Pile's) = (20°,—40°) and the reciprocity peak at (0,-,0s) = ( —40°,20°), as shown in
Fig. 3.7(d). As evident from the figures, the angular memory effect of the soil medium used
in this investigation at X-band frequencies is very similar to its counterpart at millimeterwave frequencies shown in Fig. 3.3.
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39
Figure 3.6: The top view o f the X-band bistatic target detection system . In the course o f the experiments,
the system scans over the composite target-soil medium contained in the sand box.
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40
(a) Attenuation constant
(b) Relative permittivity constant
10i------■
------■
------■
------■
------ -m
>■» 8
>
CD
Q_
I 4'
cd
—---
1 —
CD
“ 2
0I
/
8
9
10
11
7
12
,-------------,------,------,-----8
9
10
11
12
13
frequency in GHz
frequency in GHz
x 1 0 -3
(c) Soil ACF
0.02
(d) Soil ACF
4
0.015
CD
3
h
0.01
E
u_
u.
O
<c
o
0.005
0
20
40
incident angle in degrees
1
0
0
20
40
60
incident angle in degrees
Figure 3.7: Frequency dependence o f the dielectric parameters o f the soil medium used in the experiment:
(a) attenuation constant, (b) relative permittivity constant, (c) A C F magnitude along (solid line) and perpen­
dicular to (dotted line) the angular memory line for { 0 i , 9 a) = (2 0 ° , - 2 0 ° ) and (d) ACF magnitude along
(solid line) and perpendicular to (dotted line) the angular memory line for (0 i , 9 s ) = (2 0 °, —40°)
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41
3.2.2
Experimental results
To investigate the feasibility of detecting buried targets using ACF at X-band frequencies.
An approach similar to the one discussed in Section 3.1.1 for millimeter-wave frequencies
is employed - a long conducting cylinder of diameter a = 3 cm is buried under the soilair interface at a depth of d = 6 cm. By adopting scanning schemes (angular scan and
spatial scan) identical with the previous millimeter-wave experiments, both ACF and RCS
variation profiles as a function of footprint location x and frequency are obtained. The
corresponding results are presented in Fig. 3.8. The location of the buried cylinder lies
within the region of [50,60] cm.
It is clear from the figure that while the traditional RCS (intensity) approach in
Fig. 3.8(a) produces inconclusive information about the location of the buried object,
ACF measurement in Fig. 3.8(b) distinguishes itself by exhibiting one single sharp peak
at the same physical location occupied by the object. From this comparison, it is clear that
ACF technique is more effective than RCS technique in target detection through better
clutter suppression.
Finally, as a controlled demonstration on the superiority of the ACF technique over
the RCS approach, the same cylinder is removed from the medium and placed on top of
the soil surface. Intuitively, this setup should result in high target visibility using either
ACF or RCS technqiue. With the same reference and variable antenna configurations
m , 0 s) — (20°, —20°) and (0i: 0S) = (20°, —40°)) as before, the corresponding variations
of ACF and RCS signatures are recorded as a function of footprint and frequency. The
corresponding experimental results are depicted in Fig. 3.8(c) and 3.8(d), respectively.
Consistent with intuition, both ACF and RCS data produce visually conclusive information
in this case, allowing one to make unambiguous decision about the existence of the
exposed object easily.
Nevertheless, it is clear from Fig. 3.8(d) that ACF technique
detection technique, among other advantages over RCS detection technique, provides far
better spatial resolution (finer peak width) about the object location. In practice, this
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42
■10$ ) 6cm-cylinder below: (30°,-30°)
[ 10 $ ) 6cm-cylinder below: (50°,-50°)
3
c:
o
CD 6
o03
“2
ca
CO
o
o
Ll_
o
CO
■°1
CO 1
0
0
20
40
60
distance in cm
80
0
100
20
40
60
distance in cm
80
100
(d) 6cm-cylinder above: (50o,-50°)
(c) 6cm-cylinder above: (30°,-30°)
0.04
0.015
0.03
0.01
0 0 .0 0 5
<8
0
20
40
60
distance in cm
80
100
0
20
40
60
distance in cm
80
100
Figure 3.8: Object buried at 6 cm below the surface with (9i,9s ) = ( 3 0 ° ,- 3 0 ° ): (a) radar cross section
as a function o f footprint. ( 9^8,) = (3 0 ° ,—30°), (b) AC F magnitude as a function o f footprint. (d\,9'a) =
(5 0 ° ,—50°). Object placed above the surface with i.6i,9s ) = (3 0 ° ,—30°): (c) radar cross section as a function
o f footprint. (8{,9S) = (3 0 °,—30°), (d) ACF magnitude as a function o f footprint. {9'i,9s ) = (5 0 ° ,—5 0 °).
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43
improvement in spatial resolution achieved by ACF technique is an especially attractive
feature for accurate localized search for dangerous objects such as abandoned landmines
by military or civilian agencies.
3.3
Summary o f target detection using correlation technique
In this chapter, experimental results are presented for wave scattering by random media.
The results clearly illustrate the existence of angular memory effect over a wide range of
(1) scattering media (such as fine sand, rough sand, gravel, rock and garden soil) and (2)
frequencies (such as millimeter-wave (75-110 GHz) and X-band (7-13 GHz) frequencies).
The key to applying this unique correlation phenomenon in practical detection is­
sues comes from a simple observation: field correlation due to scattering by incoherent
mechanisms is completely different from that by coherent mechanisms - while low-level
correlation manifests itself in the former, high-level correlation appears in the latter at
points (represented by antenna locations) far away from angular memory line on the
sin(0-) — s in (^ ) plane. It is this high-to-low ratio that allows ACF technique to yield
higher target visibility contrast than the traditional RCS technique.
It is important to
note that this clutter-rejection property is nothing more than an inherent property of ACF
measurement, and the corresponding improvement in target-to-clutter ratio does not come
at the cost of sophisticated signal processing or expensive hardware component additions.
Therefore, in view of practical implementability, ACF technique should fit into existing
detection radar systems without too much modification.
In next chapter, the details of applying correlation technique in target imaging to pro­
duce clutter-suppressed imaging will be presented. In particular, key emphases are made
on a novel 3-D radar imaging system called circular SAR. The development, construction,
operation and application will form the core of the discussion that follows.
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Chapter 4
CORRELATION TECHNIQUE IN TARGET IMAGING
In Chapter 3, the concept of correlation is introduced in target detection applications
for clutter-suppressed measurement. It is demonstrated that by incorporating correlation
operation in data processing, enhanced target-to-clutter ratio can be obtained, leading to
a higher detection rate.
In comparison with detection applications where a mere binary decision (i.e.
ab­
sence/presence) is all that is required, imaging applications, on the other hand, require
more detailed information such as target size and/or shape. For this reason, high-resolution
detection tools were developed in the past and among different alternatives, SAR has
evolved to be the standard choice of imaging tools in existence nowadays.
Since its emergence in the late 1950’s, SAR has remained one of the most robust and
popular imaging techniques available for civilian and military applications [25], Among
these applications are terrain mapping, imaging of vegetation features (e.g. trees, rivers and
grasslands), as well as subsurface imaging of concealed military facilities in battlefields.
As shown in Fig. 4.1(a) and discussed in Chapter I, most traditional SAR-based
systems operate along a straight flight path [31], producing an equivalent linear antenna
array with an azimuthal resolution (also known as “along-track resolution” or “cross-range
resolution”) of D / 2 , where D is the aperture size of the antenna [5]. The range resolution
(also known as “across-track resolution” or “down-range resolution” ) is then determined
by the bandwidth of the transmitting pulse [5],
For brevity in the following discussion, the term linear SAR is used to refer to these
SAR systems. Note that the images produced by these linear SAR systems are projection
images only and are therefore 2 -D in nature [27]. As a result, for a given illuminated
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45
spotlight linear SAR
illumination spots
(a)
LSAR
d2 = d
d- d
/
b\
'
\
\
\
/
/
/
natural terrain
Linear SAR cannot resolve altitude ambiguity
since d1 is equal to d 2.
(b)
Figure 4.1: (a) The geometry o f spotlight-mode linear SAR. (b) Altitude ambiguity caused by propagation
paths o f equal lengths (di = do).
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46
structure such as a hill in Fig. 4.1(b), radar echoes due to those scattering centers lying
along a common wavefront but at different altitudes will all be mapped undesirably as
one single projection image pixel, with no resolution in vertical dimension. To remedy
this altitude ambiguity of linear SAR, the interferometric SAR (InSAR) technique, which
requires either multiple antennas or repeated flight paths, was developed and is widely
used within remote sensing community [10].
4.1
3 -D imaging using circular SAR
In this section, an alternative way for radar topographical (3-D) imaging is introduced.
This particular variant of SAR, known as circular SAR [26], requires the SAR sensor to
move in a circular orbit as shown in Fig. 4.2(a).
The advantages of circular SAR over linear SAR are discussed as follows. First, by
virtue of its obvious geometry circular SAR provides a full-rotation (360°) view of the
illuminated area. Second, the spotlight nature of its operation [3] makes circular SAR
capable of achieving image pixel resolution on the order of A (analogous to the diffraction
limit of a lens in optics), where A is the center wavelength of the wideband radiation
produced by the pulse-sending SAR system. Third, the altitude ambiguity encountered
by linear SAR described above can now be resolved using circular SAR measurements
made from several different azimuthal angles, as shown in Fig. 4.2(b). Because of this
altitude-resolving capability, it is possible to use circular SAR to perform 3 -D image
reconstruction of objects at microwave frequencies. Lastly, from a budget point of view
circular SAR can be implemented relatively easily by mounting SAR sensors on lowflying aircraft or helicopters without major costly modifications in existing hardware. To
summarize, the features of linear and circular SARs are briefly outlined in Table 4.1.
Note that the 3-D image reconstruction feature of circular SAR at microwave fre­
quencies finds its close counterpart at optical frequencies, namely, confocal imaging us­
ing optical confocal microscopes. Currently, such microscopes are widely available to
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47
spotlight circular SAR
illumination spot
(a)
CSAR
/
¥
A
natural terrain
Circular SAR can resolve altitude ambiguity
since d 1 is not equal to d 2.
(b)
Figure 4.2: (a) The geometry o f spotlight-mode circular SAR. (b) Altitude ambiguity resolved by propagation
paths o f different lengths (d\ ^ do).
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48
Table 4.1: Specification comparison between linear SAR and circular SAR
Features
Linear SAR
Circular SAR
Coverage
large ( ~ 1-10 km)
medium (~ 10-100 m)
Maximum spatial resolution
~ antenna size (D )
~ wavelength (A)
Image processing time
intensive
intensive
Image view
partial (one-sided)
full (360°)
3-D imaging capability
No
Yes
SAR platform
aircraft, satellite
helicopter, aircraft
achieve confocal imaging where it finds extensive applications in medical microscopy for
examination of various kinds o f translucent tissues [17, 20]. In essence, optical confocal
imaging refers to an imaging scheme in which optical sectioning is applied to a given
illuminated target. The resulting layer images so individually obtained are then stacked
up together to reconstruct the original volume image.
In addition to the inherent close similarity discussed above, it is important to observe
the difference between these imaging techniques from an image processing point of view.
In optical confocal imaging, focusing is achieved by properly positioned pinholes whereas
image formation is performed by the Fourier-transform computations inherently introduced
by light transmission through optical lenses.
Since such Fourier-transform operations
proceed almost instantaneously, real-time 3-D imaging can be achieved with ease at
optical frequencies.
At microwave frequencies, on the other hand, both focusing and
image formation processes are achieved by applying signal processing technqiues (the
processing algorithms are to be detailed in the next section) to raw measurement on
digital computing facilities. As a result, compared with its optical counterpart at, 3 D image reconstruction using circular SAR at microwave frequencies involves intensive
computation efforts and is usually a time-consuming process.
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49
In an attempt to combine the usefulness of correlation technqiues in clutter suppression
covered in Chapter 3 and the desirable feature of circular SAR in achieving 3-D confocal
image to be elaborated in this chapter, it is the goal of subsequent sections to develop the
bridge these two concepts together and explore from the resulting combined technique the
feasibility of achieving 3-D clutter-suppressed image reconstruction for remote sensing
applications.
4.2
Circular SAR processing algorithm
In general, SAR imaging involves temporal field measurement over a Finite bandwidth B
along a flight path P. Assuming the following discretization scheme
• M points over the bandwidth B and
• N points over the flight path P ,
one can construct a matrix E ( u, r ) to hold the raw frequency-domain field measurement
as follows
E(uJi,ri)
E{u)i,r2)
E(a;1:r,V- i)
E ( j j x, r N)
E{u>2 , r x)
E (ijo 1^ 2)
E ( ll)o,T n - i )
E(jJo,rx)
(4.1)
E{ u m - i , t x)
E ( u M , r i)
E { u M. u r 2)
...
E(uiM l r 2) . . .
E(u;jV/ - i , r jV- i)
E{ ujM- u r N)
E(u>M , r N_ i)
E {u M, r N)
where each matrix element is a complex number, containing both magnitude and phase
information. For typical circular SAR imaging measurement in which field measurement
is acquired at 1° increment over a bandwidth of 7-13 GHz sampled at 201 points, the
corresponding E (u, r), for instance, is therefore a 201-by-360 matrix, holding a total of
72,360 complex numbers, or 2 x 72,360 = 144,720 real numbers.
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50
Prior to focusing at a particular point r Qwithin the illumination footprint, it is necessary
to apply gating to the raw data to eliminate the direct coupling between the antennas. More
importantly, application of gating can help minimizing other undesirable contributions to
the received signal due to reflections by various parts of the experimental setup. In simple
terms, gating around r Q involves column-wise convolution of the SAR matrix E (^ . r)
with a gating matrix G {w, r;r0) whose columns contain the appropriate gating filter
responses. A typical gating matrix has the following form:
G (u > i,ri;r0)
G (uJ i,r2:ra)
& ( M , f v - i ; r 0)
G ( u J i , r ^ : r a)
G{uj2, r i : r 0)
G(ui2,r2',r0)
G{uj 2j T v- i ;To)
G(uj2, r,v ; f 0)
G ^ v r - i : n ; r o)
G(uJM , r i ; r 0)
r 2; r Q) . • G(ujm-i-. r ;V-i: r a)
G(u!\[.r2',ra)
G{ ujm-J n - f f o )
G { ^ \ { • r.V, f a )
(4.2)
Note that the convolution procedure mentioned above is generally a time-consuming pro­
cess, especially when long data sequences are involved. Fortunately, the advent of Fast
Fourier Transform (FFT) algorithms since late 60’s has made it possible to alleviate this
computation burden significantly by performing the convolution operation in its equivalent
time-domain operation, namely, multiplication between the inverse Fourier transforms of
E ( u , r ) and G (uj.r:r0). This equivalence can be summarized as follows
E (t , r ) x G ( t , r ; r a) <==>E (^,r)<g> G (c u ,r;r0)
(4.3)
where both ® and x are understood to operate in column fashion on their respective
operands. Because of this equivalence, performing time-domain gating on E {t- f ) is
thus equivalent to multiplying time-domain raw data with window functions appropriately
centered at different times. Fig. 4.3 depicts the use of time-domain window functions.
After time-domain gating and Fourier transform, the raw gated data, denoted by
Eg ( uj , f :r0), should then be phase-adjusted by a focusing matrix </> (uj. r; r Q) which com-
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51
focal point
N d2
N>
! di
\
\
\
\
r2
> 7 ~
window function at q
distance.tim e
window function at £
distance,tim e
Figure 4.3: Different window functions are used at different data-acquisition positions in order to capture
the “clean” responses from target located at the focal point.
