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Relaxation-based methods for SAR target feature extraction and image formation

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Title:
Relaxation-based methods for SAR target feature extraction and image formation
Creator:
Bi, Zhaoqiang, 1968-
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English
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xi, 109 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Ambiguity ( jstor )
Data models ( jstor )
Data ranges ( jstor )
Feature extraction ( jstor )
Image resolution ( jstor )
Imaging ( jstor )
Radar ( jstor )
Reflectors ( jstor )
Sine function ( jstor )
Three dimensional imaging ( jstor )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 104-108).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Zhaoqiang Bi.

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Full Text










RELAXATION-BASED METHODS FOR SAR TARGET
FEATURE EXTRACTION AND IMAGE FORMATION












By

ZHAOQIANG BI












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


UNIVERSITY OF FLORIDA


1999















This work is dedicated to my wife, Jing Wang, for her love and sharing my

happiness and depression in the past years; to my parents, who continue supporting

and encouraging me for my education.














ACKNOWLEDGEMENTS




I am sincerely grateful to my advisor and committee chairman, Dr. Jian

Li, not only for her academic guidance and insightful suggestions throughout the

development of this dissertation and providing an invaluable environment for the

research, but also for her financial support in the past years.

I would like to thank my graduate committee members, Dr. Jos6 C. Principe,

Dr. Fred J. Taylor, Dr. William W. Edmonson, and Dr. David C. Wilson, for their

time and interest in serving on my supervisory committee.

I would also like to thank Dr. Zheng-She Liu, Dr. Renbiao Wu, and the fellow

students, past and present, of the Spectral Analysis Laboratory (SAL) for their great

help.

Finally, I would like to gratefully acknowledge all the people who helped me

during my Ph.D program.




















iii














TABLE OF CONTENTS




ACKNOWLEDGEMENTS ................... ........ iii

LIST OF TABLES ....... .................. ......... vi

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

A BSTRACT . . . . . . . . . . . . . . . . . . x

CHAPTERS

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

1.1 Background ................... ........... 1
1.2 Scope of the W ork .......................... 4
1.3 Contributions ................... ......... 8
1.4 Dissertation Outline ................... ...... 9

2 LITERATURE SURVEY ......................... 11

2.1 2-D Super Resolution SAR Image Formation .......... 11
2.2 3-D SAR Feature Extraction ...................... 15
2.3 SAR Motion Compensation ................. ... 16

3 PARAMETRIC METHODS FOR SAR IMAGING .. ......... 18

3.1 RELAX ................................ 18
3.1.1 Review of the RELAX Algorithm ............. 18
3.1.2 Image Formation ....................... 21
3.2 RELAX-NLS .................... ......... 23
3.2.1 Review of the RELAX-NLS Algorithm ........... 23
3.2.2 Image Formation ................. ..... . 27
3.3 Experimental Results ........................ 28

4 SEMI-PARAMETRIC METHODS FOR SAR IMAGING ....... 35

4.1 Problem Formulation and Data Model ............... 35
4.2 Review of the APES Algorithm .......... ....... 37
4.3 Data Model Ambiguities and Their Effects on SAR Image Formation 38
4.3.1 Model Ambiguities ...................... 39
4.3.2 Image Formation ...... ................. 41
4.3.3 Model Ambiguity Effects on SAR Image Formation .... 43
4.4 The SPAR Algorithm ........................ 45


iv








4.4.1 Target Feature Extraction. . . . . . . .... 45
4.5 Modified RELAX-NLS Algorithm . . . . . . ... 51
4.6 Cram6r-Rao Bound (CRB) of the Semi-Parametric Data Model 52
4.7 Numerical and Experimental Results . . . . . ..... 54

5 3-D TARGET FEATURE EXTRACTION USING CLSAR ...... 63

5.1 CLSAR Data Model ......................... 63
5.1.1 1-D Data Model for High Range Resolution Radar . .. 63
5.1.2 Full Synthetic Aperture Radar . . . . . . ... 64
5.1.3 Curvilinear SAR ....................... 67
5.2 The RELAX Algorithm ..................... . 68
5.2.1 Full Aperture .... .. ..... ... .. ... .. .. 68
5.2.2 Curvilinear Aperture . . . . . . . . ... 70
5.3 Performance Analysis of Parameter Estimation via CRB . . 71
5.4 Numerical and Experimental Results. . . . . . ... . 72
5.4.1 Performance Analysis of Different Curvilinear Apertures
via C R Bs . . . . . . . . . . . . . .. 72
5.4.2 Experimental Examples . . . . . . . . ... 74

6 AUTOFOCUS IN CLSAR ........................ 79

6.1 Aperture Error Effects ................... ... 80
6.2 The Relaxation-Based Autofocus Algorithm (AUTORELAX) .83
6.2.1 Aperture Parameter Estimation .............. 83
6.2.2 Target Feature Extraction . . . . . . . ... 85
6.3 Experimental and Numerical Results . . . . . ..... 87

7 CONCLUSION ................... ......... 94

7.1 Summary ...... ... .............. ...... 94
7.2 Future W orks ................... ......... 96

APPENDIXES

A A SINGLE SCATTERER FEATURE EXTRACTION VIA SPAR .98


B APPROXIMATION OF THE RANGE R . . . . . . ... 100


C CALCULATING (x, y, z) FROM (x, y, ) ................ 102

REFERENCES ................................... .104

BIOGRAPHICAL SKETCH ...... . . ....... ..... 109









v














LIST OF TABLES



4.1 True parameters of the two dihedrals used in Figures 1 and 2 ... 44

5.1 Comparison of the CRBs (in dB) of the target features for the cases
of the full, circular, parabolic, L-shaped, Arc-1, and Arc-2 apertures
when M = 63, K = 1,w = = w= 0, a = 1, and 2 = 40. . . . 73

6.1 Estimation errors for the parameters of the highest scatterer in the
simulation example before and after autofocusing. . . . ... 89



































vi













LIST OF FIGURES



1.1 Illustration of 2-D radar imaging. . . . . . . . ... ... 2
1.2 Illustration of SAR imaging modes. (a) Spotlight-mode SAR. (b)
Strip-m ap SAR .... .. .... . .. . . .. .. ... ... 3
1.3 Possible apertures for a 3-D SAR system, a) Full aperture. b) Parabolic
aperture. (c) Orthogonal subaperture. ( 0 and 0 denote the azimuth
and elevation angles, respectively.) . . . . . . . . . 6
1.4 Mesh plots of the modulus of the RCS obtained by using 2-D FFT with
different apertures (range information suppressed for the illustration
purpose only). (a) Full aperture as shown in Figure protect 1.2(a). (b)
Curvilinear aperture as shown in Figure 1.2(c). . . . . . . 7

3.1 RELAX SAR Images by using the 40 x 40 ERIM data with K = 59. (a)
Unwindowed 2-D FFT image. (b) Windowed 2-D FFT image. (c) Un-
windowed RELAX image without background clutter. (d) Windowed
RELAX image without background clutter. (e) Unwindowed RELAX
image with background clutter. (f) Windowed RELAX image with
background clutter. ........................... 31
3.2 RELAX Images by using the 20 x 20 ERIM data with K = 41. (a)
Unwindowed 2-D FFT image. (b) Windowed 2-D FFT image. (c) Un-
windowed RELAX image without background clutter. (d) Windowed
RELAX image without background clutter. (e) Unwindowed RELAX
image with background clutter. (f) Windowed RELAX image with
background clutter. ... . . . . . . . . . 32
3.3 Target photo taken at 450 azimuth angle. . . . . . . ... 33
3.4 SAR images of the Slicy data (00 azimuth and 300 elevation angles). (a)
Original windowed 2-D FFT image. (b) Windoed 2-D FFT image with
spoiled resolution. (c) ~ (d): Windowed RELAX image without and
with background clutter with K = 36. (e) ~ (f): Windowed RELAX-
NLS image without and with background clutter with K = 9. (c) ~
(f) are obatined using the data in (b). (The vertical and horizontal
axes are for range and cross-range, respectively.) . . . .... 34








vii








4.1 Ambiguity effect on the SAR image formation in the absence of range
estimation errors. (a) True windowed FFT SAR image. (b) Windowed
FFT image of the first "scatterer". (c) Windowed FFT image of the
second "scatterer". (d) Combined windowed FFT image of the two
"scatterers". (The vertical and horizontal axes are for range and cross-
range, respectively.) ........ .................. 58
4.2 Ambiguity effect on the SAR image formation in the presence of range
estimation errors. (a) Windowed FFT image of the first "scatterer".
(b) Windowed FFT image of the second "scatterer". (c) Combined
windowed FFT image of the two "scatterers" with ( = 1 (without
extrapolation). (d) Combined windowed FFT image with ( = 2.
(The vertical and horizontal axes are for range and cross-range, re-
spectively.) . . . . . . . . . . . . . . .. . 59
4.3 Comparison of SAR images formed using different algorithms for sim-
ulated data at high SNR (u = 0.6). (a) True SAR image. (b) Win-
dowed 2-D FFT SAR image. (c) Hybrid SAR image. (d) SPAR SAR
image. (e) RELAX-NLS SAR image. (f) Modified RELAX-NLS SAR
image. (The vertical and horizontal axes are for range and cross-range,
respectively.) ... ... ........... ......... .. 60
4.4 Comparison of SAR images formed using different algorithms for sim-
ulated data at low SNR (Uc = 6). (a) Windowed 2-D FFT SAR image.
(b) Hybrid SAR image. (c) SPAR SAR image. (d) RELAX-NLS SAR
image. (e) Modified RELAX-NLS SAR image. (The vertical and hor-
izontal axes are for range and cross-range, respectively.) ...... ..61
4.5 Comparison of SAR images obtained via different algorithms for the
Slicy data hb15533.015 (0 azimuth and 300 elevation angles). (a)
Windowed 2-D FFT SAR image from the XPATCH data. (b) Win-
dowed 2-D FFT SAR image from the Slicy data. (c) Hybrid SAR
image. (d) SPAR SAR image. (e) RELAX-NLS SAR image. (f) Mod-
ified RELAX-NLS SAR image. (c) ~ (f) are all obtained from the
data used in (b). (The vertical and horizontal axes are for range and
cross-range, respectively.) . . . . . . . ..... ....... 62

5.1 Broadside spotlight-mode SAR data collection geometry. ...... . 65
5.2 Curvilinear apertures. a) The circular aperture. b) The Arc-1 aper-
ture. c) The Arc-2 aperture ...................... . 73
5.3 The CRBs of the target parameters as a function of the circular aper-
ture radius when K = 1, a = 1, w = C = c = 0, and a2 = 40. a) The
CRB of the complex amplitude a. b) The CRB of w. c) The CRB of
D. d) The CRB of C. ......................... .. 77
5.4 3-D plots of K = 8 scatterers extracted by using RELAX with the
indoor experimental data. (a) Obtained with full aperture as shown in
Figure 1.2(a). (b) Obtained with curved aperture as shown in Figure
1.2(c). .... .. ..................... ......... .. 78



viii








6.1 (a) Curvilinear apertures for the experimental example. (b) True scat-
terer distribution. (c) 3-D plot of K = 20 scatterers extracted from
the initial aperture (dashed line) in (a) with RELAX. (d) 3-D plot
of K = 20 scatterers extracted from the manually adjusted aperture
(solid line) in (a) with RELAX. ....... ............. . 90
6.2 Autofocused curvilinear aperture and scatterer distribution obtained
with AUTORELAX by autofocusing only in the elevation direction
and using K = 20 for the experimental example. (a) Autofocused
curvilinear aperture (dotted line). (b) Scatterer distribution. . . 91
6.3 (a) Manually adjusted (solid line) and autofocused (dotted line) curvi-
linear apertures for the experimental example. (b) Fitting the manu-
ally adjusted aperture to the autofocused aperture by adding a line to
the form er .. . . . . . . . . . . . . . . . 91
6.4 Autofocused curvilinear aperture and scatterer distribution obtained
with AUTORELAX by autofocusing in both the elevation and azimuth
directions and using K = 20 for the experimental example. (a) Auto-
focused curvilinear aperture (dotted line). (b) Scatterer distribution. 92
6.5 (a) Curvilinear aperture for the simulation example. (b) True scatterer
distribution for the simulation example. (c) Scatterer distribution ob-
tained from the initial aperture (dashed line) in (a) by using RELAX
with K = 20. (d) Scatterer distribution obtained via AUORELAX
with K = 20 and using the autofocused aperture (dotted line) shown
in (a). . . . . . . . . . . . . . . . . .. .. 93

























ix














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.




RELAXATION-BASED METHODS FOR SAR TARGET
FEATURE EXTRACTION AND IMAGE FORMATION

By

Zhaoqiang Bi

May 1999


Chairman: Dr. Jian Li
Major Department: Electrical and Computer Engineering

Synthetic aperture radar (SAR) has been a mature but actively researched

technology due to its day-and-night and all-weather capability of offering high res-

olution imaging for both military and civilian applications. As the foundation of

automatic target detection and recognition, SAR imaging and autofocusing continue

to attract more research interest. Non-parametric spectral estimation methods are

robust methods for SAR image formation. However, non-parametric methods cannot

be used to significantly improve the resolution of the formed SAR images since they

generally do not fully exploit the characteristics of radar targets of interests even

when such information is available.

In this dissertation, efforts have been made to form super resolution two-

dimensional (2-D) SAR images via relaxation-based parametric methods. The

relaxation-based optimization methods have been proved to be quite useful in several




x








other applications, such as radio astronomy, microwave imaging, and spectral estima-

tion. The relaxation-based methods are extended for super resolution SAR imaging

of radar targets consisting of only trihedrals or both trihedrals and dihedrals. We

have also devised a robust and computationally simple SPAR (Semi-PARametric) al-

gorithm for 2-D SAR imaging based on a flexible semi-parametric data model when

it is difficult to establish an accurate target data model in cross-range. Hence SPAR

takes advantages of both parametric and non-parametric spectral estimation meth-

ods to form enhanced SAR images. Numerical and experimental examples have been

used to demonstrate the performances of the proposed algorithms. We have observed

that the relaxation-based parametric methods provide super resolution SAR images

when the assumed data model is valid; otherwise SPAR performs better.

Three-dimensional (3-D) target features, including the height information,

radar cross section (RCS), and 2-D location (range and cross-range), provide quite

useful information for such applications as automatic target recognition. Thus, ef-

forts have also been made in this dissertation to devise an effective relaxation-based

algorithm, referred to as AUTORELAX, for both 3-D target feature extraction and

motion compensation via curvilinear SAR (CLSAR), a novel technology which is

still at its developing stage. The proposed AUTORELAX algorithm is shown to be

promising when evaluated by using both the experimental and simulated examples.

















xi














CHAPTER 1
INTRODUCTION

1.1 Background

RADAR stands for RAdio Detection And Ranging. By now, radar has been

used not only to detect and locate targets, as was originally intended, but to gen-

erate images of illuminated scenes without the restriction of weather and time as

well. Radar image is defined as a two-dimensional (2-D) mapping of the reflectivity

corresponding to the scene illuminated by a radar. More specifically, a radar image

is the reconstruction of the illuminated scene from the electromagnetic reflectivity,

which is usually treated as a 2-D function in range and cross-range. The direction

in which a radar beam propagates is referred to as the range direction, while the

one perpendicular to the line of sight is called the cross-range direction, or the az-

imuth direction (see Figure 1.1 for an illustration). Radar imaging [2] has played a

crucial role in both military applications, including reconnaissance and surveillance

[21], and civilian ones including geological and topographic map generation, weather

forecasting, scene classification, and target recognition [60, 65].

The quality of radar images depends greatly on the image resolution, which

is nominally referred to as the ability to distinguish two closely-spaced elements of a

target [48]. The higher the radar resolution, the more details of target features are

available for detection, characterization, and identification of targets of interest. As

such, much effort has been made to improve 2-D imaging radar resolutions, which

include range and cross-range resolutions. High range resolution can be achieved

by using wide bandwidth signals, such as linear frequency modulated (FM) chirp

pulses [48], since range resolution is inversely proportional to the transmitted signal


1






2


Flight Path











---------------^----/
Cross-Range


Figure 1.1: Illustration of 2-D radar imaging.

bandwidth. The cross-range resolution 6, of a conventional radar or real aperture

radar (RAR) is range-dependent and is determined by

S= R, (1.1)

where R represents the range of the target; A and L, respectively, denote the wave-

length and the size of the antenna aperture. To increase cross-range resolution (given

R), an obvious way is either to reduce the wavelength, or to increase the size of the

antenna. Using shorter wavelength requires more transmission power since electro-

magnetic signals with smaller wavelengths are more likely absorbed by the atmo-

sphere. In the meantime, it is not feasible to increase the antenna size too much,

especially when the radar is carried on board of an airplane or spacecraft. Hence

a conventional radar or RAR inherently has a limited cross-range resolution. It is

desired to achieve high cross-range resolution without reducing the wavelength and

exploiting large physical antennas. Synthetic aperture radar (SAR) is a solution and

can achieve high cross-range resolution independent of range.

The SAR technology was first proposed in the 1960s [59]. Instead of exploiting

large antennas and reducing wavelength, SAR makes use of a synthetic aperture to

achieve high cross-range resolution. A synthetic aperture is formed by coherently





3

Synthetic Aperture Synthetic Aperture
^ "- / Flight Path Flight Path
--- -" ........ _-------.....


Cross-Range

Cros-Range ---________ --- - -


(a) (b)

Figure 1.2: Illustration of SAR imaging modes. (a) Spotlight-mode SAR. (b) Strip-
map SAR.

processing radar returns from a sequence of locations when a nominally-sized antenna,

which is carried on board of an aircraft, sequentially moves along the flight path

(cross-range direction). Consequently, a synthetic antenna with an effective antenna
length proportional to the synthetic aperture can be realized even with a small real

antenna. The longer the synthetic aperture, the finer the cross-range resolution

achieved. It has been demonstrated in [29] that the cross-range resolution of SAR

depends only on the wavelength A and the synthetic aperture interval. To achieve

such a desirable cross-range resolution, many techniques have been developed to

record data, compensate motion errors, and construct high quality SAR images ever

since the 1960s. (Detailed descriptions of these techniques are well documented in

many books, e.g. [29, 11]). SAR has grown so dramatically through advances in

digital signal processing, computing power, and microwave technology that it is one
of the most important techniques for high resolution radar image formation.

Spotlight-mode and strip-map SAR are two commonly used SAR imaging

modes. In a spotlight-mode SAR, antenna beams continuously steer to the same

terrain patch (scene) illuminated by a radar when the aircraft traverses the flight

path, while the antenna steering of a strip-map SAR is fixed relative to the flight
path when the aircraft moves along the flight path (see Figure 1.2 for the difference).






4

The cross-range resolution of strip-map SAR is limited by the azimuth beam-width,

while that of spotlight-mode SAR is independent of the azimuth beam-width since the

antenna steers its beam to continuously illuminate the same scene. Hence Spotlight-

mode SAR is more suitable for high resolution SAR image formation of a small scene

than strip-map SAR.


1.2 Scope of the Work

Spotlight-mode SAR has been extensively applied to 2-D high resolution im-

age formation and target feature extraction of small scenes due to its finer resolutions

and readily controllable flight path, which is a straight line. The 2-D target features

include the radar cross section (RCS) and the 2-D location (range and cross-range) of

each scatterer. One topic of our work is super resolution 2-D image formation from

phase history data collected by spotlight-mode SAR. The super resolution herein is

defined as the resolution that is better than the Rayleigh limits determined by the

transmission wavelength and the length of the synthetic aperture of SAR. A conven-

tional approach for 2-D image formation from SAR phase history data is the fast

Fourier transform (FFT), which is computationally efficient and robust. However,

SAR images obtained via FFT suffer from poor resolution and high sidelobes. SAR

image formation can be considered as a spectral estimation problem since SAR images

are the reconstruction of scene reflectivity intensities versus space locations. Mod-

ern spectral estimation methods including parametric and non-parametric methods

have found extensive applications in improving the resolution and accuracy of the

formed SAR images. In particular, parametric spectral estimation methods are more

suitable to form super resolution SAR images when parametric data models can be

used to model the target. The super resolution of the formed SAR images can be

achieved by extrapolating the phase history data beyond observations using an appro-

priate data model and the parameter estimates. We will discuss how to form super





5

resolution SAR images by using the target features extracted via relaxation-based

methods including RELAX [39] and RELAX-NLS [42] in Chapter 3. The relaxation-

based optimization approach is one of the effective ways to minimize a nonlinear least

squares (NLS) cost function and has been proved to be quite useful in many applica-

tions, such as radio astronomy [28], microwave imaging [66], spectral estimation [23],

and sinusoidal parameter estimation and target feature extraction [39, 42]. RELAX

was originally proposed for the parameter estimation of complex sinusoids in colored

noise. Compared with other NLS algorithms, RELAX is relatively robust against

data mismodeling errors [39]. We will show that RELAX can be used to enhance

SAR images of targets consisting of point scatterers. However, the point scatterer

data model may not be valid in practice since, for example, for many man-made

targets, such as vehicles and buildings, much of the returned energy is caused not

only by trihedral corner reflectors, but also by dihedral corner reflectors of a target

[36]. We also describe how to form super resolution SAR images of targets consisting

of both trihedrals and dihedrals using RELAX-NLS [42]. RELAX-NLS is based on

a mixed data model, which models a trihedral as a point scatterer and a dihedral as

a complex sinusoid with a constant phase and a variable amplitude described by a

sinc function in cross-range. We will demonstrate the super resolution property of

the RELAX and RELAX-NLS algorithms with experimental data including MSTAR

and ERIM (Environmental Research Institute of Michigan) data.

Parametric methods may outperform non-parametric methods provided that

the assumed data models are valid. In the more likely cases where it is difficult to

establish a good parametric data model in cross-range due to a variety of physical

phenomena including glints, resonances, and motion-induced phase errors [15], a more

flexible data model is desired and non-parametric and parametric methods may be

combined together for 2-D SAR target feature extraction and image formation. In

our work, we also establish a more flexible data model for man-made targets when





6












(a) e (b) 0 (c) 0

Figure 1.3: Possible apertures for a 3-D SAR system. a) Full aperture. b) Parabolic
aperture. (c) Orthogonal subaperture. ( 0 and 4 denote the azimuth and elevation
angles, respectively.)

no accurate parametric data model is available in cross-range. Based on the flexible

data model, we present a robust and computationally simple semi-parametric (SPAR)

algorithm for target feature extraction and SAR image formation by exploiting a

hybrid of non-parametric and parametric spectral estimation methodes.

It has been shown in [62, 16] that three-dimensional (3-D) target features

including the height information as well as the 2-D target features provide very useful

information in certain applications, such as automatic target recognition (ATR). 3-D

SAR image formation and feature extraction are becoming active research areas. It

is possible to obtain 3-D information of an illuminated scene with Spotlight-mode

SAR when a non-straight line synthetic aperture is exploited, which is determined by

the azimuth angle 0 and elevation angle 0. In our work, we will also study 3-D target

feature extraction and motion compensation (autofocus) using curvilinear SAR, in

which the trajectory of a spotlight-mode SAR is a suitable curvilinear aperture.

Among existing 3-D SAR imaging techniques, interferometric SAR (IFSAR)

is now a well-developed technique that plays an important role in 3-D (as well as 2-D)

imaging. However, IFSAR suffers from ambiguity problems [11, 40]. Theoretically,

3-D target features and images can also be obtained when the trajectory of SAR is a

full aperture as shown in Figure 1.2(a) even though such an aperture is not feasible in






7









Height Cross-Range Height Cross-Range


(a) (b)

Figure 1.4: Mesh plots of the modulus of the RCS obtained by using 2-D FFT with
different apertures (range information suppressed for the illustration purpose only).
(a) Full aperture as shown in Figure protect 1.2(a). (b) Curvilinear aperture as shown
in Figure 1.2(c).

practice. Nevertheless, a suitable subset of the full aperture can be used as a trajectory

of SAR to obtain 3-D information. This technique is known as curvilinear SAR

(CLSAR). CLSAR traverses a curvilinear aperture path, which is a subset of the full

aperture (see, for example, Figures 1.2(b) and (c)), and collects a sparsely sampled

dataset, from which 3-D information can be obtained. CLSAR avoids ambiguity

problems inherent in IFSAR systems, but suffers from severe high sidelobes when

used with FFT to form SAR images due to the limited size of measurements. This

observation is illustrated in Figure 1.2. (Figures 1.2(a) and (b) show the modulus

of the RCS of a single point scatterer obtained by applying 2-D FFT to the data

collected via the full aperture as shown in Figure 1.2(a) and the curvilinear aperture

as shown in Figure 1.2(c).) Due to its high sidelobes, CLSAR is of little interest in

SAR imaging. Yet CLSAR can be used with parametric spectral estimation methods

to extract 3-D target features of small targets consisting of a finite number of isolated

point scatterers. In Chapter 5, we will establish a CLSAR data model and describe

how the RELAX algorithm [39] can be used for 3-D target feature extraction using

CLSAR with different curvilinear apertures.






8

Motion error compensation also plays a crucial role in accurately extracting

target features using CLSAR. Motion errors in 2-D SAR systems can be approximated

as phase errors in cross-range since the azimuth angle 0 is very small and phase errors

are mainly introduced by the uncertainty in the distance from the radar to the center

of the illuminated scene [29]. However, it is not the case for CLSAR since curvilinear

aperture determined by azimuth angle 0 and elevation angle is critical in obtaining

3-D information of the target. Hence motion errors in 0 and may not be negligible.

Motion compensation (autofocus) in CLSAR turns out to be more complicated than

that in 2-D SAR systems. In Chapter 6, we investigate the effects of aperture errors

on 3-D target feature extraction and present a relaxation-based autofocus algorithm,

referred to as AUTORELAX, to extract target features in the presence of aperture

errors using CLSAR.


1.3 Contributions

The significance of this research lies in that it examines the possibility of super

resolution image formation and autofocus of small targets via spotlight-mode SAR

system by using relaxation-based parametric and semi-parametric spectral estimation

methods. Super resolution SAR images obtained via the proposed methods can

be further used for SAR image segmentation and automatic target detection and

recognition. This research makes contributions in the following two areas:

(I) 2-D super resolution SAR image formation

(1) We have extended RELAX to form enhanced SAR images of targets consisting

of point scatterers;

(2) We have extended RELAX-NLS for super resolution SAR image formation of

targets consisting of both trihedrals and dihedrals;





9

(3) We have devised a novel, robust, and computationally simple Semi-PARametric

(SPAR) algorithm for SAR target feature extraction and image formation based

on a flexible data model;

(4) Experimental results using ERIM and MSTAR Slicy data show that the relaxation-

based parametric approaches including RELAX and RELAX-NLS and the semi-

parametric algorithm SPAR, can yield SAR images with higher resolution than

FFT based methods. Also SPAR is more robust and computationally simple

than RELAX-NLS.

(II) 3-D target feature extraction and motion compensation via CLSAR

(1) We have established a 3-D data model;

(2) We have performed Cram6r-Rao Bound (CRB) analyses of parameter estima-

tion using various flight trajectories, which provide more insight into the per-

formances associated with different curvilinear apertures;

(3) The effects of aperture errors on target feature estimation have been investi-

gated and a relaxation-based approach for autofocus and target feature extrac-

tion using CLSAR has been introduced.


1.4 Dissertation Outline

The remainder of the dissertation is organized as follows. Chapter 2 gives a

literature survey of emphasis on the topics considered, including 2-D SAR imaging, 3-

D SAR target feature extraction, and motion compensation (autofocus). In Chapter

3, we describe how to form super resolution 2-D SAR images via the relaxation-

based parametric spectral estimation algorithms including RELAX and RELAX-

NLS. A novel semi-parametric (SPAR) method is introduced in Chapter 4 for SAR

image formation when an accurate parametric data model in cross-range is difficult





10

to establish. In Chapter 4, we also present a modified RELAX-NLS algorithm by

using the initial conditions provided by SPAR to save the amount of computations

needed by the original RELAX-NLS algorithm. In Chapter 5, a CLSAR data model

is established and RELAX is introduced for 3-D target feature extraction via CLSAR.

Chapter 6 investigates the ambiguity problems inherent in the proposed CLSAR data

model when aperture errors are present. A relaxation-based autofocus algorithm for

3-D target feature extraction in the presence of aperture errors is also presented in

Chapter 6. Finally, Chapter 7 give the summary and conclusions of this research and

outlines the future work.














CHAPTER 2
LITERATURE SURVEY

In this chapter, we give a brief review of the topics related to the work, namely

super resolution 2-D SAR image formation, 3-D SAR target feature extraction, and

SAR motion compensation.


2.1 2-D Super Resolution SAR Image Formation

Since SAR image formation can be considered as a spectral estimation prob-

lem, modern spectral estimation methods find extensive applications in improving

the resolution and accuracy of the formed SAR images. Excellent texts on spectral

estimation are available [63, 30, 46] and historical and modern perspectives on the

general topics of spectral analysis can be found in [54, 31]. A brief review of super res-

olution SAR image formation via non-parametric and parametric spectral estimation

methods is given below.

Non-parametric Spectral Estimators

An important class of non-parametric spectral estimators are the adaptive

filterbank approaches which can be used to form SAR images with better resolu-

tion and lower sidelobes than the FFT based methods [15]. The adaptive sidelobe

reduction (ASR) approach [14] and the widely used Capon method [10] as well as

its reduced rank variations [5, 6] are members of this class. A recently introduced

matched-filter bank based complex spectral estimator that is of interest is the APES

(Amplitude and Phase Estimation of a Sinusoid) method [38]. Despite their different

derivations, it was found that both Capon and APES are members of the general

class of the matched-filterbank (MAFI) approaches to complex spectral estimation


11





12

[37]. Empirically, it was observed in [38] that Capon appears to give biased spec-

tral estimates whereas APES is unbiased. This observation was later proved in [37]

by using the second-order Taylor expansion technique. Also the statistical perfor-

mance of the Capon estimator degrades rapidly as the filter length increases, while

that of APES remains unaffected. The study in [37] gives a compelling reason for

preferring APES to Capon in most spectral estimation applications including SAR

imaging. In cases where an appropriate parametric data model of targets is not

available, APES is a good candidate for SAR imaging which provides much lower

sidelobes and narrower spectral peaks than Capon and the conventional FFT based

methods. However, significant resolution improvement cannot be obtained by using

non-parametric methods since they do not fully exploit the characteristics of targets,

especially when such information is available.

Parametric Spectral Estimators

When parametric data models can be used to describe radar targets, paramet-

ric spectral estimators provide attractive alternatives for super resolution SAR image

formation. The Burg's maximum entropy (ME) method [9] was applied to radar im-

age enhancement by Mancill in 1977 [44]. Unlike FFT methods which assume that

the signal beyond the observations is zero, ME methods make linear prediction of the

signal beyond the observation window so that they can achieve better resolution than

FFT methods. Identical to ME methods in principles, the autoregressive (AR) model

based methods [15, 13, 20, 19, 26] have attracted a lot of research attention. However,

the high resolution property of the AR based methods are highly dependent on the

presence of observation noise [31]. As the signal-to-noise ratio (SNR) decreases, so

does the resolution. On the other hand, the AR based methods are more appropriate

for SAR image formation of distributed scatterers rather than point scatterers since

the AR spectral estimator by nature is a continuous spectral estimator.





13

In recent years, many sophisticated and high resolution methods based on

point scatterer data model, such as eigendecomposition based methods [4], including

MUSIC (MUltiple SIgnal Classification) [58] and ESPRIT (Estimation of Signal Pa-

rameters via Rotational Invariance Technology) [56], and relaxation-based nonlinear

least squares (NLS) method such as RELAX [39], have also been applied to super

resolution SAR image formation. The eigendecomposition based methods are signal

or noise subspace based approaches and generally are not suitable for SAR image

formation since they require that the number of data samples be much larger than

that of the dominant point scatterers and the process of whitening the noise sub-

space (used in MUSIC) will destroy the spatial inhomogeneities of the terrain clutter

[15]. The RELAX algorithm [39] was originally proposed for sinusoidal parameter

estimation in colored noise. RELAX obtains the parameter estimates by minimizing

the NLS cost function via a sequence of FFTs, which is conceptually and computa-

tionally simple and hence robust. RELAX coincides with the maximum likelihood

(ML) method when the noise is white and is also asymptotically (for large numbers

of data samples) statistical efficient when the noise is an AR or autoregressive mov-

ing average (ARMA) process that can be described by a finite number of unknowns

[63]. Due to the iterations performed in each step of RELAX, a set of more accurate

initial estimates are available, which make RELAX converge to the global minimum

with a high probability. RELAX can also be referred to as SUPER CLEAN when

compared with two other approximate relaxation methods including CLEAN [28] and

MCLEAN (more clean) [23]. CLEAN was first proposed in radio astronomy and later

used in microwave imaging [66]. If no iteration is performed in each step of RELAX,

then it is equivalent to CLEAN. RELAX becomes MCLEAN if only one iteration

is performed in each step of RELAX. Hence RELAX is a computationally efficient

and robust algorithm with good statistical performance and is suitable for super

resolution SAR image formation of targets consisting of point scatterers (trihedrals).






14

Recently, RELAX-NLS [42] was introduced for the feature extraction of targets

consisting of both trihedrals and dihedrals based on a mixed data model rather than

the point scatterer data model. An ideal trihedral acts as a triple-bounce reflector

and can be modeled as a strong point scatterer [53]. The point scatterer data model,

however, is no longer valid in cross-range for an ideal dihedral corner reflector, which

acts as a double-bounce reflector, since its radar cross section (RCS) is described by

a sinc function, i.e., sinc(0) = sin(0)/0, where 0 denotes the angle between the radar

beam and the line perpendicular to the dihedral.

Super resolution SAR images obtained via modern spectral estimation meth-

ods provide more details required for the detection and automatic target recognition

(ATR) than the conventional FFT based methods. Quantitative results showing the

advantages of using these sophisticated spectral estimation methods in SAR appli-

cations begin to appear in the literature. For example, it was shown in [51, 50] that

a significant improvement in ATR performance can be achieved by using SAR im-

ages with enhanced resolution obtained via the reduced-rank variations of the Capon

method.

Compared with non-parametric spectral estimators, high resolution paramet-

ric spectral estimators provide great potential for SAR imaging with better resolution

and lower sidelobes when the assumed data model is valid. On the other hand, para-

metric methods are more sensitive to data model mismatch than non-parametric

methods. Hence, practically only robust super resolution parametric methods can be

used for SAR image formation. We will use robust relaxation-based algorithms in-

cluding RELAX and RELAX-NLS to form SAR images of different man-made targets

in our work. In cases where an appropriate data model of man-made targets is not

available in cross-range, robust non-parametric spectral estimators with good perfor-

mances such as APES and high resolution parametric methods may work together to

obtain satisfactory results. To the best of our knowledge, a hybrid of non-parametric





15

and parametric methods for feature extraction and image formation of SAR targets

has not been addressed in the literature before.


2.2 3-D SAR Feature Extraction

One relatively mature technology for 3-D SAR imaging and feature extrac-

tion is IFSAR [24, 55, 69, 27, 70] which exploits two different apertures. A simple

implementation of a IFSAR is to use two antenna displaced in the cross-track plane

to obtain two coherent and parallel measurements. The relative height information

can be estimated from the phase difference between the two measurements. The use

of interferometric techniques with SAR imagery to provide height information makes

IFSAR suitable for 3-D imaging since it inherits the finer resolutions (in range and

cross-range) associated with the spotlight-mode SAR and also has reasonable height

resolution. Many 2-D modern spectral estimation algorithms including Capon [10],

APES [38], and MUSIC [58] have been generalized for 3-D imaging and target feature

extraction using IFSAR [40, 47]. An NLS method that outperforms MUSIC in speed

and estimation accuracy was also reported in [40]. However, such a system cannot

resolve more than one target scatterer at the same projected range and cross-range

but at different heights [40].

Another relatively new technology for 3-D imaging is curvilinear SAR. CLSAR

traverses a curvilinear aperture path and avoids the height ambiguity problem inher-

ent in IFSAR systems. CLSAR images are useless in practice due to high sidelobes.