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52
pensates, with respect to a reference range1, the path differences between the focal point
r Q and each of the N points (i.e., r ^ r o , . . . , rN _i,r/v) along the flight path. Precisely
speaking, 0 ( w ,r ;r 0) should perform not only phase compensation but also magnitude
compensation with the following form:
r 2e 2j e I7*1 r°l
r 2e 2 j~cLl7'2 r°l
r 2 e 2i — | r i - rd
r 2e 2 j - f \ r , - r 0
r 2 e 2j
^
|r ,-r .|
r 2e2 j ^ Irx-rol
■aJOi— _
2
r^e
-•
r7v _ i e 2^ l r ^ - > - r °l
r 2v e 2 i ^ - | r w - r 0 |
r ^ !e2j^
r 2v e2J ^ ^ - ?0'
- 1“Fo'
0
r o —r 0 |
OJ-,
Mr . v - r o l
rs e
0
r ^2ve2
e ^ | r A- F ol
>■2—ro|
(4.4)
where c is the speed of light in free space and has a value of 3 x 108 m s 1. By multiplying
E g (uj,r;ra) with 0 (a/, r ; r 0), the phase-adjusted, gated matrix
( u ;,r ;r 0) results
/?g,c5 (a2, r ; r 0) = £ ’g ( u ;,r ;r 0)x 0 ( u ;,r ;r 0)
(4.5)
that is ready for coherent summation to produce focusing (a process also known as beamform ing in basic antenna theory [13]), forming the final image <
7 Sa r { t 0) at the focal point
r Q:
M
0SA R (ro)
;V
=
(4 -6 )
r
Numerically, the summation over frequency in Eq. 4.6 above is has the same effect as
uj
applying inverse Fourier transform to the coherent sum with the transform evaluate at time
t = 0
/ F(uj)eJUJtdui
t=o
= ^ 2 F( m A u ) A u ) .
(4.7)
and is equivalent to retrieving radar echo from the focal point in time domain if the
calibration plane of the SAR system has been extended to that point prior to measurement.
Finally, to obtain a complete image mapping, procedures expressed from Eq. 4.2 to
Eq. 4.6 above are repeated for each discrete point within the illumination scene. For
'S ee Appendix C
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53
the case of 3-D imaging, r Q can be any point within the illumination volume. In Sec­
tion 4.5.1, the algorithm presented here will be cast into a slightly different form for
parallel algorithmic comparison with other SAR processing algorithms.
4.3
Generalized ambiguity function o f circular SAR
From previous discussion, it is clear that SAR and confocal imaging are principally similar.
W hile SAR mainly operates at microwave frequencies for terrain mapping applications,
confocal imaging commonly performs at optical frequencies for medical sub-tissue imaging
applications. In this section, an effort is made to combine these two techniques to achieve
space-time confocal imaging using SAR on a circular, or more generally, a curved path.
The general SAR imaging formulation based on the conventional SAR technique is first
developed. Then, numerical and experimental demonstrations are presented to illustrate
that 3-D imaging capability and improved spatial resolutions (on the order o f a fraction
of wavelength) are possible with circular SARs.
4.3.1
Analytic formulation
In radar engineering, given an imaging radar system it is a common mathematical practice
to analyze its azimuth and range resolutions using the concept of ambiguity function [25].
In this section, the ambiguity function of circular SAR system, assuming Gaussian chirp
pulse input [31], is derived.
As shown in Fig. 4.4, circular SAR is an ordinary SAR sensor moving along a circular
track to obtain data over 360°. To perform confocal imaging of a point r G located on a
plane at a depth h 0 below the track, it is important to first note following the conventional
SAR formulation that the signal received by the radar at r sn( f sn = ( xsn, 0), where n =
1 ,2 ,..., N ) is given by [6, 25]
volum e
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(4.8)
54
where g n ( t , r) is the convolution of the input pulse ut (£) and the two-way Green’s function
from r s n to r and back to r s n , and S ( r ) is the volume reflectivity of the incident wave
from r s n to r . In general, S ( r ) depends also on the direction of the incident wave. In this
simplified model, however, it is assumed that S ( r ) is almost independent of the incident
direction.
This signal h n ( t ) is then filtered with the focusing filter function / „ ( i . r 0) matched to
a point target at r Q. The filter function / „ ( i , r 0) is the convolution of the gate function
u j ( t ) and the two-way delay phase factor for focusing at r c from a radar located at r s n .
Therefore, the filter output v n ( r Q ) is given by
j
V n (r0) =
f t ( t , r 0)h n (t)d t.
(4.9)
The SAR output v ( r a ) is the coherent sum of all v n ( r a ) along the entire flight track and
is therefore given by
x
v (To) = 2 2 v*(r °)-
(4.10)
n
With Eq. (4.8) and Eq. (4.9), Eq. (4.10) can be rewritten in the following form
v ( r a) = [
S ( r ) x ( r , r 0)d r
(4.11)
J v o lu m e
where
N
r
x ( r . r 0 ) = Y 1 j 9n ( t , r ) f * ( t , r 0 ) d t
(4.12)
n
is called the “generalized ambiguity function” [25]. The functions g n { t , r ) and f n ( t , r 0 )
can be expressed by their Fourier transforms, ~gn ( u i , r ) and f n ( u j , r 0 ) , respectively, as
follows
9 n(l, r)
= -^
/n((,T 0) =
f
J
(4.13)
J n( u , r 0 )e~tuJtdw
(4.14)
where g n ( u j , r ) is given by the product of the spectra o f the input pulse and the two-way
Green’s function
j n( w , r ) = i i i (w )G o(a;,rB), and
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(4.15)
55
circular flight path
/ 1
//■
/
/
SAR platform
rsn = (Xsn-°)
'On
focal point #
I'd = (x o--ho)
/
r = (x,-h)
Figure 4.4: Geometry o f a circular SAR system. Focal point r a being located at ( x 0, —h 0 ).
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56
g*~2rn
G 0( u ,r n ) -
(4.16)
— ------------
(47rrn r
and f n ( u j . f 0 ) is given by the product of the spectra of the gate function and the focusing
function
with r n
=
7 n( u , r 0) = u f (u>)Gf(w,ron), and
(4.17)
G f ( u , r o„) = el c2ro"
(4.18)
|r3n —r| and r on = |rsn —r 0|. Substituting Eq. (4.13) and Eq. (4.14) into Eq.
(4.12), the following equation results [25]
-v 1 r
-*
X ( r , r a) = ]T — / 9 n { u , r ) f n(u},r0)cLj.
„ 27t J
(4.19)
This is the general expression for the generalized ambiguity function, where 9 n ( ^ : r ) and
f n ( i u . r 0 ) are given in Eq. (4.15) and Eq. (4.17).
Denote in thefollowing the input pulse Ui(t) and the gate function u/ ( t ) . For SAR,
it is normal to use a chirp for both Uj(£) and uj (t ) [31]. The chirp is given by
-iuj0t-iyt 2
{
U| < T
°
\t\ > T 0
o
For mathematical convenience, the equations above can be combined as
Ui(t) = Uf (t ) = e-^o t-(a '+ la")l2
_ O Q < t < l30
(4.20)
The frequency is thus given by uiQ+ 2a"t and the bandwidth of the chirp for the pulse in
rT 0
-To
Thus,
/
oo
i 2
e~a
d t=
[To
Eq.
(4.20) is uii = 4a "T a. Also, one can choose a
dt
(4.21)
J -T o
-oo
i
n
a = a + ta =
7T
JJh
+ i—
(4.22)
Making use of Eq. (4.20) and Eq. (4.22), the spectra Ui(ui) and u/(cu) can be written as
/» tr
(u ~ (Jn
^
Ui(u) = Uf{uj) - -r—rs/cF e- 41“I- “ _,Q
O'
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(4.23)
57
Substituting Eq. (4.23) into Eq. (4.15) and Eq. (4.17), the generalized ambiguity function
in Eq. (4.19) becomes
l— — \
f
1
*
- i ^ ^ - Q' + ^ 2 ( r n - r o n ) ,
X (r,r0) = 2 V^ 7 2Ttt 7-00 T(47rrn)-|o;|
s
v n ~ \e
_ ^ _V 27^
V
^
i s^ - 2 ( r „ - r 0n ) - 4 r ( ^ T + ^ - ) ( r n - r o n ) 2
(4 ttr„)2
°
(4 ‘24)
where r n = |r sn — r| and r on = |r sn — T0|, u Q is the carrier frequency, Ub is the chirp
bandwidth, and T 0 is the sweep time. This is the general expression for x (r, r 0) applicable
to SAR on any curved path. The radar is located at r sn and radiates a Gaussian chirp
signal in Eq.
(4.20), and the received signal is multiplied by the complex conjugate
of focusing filter function to obtain confocal imaging at r 0. The output at r sn is then
coherently summed to give the final SAR output v(rQ).
Pixel resolution
For ideal focusing at r — r Q, Eq. (4.24) should resemble a delta function S(r — r Q)
and therefore, by examining Eq. (4.24), it is possible to determine the resolution of the
imaging system.
Although Eq. (4.24) gives a general expression which can be numerically calculated,
it is instructive to examine its approximate analytical expression for resolutions in a few
special cases. For a down-looking circular SAR focusing at a point T0 lying on the plane
2 = —h0, note that
rsn =Zsn
<pn
r
T0
— a(cos 6 nx + sin (pny)
(4.25)
= n A 0 = ^(2 7 t)
(4.26)
—x — h z
(4.27)
= Xo - hoi.
(4.28)
The resolution can be obtained by examining the normalized ambiguity function
W (r,f^) = | ^ i l | .
X{T0 , r 0)
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(4.29)
58
Altitude resolution
To consider the on-axis resolution. Let x = x Q = 0, resulting
rn
Assuming A h = \ h — h 0 \
r on = y / h , 2 + a 2
-
-
\Jh 2 + a 2.
(4.30)
h 0,
r n - r an ~
h 0A h
. -------
(4.31)
Substituting Eq. (4.31) into Eq. (4.29),
_ 1 / jr ■
\ (hgAh)“
iV(r,r0) cxe
(4 32)
Thus, the resolution A h which reduces N ( r , r a ) to e ~ l is given by
Ah = —
where 6 dp = sin
2
c
------------- -3-----
(4.33)
(—? % = ) is the depression angle. The factor of 2 in Eq. (4.33) signifies
X /13 +a2
the symmetry of N ( r , r a) about the focal point r 0 = (0. —hQ). Note also that if the sweep
time T 0 is much longer than the inverse of the bandwidth (u;^1), then, A h is approximately
given by
, ,
A/i «
2 \Z2 tt
, c
)
sin 6'dp idfj
(4.34)
Azimuthal resolutions
Next, consider the resolution in the transverse plane (z = —h. where h — h 0) where
rn — i"o n = \Jh 2 + |x sn — x \2 — \Jh 2 + |xsn — X o \ 2 . If one considers the resolution near
the axis (|x|
ha and |xQ| <§: hQ), then
rn
~ r on =
X STl • ( X o —
/
x)
’
(4.35)
Substitute Eq. (4.35) into Eq. (4.24) and note
N
r2 ir
(4 3 6 )
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59
Also note that the ^ 2 (rn — r on) term in the exponent in Eq. (4.24) makes a dominant
contribution to the summation since u/Q
N ( r , r Q)
oc
=
ujb
/
2 t t Jo
T ~ l . Therefore,
d(pel~^2('rn r°n^
Jo(— 2cos0dP\xo —x\).
c
(4.37)
Noting that the first zero of Bessel function J 0(") is approximately 2.4 and including the
symmetry of N( r , r c) about the focal point r Q = ( x0, —h a), the resolution in the transverse
plane is obtained as follows
4.8
Aa; = |a:0 - x\ w M
-2 cos (7dp
c
In the above, special cases of resolution on and near an axis are discussed.
(4.38)
It is
significant to note that the axial resolution (Eq. (4.34)) comes primarily from the band­
width while the transverse resolution (Eq. (4.38)) depends mostly on the wavelength. In
general, however, the pixel resolution at an arbitrary point r in space depends on both the
bandwidth and the wavelength.
4.3.2
Numerical calculations
In this section, the confocal imaging capability of circular SAR is demonstrated analyti­
cally using the generalized ambiguity function x ( r , r Q) derived in Section 4.3.1. Define in
the following a set of numerical parameters pertaining to a particular circular SAR system
operating at X-band frequencies
• bandwidth B = 7-13 GHz,
• sweep time Ta = 4 s. Chirp rate a" = ^
• radius a of the circular SAR flight track = 1.0 m,
• depth hQ of the plane to be confocally imaged = 1.0 m below the track.
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60
• sampling rate of SAR measurement along the flight track = every 1°.
Let
x ( x ), x(y)>
and x(2) denote the magnitude variations of x (^ ? r0) projected re­
spectively along the x - , y~, and z-axes with the focal point located at (0, - h 0). Plots of
x (^ ), x(y)> and
x(z )
are depicted in Fig. 4.5
With reference to the figure, it is apparent that both
x(x)
and
x(y)
bear functional
variation closely similar to that of a delta function, as discussed briefly for the case of
ideal focusing at the end of Section 4.3.1. In addition, at a center frequency f Q= 10 GHz
(A0 = 3.0 cm), the 3-dB peak widths
0.25Ao) of
x( x )
and
x(y)
are indeed on the order
of a fraction of wavelength. This fact should make circular SAR a competitive candidate
for high-resolution imaging applications.
Compared with
x( x ) and x(y)
in Fig. 4.5, x ( 2)’ however, shows a broader peak around
the focal point. In fact, given a fixed bandwidth the peak width of
x{z)
is approximately
given by Eq. (4.34). By setting the focal point r Q to be located at (0, —h), the peak
widths of x(^)> x(l/)> and
x ( z ) as a function
of depression angle (= ta n -1 (£)) are shown
in Fig. 4.6, which show dependence on the depression angle in a manner consistent with
Eq. (4.34) and Eq. (4.38). It is obvious from the figure that at a depression angle of
about 77.5°, circular SAR produces uniform pixel resolutions in the x - , y-, and z-axes.
Finally, it is instructive to compare the focusing capability of a circular SAR system
with that of a conventional linear SAR system. By applying Eq. (4.24) with r sn located
along a a straight line (of length 2a) on the y-axis , xC^h x(y)> and x(z) are calculated
and shown in Fig. 4.7. Unlike circular SAR for which x(^), x(y)> and
xiz)
converge
to a common resolution at about 77.5° (corresponding to an incidence angle of 12.5°),
conventional SAR achieves uniform resolution only for
x(y)
and
x{z )
at a depression
angle of 45°.
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61
IX(x/0 = ((U io))
IX<y,ro = (0,-tio))
0.8
<*>
•5
x-range in wavelength
0
5
y-range in wavelength
IX(z;o =(0,-ho))l
■5
0
5
z-range in wavelength (shifted by +h
Figure 4.5: Magnitude variations o f the generalized ambiguity function x ( r , r Q) projected along the x - , y - ,
and r-a x e s. Bandwidth: 7 -1 3 GHz. Sweep tim e = 4 s. Depth h 0 o f the focal plane = 1 m below the flight
track. Depression angle 6dp = 45°. rc = (0, - h 0).
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62
circular SAR image resolutions: X(x), Y(+), and Z(o).
10
20
30
40
50
depression angle in degrees
60
70
80
Figure 4.6: Pixel resolution (in terms o f wavelength) along the x - , y - , and r -a x e s as a function o f depression
angles Qdp for circular SA R system . Bandwidth: 7 -1 3 GHz. Sweep time = 4 s. r 0 = ( 0 . - / i ) , where
h — a x tan(0jp ).
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63
linear SAR image resolutions: X(x), Y(+), and Z(o).
10
20
30
40
50
depression angle in degrees
60
70
SO
Figure 4.7: Pixel resolution (in terms o f wavelength) along the x - , y - , and c -a x e s as a function of
depression angles 0<fp for linear SAR system. Bandwidth: 7 -1 3 G H z. Sweep time = 4 s. r 0 = (0, - h ) ,
where h = a x ta n (Odp)-
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64
4.3.3
Experimental results
In order to verify the correctness of the analytic formulation and numerical calculations
presented previously, a simple imaging experiment using circular SAR at microwave fre­
quencies (the details of the experimental setup will be discussed fully in Section 4.4
below) is conducted. In this experiment, a 63 mm conducting sphere is placed on top of a
thick absorbing sheet at a distance of 15 cm from the center of the illumination footprint.