However, CLSAR can be used in conjunction with modern spectral estimation meth-

ods to accurately extract 3-D features of small targets, which consist of a finite number

of isolated point scatterers. CLSAR is still at its exploratory stage of development

and only a few studies were reported in the literature. In [33], a coherent CLEAN

algorithm was presented by Knaell to effectively eliminate sidelobes in CLSAR im-

ages. The CLEAN algorithm was originally proposed in radio astronomy [28] and





16

later was used in microwave imaging [66]. Encouraged by the initial results in [33],

Knaell later presented in [34] a correlation technique which works in conjunction with

the coherent CLEAN algorithm to reduce sidelobe interference. Furthermore, Knaell

in [35] compared the performances of the coherent CLEAN method and two other

methods including a maximum likelihood (ML) method and a sidelobe leakage reduc-

tion algorithm [12] in extracting 3-D target features by using CLSAR datasets. We

will discuss how RELAX (SUPER CLEAN) can be extended for 3-D target feature

extraction using CLSAR in Chapter 5.


2.3 SAR Motion Compensation

Aperture errors exist due to atmospheric turbulence and platform uncertainty.

These undesirable errors may significantly degrade the quality of SAR images in sev-

eral ways. For example, aperture errors may cause image geometric distortion, spu-

rious targets, loss of resolution, and decrease in image contrast [32, 25] depending on

their natures. Therefore, effective motion compensation methods are indispensable

for SAR imaging and target feature extraction in practice. In 2-D SAR systems, mo-

tion errors mainly cause phase errors in cross-range [29]. Autofocus techniques that

compensate for phase errors using radar phase history data have been extensively

exploited by 2-D SAR systems. Of the existing autofocus algorithms, mapdrift [45]

and phase difference [68] algorithms are used to compensate for phase errors asso-

ciated with certain order polynomials, whereas the phase-gradient autofocus (PGA)

algorithm [17, 18, 67] is independent of the phase error model, thereby is more ro-

bust. Autofocus algorithms based on super resolution spectral estimation methods

also begin to appear in the literature [43, 52]. All the aforementioned 2-D autofocus

algorithms assume that phase errors change in cross-range only and are constant for

all range bins. This assumption is generally valid for the 2-D SAR systems where





17

significant phase errors are caused by motion measurement errors, such as along-track

velocity errors or line-of-sight acceleration errors [11].

As in 2-D SAR systems, motion error compensation is also indispensable for

3-D SAR systems. Autofocus in CLSAR, however, becomes more complicated since

the aperture errors can no longer be approximated as phase errors only in cross-range

since the errors introduced in 0 (azimuth angle) and 0 (elevation angle) may not be

negligible and the extracted 3-D target features are sensitive to the shape of the

curvilinear aperture (see the discussion in Chapter 6). To the best of our knowledge,

autofocus for CLSAR has not been reported in the literature, partly because CLSAR

is a relatively new technology.














CHAPTER 3
PARAMETRIC METHODS FOR SAR IMAGING

When an appropriate data model is available for radar targets, parametric

spectral estimation algorithms, which are less sensitive to the model mismatch, offer

the potential to significantly improve the resolution of the formed SAR images. In

this chapter, we consider forming enhanced SAR images with robust relaxation-based

methods including RELAX and RELAX-NLS.


3.1 RELAX

When the dominant features of targets can be approximated as point scatter-

ers, enhanced SAR images can be formed by using the target features extracted via

RELAX and the point scatterer data model. We first briefly introduce the RELAX

algorithm [39] and then discuss how to form SAR images using RELAX.

3.1.1 Review of the RELAX Algorithm

Assume that there are K dominant point scatterers or trihedral corner re-

flectors in a target. An ideal trihedral acts as a triple-bounce reflector and can be

modeled as a strong point scatterer [53] since its radar cross section (RCS) can be

considered as a constant due to the very small angle variation of the radar beam in

SAR. Then the target data model in the presence of unknown noise has the form:
K
y(n,h)= akej(ckn+Wkf) +e(n,'), n= 0,1,. *,N-1, = 0,1, -1,
k=1
(3.1)

where N and N denote the numbers of data samples in range and cross-range, respec-

tively; ak and {(k, Wk}, respectively, denote the unknown complex amplitude and 2-D


18





19

frequencies of the kth sinusoid or point scatterer, k = 1, 2, K; {e(n, n)} denotes

the unknown noise sequence. Note that {wk, wk} corresponds to the 2-D location of

the kth scatterer and ak is determined by the RCS and range of the kth scatterer.

The NLS estimates {&k, k, Wk }kk= of {ok, Wk, Wk}k= can be obtained by minimizing

the following NLS criterion [39]:
K 2
C, ({ak,wk, kI}k ) = WN(Wk)& ) (3.2)
k=1 F
where (.)T denotes the transpose; I|| IF denotes the Frobenius norm; Y is an N x N

matrix with y(n, n) being its (nn)th element;

WN(k) = 1 ewk ... e(-1) ; (3.3)

r IT
g(wk) = 1 ejk ... eik(-1) (3.4)

RELAX minimizes the NLS cost function C1 in (3.2) by using a complete relaxation-

based search, i.e., for each K, the intermediate number of scatterers, only the param-

eters of one scatterer vary while all others are fixed at their most recently determined

values. Let
K
Y =Y- i N (i) i), (3.5)
i=l,ik
where {&i, -i, 'i}i=,iW k are assumed to have been estimated. Then {&k, Ck, &k} are

obtained by

Wk (wk)Yk w k=kI (3.6)
ak ~ NN Wk=WkWk=Wk

where (-)H and (.)*, respectively, denote the conjugate transpose and complex con-

jugate, and

{(Wk, k} = argmax wH(wk)YkW (0k)12, (3.7)
WkWk
which indicates that {Ck, wk} can be obtained as the location of the dominant peak

of the periodogram wH(wk)YkW*f (k) 2 /NN, which can be efficiently calculated

via FFT. Then, &k can be calculated from the complex amplitude of the peak of





20

wH(wk)YkW* (Jk) 2 /NN. The RELAX algorithm is summarized as follows:

Step (1): Assume K = 1. Obtain {fk, k, k k=1 from Y by using (3.7) and (3.6).

Step (2): Assume K = 2. Compute Y2 with (3.5) by using {fk,2 k,&k}k=1 ob-
tained in Step (1). Obtain {fk,Wk, &kk}k=2 from Y2. Next, compute Y1 by using

{kjk, lk, &k}k=2 and then redetermine {k, Wk, &k}k=1 from Y1.
Iterate the previous two substeps until "practical convergence" is achieved.

Step (3): Assume = 3. Compute Y3 by using {Ok, k, &k =1 obtained in Step

(2). Obtain {&k, k, k&k}k=3 from Y3. Next, compute Y1 by using {Ck, Wk, &k k=2 and

redetermine {jk, k, &k} k=1 from Y1. Then compute Y2 by using {Ck, Wk, kjck=1,3
and redetermine {Wkc, k, k, }k=2 from Y2.

Iterate the previous three substeps until "practical convergence".

Remaining Steps: Continue similarly until K is equal to K, which can be deter-

mined by using generalized Akaike information criterion (GAIC) to be discussed later
on.

The "practical convergence" in the iterations may be determined by checking

the relative change e of the cost function C1 ({fk, k, &k}=l) in (3.2) between two
consecutive iterations.

It has been demonstrated in [39] that RELAX is related to CLEAN [28], which
is only an approximate relaxation-based method that minimizes the NLS criterion.

RELAX turns out to be CLEAN when no iteration is performed in each step of

RELAX. Due to the iterations performed in each step of RELAX, a more accurate

initial can be obtained, which makes RELAX converge to the global minimum with a
high probability. Meanwhile, RELAX is asymptotically (for large number of samples)

efficient, while CLEAN gives biased estimates, especially when two scatterers are

closely spaced. The RELAX algorithm, thereby, can also be referred to as SUPER

CLEAN.






21


We consider using GAIC (see [39] and the references therein for details) to

determine K, the number of sinusoids, by assuming the unknown noise and clutter

being white. The estimate K of K can be determined as an integer that minimizes

the following GAIC cost function:

(N-1N-1
GAICk = NNln 6 |(n, 2)|2 + y7n[ln(NN)](4k + 1), (3.8)
n=o n=0o
where
k
((n,n)=y(n,))- &kej3(kn+ k'n), n=0,1,--,N -1, n=0,1,.--,-1,
k=1
(3.9)

4k + 1 denotes the total number of unknown real-valued parameters (of which 4K

are for the sinusoids and 1 is for the white noise), and y is a parameter of user choice.

3.1.2 Image Formation

We can form SAR images by using the point scatterer data model and the

target features extracted via RELAX. SAR images obtained via RELAX are referred

to as RELAX SAR images or simply RELAX images. The steps of using RELAX

for SAR image formation are outlined as follows.

Step 1: Obtain the parameter estimates of {ok, Wk, kk K= via RELAX by using the

measured phase history data (see Section 3.1.1 for details).

Step 2: Generate ys(ns, ns), the simulated phase history data of large dimensions,

from the estimated parameters and based on the data model in (3.1):

K
ys(n7, n,) = eakej((kn.+ks8), n, = 0,1, .. N 1, ns = 0,1, .. CN 1,(3.10)
k=1

where ( (( > 1) denotes the extrapolation factor, which is a parameter of user

choice, and &k, jk, and &k, respectively, denote the estimates of 0k, Wk, and ak,

k = 1,2,---,K.

Step 3: Form RELAX SAR images containing only the dominant target features by





22


applying the normalized FFT to the simulated phase history data {y,(ns, s,)}, i.e.,

by computing
S(N-1 (N-1
C2NNE E ys(nsns)e-Ei(w9+^8). (3.11)
n,=O fi,=0
To suppress the sidelobes, the normalized FFT can be applied to the windowed

sequence {w( (ns, ft,)y,(ns, is)}, where the window sequence w,(n,, ns) satisfies

(N-1 (N-1
E W,(n.,Iu) = (2NN. (3.12)
n,=O is=0

The RELAX SAR images containing both the dominant target features and back-

ground clutter may be appreciated since, for example, the shadow information may

be desired for automatic target recognition. If so, the normalized FFT can be applied

to the sum of the simulated phase history data {ys(n,, i,)} and {(26(n, u)} with zero

padding to have dimensions (N and (N, where 6(n, n) denotes the estimated back-

ground clutter and is determined by (3.9) with K replaced by K. Note that scaling

the e(n, n) by a factor of (2 is needed when the background clutter is included in

the RELAX SAR images since both of its dimensions are 1/( times of those of the

simulated phase history data. Similarly, the normalized FFT can also be applied

to the sum of {ws(ns, s)ys(nris)} and {(C2W(n,, )6(n, )} with zero padding to

have dimensions (N and (N to suppress the sidelobes, where the window sequence

we(n, n) satisfies
N-1 N-1
S we (n, n) = NN. (3.13)
n=0 n=0
Note that since we cannot model the clutter effectively, its resolution cannot be

improved. We remark that the resolution of the RELAX SAR images is determine

by RELAX and ( > 1 is just used to demonstrate the super resolution property of

the RELAX algorithm for target feature extraction.





23

3.2 RELAX-NLS

When targets consist of both trihedral (point scatterers) and dihedral corner

reflectors, we can form enhanced SAR images by using the RELAX-NLS algorithm

[42]. We first give a brief review of the RELAX-NLS algorithm and then discuss SAR

image formation via RELAX-NLS.

3.2.1 Review of the RELAX-NLS Algorithm

RELAX-NLS [42] is devised to identify and obtain target features of multiple

corner reflectors (trihedral and dihedral) based on a mixed parametric data model.

An ideal dihedral corner reflector acts as a double-bounce reflector and the RCS

achieves the maximum when the radar beam is perpendicular to the dihedral and

falls off as a function of sinc(0) = sin(0)/9 when the angle between the radar beam

and the line perpendicular to the dihedral increases [49, 57]. Then dihedrals can be

modeled as
Kd
Sd(n, n) = adsinc[rbk(n- Tk)]ej( dk n+dk), n = 0, 1, -. N-1, n = 0, 1, N-,
k=1
(3.14)

where N and N denote the numbers of the available data samples in range and

cross-range, respectively; adk, {Wdki, dk}, and bk, k = 1, 2, - Kd, are, respectively,

proportional to the maximal RCS, the central location, and the length of the kth

dihedral corner reflector; Tk, k = 1, 2,. Kd, denotes the peak location of the data

sequence in cross-range and is determined by the orientation of the kth dihedral

corner reflector; finally, Kd is the number of the dihedral corners. Assume that a

target consists of Kd dihedrals and Kt trihedrals and K = Kd + Kt. Then the target

data model in the presence of noise and clutter has the form:


y(n, n) = sd(n, ) + st(n, n) + e(n, n), n = 0,1, N 1, = 0, 1, V 1,
(3.15)





24

where
Kt
St(n, i) = Zatej(tke +t, n =O,1, ,N -1, = 0,1, *, N -1, (3.16)
k=1

with {atk}k= and {Wtktk}=,K respectively, denoting the unknown complex am-

plitudes and 2-D unknown frequencies of the Kt trihedrals. The RELAX-NLS al-

gorithm proposed in [42] can be used to identify and effectively estimate the tar-

get features{adk, bk, Wdk, Wd, k 7k=l} and {a, tk, tk 1k by minimizing the following
NLS criterion:


C2 ({atkWtk, tk }k=l, dk, bk, Wdk) WdkTk}i) = ||Y-St -Sd (3.17)

where [Y]j = y(i,j), [St]ij = st(i,j), and [Sd]ij = Sd(i,j). RELAX-NLS converts
the problem of multiple corner reflector identification and feature extraction to that
of a sequence of single reflector identification and feature extraction by using the

relaxation based optimization approach. The identification and feature extraction of

a single corner reflector in RELAX-NLS is described as follows:
Feature Extraction of a Single Trihedral Corner Reflector
Let Yt denote the data of a single trihedral in the presence of unknown noise
Et. Then

Yt = atWN(Wt)W (Wt) + Et, (3.18)

where at and {wt, et}, respectively, are proportional to the RCS and 2-D location
of the trihedral. The NLS estimates {&t, t, tt} of {at, wt, @t} can be obtained by

minimizing the following cost function

Ct (at, WtCt) = IYt atwNy(Wt)WN^(wt)I F, (3.19)

and {&t, LCt, wt}, respectively, are given by

= W *(wt)Ytw ')
N l7 (3.20)
NN





25

and

{LJt, t} = argmax w(w) (3.21)

Thus {fLt, w't} corresponds to the peak location of IwH(wt)Ytw* (act)|2, which can be
calculated efficiently via 2-D FFT, and &t corresponds to the complex height of the
normalized 2-D FFT of Yt at {Pt, Wt}.

Feature Extraction of a Single Dihedral Corner Reflector
Let Yd denote the data of a single dihedral in the presence of unknown noise
Ed. Then
Yd = adWN(d)gT + Ed, (3.22)

where {Wd, Wd ad, b, 7} are the target features of a dihedral corner reflector, and


== g(o) g(1) .. (N -1) (3.23)

with g(n) = sinc[b7r(r n)]ej"a1, n = 0,1, -,N 1. The NLS estimates

{ad, dd, bW, ,} of {ad, Wd, C,b, T} are determined by minimizing the following cost
function:

Cd (a, Wd, Wd, db, r) = YYd adWN(Wd)gT (3.24)

It can be shown that &d is determined by

w( = wYdg* (3.25)

and {Wd, d, b, } can be obtained by maximizing the cost function

C3 (Wd, Wd, b, T) = Iw(wN)Ydg* 2 (3.26)

RELAX-NLS maximizes C3 in (3.26) by utilizing an alternating maximization ap-
proach [8], i.e., alternatively updating {Od, wd, 6, r} while fixing other parameter esti-
mates at their recently determined values. Note that four-dimensional search over the
parameter space is performed to estimate the dihedral corner reflector parameters.





26

Identification of Corner Reflector
It has been shown in [42] that Cd is much less than Ct when the true corner
reflector is a dihedral and Ct is almost the same as Cd when the true corner reflector
is a trihedral. Also b 26f when a trihedral is mistakenly considered as a dihedral,
where 5f is the smallest positive solution to

sin(Nflr) N
-- (3.27)
sin(6f7r) 2

Then a corner reflector can be identified by checking the estimate b and the cost
functions Ct in (3.19) (with {at, t,&t} replaced by {a&tt, ,t}) and Cd in (3.24)
(with {Wd, Ow, ad, b, r} replaced by {jd, Wd, ad, b, r}. A corner reflector is identified as
a dihedral if C- > 0.1 and b > 26.
Cd >
Steps of RELAX-NLS are outlined as follows:
Step (1): Assume K = 1. Identify and obtain the parameters of a single
corner reflector.
Step (2): Assume K = 2. Subtract the signal obtained in Step (1) from
{y(n, )} and then identify and obtain parameters of the second corner reflector.
Subtract the second signal from {y(n, n)} and re-identify and redetermine the pa-
rameters of the first signal.
Iterate the previous two substeps until "practical convergence".
Step (3): Assume K = 3. Subtract the two signals obtained in Step (2) from

{y(n, n)} and then identify and obtain the parameters of the third signal. Subtract
the second and third signals from {y(n, n)} and re-identify and redetermine the first
signal. Subtract the first signal (most recently determined) and the third signal from
{y(n, A)} and then re-identify and redetermine the second signal.
Iterate the previous three substeps until "practical convergence".
Remaining Steps: Continue similarly until K = K (K can be determined
by using GAIC to be discussed later on).






27


The "practical convergence" in the iterations can also be determined by check-

ing the relative change of the cost function C2 in (3.17) between two consecutive

iterations.

We can also determine K, the total number of trihedrals and dihedrals, by

extending the GAIC discussed in Section 3.1 and assuming white noise and clutter.

The estimate K of K can be determined as an integer that minimizes the following

extended GAIC cost function:
(N-1N-1
GAICk = NNlnn (n, )|2 + 7ln[ln(NN)](4t + 6K + 1), (3.28)
n=0 n=0
where
kd kt
e(n, n) = y(n, f) S & dsinc[7b(k rtk)]ejk+dk) ckej), (3.29)
k=1 k=1
with n = 0, 1, l N 1 and n = 0, 1, N 1; 7, as before, is a parameter of user

choice; K = Kt + d with kt and Kd denoting the numbers of trihedral and dihedral

corners, respectively, determined by RELAX-NLS given K; finally, 4Kt + 6Kd + 1

is the total number of unknown real-valued parameters (of which 4Kt and 6Kd,

respectively, are for the trihedral and dihedral corners and 1 is for the white noise).

3.2.2 Image Formation

We can form enhanced SAR images of targets consisting of both trihedrals

and dihedrals by using the extracted target features via RELAX-NLS and the mixed

data model in (3.15). The SAR images obtained via RELAX-NLS are referred to

as RELAX-NLS SAR images or simply RELAX-NLS images. To form RELAX-NLS

images, the dominant target features are first extracted by using the RELAX-NLS

algorithm (see Section 3.2.1 for details) and then Steps 2 and 3 discussed in Section

3.1.2 are used except that the simulated phase history data of large dimensions used

in Step 2 is now determined by
Kd Kt
y,(ns, ,) = C & cdkSinc[7rbk (s k)]ej(2dk dkf--)t + E &tk e(atk + n), (3.30)
k=1 k=l





28


where n, = 0, 1, - -, (N 1 and i, = 0, 1, .. CN 1 with ( being an extrapolation

factor (C > 1); &dk, dk), Wdk, bk, and 7k, respectively, denote the estimates of adk,

Wdk, 7dk, bk, and Tk, k = 1, 2, *, Kd; finally, &t,, Wtk, and 2 t, respectively, denote
the estimates of atk, wik, and wtk, k = 1,2, *. Kt. The clutter estimate 6(n, n)

used in Step 3 is now determined by (3.29) with kt and kd replaced by Kt and Kd,

respectively. Note that scaling the simulated data for the dihedrals by a factor of C

in (3.30) is needed since the sinc function goes to zero as n, increases or decreases

away from Tk. Note also that C is a parameter of user choice and ( > 1 is just

used to demonstrate the super resolution property of RELAX-NLS. Finally, since the

Fourier transforms of the sinc functions of sufficient lengths do not result in sidelobes,

when ys(ns, n,) in (3.30) is windowed, the second term of y,(ns, n,) is multiplied by

wz(ns, in) and the first term is multiplied by wi(n,), where wl(n,) is a 1-D window

sequence satisfying
(N-1
E w(n,) = (N. (3.31)
n,=o

3.3 Experimental Results

In this section, we demonstrate the image formation performances of RELAX

and RELAX-NLS with the experimental data. In the following examples, the extrap-

olation factor C = 2 is used and GAIC with y = 4 and y = 18 are used to determine

K for RELAX and RELAX-NLS, respectively. Kaiser windows with shape parameter

/ = 6 are used whenever needed.

First consider SAR image formation via 2-D FFT and RELAX by using a

portion of the 2-D data corresponding to some roof rims collected by one of the two

apertures of the ERIM's (Environment Research Institute of Michigan's) DCS inter-

ferometric synthetic aperture radar (IFSAR). Figures 3.1(a) and (b), respectively,

show the unwindowed and windowed 2-D FFT images obtained by zero-padding the

40 x 40 phase history data. The unwindowed and windowed RELAX images with and





29

without the background clutter are illustrated in Figures 3.1 (c) ~ (f) with K = 59.

Note that the RELAX images have a higher resolution than the FFT images for the

dominant scatterers. Utilizing only 25% of the 40 x 40 phase history data, i.e., using

a 20 x 20 phase history data, we form the 2-D FFT SAR images shown in Figures

3.2(a) and (b). Figures 3.2(c) ~ (f) show the RELAX images with K = 41. Compar-

ing Figures 3.2(f) and 3.2(b), we note that the two images are quite similar although

the former uses only 25% of the data used by the latter.

Consider next an example of the MSTAR Slicy data consisting of both trihe-

dral and dihedral corner reflectors collected by the SNL using the STARLOS sensor.

The field data was collected by a spotlight-mode SAR with a carrier frequency 9.559

GHz and bandwidth 0.591 GHz. The radar was about 5 kilometers away from the

ground target shown in Figure 3.3. The SAR images are obtained when the tar-

get is illuminated by the radar from the azimuth angle 0 and elevation angle 30.

The range and cross-range resolutions of the original data (54 x 54) are 0.3 meters

and 0.32 meters, respectively, and the windowed 2-D FFT SAR image (obtained

via Taylor window) is shown in Figure 3.4(a). To demonstrate the super resolution

property of RELAX and RELAX-NLS, we use part of the original data, whose di-

mension is 32 x 32. Figure 3.4(b) shows the windowed 2-D FFT images with the

spoiled resolution 0.51 x 0.54 meters obtained from this reduced size data. RELAX

and RELAX-NLS are also applied to this 32 x 32 phase history data matrix. Figures

3.4(c) and (d), respectively, show the windowed RELAX SAR images with and with-

out estimated background clutter with K = 36. We note that RELAX images have a

better resolution than the FFT images for the dominant trihedrals and resemble the

FFT images for dihedrals even though the data model in the cross-range dimension

used by RELAX is not correct for this example. The windowed RELAX-NLS SAR

images with and without estimated background clutter with K = 9 are illustrated

in Figures 3.4(e) and (f), respectively. We note again that the RELAX-NLS images






30


not only have a better resolution than the 2-D FFT images, but also more closely

reflects the characteristics of the photo shown in Figure 3.3.






31





i . .










(a) (b)






410








(c) (d)















(e) (f)
Figure 3.1: RELAX SAR Images by using the 40 x 40 ERIM data with = 59.
(a) Unwindowed 2-D FFT image. (b) Windowed 2-D FFT image. (c) Unwindowed
RELAX image without background clutter. (d) Windowed RELAX image without
background clutter. (e) Unwindowed RELAX image with background clutter. (f)
Windowed RELAX image with background clutter.





32















(a) (b)














(c) (d)














(e) (f)
Figure 3.2: RELAX Images by using the 20 x 20 ERIM data with K = 41. (a)
Unwindowed 2-D FFT image. (b) Windowed 2-D FFT image. (c) Unwindowed
RELAX image without background clutter. (d) Windowed RELAX image without
background clutter. (e) Unwindowed RELAX image with background clutter. (f)
Windowed RELAX image with background clutter.






33




































Figure 3.3: Target photo taken at 45 azimuth angle.






34















(a) (b)













(c) (d)







-* 40 A*






(e) (f)
Figure 3.4: SAR images of the Slicy data (0 azimuth and 300 elevation angles).
(a) Original windowed 2-D FFT image. (b) Windoed 2-D FFT image with spoiled
resolution. (c) ~ (d): Windowed RELAX image without and with background clutter
with K = 36. (e) ~ (f): Windowed RELAX-NLS image without and with background
clutter with K = 9. (c) ~ (f) are obatined using the data in (b). (The vertical and
horizontal axes are for range and cross-range, respectively.)














CHAPTER 4
SEMI-PARAMETRIC METHODS FOR SAR IMAGING

4.1 Problem Formulation and Data Model

Compared with parametric spectral estimation methods, non-parametric spec-

tral estimation methods are robust, but cannot be used to significantly improve the

resolution of the formed SAR images. As can be seen from the previous chapter that

relaxation-based parametric methods can be used to generate super resolution SAR

images when the data model for targets of interest is appropriate. However, the super

resolution property of parametric methods will be compromised when the appropriate

data model is not available. In this chapter, we consider the case where it is difficult

to establish a good parametric data model in cross-range for target scatterers due

to a variety of physical phenomena including glints, resonances, and motion-induced

phase errors, etc. Instead of modeling the target of interest as 2-D sinusoid (point

scatterer) in both range and cross-range, we apply herein a semi-parametric model

and simply assume that the data model in cross-range due to one or more corner re-

flectors is a complex sinusoid with an arbitrary amplitude and a constant phase. Due

to the flexibility of this data model, feature extraction and image formation meth-

ods based on this data model should be more robust compared with those based on

the approximated dihedral and trihedral data models in Chaper 3. The robustness,

however, comes at the cost of small resolution improvement in cross-range over the

conventional fast Fourier transform methods due to assuming little in cross-range. To

preserve the robustness, we use a refined non-parametric spectral estimation method

such as APES (Amplitude and Phase Estimation of a Sinusoid) algorithm [38] to

improve the cross-range resolution.


35





36

The received signal reflected from a target scatterer herein is modeled as:

s(n,h)= x()ejOej2(fnf), n= 0,1,-,N 1, n = 0,1, ,N 1, (4.1)

where N and N denote the dimensions of the available data samples in range and

cross-range, respectively; x(n) is an arbitrary unknown real-valued function of h

determined by the radar cross section (RCS) of the scatterer; 0 is a constant phase;

finally, {f, f} is the frequency pair proportional to the 2-D location (range and cross-

range) of the scatterer. This data model is essentially semi-parametric since little

parameterization is done in cross-range.

Assume that a target consists of K scatterers. Then the target data model in

the presence of noise has the form:
K
y(n, ) = Z xk()e k e3j2n(fkn+kf) +e(n, n), n = 0,1, - N-1, n = 0,1, N-1,
k=1
(4.2)
where {xk ( 0)}1- denotes the real-valued amplitude function of n for the kth scat-

terer; Ok and {fk, fk}, respectively, are the constant phase and the frequency pair

of the kth scatterer; finally, {e(n, )} denotes the unknown 2-D noise and clutter

sequence.
Since SAR images are often used in SAR applications, our problem of interest
herein is to estimate the target parameters {(k, Xk( ( k, fk= from the 2-

D data sequence {y(n, n)} and then to form high resolution SAR images with the

estimated target parameters. Before presenting the effects of the semi-parametric

data model in (4.2) on SAR image formation and the SPAR (Semi-PARametric)

algorithm for feature extraction and image formation, we first briefly introduce the

APES algorithm.





37

4.2 Review of the APES Algorithm

The APES [38] algorithm belongs to the adaptive filter-bank class, in which
the adaptive finite impulse response (FIR) filter is designed based on not only the data
to be processed, but also the interference at the frequency of interest. APES mimics
the maximum likelihood estimator of a complex sinusoid in circularly symmetric
zero-mean Gaussian noise. It has been shown in [37] that APES has better statistical
performance than Capon [10], another popularly used adaptive filter-bank approach
for SAR imagery.
Let {zn}[=0 denote the available 1-D data sequence and have the form:

Z = a(w)exp{jnw} +eN(w), n= 0,1,. N 1, (4.3)

where a(w) denotes the complex amplitude of a sinusoid with frequence w and en(w)
denotes the unmodeled noise and interference at w. Then APES can be used to obtain
the complex amplitude estimate &(w) of a(w) from {zn}o-1. It has been shown in
[38] that the least-squares estimate of a(w) using the forward and backward FIR
filter output has the form:

&() = [hH(w)Z(w) + /(w)ZH(w)h(w)], (4.4)

where h(w) denotes the impulse response of an M-tap FIR filter and has the following
form:
h(w)= hi(w) h2(w) .. hM(w) ; (4.5)

Z(w) and Z(w), respectively, denote the normalized Fourier transforms of {Zi} M
and {zi} i.e., the Fourier transforms of {zi}N- and {i,}m divided by (N-
M + 1), where zi and i, respectively, denote the overlapping subvectors of the data
vector z = [ zO z ZN-1 ] and the complex conjugated and backward-ordered

data sequence z = z*,_- Z_2 ... z* and have the form

Zi= zi z i+ Zi+M-1 i=0,1,---,N-M, (4.6)





38

and

ii= z*,_ z,_i2 iM i = 0,1, N M, (4.7)
[ ZN-i-1 ZN-i-2i
respectively; finally, /(w) = exp{-j(N 1)w}. It has been shown in Appendix C
of [38] that by using a certain conjugate symmetric condition, &(w) in (4.4) can be
reduced to
&(w) = hH(w)Z(w). (4.8)

The adaptive FIR filter hAPES(W) corresponding to APES is derived in Ap-
pendix A of [38] and has the form:

Q- (w)a(w)
hAPES() = a( ( (4.9)
aH wQ-19a^)
where the steering vector a(w) is given by

rT
a(w) ei ... ei(M-1) (4.10)

and Q(w) denotes the estimate of the covariance matrix of the noise and is given by:

Q(w) = R Z(w)ZH(w), (4.11)

with
Z(w) = [ Z(w) Z(w)] (4.12)

and R = (R + R)/2 denoting the average of the forward and backward sample
covariance matrices R and R. The forward sample covariance matrix R is obtained
from zi as follow:
N-M
R = N-M 1 (4.13)
i=O
and the backward sample covariance matrix R can be obtained similarly from zi.


4.3 Data Model Ambiguities and Their Effects on SAR Image Formation

Target feature extraction methods devised based on the data model in (4.2)
are robust against data mismodeling errors due to the model flexibility. However,





39

there are ambiguity problems associated with the semi-parametric data model. In
this section, we first analyze the possible data model ambiguities and then illustrate
their effects on SAR target feature extraction and image formation. The discussions
below will motivate the introduction of the SPAR algorithm, which will be presented
in detail in Chapter 4.4.

4.3.1 Model Ambiguities

Due to the flexibility of the data model in (4.2), there are various types of
ambiguities that may impact the feature extraction of each scatterer. Below we list
several types of the ambiguities inherent in the data model.
Type 1: Single scatterer
From (4.1), we note that ambiguity exists between and x(n) since

X(n)eo = -x(f)ej(O+") ()e (4.14)


where r(n) = -x(n) and 4 = + 7r. Ambiguity also exists between f and x(i) since

(n)e = (-1)x(i)ej2"(f-05)" )e2, (4.15)

where 1(n) = (-1)"z(n) and f = f-0.5. The above two types of ambiguities cannot
be resolved.
Type 2: Two identical scatterers located in the same range
Let fa and fb, respectively, denote the cross-range locations of two identical scatterers
and let 0a and Ob, respectively, denote their phases. Then


x(n) [e(ji+i2- + ei+i n)J 2x(fi) cos [r(h fb)f + 2 ] ej 2
e 7(ao+)^, (4.16)

which indicates that two identical corner reflectors (trihedrals or dihedrals) located
in the same range but different cross-range positions fa and fb, respectively, can be
modeled by (4.1) as one "scatterer" located at (fa + fb)/2 in cross-range with x(n)






40


modulated by 2 cos[7r(fa fb)n + (0a b)/2]. Thus the data model in (4.1) cannot

be used to describe each of the two corner reflectors in this case.

Type 3: Two different scatterers located in the same range

Assume that two different corner reflectors with parameters {4i, {xi()} f, fi }i=

are located in the same range. Then the target model in the absence of noise has the

form:
2
r(n,n)= Exi()ej'ej2 (fn++hn), n= 0,1,-,N -1, = 0,1, ,N 1.
i=1
(4.17)

With straightforward calculations, we can rewrite (4.17) as
2
r(n, n) = Zi (A)eje2 ej2i"fifn ), (4.18)
i=1

where 2 = 1 + 7i/2 with 1 denoting an arbitrary phase, fi = f2 f f, = f2 = f

with f denoting an arbitrary cross-range location,
2
x,(n) = E i() cos[27r(f- f)n + ( 1)], (4.19)
i=1
and
2
x2(n) = Xi(n) sin[2zr(f f)n + (Oi 01)]. (4.20)

Note from (4.18) that { i, {i()}N 'fi, fi =1 are the ambiguous features of

{<, {xi()}=o J, Ji=l
Type 4: Multiple scatterers located in the same range

When more than two scatterers are located in the same range, the data model in the

absence of noise can still be written as (4.18) except that
L
Zi(") = xi(A) cos[27r(fi f) + (q 1)], (4.21)
i=1
and
L
C2(l) = i(n) sin[2r(f f)[ + ( <1)], (4.22)
i=1





41

where L > 2 denotes the number of scatterers located in the same range. Thus the

L scatterers located in the same range are considered as two "scatterers" when using

the data model in (4.2).

Before we discuss the impact of the model ambiguities on SAR image for-

mation, we first describe how the image formation is done if we have the estimated

model parameters.

4.3.2 Image Formation

Assume for now that we have extracted the target features based on any of

the ambiguous data models. For notational convenience, we will use the notation

used in (4.2). Since the target data model in range is a sum of several complex

sinusoids with constant amplitudes and phases, we can use the estimated sinusoidal

parameters to simulate a data matrix with a larger dimension in range and then use

FFT to demonstrate the super resolution property of the feature extraction algorithm

we shall present. Yet we cannot extrapolate the estimate {xk(n)} of {xk(n)} since it is

assumed to be an arbitrary unknown real-valued function of n and hence FFT cannot

be used to obtain SAR images with enhanced resolution in cross-range. Instead, we

use 1-D APES [37, 38, 41] in cross-range when forming SAR images since APES

belongs to the class of matched filterbank spectral estimators and provides lower

sidelobes, narrower spectral peaks, and more accurate spectral estimates than FFT.

Let {s, (n, )} denote the simulated data sequence with a larger dimension in

range based on the estimated target features {k, {-k(n)} =0, fk, fk}= Of

{(k, {xk(n)} 0 fk, k= where K denotes the estimate of the scatterer number
K. Then
k
s9(n, F) = 'k()ejke j2"(fk +kn), = 0,1,. CN -1, = 0,1, .., N 1,
k=1
(4.23)





42

where ( denotes an extrapolation factor (( > 1) and is a parameter of user choice.

Note that the super resolution property of the so-formed SAR images is determined

by the feature extraction algorithm and C 2 1 is only used to demonstrate the super

resolution property of the target feature extraction algorithm. The estimated noise

and clutter data matrix is


(n, ) =y(n, )- ,(n, ), n = 0, 1, N 1, n = 0,1, N 1, (4.24)

which is also important in many SAR applications since, for example, important

target information such as the target shadow information is contained in 6(n, h). We

cannot extrapolate 6(n, n) in either range or cross-range since no parametric data

model is available for ((n, i).

To obtain SAR images with low sidelobes in range via 1-D FFT, we apply 1-D

windows to ,s(n, A) and &(n, n) in range. We obtain a new (CN) x N data matrix Y

as follows:

y(n, n) = s,(n, n)ws(n) + (e(n, n)we(n), n = 0, 1, ., N 1, n = 0,1, 1 ., V 1,

y(n,f) = s,(n,f)w,(n), n = N,N + 1, ,CN 1, n = 0,1, ,N 1,
(4.25)

where y(n, n) denotes the (n, n)th element of Y and w,(n) and we(n) are 1-D windows

of lengths (N and N, respectively, satisfying

(N-1
E w,(n) = CN, (4.26)
n=O
and
N-1
we,(n) = N. (4.27)
n=0
Note that scaling 6(n, A) in y(n, n) by a factor of ( is necessary since the range

dimension of ,s(n, n) is ( times of that of 6(n, n). The steps needed for SAR image

formation are as follows:

Step (1): Form Y from y(n,n) by using (4.23), (4.24), and (4.25).