By focusing the circular SAR system right at the location of the sphere, a sharp image
is obtained. The processed image is shown in Fig. 4.8. To examine the “sharpness” of
the sphere image, the data points along the x-axis that contains the peak in Fig. 4.8 were
extracted and plotted in Fig. 4.9. As evident in the figure, the 3dB peak width, which is
roughly equal to 0.25A, is indeed on the order of a fraction of the wavelength.
Up to this point, it has been demonstrated analytically, numerically and experimentally
that by using circular SAR, imagery with super-resolution can be achieved. While super­
resolution is a desirable feature in imaging, it usually entails forbiddenly long processing
time. In subsequent sections, however, attention will be directed more to the 3-D imag­
ing issues of circular SAR and its combined use with correlation technique to produce
enhanced target-to-clutter ratio for imaging in heavy-clutter environment.
4.4
Experimental studies on 3-D imaging using circular SAR
To illustrate the principle of circular SAR in performing 3-D confocal imaging, controlled
laboratory experiments are performed at X-band frequencies (7-13 GHz). In particular,
3 sets of experiments are conducted. They are, in sequential presentation order, 3-D
confocal reconstruction of (A) layers of spheres, (B) a single sphere, and (C) a palm-sized
conducting model helicopter.
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65
SARimage of a sphere infree space
x-range pixel resolution inI
Figure 4.8: 2-D circular SAR image of a conducting sphere in free space. The sphere was located at a
distance of 15 cm (or 5A at a center frequency of 10 GHz).
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66
Generalized am
biguityfunctionof circularSAR
0.35
0.25
£ 0.2
0.05
-8
-6
4
-2
0
2
4
6
8
pixel resolution inI
Figure 4.9: 1-D extraction o f data points along the x -a x is that contains the bright im age in Fig. 4.8. The
3dB peak width o f the generalized ambiguity function o f circular SAR system is approximately 0.25A.
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67
4.4.1
Experiment A: Confocal reconstruction o f layers o f spheres
X-band microwave imaging experiments are conducted to investigate the feasibility of
imaging of multiple layers of conducting spheres, as shown in Fig. 4.10.
The trans­
mitting and receiving antennas are mounted on a stationary stage and positioned in the
backscattering direction, radiating at W -polarization at a 45° depression angle.
At a slant reference range of 140.0 cm the dimension of the illuminated volume is
37.4 cm. Within the illuminated volume, spheres are supported on a rotation table by two
thick sheets of Styrofoam (a household insulation material that is nearly transparent at
microwave frequencies). The lower sheet supports three uniformly-spaced metal spheres
of different diameters (one of 5.0 cm and two of 2.5 cm, with a 10.2 cm separation) and
the upper sheet supports one metal sphere with a diameter of 6.4 cm located at the center
of rotation. The two sheets are separated by a distance of 19.0 cm. A frequency-domain
SAR measurement is made every 5° along the circular path over a frequency band of 7-13
GHz.
To process the SAR data, procedures expressed from Eq. 4.2 to Eq. 4.6 above are
used for each discrete point within the illumination volume. In particular, the illuminated
volume is first discretized into 77
x
77
x
9 cells (that is, 77
x
77 cells on each of the
9 layers at a pixel resolution of 6 points/wavelength). To each of these cells, a unique
set of time-domain, Kaiser-based gating functions is computed and applied to each of the
measurements made along the circular antenna path. The resulting gated measurements
are then magnitude-phase adjusted and coherently added for beam-forming. Since this
spotlight-mode focusing process can be done on any plane within the illumination volume,
three-dimensional images can be obtained by focusing one slice at a time and using surface
or volume rendering software on the stacked-up image slices.
The images of a few selected layers (out of the 9 layers in total) are processed and
they are presented as contour plots in Fig. 4.11. As evident from the contour plots, each
subplot addresses correctly the arrangement of the metal sphere(s) shown in Fig. 4.10.
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68
V
antenna
layer (a)
6.4cm
H
h
layer (b)
T
1
19.0cm
i
^
"
-
1
y / / / y y y / / '/{■/
2.5cm
+1\ k
r
A
W
'/'/////" /.■
, /V /y /X 'Z /
Styrofoam sh eet
2.5cm
6.4cm
z
k
r
>z1 rk
a
V
layer (c)
Styrofoam sh eet
layer (d)
Figure 4.10: Schem atic for 3 -D confocal imaging o f layers o f metallic spheres using circular SAR. Band­
width: 7 -1 3 GHz.
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69
As the focal plane descends from top to bottom, the contour plots undergo the transitions
from no sphere to one sphere, then from one sphere to three spheres and finally from
three spheres to no sphere again. From this experiment, it is clear that circular SA R can
focus at a given plane while defocusing all other planes within the illumination volume,
thus demonstrating its 3-D confocal imaging capability once available only at optical
frequencies.
4.4.2
Experiment B: Confocal reconstruction o f a single sphere
As an additional illustration, this section describes another experimental imaging setup
used to reconstruct the 3-D image of a metal sphere suspended in free space. For full
bistatic surveillance capability to be utilized in near future, the simple setup discussed in
Section 4.4.1 is modified from ground up, resulting in the circular SAR system schemat­
ically shown in Fig. 4.12. With reference to the figure, this circular SAR system is
constructed for use at X-band frequencies (7-13 GHz). The dimensions of this system
are approximately 2 m x 2 m x 2 m
and is built almost entirely of 4 ’ x 4 ’ wooden
poles. To provide full bistatic surveillance capability, the transmitting and receiving an­
tennas are mounted on two separate circular rings, each of which is individually driven by
computer-controlled stepping motors with an angular precision of approximately 0.02°.
To perform SAR imaging, a 63 mm conducting sphere is placed on top of a Styrofoam
support located at the center of the illumination footprint. At a slant range of 0.97 m and
a depression angle of 46°, an illuminated scene of size 42 cm x 42 cm is imaged by the
circular SAR system which performs frequency-domain SAR measurement at an angular
increment of 1° along the circular track. By applying similar SAR processing procedures
outlined previously, the corresponding 3-D image is obtained and shown in Fig. 4.13. Note
that the apparent dark line that connects the top and bottom contour layers is produced
by the PLOT3 routine in MATLAB and has nothing to do with the actual image. Once
again, this experiment confirms the 3-D imaging capability of circular SAR.
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0.5
-
0.2
-
x-range in m
eters
0.1
0
0.1
0.2
x-range in meters
(c)
(d)
1.1
1.1
CO
CO
Jj
® 1
1
CD
05
E
in
.S t
-0.9
t
*
CD
CO
E
CD
CD
05
cn
c
c=
20.8
20.8
>*
0.7
-
0.2
0.7
-
0.1
0
0.1
x-range in m
eters
0.2
-
0.2
-
0.1
0
0.1
0.2
x-range in meters
Figure 4.11: Experimental result for the imaging experiment shown in Fig. 4.10: (a) no sphere, corresponding
to “layer (a)” in Fig. 4.10, (b) one sphere, corresponding to “layer (b)” in Fig. 4 .1 0 , (c) three spheres,
corresponding to “layer (c)” in Fig. 4.10 and (d) no sphere, corresponding to “layer (d)” in Fig. 4.10.
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71
bearing supports
w ooden ring
antenna
antenna
w ooden ring
metal sp h ere
_ -styrofoam support
...........
at
absorber sh e e t
-.
turntable
■ ' ' / / / / / / ' ■ ' / / / / / / / / / / / / / / / / / / / / / / / r V / / ' " ‘' / / / / ' s ' /,■' ' / ■ ' / / s’ / / ' / ' / ' /
' ' / / y ///'■'■'
y/ ' / ■ '/ / .
Figure 4.12: Schematic for 3 -D confocal imaging o f a metal sphere suspended in free space using circular
SAR. Bandwidth: 7 -1 3 GHz.
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72
3D reconstruction of a sphere using circular SAR
0.07
0.06
0.05
0.04
0.03
0.02
-
0.01
-
0.01
0.02
•
-0.094
-0.096
-0.098
•
0.102
x10'3
-0.104
-0.106
-6
Figure 4.13: Experimental result for the imaging experiment shown in Fig. 4.12. The image is displayed as
a stack o f uniformly spaced (vertically) contours. The apparent dark line that connects the top and bottom
contour layers is produced by the PLOT3 routine in MATLAB and has nothing to do with the actual image.
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73
4 .4 .3
E x p e r im e n t C : C o n fo c a l r e c o n s tr u c tio n o f a m o d e l h e lic o p te r
So far both Experiment A and B deal with isotropic scatterers only (those objects that
radiate uniformly over the entire 4 it steradians). In order to demonstrate the imaging
versatility of circular SAR, an imaging experiment for a non-isotropic object is conducted
and specifically, a palm-sized toy plastic helicopter is used for this purpose, the schematic
of this experimental setup is shown in Fig. 4.14.
In this experiment, the plastic helicopter is first painted with a kind of fluid metallic
paint (known as “Nickel print”) so that it becomes conducting at microwave frequencies.
Next, it is placed on top o f the same Styrofoam support used in the previous experiment
and is then imaged by the circular SAR system which performs frequency-domain SAR
measurement at an angular increment o f 1° along the circular track. By applying similar
SAR processing procedures outlined previously in Experiments A and B, the resulting
3-D image of helicopter is reconstructed is obtained and displayed as distributed clusters
of dots in Fig. 4.15.
From the figure, it is clear to see the main features of the helicopter such as propellers,
head, body shaft and tail. In particular, it is interesting to correlate the com er structures
located at the helicopter tail with the strong reflection shown in Fig. 4.15. In addition,
note that the length of the aircraft shown in Fig. 4.14 agrees with that of the processed
image in Fig. 4.15.
4 .5
C lu tte r s u p p r e s s io n u s in g c o r r e la tio n im a g in g
Radar imaging in heavy-clutter environment has never been a trivial issue to radar design­
ers [25, 31]. The undesirable effects of clutter manifest themselves in the form of speckles
(bright spots) in the processed radar images, making image interpretation inaccurate. In
reality, this failure to characterize radar images accurately could lead to costly and/or
dangerous consequences in applications such as detection of unexposed active landmines
in abandoned battlefields.
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74
bearing supports
w o o d en nng
antenna
antenna
w ood en nng
30 cm
toy helicopter
styrofoam support
' 'sSfis
absorber s h e e t
tum tabe
Figure 4.14: Schematic for 3 - D confocal imaging o f a palm-sized model helicopter suspended in free space
using circular SAR. Bandwidth: 7 -1 3 GHz.
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^ ft}
,gure
,
tf/V.
Ae//
0.15
AS; c-
^ botM
Pcn'ty,
* * ,l r
**§0
P'CVn
ftcprl
*"7p „
a^ees > * & * * ^ /6/f ^Periff.
w tb ,L g strn„
‘h v
7/
* -//f c ,
'-TSs&sfeij
’**
^ e 0^r c° ^
'eo^/s,
S/bo
°^ 6
C°P^
er
’e r
re/0r,
°afo
ecy
0(7/
7^6
Pen^/s,
Von.
76
From the theory of scattering by random media, it is a well-known fact that the
scattered waves produced by random scattering carry almost no coherent information (e.g.
polarization, amplitude, phase, frequency, angle o f incidence and so forth) of the original
incident wave [13]. In contrast, for scattering by most man-made targets (e.g. buried
landmines or underground utility pipes which are characterized by well-defined scattering
structures) where deterministic scattering is the dominant scattering mechanism, however,
the scattered waves tend to retain a decent amount of coherence.
One way of taking advantage of this intuitive absence/presence of coherence in scat­
tered waves to achieve clutter suppression is to consider the correlation of waves, where
the weak correlation response from clutter (random scattering) is brought into sharp con­
trast with the strong correlation response from man-made targets (deterministic scattering)
- it is these distinct correlation behaviors between clutter and targets that permit enhanced
target-to-clutter ratio using correlation techniques. Obviously, the success of this corre­
lation approach depends on the dissimilarity between clutter and target responses: the
more dissimilar between clutter and target responses, the more enhanced the resulting
target-to-clutter ratio is. In Chapter 3, ACF technique has been successfully incorporated
to achieve superior detection capability over traditional cross-section (intensity) technique
in detection applications. In parallel with this development, the remaining sections in this
C hapter describes different SAR processing schemes and compare their relative figure of
merits. In particular, heavy emphasis is put on the combined use of ACF and circular
SAR. The resulting technique, known as angular-correlation SAR, is to be an effective
means of suppressing clutter for imaging in heavy-clutter environment.
Before comparing conventional circular SAR processing with other types of correlation
SAR processing, it may be helpful to examine the details of each type of these SAR
processing schemes. Specifically, two types of correlation SAR processing schemes will
be examined in addition to the conventional circular SAR processing scheme.
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77
4.5.1
Conventional SAR processing
Although the conventional SAR processing algorithm has been adequately elaborated in
Section 4.2, the expressions derived in that section are not in the most convenient forms
for comparison purposes in this section. For this reason, the same conventional SAR
processing algorithm developed in Section 4.2 will be cast into slightly different forms in
the formulation below.
In a conventional focused SAR system, raw vector frequency measurement E (u, r) is
first acquired as a function of space. Given a point r a to be focused, this raw vector data
matrix E (a;, r) is then frequency-convolved with a gating filter G (u , r : r 0) to minimize
directcoupling
between antennas and multipath interference produced by experimental
artifacts. Next,the resulting gated data matrix is
phase-compensated by d (uj,r:ra) and
then coherently summed together over frequency and space for focusing and beamforming.
The operations described above can be summarized as
f r e q space _
<rsAR(r0) = I
_
_
H [E ( ^ , r ) 0 G ( a /,r ;r 0)]x 0 («;, r: r 0)|2
uj
(4.39)
r
where crsAR{ra) is the processed cross section with the beam focusing at r 0 and ® denotes
convolution operation implemented using FFT algorithms. Note that the gating filter G is
a function of the focal point T0. In effect, this functional dependence on space of gating
allows spatial tracking in addition to focusing operation, resulting in more effective gating.
4.5.2
Frequency-correlation SAR processing
One common technique for speckle suppression in SAR processing is to perform frequency
correlation between SAR images processed at different pairs of sub-bands [cjf] and
[ujj]
over the entire measurement bandwidth. In many ways this technique is very similar to
conventional SAR processing discussed above and involves the following operations:
freq
Vu-SARifo)
=
\<
space
_
_
_
^ 2 {[E ( K ] , r ) 0 G ([w<],T;r0)]x <b ([w i],r;r0)}
KI.KI r
V i^ j
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78
{ [ E ( K ] ,r ) ® G ([o ;j],r;r0)]x 0 ([u!j\,r‘, r Q) }* > |
(4.40)
where * and () represent complex conjugation and ensemble averaging operation, respec­
tively. In effect, this correlation approach involves partitioning SAR measurement into a
number of sub-bands of SAR data over the entire measurement bandwidth. For sufficient
statistical independence among these sub-bands of SAR data, the partitioned sub-bands
should be separated by at least the decorrelation bandwidth of the measurement. Con­
ventional SAR processing is then applied to each pair of the sub-band SAR data before
correlation between images takes place. The resulting correlated image is then averaged
with other correlated images obtained at different sub-band pairs to produce the final
frequency-correlation SAR image cf^ - s a r ■ Fig- 4.16 describes this correlation scheme
for the cases of (a) 4 partitions and (b) 8 partitions. It is important to observe that the
clutter suppression achieved by this correlation approach does not come without cost. By
partitioning bandwidth into smaller sub-bands, however, the spatial resolution is degraded.
4.5.3
Angular-correlation SAR processing
In addition to frequency correlation, an alternative way of performing correlation is to cor­
relate waves scattered at different observation angles over the entire measurement band­
width. Because of its consideration on angular dependence of scattered waves, this cor­
relation approach is appropriately called angular-correlation SAR processing. It involves
the following operations:
fr e q
v*-SAR(ro)
=
| < Y,
space
J2
_
_
_
{[E ( u . T i ) ® d (u>,ri;r0) ] x d) ( u j , r i : r 0)}
V (i—j)=Att>'s
{[E
G (a;.rj;T 0)]x 0 ( u j . r j - . r ^ Y > |.
(4.41)
In contrast with frequency-correlation SAR in which a full view (360°) of SAR mea­
surement is available for bandwidth partitioning, angular-correlation SAR involves correla­
tion between spatially partitioned SAR data over the entire measurement bandwidth. With
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79
4 partitions
final image = < im a g e l, image2 >
im agel
image2
'
Y //,V / / / / . ' / / / .
(a)
8 partitions
final image = < im ag el, image2, image3, image4 >
im agel
im age2
im age3
im age4
(b)
Figure 4.16: Schem atic o f bandwidth partitioning in the frequency-correlation SAR for (a) 4 partitions and
(b) 8 partitions.
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80
the particular geometry of circular SAR, this correlation scheme is depicted in Fig. 4.17.
As shown in the figure, this scheme begins with correlational imaging at a small included
angle A 0. For instance, at an included angle A <p of 3° (as in Fig. 4.17(a)), this approach
correlates the SAR pairs at {0°; 3°}, {1°,4°}, {2°, 5°}, . . . , {357°; 0°}, {358°, 1°} and
{359°, 2°}. The correlation products are then averaged to produce an image I\. Next,
at an included angle of, say, 30° (as in Fig. 4.17(b)), the same correlation operation is
applied to the SAR pairs at {0°,30°}, {1°,310}, {2°, 32°}, .. .,{357°, 27°}, {358°, 28°}
and {359°. 29°}. The resulting correlation products are then averaged to produce another
image U. As this correlation procedure continues for larger A o ’s, an image stack con­
sisting of
l\.
In.
h , • • - ; I m - l: 7 v /
will be generated, where M is the number of correlation
angles or simple a measure of the “length” of the image stack.
Note that in general
larger M implies more clutter suppression (and hence more processing time). By taking
average over the length of the image stack, an angular-correlation SAR image is formed.
Note that the use of multiple decorrelation angles in this summation process is analogous
to performing ensemble averaging. Therefore, this compact imaging scheme effectively
combines imaging formation (indicated by coherent summation) and ensemble averaging
in one procedure.
To illustrate experimentally this useful relationship between image formation and en­
semble averaging, a 63 mm conducting sphere is placed off-center on top of an absorber
sheet below the circular SAR system. Frequency-domain SAR measurement is then ac­
quired at an 1° angular increment over 360° along the circular flight track. Both con­
ventional SAR and angular-correlation SAR processing schemes are performed for image
formation. Fig. 4.18 shows the corresponding conventional SAR image. The presence
of the bright dot in the figure suggests strongly that the conventional SAR processing
algorithm, when combined with circular SAR configuration, is indeed very effective for
high-resolution imaging. In contrast, the angular-correlation SAR image with single decor­
relation angle (2°) in Fig. 4.19 produces a relatively vague (and broad) dot compared with
that in Fig. 4.18. Besides the contribution by the faint reflections from the absorber sheet,
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81
(c)
O
*
f
0
Figure 4.17: Schematic o f angle partitioning in the angular-correlation SAR for (a) 3 °, (b) 3 0°, (c) 4 5 °, (d)
90°, (e) 120° and (f) 150°. A ngles are not drawn to scale.
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82
summation over many highly-correlated terms (due to the small decorrelation angle used
in the processing) in Eq. 4.41 is the primary mechanism leading to the bright annular
image surrounding the sphere.
To obtain an improved version of this correlation image, one could therefore make
use of multiple decorrelation angles in the processing algorithm. Based on the argument
presented in previous paragraph, a sensible choice on multiple decorrelation angles in
Eq. 4.41 will thus include large number of large decorrelation angles. Fig. 4.20 shows the
improved image with a decorrelation angle of 20°. As the number o f large decorrelation
angles increases, the image becomes more enhanced, leading to the one in Fig. 4.21. This
image demonstrates the importance of using large number of large decorrelation angles in
angular-correlation SAR processing in low-clutter environment.
As most man-made targets reflect strongly only for a relatively narrow range of obser­
vation angles, it is important to include both small and large A 0 ’s in the stacking process
discussed above. Such inclusion is especially important for imaging in heavy-clutter en­
vironment where small Aci’s help distinguishing target response from clutter response
and large A o ’s warrant low clutter response to the final angular-correlation SAR image
a < S > -S A R -
Like frequency-correlation SAR processing, the clutter suppression brought by angularcorrelation SAR processing does not come without cost. By partitioning the full 360° field
of view (FOV) into smaller sub-FOV’s of various angular widths, image of the targets
which have fine angular-dependent radiation pattern may be smeared, thus degrading
angular resolution of the final angular-correlation SAR image.
Note that the success of angular-correlation SAR processing depends heavily on the
difference between the decorrelation rates of clutter and target responses. In low-clutter
imaging environment such as previous experiment where clutter response decorrelates at
about the same rate as target response, inclusion of large number of rapidly decorrelating
terms in the coherent summation process is necessary for quality images. In heavy-clutter
imaging environment, on the other hand, clutter response tends to decorrelate far more
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83
Conventional SAR
x-range inmeters
Figure 4.18: Conventional SA R image o f a 63 mm conducting sphere on top o f absorber material.
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84
Angular-correlation SAR: 2°
-0.3
-0.2
-0.1
0
0.1
x-range inm
eters
Figure 4.19: Angular-correlation SAR image of a 63 mm conducting sphere on top of absorber material.
The image was processed with a decorrelation angle of 2°. The bright annular ring results from the
highly-correlated terms in the summation mechanism in Eq. 4.41.
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85
Angular-correlation SAR: 20°
x-range inmeters
Figure 4.20: Angular-correlation SAR image of a 63 mm conducting sphere on top of absorber material.
The image was processed with a decorrelation angle of 20°. Note that the previous bright annular image
in Fig. 4.19 has dimmed significantly as a result of using large decorrelation angle (i.e. 20°) in this case.
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86
Angular-correlation SAR: 20°, 40° and 60'
x-range inmeters
Figure 4.21: Angular-correlation SAR image of a 63 mm conducting sphere on top of absorber material.
The image was processed with multiple decorrelation angles of 20°, 40° and 60°. Note that the previous
annular image in Fig. 4.20 has disappeared almost completely. This image demonstrates the importance
of using large number of large decorrelation angles in angular-correlation SAR processing in low-clutter
environment.
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87
rapidly than target response, and inclusion o f large number o f rapidly decorrelating terms
will thus desirably de-emphasize the masking effect of clutter on the target response. In
the next section, experimental results will be presented to illustrate the effectiveness of
angular-correlation SAR processing over conventional SAR processing in heavy-clutter
environment.
4.5.4
Experimental comparison
To compare the relative merits among different SAR processing procedures (conventional
SAR, frequency-correlation SAR and angular-correlation SAR) described previously, two
sets of SAR imaging experiments are conducted at X-band frequencies (7-13 GHz). Ex­
periment A examines the relative performance of these processing schemes in mediumclutter environment whereas Experiment B focuses on the distinct effectiveness of angularcorrelation SAR processing in heavy-clutter environment.
Experiment A
With the circular SAR setup described in Section 4.4.2, two identical metal spheres
(diameter = 25 mm) are placed on top of a large Styrofoam imaging platform supported
by thick absorbing sheets. The spheres are separated from each other by 20 cm and
symmetrically located about the center of the circular flight path. To simulate a mediumclutter environment, tiny gravel particles are poured onto another layer of Styrofoam sheet
placed directly on top of the spheres. The resulting experimental setup is schematically
shown in Fig. 4.22.
Conventional SAR processing is applied to the raw data, yielding the image in Fig. 4.23.
As shown in the figure, it is clear that conventional SAR processing continues to produce
faithful image of the scene. The two spheres are indicated unambiguously by the bright
dots in the figure.
On the other hand, the corresponding image using frequency-correlation SAR process-
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88
tiny gravel particles
Styrofoam sheet
metal sphere
metal sphere
Styrofoam sheet
c ^ 'T s y ///////////////,- /, v/ / / / / / / / / / / / /
Absorber material
Figure 4.22: Experiment A: a schematic for microwave imaging in medium-clutter environment using
different kinds o f SAR processing methods. The size o f spheres (diameter = 25 mm) is much larger than
the size (mean diameter ~ 3 mm) o f the gravel particles.
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89
Conventional SAR
x-range in m
eters
Figure 4.23: Experiment A: conventional SAR processing continues to produce faithful image of the scene
in medium-clutter environment The two spheres are indicated unambiguously by the bright dots in the
figure.
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90
ing is displayed in Fig. 4.24. Again, the presence of the spheres is clearly represented by
the same two bright dots in the figure. Essentially, both Fig. 4.23 and Fig. 4.24 convey
the same information.
Finally, angular-correlation SAR technique is used to process the image from the raw
data and the resulting image is portrayed in Fig. 4.25. Coincidentally, the processed image
looks alike those depicted in Fig. 4.23 and Fig. 4.24. It appears that even in mediumclutter environment, the three processing schemes appear to produce consistent images
- an observation which finds close counterpart in low-clutter environment elaborated in
Section 4.5.
Experiment B
To study the effectiveness of angular-correlation SAR processing over other two SAR
processing schemes examined in this chapter, an imaging experiment in heavy-clutter
environment is conducted. Specifically, the setup is shown in Fig. 4.26 in which the two
spheres (diameter = 25 mm) used in the previous experiment were placed on top of a
layer of large gravel particles (mean particle diameter « 30 mm). As before, the spheres
are separated from each other by 20 cm and symmetrically located about the center of
the circular flight path. The top view of this assembly is shown in fig. 4.27. In view of
the heavy clutter introduced by the volume scattering medium made up o f large gravel
particles, it is expected that the processed images, unlike those presented in previous
section, contain a modest amount of speckles.
At a frequency bandwidth from 7-13 GHz (wavelength A = 3 mm at a center frequency
of 10 GHz), SAR measurement is performed over the circular flight path at every 1°
increment. The resulting SAR data is then processed individually using Eq. 4.39, 4.40
and 4.41, with the results being summarized in Fig. 4.28-Fig. 4.30.
In Fig. 4.28 and Fig. 4.29, it is evident that although both conventional SAR processing
(in Fig. 4.28) and frequency-correlation SAR processing (in Fig. 4.29) are capable of
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91
Frequency-correlation SAR
x-range in m
eters
Figure 4.24: Experiment A: frequency-correlation SAR processing results in an image strikingly similar
to that produced by conventional SAR processing. This mage demonstrates that the benefit of applying
frequency correlation technique is not obvious in not only low-clutter environment, but also medium-clutter
environment
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92
Angular-correlation SAR
x-range in m
eters
Figure 4.25: Experiment A: angular-correlation SAR processing results in an image strikingly similar to
that produced by conventional SAR processing. This image demonstrates that the benefit of applying
angular correlation technique is not obvious in not only low-clutter environment, but also medium-clutter
environment
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93
metal sphere
metal sphere
large gravel particles
Absorber material
Styrofoam sheet
Figure 4.26: Experiment B: a schematic for m icrowave imaging in heavy-clutter environment using different
kinds o f SAR processing methods. The size o f spheres (diameter = 25 mm) is about the same as that (mean
diameter « 30 mm) o f the gravel particles.
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94
metal sp h eres
SAR s e n so r
i
i
i
i
Figure 4.27: Experiment B: a top view o f Fig. 4.26.
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95
capturing the image of one sphere (appearing as bright dots in the figures), both o f them
fail to depict the image of the other sphere (located at +10 cm along the y-axis), which
may have been half buried in the gravel medium. In fact, the clutter in this experiment is
so strong that even the frequency-correlation SAR processing does not bring about much
improvement over conventional SAR processing - the two images are almost identical.
The angular-correlation SAR processing algorithm expressed in Eq. 4.41, on the other
hand, produces the image shown in Fig. 4.30. With reference to the figure, the processed
image correctly depicts the existence of the two spheres with reasonably low level of
clutter. From this experiment, one can conclude that in heavy-clutter environment, the use
of ACF technique is an effective tool of achieving a higher degree of clutter suppression
compared with the conventional technique, resulting in a higher target-to-clutter ratio.
4.6
Summary o f target imaging using correlation technique
In this chapter, a novel approach for 3-D imaging in heavy-clutter environment at RF
frequencies is described. In essence, this approach involves the use of circular SAR. As
a slight modification on flight-path geometry of the conventional linear SAR, the circular
SAR has the advantages of offering a full-view (360° image and higher spatial resolution.
In addition to these advantages, it is illustrated, both analytically and experimentally, that
circular SAR is also capable of performing decent 3-D imaging. Experimental studies are
conducted for imaging of (1) layered structure, (2) single sphere and (3) toy helicopter,
showing promising results.
On the issue of clutter suppression, on the other hand, due attention is paid to the
investigation into relative performance among three SAR processing algorithms which are
(1) conventional SAR, (2) frequency-correlation SAR and (3) angular-correlation SAR.
Algorithmic formulations are given to each of these SAR processing schemes and ex­
perimental results are included to examine the relative merits o f these schemes.
It is
found that in low-clutter imaging environment all three types of processing methods es-
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96
Conventional SAR
x-range inm
eters
Figure 4.28: Experiment B: conventional SAR processing fails to display the correct image of spheres in
heavy-clutter environment Sphere size = 25 mm, mean gravel particle size « 30 mm, bandwidth = 7-13
GHz.
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97
Frequency-correlation SAR
x-range inmeters
Figure 4.29: Experiment B: frequency-correlation SAR processing results in an image strikingly similar
to that produced by conventional SAR processing. Again, frequency-correlation SAR processing fails
to display the correct image of spheres in heavy-clutter environment. Sphere size = 25 mm, mean gravel
particle size ~ 30 mm, bandwidth = 7-13 GHz. This image demonstrates that in heavy-clutter environment,
frequency-correlation SAR processing may not be an effective means for clutter suppression.
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98
Angular-correlation SAR
-0.1
0
0.1
x-range in meters
Figure 4.30: Experiment B: angular-correlation SAR processing results in an image that correctly accounts
for the presence of the spheres. In fact, it is the only means among other SAR processing schemes examined
in this investigation that brings clutter level down to a reasonably low level, leading to clear visibility of
spheres in the presence of strong clutter. Sphere size = 25 mm, mean gravel particle size ~ 30 mm,
bandwidth = 7-13 GHz. This images demonstrates that in the presence of strong clutter, angular correlation
SAR processing is an effective tool of achieving a higher degree of clutter suppression compared with the
conventional SAR technique, resulting in a higher target-to-clutter ratio.
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99
sentially converge at the same image quality, whereas in heavy-clutter imaging environ­
ment, angular-correlation SAR processing greatly outperforms both conventional SAR and
frequency-correlation SAR, resulting in a higher degree of clutter suppression and thus an
enhanced target-to-clutter ratio.
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Chapter 5
CONCLUSION
5.1
Summary
In this dissertation, experimental studies on radar detection and imaging of targets em­
bedded in clutter environment using correlation techniques are presented. First, a brief
analytical treatment of angular memory effect is given in Chapter 2, where it is emphasized
that the effect becomes observable when the scattering mechanism is random in nature.
One of the implications of this sole dependence on random scattering is that the effect,
together with its applications, is observable over a wide range of circumstances in natural
remote-sensing problems. In simple terms, angular memory effect states that a non-zero
correlation exists among scattered waves observed in different directions as a result of
a change in the direction of incident waves. Included in the chapter is an analytical in­
troduction on the concept of ACF and its properties. It is shown that angular memory
effect exhibits little dependence on observation orientation and manifests itself in 9 - or
©-scattering planes, or both. Angular memory effect is thus a rather universal scattering
phenomenon. In fact, it is this apparent “universality” (lack of dependence on scattering
media and observation orientations) that makes angular memory effect applicable in a
wide variety of detection and imaging issues to be examined in this research.