43

Step (2): Apply the normalized 1-D FFT to each column of Y to obtain

an intermediate matrix and then apply 1-D APES to each row of the intermediate

matrix. (See [41] for the efficient implementation of APES.) Note that the normalized

1-D FFT has the form
1N-1
S y(n, )e-j2"fn, = 0,1,- ,N 1. (4.28)
n=O

4.3.3 Model Ambiguity Effects on SAR Image Formation

All of the aforementioned types of ambiguities will have no effect on the SAR

image formation if no parameter estimation errors exist since the scatterers will be

perfectly reconstructed by using any of the possible ambiguous data models. For

example, when there are two identical scatterers located in the same range, the data

model in (4.2) can still be used for SAR image formation since the two scatterers are

now described as one "scatterer" with (4.16), which still fits the data model of (4.2)

with K = 1. Hence the original SAR image can still be reconstructed by using the

parameters of the one "scatterer" described by the right side of (4.16).

In the presence of parameter estimation errors due to the presence of noise

and clutter, however, Types 1 and 2 ambiguities discussed in Section 3.1 will have

little effect on SAR image formation, whereas Types 3 and 4 ambiguities can result in

artifact problems for the high resolution SAR image formation. Generally speaking,

the higher the SNR, the more accurate the parameter estimates and hence the less

significant the artifact problem. The effect of Type 3 ambiguity on SAR image

formation in the presence of estimation errors is illustrated in Figures 4.1 and 4.2.

(The effects are similar for Type 4 ambiguity.) Figure 4.1 is obtained by assuming

no parameter estimation errors. Figure 4.1(a) shows the FFT image of a target

consisting of two dihedrals of different lengths located in the same range. We use


Xk(n) = aksinc[bk7r(in Tk)], k = 1,2, n = 0,1, ... 31, (4.29)





44


k ak k fk fk bk Tk

k=l 9.6 0 0.1 -0.3 0.3 18.6

k=2 6.4 0 0.1 0.1 0.2 18.6


Table 4.1: True parameters of the two dihedrals used in Figures 1 and 2.

to simulate the dihedrals, where ak and bk, respectively, are proportional to the

maximal RCS and the length of the kth dihedral corner reflector and Tk denotes the

peak location of the data sequence and is determined by the orientation of the kth

dihedral. The size of the simulated data matrix is 32 x 32 (i.e., N = N = 32).

The parameters for the two dihedrals are given in Table 1. An ambiguous set of

target features can be obtained by choosing fi = f2 = 0.1, fl = f2 = f = 0.2,

01 = 0, and 02 = 7r/2 in (4.19) and (4.20). The windowed FFT SAR images of
the two "scatterers" are shown in Figures 4.1(b) and (c), respectively, which differ

considerably from the two dihedral scatterers in Figure 4.1(a). The combined SAR

image of the two "scatterers" is given in Figure 4.1(d), which is exactly the same as

the true image shown in Figure 4.1(a). However, due to the presence of noise and

clutter, parameter estimation errors are inevitable. The errors in range are the main

cause of the artifact problem in the high resolution SAR image formation. In Figure

4.2, we assume that all parameters are accurate except that f2 = f2 + 0.01 = 0.11.

Figures 4.2(a) and (b), respectively, show the windowed FFT images of the two

aforementioned "scatterers" in the presence of estimation errors and Figure 4.2(c)

shows the combined SAR image. By comparing Figures 4.1(a) and 4.2(c), we note

that an extra line (artifact) shows up next to the short dihedral. The reason is that

due to the estimation errors, f, 5 f2. Hence the two "scatterers" in Figures 4.2(a)

and (b) are not exactly in the same range and cannot be "combined" perfectly to

obtain the two dihedral lines in Figure 4.1(a). This problem becomes even worse

when super resolution SAR images are formed via data extrapolation in range. The





45


larger the extrapolation factor (, the more significant the artifact problem since the

difference between fl and f2 is exaggerated C times. Figures 4.2(d) shows the SAR

image obtained with ( = 2. By comparing Figures 4.2(d) and (c) (here ( = 1,

without extrapolation), we note that the artifact next to the short dihedral becomes

more significant. Severe artifacts may exist at low SNR since the accuracy of the

parameter estimates is poor. The SPAR algorithm we present below attempts to

avoid this problem by using windows to isolate the multiple scatterers located in the

same range.


4.4 The SPAR Algorithm

SPAR can be summarized by the following two steps:

Step 1: Scatterer Isolation based Target Feature Extraction: See Chapter

4.4.1 below for details.

Step 2: SAR Image Formation: See Chapter 4.3.2 for details, where the

estimated target features are obtained by using Step 1.

4.4.1 Target Feature Extraction

The basic idea behind SPAR is to extract the features of each scatterer sepa-

rately. Before we present the target feature extraction algorithm, we first summarize

the steps needed for the feature extraction of a single scatterer as a preparation. The

generalized Akaike information criterion (GAIC) is also introduced to estimate K,

the number of scatterers, at the end of this subsection.

Feature Extraction of a Single Scatterer

The data model of a single scatterer in the presence of noise has the form:


ys(nr ) = s(n,) + e,(n,n), n = 0,1,. *, N 1, n = 0,1, * -, N 1, (4.30)





46

where s(n, ) is given in (4.1) and {es(n, )} denotes the unknown 2-D noise and
clutter sequence. Let

WN(f)= 1 ej2rf ... e2if(N-1) (4.31)

and
(f) = 1 ej2l ... ej2 (R-1) (4.32)

where (.)T denotes the transpose. Let D(f) denote the following diagonal matrix:

D(f) = diag 1, ej2f, ..., ej2r(N-) (4.33)

Define
x = [(0) x(1) .. x(SN -1) (4.34)

Let Y, be an N x N matrix with its (n, n)th element being y,(n, n). Then we can
rewrite (4.30) as:
Y, = ejOG(x, f, f) + E,, (4.35)

where
G(x, f, f) = WN(f)xD(f), (4.36)

and E, denotes an N x N matrix with e,(n, f) being its (n, n)th element. Let y,,
S= 0, 1, N 1, denote the nth column of Y, and define

,(f)= Y w(f), (4.37)

where (.)* denotes the complex conjugate. Then the NLS estimates f{, If, ff of

{x, 4, f, /} are (see Appendix A for the detailed derivations):

S= tRe [e-jy,(f) 0 wL(f)], (4.38)

where Re(x) denotes the real part of x and 0 denotes the Hadamard matrix product
or the element-wise product of two matrices;

= arg [y *(f)]2 -j2(21) (4.39)
I=0





47

where arg(x) denotes the argument of a complex variable x; finally,

{f, f} = argmax C4(f, f), (4.40)

where
N-1 N-1
C4(f,f) = y yw(f)2 N [] e-j2(2f)" (4.41)
ft=0 1=0
The steps needed to obtain the NLS estimates of a single scatterer are summarized
as follows:
Step (I): Use (4.37) to obtain y(f) and obtain the cost function C4(f, f)
according to (4.41). Determine {f, f} by maximizing C4(f, f) using the method given
in Appendix A.
Step (II): Calculate according to (4.39) with {f, } replaced by {f, }
obtained in Step (I).
Step (III): Calculate x via (4.38) with {(, f,f} replaced by ({, f,f} deter-
mined in Steps (I) and (II), respectively.

Feature Extraction of Multiple Scatterers

When a target consists of multiple scatterers, we can obtain the NLS esti-
mates of the target features based on (4.2) by using a relaxation-based optimization
approach. Let

xk= xk(0) xk(1) .,k (N-1) (4.42)

and let Y and E be N x N matrices with their (n, i)th elements being y(n, n) and
e(n, a), respectively. Then we can rewrite (4.2) as
K
Y = E ei Gk(xk, fk, fk) + E, (4.43)
k=1
where Gk(Xk, fk, fk) has the same form as the G(x, f, f) in (4.36) except that x, f,
and f are replaced by xk, fk, and fk, respectively. Let yn, f = 0, 1, N 1, be the





48

Ath column of Y. Then the estimates {Ik, Ak, f, k}j=1 of {k, Xkk, fk k=1 can be
obtained by minimizing the following NLS cost function:
K 2
C5 ({k, xk, Ak, fk}k=) = Y- e3 Gk(xk, fk,k) (4.44)
k=1 F
where 1| pIF denotes the Frobenius norm [22]. The minimization of

C5 ({, Xkf, fk, f}=) in (4.44) is a very complicated optimization problem. The
proposed SPAR algorithm performs a complete relaxation-based search by letting
only the parameters of one scatterer vary and freezing all others at their most re-
cently determined values for each assumed number of scatterers K. Let
K
Yk = Y- e Gi(i, fi, f), (4.45)
i=l,i k

and assume that {i, ^i, fi, i}l,1,i k are given. Then the NLS estimates

{k, Xfk, fkk=1 f {k, Xk, k, fk}k= can be obtained by minimizing
C6 (k, Xk,fk, fk), where

C6(k,Xk, ffk) = Yk ejkk (Xk, fk, k) F, (4.46)

and using the method presented in the previous subsection for the feature extraction
of a single scatterer. However, when multiple scatterers are located in the same
range, the minimization of C6 k, Xk, fk, fk) has numerous ambiguous solutions that
may lead to the artifact problem in the high resolution SAR image formation.
SPAR attempts to avoid the ambiguity problem by isolating out the most
dominant scatterer in Yk by using a 2-D rectangular window, which is determined
from and applied to the 2-D FFT of Yk. The isolation process has the following
steps:
Step (i): Obtain Vk, the 2-D FFT of Yk, without zero padding.
Step (ii): Determine the 2-D window w(n, n) from Vk. We first locate the
peak location (n+, l+) of the magnitude of Vk. We then fix n to n+ and search for





49

the interval nl <_ n+ < n2 so that the magnitude of Vk is above a certain threshold,
say Ti, within the interval. Similarly, we can fix n to n+ and search for the interval

nil < n+ < in2. Then the N x N rectangular window w(n, n) has unit value for
nl < n < n2 and nl < n < n2 and zero value elsewhere. The threshold Ti we use in
our numerical and experimental examples is 10% of the peak value of the magnitude
of Vk.
Step (iii): Determine Yk by applying 2-D inverse FFT (IFFT) to Vk 0 W,
where the (n, n)th element of W is w(n, n).
Instead of minimizing C6(k, Xk, fk, fk), we now minimize

C7(k,Xk, fkk)== Yk -e 3kGk(xk,fk, fk) 2, (4.47)

where Yk is used to replace Yk in C6G(k, k, fk, fk), by using the method presented
in Section 4.1.1.
With the above preparations, now we provide the steps of the scatterer isola-
tion and relaxation based NLS algorithm, which are the substeps of Step 1 of SPAR.
Step I: Assume K = 1. Calculate Y from Y by using the isolation process.
Obtain {&, xk, fk, fk}k=1 from Y.
Step II: Assume K = 2. Compute Y2 with (4.45) by using {(k, k, fk, fk}k=1
obtained in Step I. Calculate Y2 from Y2. Obtain {(,k, k, k, fk k k=2 from Y2. Next
compute Y1 with (4.45) by using {k, Xk, k, fk}kk=2, calculate Y1 from Y1, and then
redetermine { k, fk, fk}kk=1 from Y'.
Iterate the previous two substeps until "practical convergence" is achieved (to
be discussed later on).
Step III: Assume K = 3. Compute Y3 with (4.45) by using {Jk, Xk, k, fk} =1
obtained in Step II. Calculate Y3 from Y3. Obtain {(, k, fk, 1fk}k=3 from Y3. Next,
compute Y1 with (4.45) by using {(k, Xk, fk, fk} =2, calculate Y1 from Y1, and then
redetermine {(,Ck,^ k, fk) }k= from Y1. Then compute Y2 with (4.45) by using





50


{ kc,x k, fk}ck=1,3, calculate Y2 from Y2, and then redetermine {(k,k, xk, f fk}k=2
from Y2.

Iterate the previous three substeps until "practical convergence".

Remaining Steps: Continue similarly until K is equal to the desired or

estimated number of scatterers.

The "practical convergence" in the iterations of the above relaxation-based
NLS algorithm may be determined by checking the relative change e of the cost

function C5 ({I kf, fk, fk}k=i) in (4.44) between two consecutive iterations. Our

numerical and experimental examples show that the algorithm usually converges in

a few iterations.

We can determine K, the number of scatterers in (4.2), by extending the

generalized Akaike information criterion (see [61] for details). By assuming that the

noise is white, the estimate K of K is determined as an integer that minimizes the

following extended GAIC cost function:

GAICk = NNln E 1: (n, n)2) + -yn[lny(NR)][((N + 3) + 1], (4.48)
Sn=0 ni=0
where y is a constant of user choice,
K
ek(n, n) = y(n, n)-- -xSk(n)ekej2 (fk+f), n = 0,1, .., N-l, f = 0,1, .., ,-1,
k=1
(4.49)
and K(N + 3) + 1 is the total number of real-valued unknown parameters (of which

K(N + 3) is for the scatterers and 1 is for the white noise variance).

Note that the NLS estimates of {(k, Xk, fk, kK=1 can also be estimated from

{Yk}k=1 determined in (4.45), rather than {Y}1,k=, via the above relaxation-based
optimization algorithm. We refer to this approach as the Hybrid method. When

multiple scatterers are located in the same range, Hybrid may be computationally

more efficient than SPAR since Hybrid does not isolate the scatterers so that multiple
scatterers located in the same range can be more efficiently described as at most two





51

"scatterers". However, the Hybrid SAR images may suffer from more severe artifact

problem than SPAR, especially at low SNR. When no multiple scatterers are located

in the same range, SPAR and Hybrid perform similarly.


4.5 Modified RELAX-NLS Algorithm

RELAX-NLS [42] is a parametric approach for the feature extraction of targets

consisting of both trihedrals and dihedrals. It is based on a mixed data model in

which x(n) is modeled as a real-valued constant for a trihedral or a sinc function of f

for a dihedral. Like SPAR, RELAX-NLS extracts the target features by minimizing

an NLS cost function via a relaxation-based approach. However, RELAX-NLS is

computationally expensive since a 4-D search over the parameter space is required

for dihedral corner reflectors. Since SPAR is more robust and computationally more

efficient than RELAX-NLS, the former can be used to provide the initial conditions

needed by the latter.

Let {(k, Xk, k kfk}k=1 denote the parameter estimates obtained via SPAR

according to the data model in (4.2), where K is the estimated number of scatterers

obtained via GAIC. The SPAR estimates {0k, xk, fk, fk}k= cannot be used directly

as initial conditions for RELAX-NLS. The initial conditions are obtained by applying

the first step of RELAX-NLS [42] to each Uk, k = 1, 2, ... K, where

K
Uk = ejiGi,(xii,,fi), k = 1,2,---,K. (4.50)
i=l,i k

Once we have the initial conditions, we can use the last step of RELAX-NLS [42] to

obtain the dihedral and trihedral parameter estimates, which are then used for SAR

image formation [7]. We refer to this approach as the modified RELAX-NLS.





52

4.6 Crambr-Rao Bound (CRB) of the Semi-Parametric Data Model

Let Y be an N x N matrix with its (n, n)th element being y(n, n) with
K
y(n, 5) = E k (n)ejkej2(fkn+n) +e(n, ), n = 0,1, **,N-1, = 0,1, N-1,
k=1
(4.51)
where K denotes the number of scatterers; {xk(n)}0=o denotes the real-valued am-
plitude function of f for the kth scatterer; 'k and {fk, fk}, respectively, are the
constant phase and the frequency pair of the kth scatterer; finally, {e(n, f)} denotes
the unknown 2-D noise and clutter sequence. Then we can rewrite (4.51) as:
K
Y e= eJk (fk)xTD(fk) + E, (4.52)
k=l
where E denotes an N x N matrix with e(n, f) being its (n, f)th element,

WN(fk)= 1 e j2fk e2 jfk(N-) (4.53)


(fk) = 1 ej2fk .. ej2*(7 -) (4.54)

with (.)T denoting the transpose, D(fk) denoting the following diagonal matrix:

D(fk) =diag{ 1, ej2f *.., ej2k(N-1) (4.55)

and

Xk = [ xk(0) xk(1) *.. xk(i-1) (4.56)

Let y = vec(Y) = [y yT ... y ]T with yn(A = 1,2,..-,N) being the Ath
column of Y. Then the data model can be rewritten as follows:

y = A/ + e, (4.57)

where

A= [ [(fl) 0 X1] WN(fl) *.. [N(JK) YXKI WN(fK) (4.58)





53

with 0 and , respectively, denoting the element-wise product and Kronecker prod-
uct of two vectors,
p = e 1 ... eOK (4.59)
and e = vec(E).
Next, we derive the CRB matrix for the data model in (4.57) where the noise
covariance matrix is assumed to be arbitrary and unknown. Let Q = E{eeH} be
the noise covariance matrix with (.)H denoting the conjugate transpose. Then the
extended Slepian-Bangs formula for the ijth element of the Fisher information matrix
(FIM) has the form [3, 64]:

{FIM}ij = tr (Q- Q-'Q, + 2Re [(HAH) Q-1 (At)j] (4.60)

where X' denotes the derivative of X with respect to the ith unknown parameter,
tr(X) denotes the trace of X, and Re(X) denotes the real part of X. Note that FIM
is block diagonal since Q does not depend on the parameters in (Api), and (Api)
does not depend on the elements of Q. Hence the CRB matrix for the target features
of interest can be calculated from the second term on the right side of (5.31). Let

S= O T fT T xe (4.61)

where
0= 01 02 O/K ] '

f= [f f2 fK ]
]T

L= f 2 ... / K 7
and
xe= [xT xi .. xK (4.62)

Let FI, F2, and F3 denote NN x K matrices whose kth columns are determined,
respectively, by
jei"" {[CW (fk) 0 Xk] WN(fk)} (4.63)





54

j27re k { [k~(f) 0 Xkj (WN(f) O dN)} (4.64)

with

dN = 0 1 N-1 (4.65)

and

j27re k {[R (fk) 0 Xk 0 dN] 0 WN(fk)} (4.66)

with

d = 0 1 .. -1 (4.67)

Let F4 denote an NN x KN matrix whose [(k 1)N + n]th column is determined

by

e jk {[wf,,0 (f)]r wN(fk)}, k=1,2,...,K,=l, ,2,...,N, (4.68)

where wn is an N x 1 vector with the nth element being unit and all other elements

being zeros. Let

F= F1 F2 F3 F4], (4.69)

Then the CRB matrix for the parameter vector qr is given by:

CRB(l) = [2Re(FHQ-F)]-1. (4.70)


4.7 Numerical and Experimental Results

We demonstrate and compare the SAR image formation performances of
SPAR and the modified RELAX-NLS algorithm with both simulated and experi-

mental examples. The algorithms are also compared with Hybrid and RELAX-NLS.