In Chapter 3, applications of angular memory effect (observed in the ^-scattering
plane) in target detection are considered. Specifically, wideband radar experiments on tar­
get detection in natural geophysical clutter environment are considered at millimeter-wave
(75-110 GHz) and X-band frequencies (7-13 GHz). The purposes of these controlled
experiments are twofold. First, they illustrate the existence of angular memory effect due
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101
to the combined scattering (surface and volume) o f natural random scattering media (such
as fine sand, rough sand, gravel, rock and garden soil). Second, they serve as realis­
tic studies for obtaining higher target-to-clutter ratio using practical ACF measurement
in practice. To compare the relative performance between ACF technique and traditional
RCS technique in real-life, strong-clutter environment, metallic objects of appropriate size
are embedded in various natural “noisy” geophysical media and tested with both ACF and
RCS detection schemes. With reference to the results detailed in Chapter 3, it is found
that ACF technique generally works better than traditional RCS technique, resulting in
higher target visibility in strong-clutter environment. In ACF technique, clutter suppres­
sion is achieved by means of correlation between two apparently independent random
variables. In the RCS approach, on the other hand, clutter suppression is achieved by
means of ensemble averaging over many scalar intensity measurement samples. Although
this ensemble average operation helps smooth out the fluctuation of clutter, it does not
reduce the “dc le v e r of clutter as effectively as ACF technique does.
From previous discussion, application of this unique correlation phenomenon in practi­
cal detection problems rests on an understanding of a simple observation: field correlation
due to scattering by an incoherent mechanism is very much different from scattering by
a coherent mechanism. While low-level correlation appears in the case of incoherent
scattering, high-level correlation appears in the case of coherent scattering.
It is this
high-to-low correlation ratio that provides the high target-to-clutter ratio achieved using
ACF technique. From a wave scattering point of view, it is important to realize that this
clutter-suppression property is an inherent property of wave correlation implicit in ACF
measurement. The corresponding improvement in target visibility does not come at the
cost of sophisticated signal-processing algorithms or expensive hardware component addi­
tions. In view of practical implementability, ACF technique, therefore, should fit without
too much modification into existing detection radar systems.
In target imaging applications, angular memory effect in the o-plane is considered. In
particular, considerable attention is paid to 3-D imaging using a novel SAR configuration
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102
known as circular S AR. It is illustrated, both analytically and experimentally, in Chapter 4
that such SAR configuration, apart from providing down-range and cross-range resolutions
in horizontal directions, is capable of resolving altitude information in a vertical direction.
Furthermore, the unique circular geometry of circular SAR’s permits a finer theoretical
resolution than that achievable by traditional SA R ’s. These two features, namely, 3-D
imaging ability and improved imaging resolution, should make circular SAR a suitable
candidate for localized 3-D imaging purposes.
The 3-D imaging capability is rigorously pursued in Section 4.3, in which a general
expression for the generalized ambiguity function is derived. W hile the general expression
is given in Eq. (4.19), the expression for chirp input signal, as a special case of Eq.
(4.19), is given in Eq. (4.24). The axial resolution
x{z)
as given in Eq. (4.33) and Eq.
(4.34) shows that it depends primarily on the bandwidth of the system. On the other
hand, the transverse resolutions
(x{x)
and
x(y))
as given by Eq. (4.38) show that they
depend primarily on the wavelength. Numerical calculations and microwave experiments
are presented in Section 4.3.2 and Section 4.3.3, respectively, to verify the theoretical
predictions. It should be noted, however, that detailed as the mathematical formulation
appears, it is only a simplified theoretical model. Further studies are needed to include the
polarization characteristics of targets. It is also noted that the volume reflectivity in Eq.
(4.8) is applicable for single scattering or Bom approximations, and therefore it requires
further study to include diffraction and multiple scattering effects.
On the issue of clutter suppression discussed at the end of Chapter 4, on the other
hand, due attention is paid to the investigation into relative performance among three SAR
processing algorithms. They are, namely, (1) conventional SAR, (2) frequency correlation
SAR and (3) angular correlation SAR. Algorithmic formulations are given for each of these
SAR processing schemes, and experimental results are presented to examine the relative
merits of these image-processing schemes. It is found that in a weak-clutter imaging
environment all three types of these processing methods essentially converge at the same
image quality, whereas in a strong-clutter imaging environment, angular correlation SAR
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103
processing greatly outperforms both convention SAR and frequency correlational SAR
processing methods, resulting in a higher degree of clutter suppression and hence target
visibility.
5.2
Further studies
Although this dissertation has provided an adequate glimpse of how the concept of field
correlation can be applied in practical detection and imaging applications, there is little
doubt that many technical areas in this research effort still remain underexplored and
unexplored. In view of Section 2.5, for instance, there is much room for further research
on other forms of correlation techniques that make use of polarization or time measurement.
Although a “universal” correlation tool for detection and imaging, is unlikely to work well
in all practical circumstances, a systematic study in the future on a proper combination of
these correlation concepts would definitely be a great aid to researchers.
In a practical remote-sensing environment, scattering loss (due to scattering by, for
instance, raindrops, snow particles, turbulence, etc.) alone can not adequately account for
the signal attenuation mechanism. To obtain a picture that is closer to reality, one must
consider the dissipative effect (due to absorption by vegetation, moist soil, etc.) of the
media. Although this important aspect is largely ignored in this research for simplicity
reasons, a careful relational study on wave scattering and moisture content should enhance
current understanding on the range of validity for the correlation techniques examined in
this dissertation.
On the issue of 3-D image reconstruction, further studies are required to fully char­
acterize the imaging capability of circular SAR. To proceed in this direction, one could
start with imaging of 3-D dielectric targets of arbitrary shape, rather than exclusively
conducting bodies as examined in this dissertation. As a result of microwave energy
penetration in the same manner as light passing through translucent materials, imaging of
these dielectric targets at microwave frequencies should serve as a useful benchmark to
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104
evaluate the sectioning characteristic of circular SAR.
To extend the usefulness of correlation technique in clutter-suppressed imaging men­
tioned in Section 4.5, additional refinement on the partitioning scheme in angular correla­
tion SAR processing is desired. Although it is found that the inclusion of both small and
large decorrelation angles A (p’s is necessary to determine weak target from strong clutter
response (see Section 4.5.3), it remains unresolved at this point what precise steps lead
to a workable selection of different A 0 ’s for a particular imaging problem. An adaptive
approach seems to be a viable avenue to proceed in this regard.
Seldom can a study be considered as “final”. This short but truthful statement also
applies to the work outlined in this dissertation. It is the author’s honest hope that this
work will serve as a useful reference to those who pursue this exciting research topic at a
greater depth in the future and stretch the range of applications of correlation techniques
in related scientific disciplines to its farthest extent.
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on Image Processing, 5(8): 1252-1265, Aug 1996.
[27] B. D. Steinberg and H. Subbaram. Microwave Imaging Techniques. John Wiley &
Sons, 1991.
[28] L. Tsang, J. A. Kong, and R. Shin. Theory o f Microwave Remote Sensing. WileyInterscience, 1985.
[29] L. Tsang, G. Zhang, and K. Pak. Detection of a buried object under a single random
rough surface with angular correlation function in em wave scattering. Microwave
and Optical Technology Letters, 11(6):300-304, Apr 1996.
[30] E T. Ulaby and C. Elachi. Radar Polarimetry fo r Geoscience Applications. Artech
House, 1990.
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108
[31] F. T. Ulaby, R. K. Moore, and A. K. Fung. Microwave Remote Sensing: Active and
Passive. Addison-Wesley Publishing Company, 1982.
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Appendix A
SYSTEM CALIBRATION
As in any measurement process of S parameters [9], the measuring system must
first be calibrated using some known transmission/reflection standards.
In this study,
the millimeter-wave system was calibrated using the reflection response of a large flat
conducting plate. The observed specularly reflected signal was then used to establish the
correct reference plane position and frequency response of the system accordingly. To
ensure the working condition of the system, the ACF of the plate was measured along the
specular memory line defined by &\ —0* = d's —0 3 on the sin 0^-sin d's plane for 0 L = 20°
and 9S — —40°. As apparent from the corresponding ACF response shown in Fig. A. 1, the
measured correlation attains a level close to unity along the entire memory line. Because
of the finite physical size of the conducting plate, however, the measurement also shows a
slightly stronger correlation when both the transmitter and receiver arms are perpendicular
to the plate, as manifest in the small peak at 0\ = 30°.
It is important to realize that in the actual measurement process, it is S parameters that
were being measured. Therefore the process of retrieving measurements of scattering am­
plitudes fij, and hence ACF, from S parameter measurements deserves detailed treatment
and is considered below.
On the basis of the image method, the following radar equation can be established for
the conducting plate calibration process mentioned above:
n _ p
A2
G°lGo2
°
‘ (4tt)2 ( r „ |+ r„2)*
(A' °
where Pa is the received power due to the flat plate, Pt is the transmitted power, and (rol,
r o2) and (G0i, G 02 ) represent the distances of the transmitter and receiver, respectively,
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110
ACF signature due to a large flat plate tilted at 30°
2
1.6
1.4
,12
CD
o)
1
co 1
E
li­
es
* '
0.8
0.6
0.4
02
0
0
10
20
30
angle of incidence 9.
40
50
60
Figure A. 1: ACF signature measurement o f a large tilted metal flat plate. The way the antennas are moved
describes an angular memory line for ( 9 i , 9 s ) = ( 3 0 ° ,- 3 0 ° ) on the s in 9l -sinOs plane. As expected, the
correlation level approaches to 1 over a wide range o f variable incident angles 9i
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Ill
from the plate and their boresight gains. With r ol = r o2 = r Q, Eq. A.l reduces to
P =Pt
G olG °2
°
£ (4 tr)2 4r2
(A 2)
(A ’2)
Since P 0 oc |V^|2 and Pt oc |Ut |2, it follows from Eq. A.2 that
w
= |Vil I
< A J)
In phasor notation, VQ can be expressed as
Vo =
IK
.I e ~ i+
(A.4)
where o — 2 k 0 r 0 represents the phase introduced by wave propagating in free space over
a distance of r Q.
Substituting |K>| in Eq. A.3 into Eq. A.4,
v; = ^
s
\ / ^
(A -5)
In parallel with the derivation developed above, the radar equation for extended targets
assumes the following form:
Pr = Pt l(4F tt)
la
J[a
r;r;
(A-6)
where Pr and Pt are the received and transmitted power, respectively, a is the bistatic
cross section to be measured, and (rr , r t) and (G'r , G t) represent the distances of the
transmitter and receiver, respectively, from the target and their gains. Making use of the
arguments in transition from Eq. A.2 to Eq. A.5 and noticing that with r r = r t — r and
a = 4 7 t|/|2, where / represents the scattering amplitude, one can write
v' =W*;L\l-7r-e~m’rfds
Using Eq. A.5 and Eq. A.7, it follows that
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( A
' 7 )
112
4r 02 _,2. r„ r
ej2fcar° J a
Vr
V
V0
J
G a\Go2
lG rGt _
- Z p - e - j2k°rf d S
(A.8 )
The ensemble-averaged correlation of voltages is therefore
<
>
4 r°
GoiG0o
GrG t
< f j o > dS
r4
Il A,
Va V*
Vri Ko
_
_
(A.9)
4 r 02
/
^
dS
(A.10)
Since the half-power beamwidth of the receiver is smaller than that o f the transmitter and
is fairly narrow, r remains nearly constant over the illuminated area A , and therefore the
integral in Eq. A. 10 can be approximated as [24]
GrGt
—
.-i
rj ~ d S
/,
~
G rG t JO ^
—
4 dS «
I>A,
r4
1
Go\G0o
Zi
r‘
,
G 01G 02 * r lA 0 r 2
----- 7----- z------- 57r 4 4 co s0 s
(A -n )
(A. 12)
where A 0 r is the half-power beamwidth of the receiver and is roughly equal to 6° over
95-100 GHz for the experimental setup in this study and 9's refers to the scattering angle
at which the receiver is looking.
With Eq. A. 12, Eq. A. 10 can be simplified as
=
N
< h i;- >
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(AI4)
113
where N is the normalization factor which relates measurements of S parameters, which
are the quantities actually measured by the system, to scattering amplitude measurements.
Therefore, by performing angular correlation on measured data and multiplying the result
with the inverse of N , absolute measurement of angular correlation between scattering
amplitudes can be obtained. As a result of the asymmetry between the sizes of the spots
projected by the antennas on the surfaces, however, N in fact depends on the amount of
overlapped footprints of the two antennas. This amount of overlapped area is in turn a
function of antenna positions (Q\, d's). A simplified treatment from projection geometry
was included in N in Eq. A. 14 to incorporate its dependence on antenna positions for
final presentation of experimental data in this experimental work [16].
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Appendix B
CHIRP RADAR
Chirp radars are used when the length of the time-domain pulse required for a pulse
radar is so short that the pulse must have a very high peak power. From a hardware,
implementation point of view, it is difficult to generate a high peak-power pulse within
very short duration. Therefore, a method has been devised to address this difficulty.
Basically, this method involves the use of some kind of longer pulse with a modulation
interior to the pulse, thus allowing fine resolution associated with the wider bandwidths
of the modulation. Although the resulting modulated pulse will possess the same amount
of power as the unmodulated one does, the power is spread in time (and hence space)
with the peak power level being reduced. In this appendix, only FM-modulated pulse will
be considered since it is the modulation type employed by the Hewlett-Packard’s network
analyzer (HP8720) used in this radar imaging study.
The operation of a FM chirp radar can be understood from the spectrogram shown in
Fig. B .l. The figure shows the instantaneous frequency of the transmitted pulse (linear
upward sweep of frequency as a function of time), the instantaneous frequency of the
received pulse ( ideally from a point target) and the effect of “de-chirping” such a pulse
by passing it through a filter whose time delay is a function of frequency. As shown in the
figure, the pulse duration is denoted by r and the modulation bandwidth by B. Intuitively,
the filter should have a time delay characteristic such that the frequency transmitted first
(the low end of B) is delayed long enough so that it arrives at the filter output at the
same time as the frequency transmitted last (the high end of B), as in Fig. B.2. For
ideal de-chirping, all the frequencies in-between should also arrive at this time, so they
are superimposed at a single time instant at the filter output. Ideally, this results in a
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115
d'-function de-chirped pulse, which is not possible with the finite bandwidth in practical
frequency-sweeping radar systems. Instead of being a vertical straight line shown in the
rightmost part o f the figure, the received signal will smear into a finite-duration pulse.
With a bandwidth B the approximate pulse width at the filter output is 1 /B , and if
the transmitted amplitude is kept constant during the pulse, the resulting pulse takes the
shape of sin(:r) / x dependence. The net effect of this chirping/de-chirping process is to
compression of a r-long pulse into a 1/ZMong pulse, allowing fine spatial resolution
without injection of excessive power into a short 1 /B -lo n g pulse before modulation.
A brief analysis is presented below for two cases. Case A assumes infinite pulse
duration whereas case B assumes finite pulse duration. The former aims at projecting
physical insights from idealized settings whereas the latter deals with practical consider­
ations encounter in real-life chirp radars.
B .l
C a s e A : I n fin ite p u ls e d u r a tio n
If the pulse were allowed to have infinite duration, the subsequent analysis for the de­
chirped pulse would be easy. As mentioned before, the resulting waveform at the filter
output will have a J-function shape.
Assume the following waveform for the transmitted pulse
vr {t) = eJ'(tt,ot+*ata)
(B .l)
where a is the “chirp rate” cbj/dt. Its Fourier spectrum will be
Vr{ u ) = [ +°° eiluot+?at2 - ut)dt.
J — CO
(B.2)
By completing the square in the exponent, the above equation reduces to
V rH =
)2/ 2a_
(B J)
V Ja
From matched-filter theory [14], the response F{u>) o f a filter that matches this transmitted
pulse with maximum signal-to-noise (SNR) ratio should be the complex conjugate of the
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116
freq u en cy
il
transmitted pulse
received pulse
T
I
de-chirped pulse
1
B
T
time
Figure B.L: Frequency-time plot o f transmitted pulse, received pulse, and the de-chirped pulse. The net
effect o f this chirping/de-chirping process is to com pression o f a T-Iong pulse into a 1 /B -lon g pulse,
allow ing fine spatial resolution.
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117
delay time
f,
f2
freq uency
Figure B.2: Delay-time characteristic o f a de-chirping filter matched to the transmitted chirp pulse, assuming
linear FM modulation.
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118
above spectrum with its amplitude inversed:
(B.4)
Therefore, the spectrum of the filter output will be the product of Vr {uj) and F { uj), which
is
I/crut{u !) — 1-
(B.5)
The inverse Fourier transform of V0 ut{uj), which is equal to the time-domain variation of
the filter output, is
Vout(t) = S {t).
(B.6 )
Note that the an infinite-duration sweep implies an infinite bandwidth, so that the infinite
time resolution (and hence spatial resolution) offered by the 5-function filter output cannot
be obtained in practice.
B.2
Case B: Finite pulse duration
In practice, the transmitted pulse must have finite duration and bandwidth, so the above
simple analysis does not quite apply. Real waveforms rather than complex waveforms
will be used in the following analysis to demonstrate the effect of a finite-duration-finitebandwidth on pulse broadening at the filter output. It is assumed that the received pulse
comes from a point target and has the same waveform as the transmitted pulse. Note that
in the following analysis the time axis is backward-shifted by r / 2 to achieve symmetric
properties of Fourier transforms for simpler mathematics.
Define the following window function n T/2:
(B.7)
0
otherwise
Assume the following waveform for the received pulse v r (t):
vr(t) = n 7 / 2(<) cos(u;0t + ^ a t2).
6
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(B.8 )
119
Note that the phase 4> of v r{t) is given by
1
9
(f) — uj0t + —at ,
(B.9)
so that the instantaneous angular frequency v is
dcj)
u} = — = uj0 + at.
at
(B.10)
In order to cancel the quadrature term introduced in FM modulation on the transmitted
pulse (and hence the received pulse vr(t)), the de-chirping matched filter should have an
impulse response f ( t ) given by
f ( t ) = q n T / 2 (t)c o s (u 0t - ^
2),
(B.l 1)
where q is so selected so that the filter produces unity gain at its center frequency. The
time delay td of thisfilter response is given by the derivative of the filter phase response
tp with respect to angular frequency v
dib
d .
1
td = — = — [uiQt - -a t-] .
cLj
dui
2
(B .l2)
From Eq. B.10, t can be obtained as
t = ~— — .
a
(B.13)
Substitute this t into Eq. B.12, td can be expressed as
td —
d fid0(aJ — JJa)
(id — ^o)",
------------------------------=
dui
a
2a
—
2 iU0
— UJ
.
a
(D.14)
Note that td is a decreasing function of uj. This means that longer time delay is associated
with the higher end of B and shorter time is associated with the lowerend
of B, as
expected from previous discussion.
The output of the filter is given by the tme-convolution of the received signal with the
filter response:
/
+0O
Vr( t ) f ( t - x )d x
(B.15)
-O O
/ +oo
n r / 2 ( 2 ) n r /2 (I ~ x) cos (
l j
-O O
0
x
+ - a x 2)
2
x cos[u0(t — x) — ^ a ( t — x ) 2 ]dx.
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(B.16)
120
Make use of
cos A cos B = ^[cos(A 4- B ) + cos(A. — 5 )],
(B. 17)
the cosine product in the integrand in Eq. B.16 can be expressed as
^ cos[u;0x 4- ^arr2 4- uja(t —x) — ^a(£ - a:)2]
4-
^ cos[u;0:r 4- ^ ax2 —uja(t — x) + ^ a (t —
=
cos[aa0f —- a t 2 + atx\ 4- cos[2a;0a; 4- a x 2 — ujat — a tx 4- ^ a t2].
x )2]
(B .l8)
The second term will disappear after filtering, leaving Eq. B.16 as
'
^cmi( 0 —
I
t _
rt+ 772
1
.2
o I - t / 2 ~ 9 c° s [ua0£ — ^a t 4- a tx \d x
for —T < t < 0
k ft'-T /2 <lC0S[uot — \ a t 2 4- atx]dx
for 0 < t < T
(B.19)
Make use of the following identity
sin A — sin B = 2 sin ^(4 . - B) cos ^(.4 4- B ).
(B.20)
in Eq. B.19, the filter output can be written as
=
^ s in [ ^ ( T ’ 4- t)\cosu>0t
for —T < t < 0
^ s in [ 7r(T — i)] cosujat
for 0 < t < T
i si n [ f ( r - | f | ) ] for —T < t < T .
After EFfiltering, the cosine term in Eq. B.21 disappears. Therefore,
(B.21)
the filter output
Vout(t) becomes
Vaat{t) =
for - T < t < T .
Note that a = 2-k B / T (from Eq.
B.10). The terms of the sine argument inEq. B.22
become
aTt
—
(B.22)
=
at2
2T =
*Bl
at t
t
~2 T = W t T
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121
For large values of B T , which is often the case for practical wideband chirp radars, the
quadratic term is negligible compared with the linear term near the origin. Thus, the
location of the first null is roughly the same as it would be in the absence of the quadratic
term. Hence,
JfB t-u rid ih
—
(■width =
7T
1IB ,
(B.23)
so that the null-to-null time width of the filter output is 2/ B , which also defines the finest
spatial resolution achievable by this chirping/ de-chirping scheme.
To summarize, the effect of applying FM chirp and compression filter is to take a long
pulse of duration T and convert it at the filter output to a short pulse having null-to-null
width 2/ B and approximate effective width l / B . A plot o f Eq. B.22 is shown in Fig. B.3
with T = 4 s and B — 6 GHz. The 3dB width of the de-chirped pulse is shown to be
1.057/5.
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122
De-chirped signal at compression filter output with T = 4s and B = 6 GHz.
= 02
-02
•0.4
-
0.8
0.6
•
-0.4
-02
0
02
0.4
0.6
0.8
time in ns
Figure B.3: Plot o f Eq. B.22 with sweeping time T = 4 s and chirp bandwidth B = 6 G H z. The 3dB width
o f the de-chirped signal is about 1.057/B .
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Appendix C
DETERMINATION OF REFERENCE SLANT RANGE
As noted earlier in Section 4.2, the quality of focusing depends critically on the
accuracy in the determination of the reference slant range shown in Fig. C .l. Ideally,
this reference slant range (represented by the dotted line in the figure) should intersect the
base of circular SAR system at a point cv = ce (ignoring the altitude difference), where Zv
and q refer to the center of rotation and the center of illumination, respectively.
In practice, however, it is difficult to construct a circular SAR system with the con­
dition c,. = ce satisfied. In order to study the spatial discrepancy between the two centers
(and hence evaluating the quality of focusing capability of the system), a controlled exper­
iment is derived to measure their relative locations indirectly. Specifically, the experiment
involves placing a conducting sphere (63 mm in diameter) on top of a thick absorber sheet.
The level of the absorber sheet is maintained at the same altitude of the base of the SAR
system. The location of the sphere relative to the center of rotation is precisely noted. The
top view of this experiment is shown in Fig. C.2. Note that in the figure the location of
the sphere has been exaggeratedly misplaced so as to illustrate the variables-to-be-solved
like p and 0. Also, it is understood that the 2-D plane shown in the figure is at an altitude
of h below the SAR flight track. Consequently, the variable h is suppressed in the figure.
C .l
Analytic formulation
With reference to Fig. C.2 and making use of the cosine rule, the following independent
equations can be developed:
r\
=
a 2 + p 2 + hr + 2ap cos 0
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(C .l)
124
r2 =
dr + p 2 + dr + la p sin 0
(C.2)
rf
a 2 + p 2 + hr —2 a p co s0
(C.3)
1 (r r 2t +, r 2\
p 2 +, /il2 = —
3) - a 2 = k 2.
(C.4)
=
(Eq. C. 1 + Eq. C.3) gives rise to
Making use of the notation in Eq. C.4 and the identity cos2 0 + sin2 0 = 1 in Eq. C. 1 and
Eq. C.2, I have
cos
9
/ 9
9
9\ 9
(rr — a“ — « r r
. 9
+ sin 0 = —■■— —
Aal pr
0
/ 9
(r^ —
>- + v 2
9
9\ 9
— k t r
-----,---- ,- = l.
Aazp2
_
(C.5)
Solving Eq. C.5, Eq. C. 1 and Eq. C.4 in sequential order, I have
With p,
0
^
r
*7
p
=
0
= sin- l C 2
h
=
O
9 \9
a~
la p
K~)
\ O
/
To
O\
^ V (rr - a- - « -)- 4- (r-5 - a- - /c-)-
^ k 2 - p2.
and /i, the reference slant range /?Q and the depression angle
(C.6)
(C.7)
(C.8)
can be
expressed as
R 0 = %/a2 + /i2
0<Wi = ta n _1( - )
a
C.2
(C.9)
(C.10)
Experimental results
From the analytic development in previous section, it is clear that in order to measure the
reference slant range R Q, it suffices to obtain the triplet { rl7r 2. r 3} and radius a of the
circular SAR flight track (a = 0.7 m from direct measurement). Note that the value of
triplet depends only on the SAR geometry and has nothing to do with the scene to be im­
aged. Therefore, in designing experiments for measuring { r i ,r 2, r 3} one is free to choose
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125
appropriate scenes for convenience. In this regard, I carried out two independent “sphereon-axis” experiments and selected measurement points to be located on the quadrant edges
shown in Fig. C.2, leading to the simple relationships expressed by Eq. C. 1-C.3. In prin­
ciple, these two experiments would yield the same result if the measurement system were
to have infinite bandwidth.
Experiment A: Single sphere along x-axis
C.2.1
As the first attempt to obtain { r i .r 2, r 3}, a 63 mm conducting sphere was placed on top
of a thick absorber sheet at a distance of 15 cm from the center of rotation cr along the
x-axis. The level of the absorber sheet was maintained at the same altitude h of the
base of the SAR system. Reflection measurements were performed at locations shown in
Fig. C.2. By going through the analytical derivation developed early in this Appendix, I
have
• reference slant range R 0 = 0.94372 m,
• depression angle 9l{own = 42.1158° below the horizon,
• p = 0.1493 m,
•
0
= —3.1798°, and
• h = 0.63291 m below the circular SAR flight track.
The center of illumination q is thus off from the center of rotation cv by 0.1539 m 0.1493 m, or 0.7 mm. at an angle of 180° —3.1798°, or 176.8202° from the x-axis.
C.2.2
Experiment B: Single sphere along y-axis
As the second attempt to obtain { r i , r 2 . r 3}, a 63 mm conducting sphere was placed on
top of a thick absorber sheet at a distance of 15 cm from the center of rotation £v along
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126
the y-axis. The level of the absorber sheet was maintained at the same altitude h of the
base of the SAR system. Reflection measurements were performed at locations shown in
Fig. C.2. By going through the analytical derivation developed early in this Appendix, I
have
• reference slant range R a = 0.93989 m,
• depression angle Odoum = 41.8576° below the horizon,
• p = 0.15391 m,
9 0
= 84.0305°, and
• h = 0.62718 m below the circular SAR flight track.
The center of illumination q is thus off from the center of rotation ev by 0.1539 m 0.1500 m, or 3.9 mm. at an angle o f 84.0305° from the x-axis.
Based on the results from these experiments, the average spatial discrepancy between
eg a n d c r is therefore ^(0.7/176.8202°+ 3.9/84.0305°) = 1.9643/94.2819° mm, which is
negligible (equivalent to about j~X at a center frequency of 10 GHz) at X-band frequencies.
The calculated center of illumination ce and the ideal center of illumination eg (= cT, the
center of rotation) are shown in Fig. C.3. These experiment demonstrate that the circular
SAR system used in this experimental study possesses superb construction accuracy.
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127
SAR platform
center of rotation
center of illumination
illumination footprint
Figure C .I: Experimental determination o f reference slant range. Ideally, the center o f rotation should be
the sam e as the center o f illumination (except for the obvious altitude difference).
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128
metal sphere
radius = a
center of rotation
Figure C.2: Experimental setup for determination o f reference slant range in which a 63 mm conducting
sphere, located at an altitude o f h below the circular SAR flight track, was placed at a distance o f 15 cm
along the x -a x is relative to the center o f rotation. Three independent reflection measurements were made
to solve for variables p , <b and h.
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129
ideal cen ter of illumination
= center of rotation
center of illumination (calculated)
Figure C.3: The relative locations o f the calculated center o f illumination and the ideal center o f illumination.
The negligible spatial discrepancy verifies the superb construction accuracy o f the circular SAR system used
in this experimental research.
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Appendix D
MATLAB CODES FOR CIRCULAR SAR PROCESSING
With the advent of highly efficient computation packages like MATLAB for matrix
manipulations, coding for SAR processing, which inherently takes advantage o f using
matrices as the primary data structure, can be very simple. Although in general MATLAB
codes are not as optimized as other contemporary scientific programming languages such
as FORTRAN or C/C++ in terms of run-time speed, its easy o f use and high productivity
(due to the availability of a large built-in collection o f efficient algorithms and graphics
routines) are unparalleled compared with its rivals. Furthermore, MATLAB’s high degree
of programming flexibility and interactive computation workspace introduce significant
shortcut to users whose primary concern is shorter code-development cycle in a limited
time frame.
As a result of the considerations presented above, MATLAB was chosen to be
the primary processing language for this research.
In this Appendix, I will outline
briefly the structure of the main computation codes collectively grouped in the program
CORRSAR.M. Then, I will append the MATLAB source codes for it and all other relevant
subroutines.
D .l
Main program
The structure of CORRS AR.M is shown in Fig. D. 1. The source codes of CORRS AR.M
and its subroutines are included are appended in Fig. D .2-Fig. D .l7
D.2
Other subroutines
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
131
C
O
RRSAR
G
E
T
D
A
T
A
retrieve raw data, put it in freqsp a ce matrix representation
1
V
O
LSIZE
FO
CUSG
RID
compute dimension of the
HPBW volume
generate grid points on a focal
plane
1
LEN
SG
R
ID
generate grid points along the
circular SAR flight track
I
R
A
D
IX
up-sample raw data over
frequency
1
SCREENINFO
display useful processing
parameters on screen
1
G
A
T
IN
G
construct gating matrix with
respect to a focal point
V
C
H
O
PBA
N
D
perform freq correlation SAR by
bandwidth partition
T
IM
E
_A
C
F
perform angular correlation
SAR by spatial partition
N
O
R
M
A
L
normalize p ro cessed images,
make them have magnitude = 1
Figure D .l: The structural com ponents o f CORRS AR.M.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
132
36C
O
R
R
SA
Rperforms spotlight-mode SAR-processing for circular SA
Rsystem
36using 1.) C
O
R
R
E
L
A
T
IO
N(new) technique and 2.) C
O
N
V
E
N
T
IO
N
A
L(old) technique.
36
36C
O
R
R
SA
Rrequires the following m-files:
36
35
36C
H
O
PB
A
N
D
: Perform freq-SARimage processing.
36FO
C
U
SG
R
ID
: Construct in spatial domain the grid points for the focus (object) plane.
36freq_A
C
F: Perform FR
EQ
-A
C
Fimage processing.
36G
A
TIN
G
:
construct in time domain the gating matrix for agivenfocusplane.
36getdata: Load radar data and measurement parameters fromthe input datafile.
36K
A
ISER
:
Construct Kaiser windowused by G
A
TIN
G
. FromM
A
T
L
A
B toolbox.
36lensgrid: construct in spatial domain the grid points for the lens (source) plane.
36normal:
Normalize processed images to have magnitude of 1.
36R
A
D
IX
:
Adjust the num
ber of rowof the original rawFDdata for faster fft.
36SCREENINFO:Display useful processing parameters on the screen.
36TIM
E_A
C
F: Perform time-aC
Fimage processing.
36VO
LSIZE: Com
pute volume dimension fromparameters notincluded in input .INF file.
36
36C
O
R
R
SA
Ris optimized for M
A
T
L
A
B5‘s newfeatures. It requires slight modifications
36for it to run under matlab 4.
36
36M
odify the value of M
A
X
_C
U
B
E
_H
E
IG
H
Tto accommodate for 3-D imaging. Avalue of
36m
a
x
_C
U
BE
_H
E
IG
H
T= 0 m
eans 2-d imaging only.