In the following examples, the dimensions of the original SAR phase history data

matrix are N = N = 32 and GAIC with y = 5.5 is used to determine K for the
relaxation-based feature extraction algorithms of SPAR and Hybrid. The threshold

Ti used in the isolation process of SPAR is 10% of the peak value. The maximization

of C4(f, f) in (4.41) is done in two steps. First, initial frequency estimates f and f





55


are obtained via 1-D FFT with zero-padding to a total length of 128 in range and to a

total length of 64 in cross-range. Next, these initial estimates are refined by using the

FMIN function in MATLAB alternately, i.e., by updating f while fixing f at its most

recently determined value and vice versa, until "practical convergence", which is de-

termined by checking the relative change of the cost function C4(f, f). We have used

10-3 to determine the convergence of this fine search as well as the relaxation-based

algorithm. The extrapolation factor ( = 8 is used in range for SPAR and Hybrid

and in both range and cross-range for RELAX-NLS and the modified RELAX-NLS.

Both 1-D and 2-D Kaiser windows with shape parameter 3 = 6 are used whenever

needed.

First consider a simulated example with high SNR. The SAR phase history

data matrix is simulated by assuming that there are four trihedrals and three di-

hedrals in the presence of zero-mean white complex Gaussian noise with variance

a, = 0.6. The amplitude functions for the four trihedrals are generated as follows:

Xk(n) = k =1,2,3, = 0, 1, ,N-1, (4.71)

and

x4(t})=2, =0, 1, -,N- (4.72)

The amplitude functions for the three dihedrals are

x5(n) = 9.6sinc[0.3r(f 18.6)], n = 0, 1, N 1, (4.73)

and

Xk(r) = 6.4sinc[0.27r( 18.6)], k =6,7, f = 0,1, N 1, (4.74)

where sinc(x) = sin(x)/x. Figure 4.3(a) shows the modulus of the true SAR image.

Note that two of the dihedrals are located in the same cross-range and are closely

spaced in range and two of them are located in the same range. Of the four trihe-

drals, two of them are closely spaced in range and the other two are located in the






56

same range. In this example, Hybrid and SPAR have the same estimated number of

scatterers, K = 7. Figure 4.3(b) shows the windowed 2-D FFT SAR image obtained

by applying the normalized 2-D FFT to the windowed data matrix. SAR images

formed via Hybrid, SPAR, RELAX-NLS and the modified RELAX-NLS algorithms

are shown in Figures 4.3(c) through 4.3(f), respectively. We note that at high SNR,

the Hybrid image is similar to the SPAR image. Both of the parametric RELAX-NLS

and the modified RELAX-NLS algorithms outperform their semi-parametric coun-

terparts SPAR and Hybrid since the data model used by the parametric methods is

correct rather than approximate. For this example, our simulations show that the

ratios between the MATLAB flops needed by Hybrid, SPAR, the modified RELAX-

NLS, and RELAX-NLS over the flops needed by the windowed FFT method are 27.4,

28.4, 50.1, and 70.8, respectively. Note that both SPAR and Hybrid are computa-

tionally more efficient than RELAX-NLS and the modified RELAX-NLS, with the

modified RELAX-NLS being more efficient than RELAX-NLS.

Consider next a simulated example with low SNR. The SAR phase history

data is the same as in the above example except that the noise variance is increased

to a- = 6. SAR images obtained by using the windowed 2-D FFT, Hybrid, SPAR,

RELAX-NLS, and the modified RELAX-NLS are shown in Figures 4.4(a) through

4.4(e), respectively. Note that the artifact problem starts to show up in the Hybrid

image in Figure 4.4(b) due to large parameter estimation errors. By comparing

Figures 4.4(c) with (b), it can be seen that SPAR can effectively mitigate the artifact

problem.

Finally, consider an experimental example of SAR image formation by using

the MSTAR Slicy data collected by imaging an object consisting of both trihedral

and dihedral corner reflectors, which is shown in Figure 3.3. The data was collected

by the Sandia National Laboratory (SNL) using the STARLOS sensor. The field

data was collected by a spotlight-mode SAR with a carrier frequency of 9.559 GHz





57

and bandwidth of 0.591 GHz. The radar was about 5 kilometers away from the

ground object. The data was collected when the object was illuminated by the radar

from approximately the azimuth angle 0 and elevation angle 30. To cross-validate

the experimental results given below, XPATCH [1], a high frequency electromagnetic

scattering prediction code for complex 3-D objects, was used to generate very high

resolution phase history data for the object shown in Figure 3.3. The data generated

by XPATCH has a resolution of 0.038 meters in both range and cross-range, and the

corresponding windowed FFT SAR image is shown in Figure 4.5(a). (We have used

the log scale for all of the images shown in Figure 4.5.) The original experimental

Slicy data has a resolution of 0.3 meters in range and 0.32 meters in cross-range.

The 32 x 32 data matrix we used to demonstrate the performance of our algorithms

has a spoiled resolution of 0.51 meters in range and 0.54 meters in cross-range. The

windowed 2-D FFT SAR image of this data matrix is shown in Figure 4.5(b). Figure

4.5(c) shows the Hybrid image with K = 7 (obtained via GAIC). Figure 4.5(d)

shows the SPAR image with K = 7 (obtained via GAIC). Figures 4.5(e) and (f)

show the SAR images obtained via RELAX-NLS and the modified RELAX-NLS

with K = 7, respectively. Note that Hybrid has a more severe artifact problem

than SPAR. Note also that the SPAR image shown in Figure 4.5(d) appears to fit

Figure 4.5(a) and the characteristics of the object in Figure 3.3 well. However, the

parametric algorithms are not as robust as SPAR since the parametric images shown

in Figures 4.5(e) and (f) do not fit Figure 4.5(a) as well with one of the scatterers

mis-identified and mis-located. For this experimental example, the ratios between the

MATLAB flops needed by Hybrid, SPAR, the modified RELAX-NLS, and RELAX-

NLS over the flops needed by the windowed FFT method are 29.2, 16.7, 32.8, and

43.1, respectively. Note that SPAR can sometimes be faster than Hybrid!




58





















(a) (b)















(c) (d)

Figure 4.1: Ambiguity effect on the SAR image formation in the absence of range
estimation errors. (a) True windowed FFT SAR image. (b) Windowed FFT image
of the first "scatterer". (c) Windowed FFT image of the second "scatterer". (d)
Combined windowed FFT image of the two "scatterers". (The vertical and horizontal
axes are for range and cross-range, respectively.)





59





















(a) (b)
















(c) (d)
Figure 4.2: Ambiguity effect on the SAR image formation in the presence of range
estimation errors. (a) Windowed FFT image of the first "scatterer". (b) Windowed
FFT image of the second "scatterer". (c) Combined windowed FFT image of the
two "scatterers" with C = 1 (without extrapolation). (d) Combined windowed FFT
image with ( = 2. (The vertical and horizontal axes are for range and cross-range,
respectively.)





60














(a) (b)










(c) (d)











(e) (f)
Figure 4.3: Comparison of SAR images formed using different algorithms for simu-
lated data at high SNR (oa = 0.6). (a) True SAR image. (b) Windowed 2-D FFT
SAR image. (c) Hybrid SAR image. (d) SPAR SAR image. (e) RELAX-NLS SAR
image. (f) Modified RELAX-NLS SAR image. (The vertical and horizontal axes are
for range and cross-range, respectively.)






61














(a) (b)










(c) (d)











(e)
Figure 4.4: Comparison of SAR images formed using different algorithms for simu-
lated data at low SNR (ou = 6). (a) Windowed 2-D FFT SAR image. (b) Hybrid
SAR image. (c) SPAR SAR image. (d) RELAX-NLS SAR image. (e) Modified
RELAX-NLS SAR image. (The vertical and horizontal axes are for range and cross-
range, respectively.)





62














(a) (b)










(c) (d)











(e) (f)
Figure 4.5: Comparison of SAR images obtained via different algorithms for the
Slicy data hb15533.015 (0 azimuth and 300 elevation angles). (a) Windowed 2-D
FFT SAR image from the XPATCH data. (b) Windowed 2-D FFT SAR image from
the Slicy data. (c) Hybrid SAR image. (d) SPAR SAR image. (e) RELAX-NLS
SAR image. (f) Modified RELAX-NLS SAR image. (c) ~ (f) are all obtained from
the data used in (b). (The vertical and horizontal axes are for range and cross-range,
respectively.)














CHAPTER 5
3-D TARGET FEATURE EXTRACTION USING CLSAR


5.1 CLSAR Data Model

We describe how to obtain 3-D target features via the full synthetic aperture

shown in Figure 1.2(a), which makes a preparation for the discussion on CLSAR. We

start by establishing one-dimensional (1-D) point scatterer data model of high range

resolution radar as a preparation for the establishment of 3-D data model herein.

5.1.1 1-D Data Model for High Range Resolution Radar

The range resolution of a radar is determined by the radar bandwidth. To

achieve high resolution in range, the radar must transmit wide band pulses, which

are often linear FM chirp pulses [29, 49]. A normalized chirp pulse can be written as

s(t) = e-j(2fot+7t), ItI To/2, (5.1)

where fo denotes the carrier frequency, 27 denotes the FM rate, and To denotes the

width of the pulse. We assume that fo, 7, and To are known. The signal returned by

a scatterer of a target has the form

r(t) = 6,e-j[2fo(t-7)+(t-7)2, (5.2)

where 6, is determined by the RCS of the scatterer and r denotes the round-trip

time delay. The demodulated signal d(t) is obtained by mixing r(t) with s*(t o)

for some given T0 = 2Ro/c, where (.)* denotes the complex conjugate, c denotes the

propagation speed of the transmitted electromagnetic wave, and Ro denotes the range

distance from radar to the center of the illuminated patch.

d(t) = 6,ej2(rfo-7~o)(T-7To) e- -7(r0)2 ej27(y70)t. (5.3)

63





64

The term e-j3('-T0)2 in (5.3) is usually close to a constant for all Tmin T, T0 Tmax,
where Tmax and Tmin correspond to the maximum and minimum values, respectively,

of the round-trip time delays between the scatterers of a target and the radar. This

term can also be partially removed [49]. Let D(w) denote the Fourier transform of

d(t). Then the inverse Fourier transform of D(w)eT will have the term e-jr('-ro)2

removed. Yet this removal can only be approximate since d(t) is not known for all
t and hence D(w) is not known exactly. The closer e7-j(T-7o)2 is to a constant for

Tmin < T < Tmax, the better its removal. With this removal, we have

d(t) = 6,ej2(fo-77Y)(T-7)ej2(--7t )t, (5.4)

which is a complex sinusoid with frequency 27(7-7o) and amplitude 6,ej2(fo-y770)( --).
We know Tmax and Tmin approximately since we assume that the altitude, antenna

beamwidth, and grazing angle of the radar are known. We also assume that (Tmax -

Tmin) < To. Then for -To/2 + Tma, < t < To/2 + Tmin, the scatterers of the target at
different ranges correspond to different frequencies of the signal d(t), while the RCS's
of the scatterers are proportional to the amplitudes of the corresponding sinusoids.
The ranges and RCS's of the target scatterers are the 1-D target features.

5.1.2 Full Synthetic Aperture Radar

A broadside data collection geometry in a spotlight-mode SAR is shown in
Figure 5.1 [29]. The XYZ coordinate system is centered on a small patch of ground,
where a target is located. The ground is illuminated by a narrow radio frequency

(RF) beam from the moving radar that rotates (with radius Ro) around the coordinate

origin. In Figure 5.1, R denotes the distance between the radar and a scatterer at

the position (x, y, z), and 0 and are the azimuth and elevation angles of the radar
relative to the XYZ coordinate system. We assume that 0, , and Ro are known.

The range R of the scatterer located at (x, y, z) can be written as

R = [(Ro cos cos 0 x)2 + (Ro sin cos y)2 + (R sin )2 1/2 (5.5)





65


Radar
R Scatterer Z
(x,y,z)



RO

IX 0


Y


Figure 5.1: Broadside spotlight-mode SAR data collection geometry.

It has been shown in Appendix 3B that R can be approximated as

R Ro x cos 0 cos y sin 0 cos q sin (5.6)

where
o = Ro + [(x2 z2) sin 0o cos o 2xz cos2 Q0] ( o) (5.7)
Ro = Ro + (5.7)
2Ro
-(y2 + 2) cos + 2xz sin (5.8
x= x +2R' (5.8)
2Ro
xy cos o0 + yz sin (o
y = Y + (5.9)
Ro
and
(x2 + y2) sin (o
z = z 2 (5.10)
2Ro
Note that the second terms of the right sides of (5.8), (5.9), and (5.10) are due to

the range and elevation curvature effects and can be neglected for large Ro. Let TO
= 2Ro. Since 7 = 2, then from (5.4), we have
C C

d(t, 9, ) = x,y,z exp Ij4(7rf f-+-+t) (Ro R o cos cos y sin 0 cos

-i sin )] (5.11)






66


where 5,y,z is proportional to the RCS of the scatterer located at (x, y, z). For

4(7rfo 770 + yt)(Ro Ro)/c < 27, where It T0o < To, we can write (5.11) as

d(t, 0, ) 6Xy,zej(tx+ty+itz), (5.12)


where
4(7rfo 770 + 7t) cos 0 cos (5
tx = (5.13)

S4(7rfo 770 + yt) sin 0 cos (5
ty = (5.14)
C

and
S 4(r fo 7To + yt) sin (5.15)
c
Note that d(t, 0, ) is a 3-D complex sinusoid. The frequencies of the 3-D sinusoid

correspond to the 3-D location (J, y, 2) of the scatterer, while the amplitude is propor-

tional to its RCS. Note that (2, y, 5) is not the true location (x, y, z) of the scatterer,

but is close to (x, y, z) for large Ro. (See the Appendix 3C on how to calculate (x, y, z)

from (2, y, 2).) When a target has multiple scatterers, d(t, 0, ) in (5.12) will be a sum

of sinusoids. The 3-D locations and RCS's of the target scatterers are the 3-D target

features. Since usually the samples on the t, 0, and 4 axes are uniformly spaced, the

samples of tx, ty, and tz occur at the points of a polar grid. Hence Polar-to-Cartesian

interpolation may be needed for the data samples to occur at rectangular grid points.

(See Section 5.4.2 for an alternative approach.)

After Polar-to-Cartesian interpolation and sampling, the signal obtained by

the 3-D full aperture SAR in the presence of noise can be written as:
K
y(n, 5) = akej(ikn+k+k ") + e(n, i, i), (5.16)
k=1

where n = 0,1,-*, N 1, n = 0,1, N 1, and n = 0,1, .., N 1, with

N, N, and N denoting the numbers of available data samples in the three dimen-

sions; K denotes the number of sinusoids; ak and {Wk, ck, CO}) denote the unknown





67

complex amplitude and 3-D frequencies of the kth sinusoid, k = 1,2, K; fi-

nally, {e(n, i, n)} denotes the unknown noise sequence. The sinusoidal frequencies

{Wk = 27rfk, k = 27rfk,, k = 27rfk} correspond to the 3-D location of the kth
scatterer of a radar target and ak is determined by its RCS.

Let

Y = [y(0,0,0) y (1,0,0) y(N 1,0,0) y(0,1,0) y(l,l,0) ...

y(N-l,1,0) ...... y(0,N-1,N-l) y(1,N-1,N-1)
T
... y(N-1,N-1,7N-1)
(5.17)

Let e be defined similarly from e(n, n, n) as y from y(n, f, n). Then the data model

in (5.16) can be rewritten as:

y = Aa + e, (5.18)

where

S= a, C2 * K (5.19)

and

A = a[, i 0 a ... aK K aK (5.20)

where denotes the Kronecker product [22] and

1 1 1

ejwk ejCk eiwk
ak = a, k = k = (5.21)


ej(Y--1)wk ej( --1)k ej(r-l)1)k
5.1.3 Curvilinear SAR

Assume that the curvilinear aperture used in CLSAR consists of M different

radar viewing angles. Then the received data vector y, in CLSAR, is an (MN) x 1

subvector of y in (5.18). Let Ic denote an M x (NN) matrix with each column and





68

row containing only one non-zero unit element corresponding to the locations of the

available data samples. Then

Yc = Aca + e, (5.22)

where

Ac= {Ic(ai 9 )} a *.. {Ic(aa K a )} K aK (5.23)

The unknown sinusoidal parameters {wk, wk, k, Ok, ak K=1 are the features of our

interest and are to be estimated from the y, collected by CLSAR. In the following

sections, we will consider parametric 3-D target feature extraction using CLSAR.


5.2 The RELAX Algorithm

The RELAX algorithm [39] can be extended to extract 3-D target features. We

first consider using RELAX with the full aperture for 3-D target feature extraction

and then extend this approach to the case of curvilinear apertures.

5.2.1 Full Aperture

The RELAX algorithm [39] minimizes the following NLS cost function:

C4 = ly Aal|2, (5.24)

where |-|| denotes the Euclidean norm. When the noise e(n, i, n) is the zero-mean

white Gaussian random process, the NLS estimates obtained with RELAX coincide

with the ML estimates of the target features. When the noise is colored, the NLS

estimates are no longer the ML estimates, but they still possess excellent statistical

performance [39].

We present below the relaxation-based minimization approach that leads to a

conceptually and computationally simple method. For each fixed K, the intermediate

number of scatterers, we perform a complete relaxation-based search by letting only

the parameters of one scatterer vary and freezing all others at their most recently

determined values. In this way, we will also take advantage of the fact that the





69


parameter estimates for the first K 1 scatterers can be used to initialize the search

for the parameters of the Kth one. We first briefly make a preparation for the RELAX

approach. Let
K
yk =y Y di (i,- ( ), (5.25)
i=1,i k
where fi, aj, and ai are formed, similarly to those in Equation (5.21), from ji, wi,

and ij, respectively, and {ci, S, 2i, di} l,ifk are assumed to have been estimated.

Then minimizing C4 in (5.24) with respect to ak yields the estimate &k of ak:

[ak 0 k 0 aka Yk
k = [ k ak]Hyk (5.26)
NNN
Wk =Wk ,k =Wk ,Wk =Wk

and
2
Wk, Wk, k = arg max [a&k ( ak a k (5.27)
Wk ,Wk ,Wk
Hence {Wk, wk, wk} can be obtained as the location of the dominant peak of the 3-D

periodogram,
2
[ak O ak ak]H Y
NNN
which can be computed efficiently with 3-D FFT. Note that padding with zeros

for the 3-D FFT is necessary to achieve high accuracy. An alternative approach is

to find an approximate location corresponding to the global maximum with the 3-D

FFT without much zero-padding and then use the approximate location as the initial

condition to find a more accurate position via a multidimensional search method, such

as the FMINV function in PV-WAVE. We used the latter approach in our examples

presented in Section 5.4.2. Note that &k is easily computed from the complex height

of the peak of [ak afk 0 ak]H yk/(NNN).

With the above preparations, we now proceed to present the steps of the

RELAX algorithm for 3-D target feature extraction with the full aperture SAR.

Step (1): Assume K = 1. Obtain {Lk,WkWk,kk}k=1 from y by using (5.27) and

(5.26).





70

Step (2): Assume K = 2. Compute y2 with (5.25) by using {)k, k,Wk, 7 k}k=1
obtained in Step (1). Obtain { k, k, k, ,k}k=2 from Y2. Next, compute yl by using

{Wk,i k, k, & k}k=2 and then redetermine {W(k, Wk, k, &k}k=1 from yl.
Iterate the previous two substeps until "practical convergence" is achieved (to

be discussed later on).

Step (3): Assume K = 3. Compute Y3 by using {Ok, W k, Wk, ak} =1 obtained in Step

(2). Obtain { k, k, &k}k k=3 from y3. Next, compute yj by using {,k, w, wk, &k, l =2

and redetermine {Ik, Wk, k,&k}k=1 from yi. Then compute Y2 by using

{wk, Wk, Wk, &k}k=1,3 and redetermine {Wkk, Wk, k, k}k=2 from Y2.
Iterate the previous three substeps until "practical convergence".

Remaining Steps: Continue similarly until K is equal to the desired or estimated

number of sinusoids. (Whenever K is unknown, it can be estimated from the available

data, for instance, by using the generalized Akaike information criterion (GAIC) rules

which are particularly tailored to the RELAX method of parameter estimation. See,
e.g., [39].)

The "practical convergence" in the iterations of the above RELAX method

may be determined by checking the relative change e of the cost function

C4 ({CJk, W k, &k }=1) in (5.24) between two consecutive iterations. Our numerical
examples show that the iterations usually converge in a few steps.

5.2.2 Curvilinear Aperture

The RELAX algorithm for the curvilinear aperture is similar to that of the

full aperture except that Equations (5.26) and (5.27) above are replaced by

k [{Ic(ak ak)}M ak]H Yck
k =, (5.28)
MN
wk =cWk =W k ,Wktk =k
and
I 7 2
{Ck, kk} = arg max [{Ic(ak 0 ak)} ak] yCk (5.29)
Wk ,Wk ,Wk





71

respectively, where


Yck =Yc- i [{Ic( o )} (5.30)
i=l,i k
Let 'k, be similar to Yk except that the elements in Yk that are missing in Yck are
replaced with zeros in YCk. Then the right hand side of (5.29) can also be computed

by applying 3-D FFT to Sc.

5.3 Performance Analysis of Parameter Estimation via CRB

Consider first the case of the full aperture. Let Q = E{eeH} be the noise

covariance matrix, which is arbitrary and unknown. The extended Slepian-Bangs
formula for the ijth element of the Fisher information matrix (FIM) has the form
[3, 64]:

{FIMj}i = tr (Q-'QQ- Q;) + 2Re [(aHAH Q (Aa)] (5.31)

where X' denotes the derivative of X with respect to the ith unknown parameter,
tr(X) denotes the trace of X, and Re(X) denotes the real part of X. Note that FIM
is block diagonal since Q does not depend on the parameters in (Aa), and (Aa)
does not depend on the elements of Q. Hence the CRB matrix for the target features

of interest can be calculated from the second term on the right side of (5.31). Let

[ = Re(ca) ImT(a) WT ~T T], (5.32)

where
W = W1 U02 '-" WK (5.33)

r YT
S= 1 2 * K (5.34)

and
C = 01 o2 ... I K (5.35)





72

Let

F= A jA D,) Do, DI ,] (5.36)

where A is defined in (5.20), the kth columns of D,, Dc, and D@ are 9O[k 0 ak 0

ak]/owk, 9[ak 0 ak 0 akl]/&k, and O[ak 0 ak 0 ak]/aw3k, respectively, and

S=diag{ al, o2, *.** K }. (5.37)

Then the CRB matrix for the parameter vector l is given by:

CRB() = [2Re(FHQ-1F)]-1. (5.38)

For the case of curvilinear aperture, the CRB matrix for the target parameters

is similar to the one in (5.38) except that the A in (5.38) is now replaced by Ac.


5.4 Numerical and Experimental Results

We first use CRBs to investigate the performances of different curved apertures

for target feature extraction and then demonstrate with an experimental example the

performance of the RELAX algorithm for 3-D target feature extraction using CLSAR.

5.4.1 Performance Analysis of Different Curvilinear Apertures via CRBs

Since CRBs are the best unbiased performance any asymptotic estimator can
achieve, we not only compare CRBs of target features when different curved apertures

are used for 3-D target feature extraction, but compare them with the CRB of the

full aperture. Without loss of generality, let us consider the case of a single scatter

with a = 1 and w = D = c = 0. The additive noise is assumed to be zero-mean
white Gaussian with variance a2 = 40. The full aperture data is generated according

to (5.16) with N = N = N = 32.

Consider first the example of M = 63. The curved apertures we consider

include the parabolic one in Figure 1.2(b), the L-shaped one in Figure 1.2(c), and
the ones in Figure 5.2. For the arc apertures, the Arc-1 aperture is a quarter of a






73



4 .** e. A **.

Radius % *
.
'*
S



(a) (b) (c)

Figure 5.2: Curvilinear apertures, a) The circular aperture. b) The Arc-1 aperture.
c) The Arc-2 aperture.

Aperture CRB(a) CRB(w) CRB(&) CRB(D)

Full -5.9875 -51.4511 -51.4511 -51.4511

Circular 5.3663 -39.3415 -40.8469 -40.8788

Parabolic 5.8693 -39.3415 -40.4915 -39.6521

L-shaped 4.8584 -39.3415 -38.2414 -38.2414

Arc-1 11.7482 -39.3415 -31.7272 -31.7272

Acr-2 17.2473 -39.3415 -26.8015 -27.4636


Table 5.1: Comparison of the CRBs (in dB) of the target features for the cases of the
full, circular, parabolic, L-shaped, Arc-1, and Arc-2 apertures when M = 63, K = 1,
w = = = 0, a = 1, and a2 = 40.


circle whose center is at the upper right corner of the full aperture shown in Figure

1.2(a) and whose radius is 31, as shown in Figure 5.2(b), and the Arc-2 aperture

is one half of a parabolic aperture whose vertex is at the lower right corner of the

full aperture and who starts at the upper left corner of the full aperture shown in

Figure 1.2(a), as shown in Figure 5.2(c). These curvilinear apertures are subsets

of the full aperture in Figure 1.2(a) and are made to be as large as possible. For

example, the radius of the circular aperture is 15.5 in this example. Table 1 shows

the CRBs of the target parameters. As expected, the CRB for w is the same for





74


all of the curvilinear apertures since w is in the range direction, which is normal

to the plane of the curvilinear apertures. Comparing the CRBs for the L-shaped,

Arc-1, and Arc-2 apertures, we note that as expected, the more curved the aperture

is, the lower the CRBs for the target features. Since the Arc-2 aperture is the least

curved, it has the largest CRBs. When the aperture becomes one straight line and

hence no longer curved, the CRBs go to infinity since we can no longer extract 3-D

target features. Note also that from the top to bottom of Table 5.4.1, the CRBs

for & and & increase since the aperture length decreases. The CRB for a, however,

does not always increase. Consider next the circular aperture in Figure 5.2(a) when

w = & = C = 0. Figure 5.4.2 shows the CRBs of the target parameters as a function

of the radius of the circular aperture for different M. Note that as expected, the

larger the radius and/or M, the lower CRBs of the target parameters. Note also

that, as expected, the CRB for w does not change with the radius for the fixed

M since again w is in the range direction, which is orthogonal to the plane of the

curvilinear aperture.

5.4.2 Experimental Examples

Before we present an experimental example to demonstrate the performance

of the RELAX algorithm for 3-D target feature extraction with a CLSAR, we first

introduce an alternative to the Polar-to-Cartesian interpolation. Instead of Polar-

to-Cartesian interpolation, we created an N x N x N rectangular grid and mapped

the data sample at each (tx, ty, tz) to the nearest grid point, where t,, ty, and tz are

functions of t, 0, and (see Equations (5.13) to (5.15)). Let N denote the number of

available data samples. Then for both full and curvilinear apertures, the data model

can be written as

y, = I,Aa + e, (5.39)






75


where Ir denotes an N x (NNN) matrix with each column and row containing

only one non-zero unit element corresponding to the locations of the available data

samples in the rectangular grid. Note that the larger the dimensions of the rectangular

grid, the more accurate the mapping approximation. For large enough dimensions,

however, the noise will become the dominant source of error. For our examples, we

chose N = N = N = 128. As compared to Polar-to-Cartesian interpolation, the grid

mapping approach avoids the interpolation step needed by the former, which could

be complicated for an arbitrary curvilinear aperture. The former approach, however,

is computationally more efficient due to its smaller data dimensions.

The steps of applying RELAX to y, are similar to those for the case of full

aperture except that Equations (5.26) and (5.27) above are replaced by

&k = [,( ak (9 ak)] Y (5.40)
Wk =Wk ,k =Wk ,Wk =Wk
and
2
{jk, k, k}) = arg max Ir[I(a k ( ak)]Hrk 2, (5.41)
Wk ,Uk @Wk

respectively, where
K
Yrk = yr- C iIra, (ai a) (5.42)
i=1,i k
Let ,frV be similar to Yk except that the elements in Yk that are missing in yk are

replaced with zeros in k,,. Then the right hand side of (5.41) can also be computed

by applying 3-D FFT to kYk. Since the dimensions of the rectangular grid are larger

than the dimensions of the original data due to zero-filling, the 3-D FFT of rk is

approximately periodic and we should limit our attentions to only one period. For our

examples, the former is four times as large as the latter and hence the peak searching

in (5.41) is limited to the frequency intervals [ 27r x (-0.125) 27r x 0.125 ]. Note

that the computational advantages of FFT over DFT (discrete Fourier transform)

and backprojection diminish as the dimensions of the rectangular grid increases,

especially for curvilinear apertures.





76

We now present an experimental example to demonstrate the performances

of the RELAX algorithm for 3-D target feature extraction with a CLSAR. We use

e = 0.01 to test the convergence of the RELAX algorithm in this example. The

indoor data used herein was obtained by the Radar Signature Branch, Naval Air

Warfare Center, Mugu, California. The radar carrier frequency is 9.968 GHz and the

bandwidth 1.524 GHz. The 32 x 32 x 32 data set was obtained with the full aperture

shown in Figure 1.2(a) with 32 samples in each dimension and the angular increments

were 0.28'. The target consists of K = 8 corner reflectors with a cubic configuration

and was about 15 meters away from the radar. The scatterers were about 0.5 meters

apart.

Figure 5.4.2 shows the extracted scatterers when K = 8 is used with RELAX.

Figure 5.4.2(a) is obtained when the full aperture shown in Figure 1.2(a) is used to

extract the 3-D target features and Figure 5.4.2(b) is obtained when the L-shaped

aperture shown in Figure 1.2(c) is used. The centers of the circles denote the locations

of the extracted scatterers in 3-D space and the radius of each circle is proportional

to the modulus of the RCS of the corresponding scatterer. The triangles show the

projections of the scatterer locations onto the horizontal plane and their sizes are also

scaled according to the RCS's of the scatterers. Note that Figure 5.4.2(b) is similar

to Figure 5.4.2(a) even though the former is obtained by using only 6.15% amount

of data used by the latter.








77
















20I I I I -36-

Full Aperture
15-" .. M=32 -3
1 "0,... M=64 .......................................................4
""' "'^ ............... -' M=80 ... ____.,
10--42-

S-- .... .44- Full Aperture

S-46-M=
S0 .. M--4
U S -- M=a0
0- -4 8-

.50-
-5-5

-52-
-1 -54 I I I
4 6 8 10 12 14 16 4 6 8 10 12 14 16
Radius Radius


(a) (b)
-25 -25
Full Aperture Full Aperture
.. M=32 ]...... 1=M32
-M30 *' "". M -30 ., M- 32
... - M=80 .* **..... -- M.80
0 -35- -351- ..
S................
a 4o ,'- -40-
-4 0


-45- -45


-50 -50

-5sI I I -55. I
4 6 8 10 12 14 16 6 8 10 12 14 16
Radius Radius


(c) (d)



Figure 5.3: The CRBs of the target parameters as a function of the circular aperture

radius when K = 1, a = 1, w = C = C = 0, and a2 = 40. a) The CRB of the

complex amplitude a. b) The CRB of w. c) The CRB of a. d) The CRB of C.






78






0.4

0.2 -





E -0.2 4 -


-06



-A--






(a)





0.4-

0.2



E -0.2 -















(b)
-0.6
















Figure 5.4: 3-D plots of K 8 scatterers extracted by using RELAX with the indoor
experimental data. (a) Obtained with full aperture as shown in Figure 1.2(a). (b)
Obtained with curved aperture as shown in Figure 1.2(c).
0.4 ^ ^- '= ^
^- ^ ^ c>^



































Obtained with curved aperture as shown in Figure 1.2(c).














CHAPTER 6
AUTOFOCUS IN CLSAR

To analyze the effect of CLSAR aperture errors, we first rewrite the CLSAR

data model in (5.12) with respect to radar viewing angles. Assume that a curvilin-

ear aperture consists of M different viewing angles and let {Mm, 0m}=I denote the

elevation and azimuth angle pairs of the M look angles of the radar. Let y(n, m),

n = 0, 1, ... N 1, denote the one-dimensional (1-D) data samples obtained after

dechirping from the mth viewing angle of the radar. Let

4(7rfo Yro + Ytn)
4t = (rfo,, n 0, 1, .,N- 1, (6.1)
c

where t, denotes the time samples. Let tm (n), tym (n), and t, (n),respectively, denote

the time samples of the mth look angle, where

txm (n) = tn cos(0m) cos(m), (6.2)

tym (n) = in sin(0m) cos(0m), (6.3)

and

tzm (n) = t, sin(0m), (6.4)

with m = 1, 2, *, M and n = 0, 1, *, N 1. Then y(n, m) has the form
K
y(n, m) = O ake2[fkm (n)+fkum ()+fk*"()l z(n,m)+e(n,m), m= 1,2, -, M,
k=l
(6.5)

where e(n, m) denotes the noise and clutter and K is the number of scatterers.

We have assumed that {Om, em}m=1 and Ro(m) = Ro, m = 1,2, -, M, are

known exactly in the CLSAR data (6.5). For a practical curvilinear SAR system, how-

ever, the radar positions relative to the XYZ coordinate system may not be known


79






80

exactly due to atmospheric turbulence and platform position uncertainty. In 2-D SAR

imaging, it is generally assumed that the errors in {0m, 0m}m=1 are negligible and the

errors in {Ro(m)}m=1 cause phase errors along the synthetic aperture. In CLSAR,

however, the errors in {Om, 0m}m=1 may no longer be negligible since the aperture

shape is critical for 3-D target feature extraction. Our problem of interest herein is to

compensate for the curvilinear aperture errors in {Ro(m)}= and {Om, 0m}4m=i and

extract the 3-D target parameters {ok, fk, fk, fkK= from {y(n, m)}, n = 0, 1, - -, N-

1, m = 1, 2, *. M. We first analyze aperture errors in CLSAR and then present an

autofocus algorithm, referred to as AUTORELAX, for 3-D target feature extraction

in the presence of aperture errors.


6.1 Aperture Error Effects

We will consider the approximations and the ambiguity problems in our data

model in the presence of aperture errors. We omit the dependence on n in the

following analysis For the sake of notation convenience. For the broadside data

collection geometry shown in Figure 5.1, 0m is very small. For very small 0m, we have

sin(0m) m 0m and cos(0m) 1. These approximations also hold for the true look

angles of the radar. Then according to (6.2), (6.3), and (6.4), respectively, we have
o 0
txm t(cos 0m), (6.6)
0 0 o
ty,~ t(cos 0m) Om, (6.7)

and
o 0
tzm, t(sin a), (6.8)

where {Om, )a}M=1 denote the true look angles of the radar and

= 4(7rfo 7ro + 7t)
c
Consider first the errors in {m}mf=1 when {0m =0m,Ro(m) = Ro}m=r. Let

(m =eL +Adm. (6.9)






81


Let

to = 4rf (6.10)
c

It can be shown that for very small A0m and 7rfo > 7(t To), where |t To| < To/2,

we have
0 0
t, m t cos Om -to(sin 0o)Aom, (6.11)

tym i t(cos 0o) Om, (6.12)

and
o O
tm tisin Cm +io(cos 0o)AOm, (6.13)

where 0o is the average of all 0m, m = 1, 2, - M, and hence is a constant. Then

{ -[ r o0
ak exp j2rf(fktxm ky + fktz) + ak exp j27r [fktcos +fkto cos 0 0Om

+/ftsin ,mI } exp j27r -fko sin +fi0o cos 0o Am (6.14)


Equation (6.14) shows that if A4m is a constant, then the phase error due to Aom

and the phase of ak are ambiguous. Hence the phase of ak can never be determined

exactly in the presence of Aom. If Aom is a linear function of Om, then fk and the

linear phase error due to Aom are ambiguous and cannot be determined exactly.
0
Similarly, we can analyze the errors in {0m}fI=i when {qm =m, Ro(m)

Ro0}m=1. Let
0~ =0m +AZM. (6.15)
0
For very small Om and AOm and for 7rfo > (t TO), where It Tro < To/2, we have


0 0
tX' __t COS 0m, (6.16)


tyu too cos 0o (Om +A0t), (6.17)

and
Stsin (6.18)
tZ t sin m (6.18)





82

Then

Ck exp j2r(fktxm + fktym + fktzm) aik exp j27r fktcos (m +fkt0 COS 00o m

+fktsin m} exp j27rfkto cos 0o A (6.19)

which shows that a constant AOm also results in the ambiguity between the phase

error due to AOm and the phase of ak. Hence the phase of ak can never be determined
0
exactly in the presence of AOm. Also, if A0m is a linear function of Om, then fk and

the linear phase error due to AOm are ambiguous and cannot be determined exactly.
0 0
Finally, consider the errors in Ro(m) when Om =0m and Om =Om, m =

1,2,---,M. Let

Ro(m) = Ro + ARo(m). (6.20)

Replacing Ro in (5.7) with Ro(m) and for small ARo(m), large Ro(m), and 7rfo >

y(t To), where t Tol < To/2, we have from (5.11)

d(t, 0, () 6xy,zeJ(tXt +Yt+itz)ejtjoAR(m). (6.21)

Hence the errors in {Ro(m)}mM=1 result in phase errors along the synthetic aperture,

which is consistent with the analysis in [29, 43]. Then the z(n, m) in (6.5) should,

for this case, be replaced by

x(n, m)= "cgkej2[fkt )+fktym ()+ fktzm()] }e'J, (6.22)


where rim = toARo(m) is the phase error caused by ARo(m). Note that this phase

error differs from the phase errors in (6.14) and (6.19) since it does not depend on the

parameters of the kth scatterer and is easier to deal with. If ARo(m) is a constant,

then the phase error due to ARo(m) and the phase of Ok are ambiguous. Hence

the phase of ak can never be determined exactly in the presence of ARo(m). If
0
ARo(m) is a linear function of Om, then fk and the linear phase error due to ARo(m)

is ambiguous and cannot be determined exactly.