36
36Last update: 03/30/98
36Declare constants
clc; clear;
c = 3e8;
rad = pi/180;
cm
ax = 0;
tol = 1.0e-3;
0 = 0.15;
max_cube_height = 0;
xstep = 0.010;
ystep = 0.010;
zstep = 0.010;
36 speed of light in free space
36 conversion factor fromdegrees to radians
36initial m
axim
umof colorscale
36 tolerance for user's input
36 aperture size of the antennasinmeters
36height of the illumination cube around zcentroid
36resolution inx- direction in
meters
36resolution in y- direction in
meters
36resolution in z- direction in
meters
36 [getdata] Read raw radar data
disp(['Spotlight-mode circular sar data processor']); dispC ');
basename = input(['enter the basename (<= 7 characters) of the input .asc *monostatic*
data file here: '] , 's');
extension = '.asc';
calibration = 'n';
disp(['Loading radar data...']);
[TRUE_FD_raw,startf,stopf,TRUE_npoint,TRUE_npair,TRUE_antang,nsam
ple, ref_R,dnangle] =
getdata(basename,extension);
disp(['Done.']);
resultname = basename;
36workspace nam
e to be the sam
e as filename
dnangle = 90 - dnangle;
36original "dnangle" measured fromvertical!
B= stopf - startf;
36measurement bandwidth in hz
wavelength = c/(mean([startf stopf])); 36wavelength in meters
hpbw= wavelength/D;
36half-power beam
width in radian
deltaT = 1.057/B;
36approximate pulsewidth in seconds [Ulaby v2]
refji = ref_R * sin(dnangle *rad);
36altitude of the plane being focused in meters
ref_a = sqrt(ref_RA2 - ref_hA
2);
36radius of the circular orbit in meters
clear getdata;
Figure D.2: MATLAB code o f CORRSAR.M.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
133
* [volsize] Com
pute the dimension of illuminated volume
% xspotlength = 1.50*Dx; yspotlength = 1.50*Dy; zspotlength = 1.50*Dz;
[dx,dy,dz,Dx,Dy,Dz] = vol si ze(B, hpbw,ref_R,dnangle);
clear volsize;
xspotlength = 2.15*Dx; yspotlength = 2.15*Dy; zspotlength = 2.15*dz;
% [FO
C
U
SG
R
ID
] Construct the focal-plane matrix
[FO
CUS,xgridaxis,ygridaxis,zgridaxis] =
focusgrid(ref_R,dnangle,xspotlength,yspotlength,zspotlength,xstep,ystep,zstep);
clear fcusgrid;
xgridlength = length(xgridaxis);
ygridlength = 1ength(ygridaxis);
zgridl ength = length(zgridaxis);
zcentroid = -ref_R * sin(dnangle * rad); % plane z = 0 coincides with antenna level
% [lensgrid] Construct the lens-plane matrix
disp([' ’]);
wantANG= inputC'Apply angle interpolation [linear method]? [n]: '.'s');
if wantANG= 'y'
iNTfactor = inputCEnter the up-sampling factor [2]: ');
else
INTfactor = 1;
end;
[lens,txlens,rxlens] = lensgrid(ref_a, TRUE_antang,INTfactor);
clear lensgrid T
X
L
EN
SR
X
L
E
N
S;
% txlens and rxlens are redundant in monostatic processing
% [RADIX2] Perform frequency interpolation for faster FFT
disp([' ']);
% FFTis most efficient when word length = 2*n (n: integer)
wantFFT = inputC'Apply frequency interpolation [linear method] for faster FFT? [y]:
V s ');
[FD_raw,npoint] = radix(TRUE_FD_raw,TRUE_npoint,wantFFTtlNTfactor);
clear radix;
% [SCREENINFO] Display the processing parameters for C
O
R
R
SA
R
screeni nfo(ref_R,dnangle,xgridlength,ygridlength,zgridlength,xspotl ength,yspotl ength,zsp
otlength,Dx,Dy,Dz,dx,dy,dz,xstep,ystep,zstep);
clear screeninfo;
% Pre-allocate memory for faster processing
pack;
npair = INTfactor * TRUE_npair;
k = repmat(2*pi/c*linspace(startf,stopf.npoint).', [1 npair]);
viewocm= 0.50*100*mean( [ ( max(xgridaxis) - min(xgridaxis) ) ( max(ygridaxis) min(ygridaxis) ) ] );
viewo = viewOcm/100;
halfw_index = round(viewD/c/deltaT);
filtershape = kaiser(2*halfW_index+l,10);
% Initialize all arrays
r = zeros(npoint.npair);
G
A
T
E
M
X= zeros(npoint,npair);
TD_raw= zeros(npoint,npair);
FD_gated_focused = zeros(npoint,npair);
TD_gated_focused = zeros(npoint,npair);
SA
R= zeros(xgridlength,ygridlength);
freqSAR = zeros(xgridlength,ygridlength);
tim
eACF = zeros(xgridlength,ygridlength);
freqACF = zeros(xgridlength,ygridlength);
Figure D.3: MATLAB code o f CORRS A R .M , continued from Fig. D .2
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134
% This Xconsiders both +ve z and -ve z
I = sort(find(abs(zgridaxis - zcentroid) <= max_cube_height * wavelength));
% This I considers only +ve. SO
R
Tm
akes sure that layer images are calculated in
ascending-z sequence
% I = sort(find( ( (zgridaxis - zcentroid) <= max_cube_height * wavelength ) & (
(zgridaxis - zcentroid) >= 0 ) ) );
* Correlation SA
Rprocessing below
totaltime = clock;
TD_raw= ifft(FD_raw);
for z = min(l):max(l);
% layer images arecalculated in bottom-to-top order
SARimagez = [resultname 's' num2str(z)];
freqSARimagez = [resultname 'cFs' num2str(z)];
timeACFimagez = [resultname 'cT' num2str(z)];
freqACFimagez = [resultname 'cf' num2str(z)];
planedepth = zgridaxis(z);
R
O= ref_R;
% reference slantrange forfocusing
dispC ');
disp([’Processing the ' num2str(z-min(l)+l) 'th focal slice out of a total of '
num2str(length(X)) ' slice(s)']);
for m= 1:xgri dlength; t_row = clock; m
;
for n = l:ygridlength;
r = repmat(sqrt(abs(LENS - FOCUS(m,n)).A2 + pianedepthA2),[npoint 1]);
%
T
IM
E
-D
O
M
A
ING
A
T
IN
GU
SIN
G
FFTT
OR
E
PL
A
C
E
D
IR
EC
TFR
E
Q
U
E
N
C
Y
-D
O
M
A
INC
O
N
V
O
L
U
T
IO
N
G
A
T
E
M
X= gating(r(l,:), r(l,:) ,viewOcm,deltaT,npoint,npair.filtershape);
clear gating;
%
%
%
%
%
FR
E
Q
U
E
N
C
Y
-D
O
M
A
INFO
C
U
SIN
G
FD_gated_focused =
(r .a 2) .* exp(2 * i * k .* ( r - ref_R ) ) .* fft(G
ATEM
X .* TD_raw);
TD_gated_focused = i fft(FD_gated_focused);
TR
A
D
IT
IO
N
A
LSAR's
SAR(m
.n) = abs( m
ax( ifft( sum
( FD_gated_focused.' ) ) ) )A2;
freqSAR(m.n) = chopband(FD_gated_focused);
C
O
R
R
E
L
A
T
IO
NM
E
T
H
O
DB
A
SE
DO
N
A
N
G
U
L
A
R
C
O
R
R
E
L
A
T
IO
N
FU
N
C
T
IO
N
target_phi =03; npizza4target = 30;
cluter_phi =10; npizza4cluter = 0;
freqACF(m.n) = freq_acf(FD_gated_focused,target_phi ,cluter_phi,
npizza4target,npizza4cluter);
timeACF(m.n) = time_acf(TD_gated_focused,target_phi ,cluter_phi,
npizza4target,npizza4cluter);
timeACF(m,n) = SAR(m
,n) - 2*real(
time_acf(TD_gated_focused,target_phi,cluter_phi,
npizza4target,npizza4cluter) ) time_acf(TD_gated_focused,0, cl uter_phi ,
1,npizza4cluter);
freqACF(m.n) = SAR(m
,n) - 2*real(
freq_acf(FD_gated_focused,target_phi ,cluter_phi,
npizza4target,npizza4cluter) ) freq_acf(FD_gated_focused,0, cl uter_phi ,
1,npizza4cluter);
clear time_acf freq_acf chopband
end; 3S
n
t_row = etime(clock,t_row);
disp([num2str(m) '). This image rowtakes ' num2str(t_row/60) ' minutes.']);
end; %m
Figure D.4: MATLAB code o f CORRSAR.M , continued from Fig. D.3
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135
eval([SARimagez ■= SAR;']);
eval([freqSARimagez ' = freqSAR;']);
eval([timeACFimagez ' = timeACF;']);
eval([freqACFimagez ' = freqACF;']);
layertime = xgridlength*t_row;
disp([’This layer takes ' num2str(layertime/3600) 1 hours.']);
end; %z
% [normal] Normalize the processed images
nSA
R= normal(SAR);
nfreqSAR = norm
al (freqSAR);
ntimeACF = norm
al (timeACF);
nfreqACF = normal(freqACF);
disp(' ’);
disp(['lmage plot variables are stored in "portable.mat''.']);
save portable xgridaxis ygridaxis zgridaxis nSA
RnfreqSAR ntim
eACF nfreqACF
disp(['All workspace variables are stored in ' resultname '.mat']);
eval(['save ' resultname]);
figure(l);
colormap(jet);pcolor(xgridaxis,ygridaxis,abs(nSAR.') );
colorbar('vert');caxis([0 1]);axis('equal');shading interp;
titleCConvention SAR');
xlabel('x-range in meters');
ylabel('y-range in meters');
figure(2);
colormap(jet) ;pcolor(xgridaxistygridaxis,abs(ntimeACF.'));
colorbar('vert');caxis([0 1]);axis('equal');shading interp;
title('Angular Correlation SAR');
xlabel('x-range in meters');
ylabel('y-range in meters');
figure(3);
col orm
ap(jet);pcolor(xgridaxis,ygridaxis,abs(nfreqSAR.') );
colorbar('vert');caxis([0 1]);axis('equal');shading interp;
titleC Frequency Correlation SAR');
xlabel('x-range in meters');
ylabel('y-range in meters');
Figure D.5: MATLAB code o f CORRS AR.M , continued from Fig. D.4
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
136
function
—
”
[TRUE_FD_raw,startf,stopf,TR
U
E_npoint,TRUE_npair,TRUE_antang,nsam
ple , ref_R, dnangle] =
getdata(basename,extension);
1
5G
E
T
D
A
T
Agets radar data and reshape the input column data into
>
5a matrix of dimension TRUE_npoint x TRUE_npair x nsample matrix.
“
lATfile = [basename 'z.mat'];
if exist(MATfile) = 2
eval(['load ' MATfile]);
else
iNFfile = [basename '.inf'];
iNFid = evalC['fopen(’" INFfile
frewind(INFid);
DUM
text = fscanf(INFid,'55s',1); DUM
text = fscanf(lNFid,'55s\n',1);
DUM
text = fscanf(INFid, '55s' ,1); DUM
text = fscanf(lNFid,'55s\n',1);
DUM
text = fscanf(INFid,'55s',1); startf = fscanf(lNFid,'55f\n',l)*le9;
DUM
text = fscanf(INFid,'55s',1); stopf = fscanf(lNFid, '55f\n' ,l)*le9;
DUM
text = fscanf(lNFid,'55s',1); TRUE_npoint = fscanf(lNFid, '55f\n' ,1);
DUM
text = fscanf(INFid,'55s',1); dumnyTX_startang = fscanf(iNFid, '55f\n' ,1);
DUM
text = fscanf(INFid, '55s' ,1); dum
nyTX_stopang = fscanf(INFid, '55f\n' ,1);
DUM
text = fscanf(lNFid,'55s',1); dum
nyTX_incang = fscanf(lNFid,'55f\n',1);
DUM
text = fscanf(lNFid, '55s' ,1); dumnyRX_startang = fscanf(lNFid,'55f\n',1);
DUM
text = fscanf(INFid,'55s',1); dum
nyRX_stopang = fscanf(INFid, '55f\n' ,1);
DUM
text = fscanf(INFid,'55s',1); dum
nyRX_incang = fscanf(INFid, '55f\n' ,1);
DUM
text = fscanf(lNFid,'55s',1); ref_R = fscanf(INFid, *55An' ,1);
DlM
text = fscanf(INFid, '55s' ,1); dnangle = fscanf(INFid, '55f\n' ,1);
DUM
text = fscanf(iNFid, '55s' ,1); nsample = fscanf(lNFid,'55f\n',1);
DUM
text = fscanf(INFid,'55s',1); sam
plingRANG
E = fscanf(INFid,'55An',1);
DUM
text = fscanf(INFid,'55s',1); TRUE_npair = fscanf(INFid, '5!An' ,1);
TRUE_antang = zeros(TRUE_npair,l);
for n = l:TRUE_npair;
tmp_r = fscanf(INFid, '55f' ,1);
tmp_i = fscanf(INFid, '55An', 1);
TRUE_antang(n) = tmp_r + i * tmp_i;
end;
DATAfi1e = [basename extension];
D
A
TA
id = eval(['fopen("' D
A
TA
fi1e '")']);
total = TRUE_npair * TRUE_npoint * nsample;
duirniy = zeros(total ,1);
for n = 1:total;
tmp_r = fscanf(DATAid, '5Sf' ,1);
tmp_i = fscanf(D
A
TA
id, '55An', 1);
dummy(n,l) = tmp_r + i * tmp_i;
end;
TRUE_FD_raw= reshape(dummy,TRUE_npoint*nsample,TRUE_npair);
eval (['save ' M
ATA'le]);
end
Figure D.6: MATLAB code o f GETDATA.M
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137
Function [dx,dy,dz,Dx,Dy,Dz] = volsize(B,hpbw,ref_R,dnangle)
KThis function calculates the dimension of the illumination volume based on
Kthe user-specified values of
*
S
(1) depression angle, and
K
(2) reference slant range.
%
I&dnangle: depression angle in degrees.
K
8 ref_R: reference slant range in meters.
K
Kdx, dy, dz: spatial resolution in x-, y-, and z-directions, respectively.
£
KDx, Dy, Dz: dimension of the illumination volume in x-, y-, and zKdirections, respectively,
rad = pi/180;
c = 3e8;
%
speed of light in m
dnangle = dnangle*rad;
M = c/2/B;
% pulse width in min
aropagation direction
dx = pw/cos(dnangle);
% x-range resolution inm
dy = dx;
% y-range resolution inm
dz = pw/sin(dnangle);
% z-range resolution inm
ax = ref_R*sin(0.5*hpbw)*(l/sin(dnangle+0.5*hpbw) + l/sin(dnangle-O.S*hpbw));
3y = Dx;
az = ref_R*sin(0.5*hpbw)*(l/cos(dnangle+0.5*hpbw) + l/cos(dnangle-0.5*hpbw));
Figure D.7: MATLAB code o f VOLSIZE.M
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138
function [FOCUS.xgridaxis.ygridaxis.zgridaxis] =
fcusgri d(ref_R,dnangle,xspotlength,yspotlength,zspotlength,xstep,ystep,zstep)
1
5This function computes the xy coordinates of the grid points on
1
5focal planes:
1
5
1
5
x
1
5
a
«
|
e
%
I
+-I -+
«
I I I
£ y<
j-z-j
1
5
|
|
Note that FOCUS(i,j) = xgridaxis(i) + sqrt(-l)*ygridaxis(j)
1
5
+—+
rad = pi/180;
xcentroid = 0;
/centroid = 0;
zcentroid = -ref_R * sinCdnangle * rad);
lumxhalf = ceil(xspotlength/2/xstep);
lumyhalf = ceil(yspotlength/2/ystep);
fiumzhalf = ceil (zspotlength/2/zstep);
xgridaxis = [ numxhalf*xstep : -xstep : -numxhalf*xstep] .';
ygridaxis = [ numyhalf*ystep : -ystep : -numyhalf*ystep].';
zgridaxis = [-numzhalf*zstep : zstep : numzhalf*zstep].' + zcentroid;
xdum= xgridaxis;
ydum= ygridaxis.';
FX= xdum(:,ones(l,length(xgridaxis)));
f y = ydum
(ones(length(ygri daxis),1),:);
FO
C
U
S = fx + i*FY;
Figure D.8: MATLAB code o f FCUSGRID.M
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139
function [LENS,TX
LEN
S,rxlens] = 1ensgrid(ref_a,TRUE_antang,iNTfactor)
KThis function m
aps T
Xand R
Xangles to xy coordinates
ISusing forward scattering convention of sign,
rad = pi/ISO;
ntUE_npair = 1ength(TRUH_antang);
rxang = realC interpl(
[l:TRUE_npair],TRUE_antangIlinspace(l,TRUE_npair,lNTfactor*TRUE_npair) ) ) + 180;
R
X
ang = im
agC interplC
[l:TRUE_npair]ITRUE_antang,linspace(lITRUE_npair,lNTfactor*TRUE_npair) ) );
rX
L
E
N
S = ref_a * C cos(TXang*rad) + i*sin(Txang*rad) );
rxlens = ref_a * ( cos(Rxang*rad) + i*sin(Rxang*rad) );
L
E
N
S = TX
LEN
S;
Figure D.9: MATLAB code o f LENSGRID.M
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140
Function [FD_raw,npoint] = radix(TRUE_FD_raw,TRUE_npoint,wantFFT,iNTfactor)
£R
A
D
IXinterpolates the rawdata over frequency domain so that the
Klength of the interpolated data vector is optim
umfor FFT
fispCr ']);
disp([’Existing num
ber of frequency-sampling points: ’ num2str(TRUE_npoint)]);
if wantFFT== 'y'
TRUE_npoint_FFT= 2Aceil(loglO(TRUE_npoint)/loglO(2));
disp([' ']); disp([’Closest num
ber of radix-2 points: 1 num2str(TRUE_npoint_FFT)]);
disp([' ']):
newnpoint = input(['=> C
H
A
N
G
E(if no aliasing) the existing num
ber of points to
(e.g. 512, 256, or 128) [128]: ']);
npoint = newnpoint:
dispC
Frequency interpolation...');
FD
_raw
_FR
EQ=
interpl(l:TRUE_npoint,TRUE_FD_raw,linspace(l,TRUE_npoint,newnpoint), 'linear') ;
else
disp([' ']);
newnpoint = input(['=> c h a n g e (if no aliasing) the existing num
ber of points to
(e.g. 401, 201, or 101) [101]: ']);
dfreq = round((TRUE_npoint-l)/(newnpoint-l));
npoint = newnpoint;
FD
_raw
_FR
EQ= TRUE_FD_raw(l:dfreq:TRUE_npoint,:);
end;
if INTfactor ~= 1
dispC
Angle interpolation...');
end;
dispC
Done.');
TRUE_npair = size(TRUE_FD_raw,2);
FD_raw= ( interpl(
[l:TRUE_npair], (FD_raw_FREQ
) •' ,linspace(l,TRUE_npai r,iNTfactor*TRUE_npair) ) ).';
Figure D.10: MATLAB code o f RADIX.M
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141
Function screeni nfo(ref_R,dnangle,xgridlength,ygridlength,zgri dlength,
xspotl ength,yspotlength,zspotl ength,Dx,Dy,d z ,dx,dy,dz,
xstep,ystep,zstep);
£ SC
R
EEN
IN
FOsimply display the processing parameters for the
£ main programC
O
R
R
SA
R
iispC ’);
diSp(’
1
-■
-
dispC' ');
disp(['Radar looking
disp(['Matrix size =
disp(['Actual size ='
disp(['xbecunwidth =
disp(['ybeamwidth =
disp(['zbeanwridth =
disp([’xresolution =
disp(['yresolution =
disp(['zresolution =
disp(' '); dispC ');
------
‘) ;
at ' num2str(ref_R) 'm
'' num2str(dnangle) ’ deg. down.' ]);
'num2str(ygridlength) ' x ' num2str(xgridlength)])
num2str(yspotlength*100) 'em x ' num2str(xspotlength*100) 'an'])
'num2str(Dx) 'm=> xspotlength = ' num2str(xspotlength) 'm.']);
'num2str(Dy) 'm=> yspotlength = ' num2str(yspotlength) 'm.']);
'num2str(Dz) 'm=> zspotlength = ' num2str(zspotlength) 'm.']);
'num2str(dx) 'm=> xresolution = xstep = ’ num2str(xstep) 'm.’]);
'num2str(dy) 'm=> yresolution = ystep = ' num2str(ystep) 'm.’]);
'num2str(dz) 'm=> zresolution = zstep = ' num2str(zstep) 'm.']);
disp(['Hit [R ETU R N ] to start processing ...']); pause;
Figure D. l l : MATLAB code o f SCREENINFO.M
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142
function w= kaiser(nn,beta)
&
AISERKAiSER(N.beta) returns the BETA-valued N-point Kaiser window.
1
5 Author(s): L. Shure, 3-4-87
1
5 Copyright (c) 1988-97 by The M
athworks, inc.
1
5 jRevision: 1.8 $ $Date: 1997/02/06 21:55:10 5
= round(nn);
:es = abs(besseli(0, beta));
3dd = rem(nw,2);
<ind = (nw-l)A2;
i = fix((rwH-l)/2);
<i = (0:n-l) + .5*(1-odd);
<i = 4*xi .*2;
n = besseli(0,beta*sqrt(l-xi/xind))/bes;
n = abs([w(n:-l:odd+l) w])';
tw
Figure D.12: MATLAB code o f KAISER.M
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143
function GATEMX = gating(tmp_rl,tmp_r2,viev\®cm,deltaT1npoint,npair, filtershape)
1
5This function constructs a gating matrix for time-domain gating on
1
5the original frequency-domain sar measurement.
1
5
&tmp_rl: distances between T
Xpositions and the focal point
K
1
5tmp_r2: distances between R
Xpositions and the focal point
1
5
1
5vieitfjcm: diameter(cm) of the sphere of view around the focal point
1
5
S
5deltaT: time resolution, determined by the measurement bandwidth
5
5
£ npoint: num
ber of frequency points
1
5
1
5npair:
num
ber of bistatic/monostatic angle combinations
1
5
1
5filtershape: Kaiser window shape
1
5
1
5G
A
T
E
M
X
: gating matrix used for time-domain multiplication (filtering)
1
5
with the original inverse-FOurier-transformed SA
Rmeasurement.
1
5
GATEMX contains colum
ns of appropriately shifted finite-width
1
5
shaped sequences. The gate centers are determined by both
1
5
tmp_rl and tmp_r2. The gate width is determined by viewocm.
: = 3e8;
riewD= viewDcm/100;
centerindex = round((tmp__rl + tmp_r2)/c/deltaT);
ialfw_index = round(viewD/c/deltaT);
Filterindex = zeros(2*halfw_index + 1,1);
3A
T
E
M
X= zeros(npoint,npai r);
For i = l:npair;
filterindex = (centerindex(i) - halfw_index : centerindex(i) + halfw_index) +
(i-l)*npoint;
GATEMX(filterindex) = filte r s h a p e :
end;
Figure D.13: MATLAB code o f GATING.M
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144
kinction acf = chopband(FD_gated_focused)
KC
O
R
K
:
8 Frequency decorrelation by bandwidth partitioning
condirioning_factor = 100;
ipartition = 16;
%must be power of 2
irow= size(FD_gated_focused,l);
%must be power of 2
icol = size(FO_gated_fdcused,2);
%must be an even number
fVridth = round(nrow/npartiti on);
subSARjlD= zeros(npartition, 1);
subSAR_2D= zeros(npartition,npartition);
^CFarray = zeros(npartition*(npartition+l)/2,l);
for ipartition = l:npartition
row_i = 1 + (ipartition - 1) * fvridth;
row_f = ipartition * fvridth;
subSAR_lD(ipartition) = m
ax( ifft( sum
( FD_gated_focused(row_i:row_f, ' ) ) );
and;
subSAR_2D= subSAR_lD * conj( subSAR_lD).';
ivCFarray = subSAR_2D( find(triu( subSAR_2o ) -= 0) );
^C
F= (l/conditioning_factor) * ( prod( conditioning_factor * ACFarray ) )*( l/(
length(ACFarray) ) );
Figure D.14: MATLAB code o f CHO PBA N D .M
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145
function ACF = chopband(FD_gated_focused)
* CORR:
ISFrequency decorrelation by bandwidth partitioning
conditi oni ng_factor = 100;
npartition =16;
* must be power of 2
nrow= size(FD_gated_fbcused,l);
%must be power of 2
ncol = size(FD_gated_focused,2);
%must be an even number
fvridth = roundCnrow/npartition);
subSAR_lD = zeros(npartition.l);
subSAR_2D = zeros(npartition, npartition);
iXCFarray = zeros(npartition*(npartition+l)/2,l);
For ip a rtitio n = 1: npartiti on
row_i = 1 + (ipartition - 1) * fvridth;
row_f = ipartition * fvridth;
subSAR_lD(ipartition) = m
ax( ifft( sum( FD_gated_focused(row_i :row_f,:).' ) ) );
end;
subSAR_2D = subSAR_lD * conj( subSAR_lD ) . ’;
Farray = subSAR_2D( find(triu( subSAR_2D ) ~= 0) );
<
V
C
F= (1/conditioning_factor) * ( prod( conditi oning_factor * ACFarray ) ) a ( i/(
1ength(ACFarray) ) );
Figure D.15: MATLAB code o f FREQ-ACF.M
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1 46
Function acf =
rime_jacfCTD_gated_focused,target_phi ,cluter_phi,npizza4target,npizza4cluter);
IStime_acf:
ISThis subroutine takes in the gated and focused data as
ISinput. A
n arbitrary angular separation is chosen for
ISangular correlation in -time domain-.
IS
ISsmall target_phi is good for target detection, but badfor clutter s
uppression
ISlarge C
LurER_PH
i is good for clutter suppression, but bad fortarget detection
IS
irow = size(TD_gated_fbcused,l);
icol = size(7t>_gated_fbcused,2);
tm
p = zeros(nrow.ncol);
For i i = l:npizza4target;
delta_ang1e = ii*target_phi;
tm
p = tm
p + TD_gated_focused .* conj( TD_gated_focused(:,[ (delta_angle+l):360
L:delta_angle ]) );
end;
for ii = l:npizza4cluter;
delta_angle = ii*cluter_phi;
tm
p = tm
p + TD_gated_focused .* conjf TD_gated_focused(:, [ (deltcL_angle+l):360
L:delta_angle ]) );
end;
rawacf = tmp/(npizza4target + npizza4cluter);
\CF = m
ax( sum
( ( rawacf ).' ) );
Figure D.16: MATLAB code of TIME-ACF.M
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147
function nomuimage = norm
al (org_image)
ISn o rm a l:
ISt o normalize a given matrix ORGLXMAGE,shifting the m
inlevel of
ISorgjcm age back to zero level. Thennormalize theshifted matrix
ISto 1.00
nin_level = min( min( org_image ) );
nax_level = max( max( org_image ) );
nag_level = abs( max_level - min_level );
iorm_image = ( org_image - min_level )/mag_level;
Figure D.17: MATLAB code of NORMAL.M
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VITA
Tsz-King Chan was bom in Hong Kong in 1971. He received his Bachelor degree
in Electrical Engineering (with higher honors) in 1993 from Portland State University
in Portland, Oregon, USA. After graduation he continued for further education and in
1995, he completed his Master degree in Electrical Engineering from the University of
Washington in Seattle, Washington, USA. The title of his Master thesis is “MillimeterWave Experiments on Bistatic Scattering and Angular Memory Effect of Two-Dimensional
Conducting Random Rough Surfaces” . From September 1995 to June 1998, he was a
research assistant at the Electromagnetics and Remote Sensing Laboratory at the University
of Washington, where he was devoted to his doctoral research under the guidance of
Professor Kuga on radar detection/imaging using correlation techniques.
His research
interests include R&D in all areas of RF electromagnetics with emphasis on medical
imaging, broadband system design, wireless communications, implementation o f targetlocating system in multipath environment and development of clutter-suppression software
algorithms.
Book Chapter
1. A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers, and T. K. Chan, “Polarimetric
scattering theory fo r large slope rough surfaces,” Progress in Electromagnetics
Research, 14, EMW Publishing, Cambridge, Massachusetts, 1996.
Journal Articles
1. T. K. Chan, Y. Kuga, A. Ishimaru and K. Pinyan, “Confocal Imaging in Clutter En­
vironment Using Circular-Correlation Synthetic Aperture Radar,” IEEE Geoscience
and Remote Sensing, submitted 1998.
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149
2. A. Ishimaru, T. K. Chan and Y. Kuga, “An Imaging Technique Using Confocal Cir­
cular Synthetic Aperture Radar,” IEEE Geoscience and Remote Sensing, submitted
1997.
3. T. K. Chan, Y. Kuga and A. Ishimaru, “Subsurface Detection of a Buried Object
Using Angular Correlation Function Measurement,” Waves in Random M edia, 7(3),
pp. 457-465, 1997.
4. T. K. Chan, Y. Kuga and A. Ishimaru, “Angular Memory Effect of Millimeter-Wave
Scattering from Two-Dimensional Conducting Random Rough Surfaces,” Radio
Science, 31(5), pp. 1067-1076, 1996.
5. T. K. Chan, Y. Kuga, A. Ishimaru and C. Le, “Experimental Studies of Bistatic Scat­
tering from Two-Dimensional Conducting Random Rough Surfaces,” IEEE Geo­
science and Remote Sensing, 34(3), pp. 674—680, 1996.
6. A. Ishimaru, C. Le, Y. Kuga, L. Ailes-Sengers and T. K. Chan, “Polarimetric Scatter­
ing Theory for High Slope Rough Surface: Summary,” Journal o f Electromagnetics
Waves and Applications, 10(4), pp. 489^491, 1996.
Conference Papers
1. A. Ishimaru, T. K. Chan and Y. Kuga, “Confocal Imaging Using Circular Synthetic
Aperture Radar,” National Radio Science Meeting, Boulder, Colorado, USA, 1998.
2. T. K. Chan, Y. Kuga and A. Ishimaru, “Feasibility Study on Localized Subsurface
Imaging Using Circular Synthetic Aperture Radar and Angular Correlation Function
Measurement,” International Geoscience And Remote Sensing Symposium, Singa­
pore, 1997.
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150
3. T. K. Chan, Y. Kuga, and A. Ishimaru, “Detection o f a Buried Target Based on An­
gular Correlation Function Measurement,” Progress in Electromagnetics Research
Symposium, Hong Kong, 1997.
4. T. K. Chan, Y. Kuga and A. Ishimaru, “Detection of a Target in a Homogeneous
Medium Using Angular Correlation Function,” International Geoscience A nd Re­
mote Sensing Symposium, Lincoln, Nebraska, USA, 1996.
5. A. Ishimaru, C. Le, Y. Kuga, J. H. Yea, K. Pak, and T. K. Chan, “Interferometric
Technique of Determining the Average Height Profiles of Rough Surfaces” , Inter­
national Geoscience And Remote Sensing Symposium, Lincoln, Nebraska, 1996.
6 . Y. Kuga, T. K. Chan and A. Ishimaru, “Applications of Angular Correlation Func­
tion Measurement in Target Detection,” International Symposium on Antennas and
Propagation, Japan, 1996.
7. C. Penwell, T. K. Chan, and Y. Kuga, “Detection of Buried Objects Using Xband Radar and Angular Memory Effect,” Progress in Electromagnetics Research
Symposium, Baltimore, Washington D. C., USA, 1996.
8. T. K. Chan, Y. Kuga and A. Ishimaru, “Detailed Experimental Studies on Scattering
from Two-Dimensional Conducting Rough Surfaces,” Progress in Electromagnetics
Research Symposium, Seattle, Washington, USA, 1995.
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IMAGE EVALUATION
TEST TARGET (Q A -3 )
15 0 m m
IIW IG E .In c
1653 East Main Street
Rochester, NY 14609 USA
Phone: 716/482-0300
Pax: 716/288-5989
0 1993. Applied Image. Inc.. All Rights Reserved
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