83

6.2 The Relaxation-Based Autofocus Algorithm (AUTORELAX)

The AUTORELAX algorithm obtains the estimates {Om, mn, )m}=1 and
K 0
<^ o o
{J&k, fc, fk, k}k=l, respectively, of the true values {Om, 7, m}m=l and

{ak, fk, fk, fk}k=1 by minimizing the following NLS criterion:
K M N-1
C5 O{k, fk k k ,{ m, m 7m}, m =1l y(n,m)- x(n, m)2, (6.23)
m=1 n=o
where

x(n, m) = ackeJ2 [fktm(n)+fktym(n)+fktzm(n)] } m, (6.24)
Sk=l 1
with tm (n), tym(n), and tz((n) defined in (6.2), (6.3), and (6.4), respectively. Note

that as shown below, we will determine Om and (m via a search method. Hence there

is no need to use the approximations such as those in (6.11), (6.12), and (6.13) for

txm (n), tm (n), and tzm (n), respectively. We use an alternating optimization approach
to minimize C5 in (6.23) and the AUTORELAX algorithm is outlined as follows:

Step 1: Extract the target features {&k, ,k fk fk}k= with the RELAX algorithm

from an initial curvilinear aperture by assuming that {im = 0}m= .

Step 2: Update the curvilinear aperture {0m, im}m=i and the phase error {7m}m=

with Steps (1) (4) (to be discussed in Section 6.2.1). (If {7rm}=, is known to be

negligible, then only update {Om, m}m=1 with Step (1).)

Step 3: Redetermine the target parameters with the RELAX algorithm by using

the curvilinear aperture {Om, im}m=1 and {r7m}m=l obtained in Step 2.

Step 4: Repeat Steps 2 and 3 until "practical convergence".

The "practical convergence" of AUTORELAX is determined similarly to that

of the RELAX algorithm by checking the relative change of the cost function C5 in

(6.23) between two consecutive iterations.

6.2.1 Aperture Parameter Estimation

Assume that the target feature estimates {&k, fk, fk, fk}k=i are given. Then

Om, (m, and fm can be determined by minimizing the following Cm with respect to






84


{Om, m, rim}, where
N-1
Cm,(Om,rm, rm) = E y(n,mm) (nm)12, n=0,---,N-1, m=l,2,---,M,
n=0
(6.25)

where xl(n, m) has the same form as x(n, m) in (6.24) except that {aLk, fk, fk, kf=1

are replaced by {&k, k, fk, ffk}k=l1 To simplify the optimization of Cm, we determine

{Om, em}m=1 and {rm}m=, iteratively as follows:

Step (1): Obtain {Om, ,m}, by minimizing the following Cmi with respect to {Om, mr},

where
N-1
Cmi (0m, Om) = I Uy(n, m) l(n, m)2 n = 0, 1, m = 1, --- M,
n=0
(6.26)

where 1(n, m) has the same form as z(n, m) in (6.5) except that {ak, fk, k,fk}k=1

are replaced by {&k, fk, fk, fk}K=1. When there are errors in both the elevation and
0 0
azimuth directions, we can estimate Cm and Om by the alternating minimization

approach, i.e., by iteratively fixing the estimate km of 0m and minimizing Cm, with
0
respect to Om, and then fixing the estimate 0m of 0m and minimizing Cm, with respect

to Om until "practical convergence". The "practical convergence" in the alternating

minimization approach is determined by the relative change of the cost function Cmi.

In the numerical examples, we terminate the iteration when the relative change of the

cost function Cmi between two consecutive iterations is less than 10-3. When there

are errors in only one angular direction, i.e., either 0m or qm is to be determined,

then the minimization of Cmi is a simple one-dimensional search problem.

Step (2): Determine phase errors {r m}m=" by minimizing the following cost func-

tions Cm2:
N-1
Cm2 (m) = y(n, m) 2(n, m)iem 2 (6.27)
n=0





85

where
K
z2(n, m) = &kej"r2[fklr t(n)+k t'm (n)+fktm(n), n = 0, 1, ., N- 1, m = 1, 2, -, M,
k=l
(6.28)
and im (n), ym (n), and tzm(n) are the same as tx,(n), ty (n), and tzm(n), respec-
tively, except that {Om, qm}m=1 are replaced by {Om, qm^} 1, obtained in Step (1).
tivelyera, obtained in Step (1).

This step is similar to [43] and we have

im = angle(Hmym), m 1,2,. ,M, (6.29)

where

Ym= y(0,m) y(l,m) ", y(N-l,m) (6.30)

and

2m 2 (0, m) 2(1, ) = 2(N-l ,m) (6.31)

Note from (6.29) that we do not need the search over parameter space to determine

{ m}m=1 and hence the errors in {Ro(m)}m=1 are easier to deal with than those in

{Om, m}m=1
Step (3): Repeat Step (1) by replacing y(n,m) with y(n,m) = y(n,m)e-j"m,

m = 1, 2, M, where {rim}m=1 are determined in Step (2).

Step (4): Repeat Steps (2) and (3) until "practical convergence", which is deter-

mined by the relative change of the cost function Cm in (6.25) between two consec-

utive iterations. In the numerical examples, we terminate the repetition of Steps (2)

and (3) when the relative change of Cm is less than 10-3 between two consecutive

iterations.

Note that if the errors in {Ro(m)}m=1 are known to be negligible, then Step

(1) alone is sufficient for the autofocusing.

6.2.2 Target Feature Extraction

Assume aperture parameter estimates {Om, im, 'lm}m= are given. Then the

problem becomes the target feature extraction problem considered in Chapter 5. As





86


shown in the previous chapter, the estimates {&k, fk, kfk}=i of {Ok, fk, k k=

can be obtained with the RELAX algorithm by equivalently minimizing the following

NLS criterion:

SK ) M N-1
F ({k, ,kk, k=1 = (Yn, m) 3(n,m)2, (6.32)
m=1 n=o0

where y(n,m) = y(n,m)e-j1m and 23(n,m) has the same form as z(n,m) in (6.5)

except that tm(n), tyu(n), and tzm(n) are replaced by i, (n), ym(n), and i (n),

respectively, which are determined by {Om, )m}=i instead of {Om, Om}m=1. Let
M N-1
Fk (&k, A, 1k, .k) = 9k (n, km) i2k[fk mHR)++fm)+fkzm")1 (6.33)
m=1 n=O

where t (n), ym (n), and tm (n) are the same as t, (n), tu (n), and tz (n), respec-

tively, except that {0m, 0m}m=1 are replaced by {0m, m}1=, and
K
k(n, m) = (n, m) 4 j2[if(6(")+ .i()+./^(n)], (6.34)
i= ,ifk

Then minimizing Fk in (6.33) with respect to ck yields the estimate &k of ak:
mM IN-01k (m)e-j2r-fk am {n)+ k v^mn) +hk
Zk =k(n, m)e=-j2r[o fk (n)+fk(,(n)+fkZm(n)] -,-. ---=, (6.35)
MN fk=fk,fk=fk,fk=fk

and
M N-I 2
{fk,, fk k = arg max k (n, m)ei2 [f- (n)+Af(m)+ft()] (6.36)
fk,fkfk m=1 n=O

Herein, we omit the detailed steps used for the 3-D target feature extraction

since they are the same as those in the previous chapter except that yk in (5.25)

is replaced by {(k(n,m)} in (6.34). The "practical convergence" in the iterations

of the RELAX algorithm now is determined by checking the relative change of the

cost function F ({&k, k, k fk}k=,) in (6.32) between two consecutive iterations. To

speed up the RELAX algorithm via utilizing FFT, a Cartesian grid mapping approach

is introduced in Chapter 5 to approximately map {tf (n), iym (n), m (n)}M= on a





87


Cartesian grid. When the Cartesian grid is fine enough, the errors introduced by the

approximation are negligible. In this chapter, we use a coarse Cartesian grid and the

FFT method to obtain an initial estimate of the parameters and then minimize (6.36)

via a multidimensional search method, such as the FMINV function in PV-WAVE.

The latter approach uses less computer memory.

6.3 Experimental and Numerical Results

We first present an experimental example to show the performance of the

AUTORELAX algorithm. The field data was obtained by the Deployable Signature

Measurement System (DSMS), Carderock Division, Naval Surface Warfare Center,

Bethesda, Maryland. The radar was carried on board of a helicopter. The radar

carrier frequency is 9.449 GHz and the bandwidth 0.498 GHz. The data set was

obtained with a curved aperture not exactly known but is roughly the same as the

dash line shown in Figure 6.1(a), where there are 64 look angles and 64 samples per

look angle. The radar was about 300 meters away from the ground target. The ground

target consists of 13 corner reflectors on the ground plane and 7 corner reflectors

mounted on a wooden tripod that is about 2.65 meters tall. The true distribution of

the scatterers is shown in Figure 6.1(b), where the centers of the squares denote the

locations of the scatterers in 3-D space and the length of each square is proportional

to the modulus of the RCS of the corresponding scatterer. The triangles show the

projections of the scatterer locations onto the ground plane and their sizes are also

scaled to be proportional to the RCS's of the scatterers.

Figures 6.1(c) and (d), respectively, show the scatterer distribution obtained

with the RELAX algorithm from the initial and an manually adjusted apertures

shown in Figure 6.1(a). We note that as compared to Figure 6.1 (b), the results in

Figure 6.1(d) are obviously better than those in Figure 6.1(c).

We now first consider autofocusing only in the elevation direction by as-

suming that no errors exist in both the azimuth angles {0m}B=1 and the distances





88

{Ro(m))}m,. The AUTORELAX algorithm converges after 6 iterations. Figure 6.2

shows the target scatterers extracted by AUTORELAX with the search interval for

{Aqm}m=1 being 0.34'. Comparing Figures 6.2(b) and 6.1(d) with Figure 6.1(b),
we find that AUTORELAX works well and the AUTORELAX results are slightly

better than those obtained by using the manually adjusted aperture. Figure 6.3 shows

that the manually adjusted and autofocused apertures fit quite well after adding a

line to the former. Since the results in Figure 6.2(b) are better than those in Figure

6.1(d), it appears that a linear phase error (in Om as a function of Om) was introduced

in the manually adjusted aperture. The linear phase error can cause shifts in fk and

the amount of shift is different for different scatterer k, as can be seen from (6.14).

Consider next autofocusing in both the elevation and azimuth directions by as-

suming that no errors exist in {Ro(m)}^m=. The AUTORELAX algorithm converges

after 7 iterations. Figure 6.4 shows the results obtained with the search intervals for

{Aom}m= and {Alm}m= being 0.340 and 0.0060, respectively. Compared with
the results obtained by autofocusing only in the elevation direction, we find that aut-

ofocusing in both directions provides little further improvement to the accuracy of

the estimated target parameters in this example. Hence for this example, it appears

that the aperture errors mainly occur in the elevation direction.

For this experimental example, we have also used AUTORELAX to extract

target features when we assumed that errors exist in both {Om, Om}m= and

{Ro(m)}M=1. Again, as compared with Figure 6.2(b), we have noticed little change

in the extracted target parameters. It could be that DSMS has already done a good

job compensating for the errors in {Ro(m)I}^M= with some traditional method so that

they are now negligible as compared with the errors in {(m} i=1

Finally, we use a simulation example to show that the accuracy of target

feature extraction via CLSAR is very sensitive to the accuracy of the curvilinear

aperture. We assume that there are 20 scatterers with a similar distribution to that





89


Estimation Error Magnitude Range (f) Cross-range (f) Height (f)

AUTORELAX 0.015 0.00288 0.00179 0.00348

RELAX 0.296 0.31544 0.01858 0.24444


Table 6.1: Estimation errors for the parameters of the highest scatterer in the simu-
lation example before and after autofocusing.

in the experimental example above. The true aperture is the dot line shown in Figure

6.5(a) and the scatterer distribution is shown in Figure 6.5(b). Here we simulate the

case where the aperture errors exist only in the elevation direction. Figure 6.5(c)

shows the scatterer distribution obtained with the RELAX algorithm from the initial

aperture (dash line) shown in Figure 6.5(a). We note that the scatterer distribution

in Figure 6.5(c) is quite different from the true one in Figure 6.5(b). When used

with the simulation data, AUTORELAX converges after four iterations. The auto-

focused aperture is the dot line shown in Figure 6.5(a) and the scatterer distribution

obtained with AUTORELAX is illustrated in Figure 6.5(d). We see that the scat-

terer distribution obtained by AUTORELAX is almost the same as the true one. We

also notice that the shape of the autofocused aperture is closer to the true one than

the initial aperture. The constant difference between the autofocused and the true

apertures will cause phase errors in {Oak K=1, which cannot be eliminated due to the

ambiguity problems discussed in Section 6.1.To quantitatively illustrate the accuracy

improvement of the parameter estimates via AUTORELAX, we compare the esti-

mation errors of the highest scatterer in Figure 6.5(b) before and after autofocusing.

Table 6.3 shows the differences between the true values and their estimates obtained

via RELAX and AUTORELAX, respectively.




Full Text
TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT x
CHAPTERS
1 INTRODUCTION 1
1.1 Background 1
1.2 Scope of the Work 4
1.3 Contributions 8
1.4 Dissertation Outline 9
2 LITERATURE SURVEY 11
2.1 2-D Super Resolution SAR Image Formation 11
2.2 3-D SAR Feature Extraction 15
2.3 SAR Motion Compensation 16
3 PARAMETRIC METHODS FOR SAR IMAGING 18
3.1 RELAX 18
3.1.1 Review of the RELAX Algorithm 18
3.1.2 Image Formation 21
3.2 RELAX-NLS 23
3.2.1 Review of the RELAX-NLS Algorithm 23
3.2.2 Image Formation 27
3.3 Experimental Results 28
4 SEMI-PARAMETRIC METHODS FOR SAR IMAGING 35
4.1 Problem Formulation and Data Model 35
4.2 Review of the APES Algorithm 37
4.3 Data Model Ambiguities and Their Effects on SAR Image Formation 38
4.3.1 Model Ambiguities 39
4.3.2 Image Formation 41
4.3.3 Model Ambiguity Effects on SAR Image Formation .... 43
4.4 The SPAR Algorithm 45
IV


22
applying the normalized FFT to the simulated phase history data {ys(ns, ns)}, i-e.,
by computing
CN-1C-1
75jvjv E E (3.11)
ns= 0 na=0
To suppress the sidelobes, the normalized FFT can be applied to the windowed
sequence {ws(ns,s)ys(ns,s)}, where the window sequence ws(ns,s) satisfies
CAT-1 C 1
y; y ws(ns,ns) = £2_/VIV. (3.12)
ns=0 s=0
The RELAX SAR. images containing both the dominant target features and back
ground clutter may be appreciated since, for example, the shadow information may
be desired for automatic target recognition. If so, the normalized FFT can be applied
to the sum of the simulated phase history data {ya(ns, ns)} and {£2(n, )} with zero
padding to have dimensions (N and £1V, where (n, ) denotes the estimated back
ground clutter and is determined by (3.9) with K replaced by K. Note that scaling
the (n, ) by a factor of £2 is needed when the background clutter is included in
the RELAX SAR images since both of its dimensions are 1 /£ times of those of the
simulated phase history data. Similarly, the normalized FFT can also be applied
to the sum of {ws(ns, s)ys(ns, s)} and {(2we(n, )(n, )} with zero padding to
have dimensions (N and £ to suppress the sidelobes, where the window sequence
we(n,) satisfies
N-l -1
y y We{n,) = N. (3.13)
n=0 =0
Note that since we cannot model the clutter effectively, its resolution cannot be
improved. We remark that the resolution of the RELAX SAR images is determine
by RELAX and £ > 1 is just used to demonstrate the super resolution property of
the RELAX algorithm for target feature extraction.


48
nth column of Y. Then the estimates {fa, xfc, fk, fk}k=1 of {fa, xfc, fk, fk}k=1 can be
obtained by minimizing the following NLS cost function:
C5 ({fa,^k, fk, fk}k=i) =
K
Y-£e*G k{xk,fkJk)
k=1
(4.44)
where || \\p denotes the Frobenius norm [22], The minimization of
C5 {{fa, xfc, fk, fk}k=1) in (4.44) is a very complicated optimization problem. The
proposed SPAR algorithm performs a complete relaxation-based search by letting
only the parameters of one scatterer vary and freezing all others at their most re
cently determined values for each assumed number of scatterers K. Let
K
Y*=Y- £ ,(S¡,/¡,/i),
i=l,i^k
(4.45)
and assume that {fa, x, /, fi}?=iare given. Then the NLS estimates
{fa,^k,fk,fa}k=i of {h,^k,fk,fk}k=i can be obtained by minimizing
C6{fa,Xk,fk,fk), where
C6{fa,xkJk,fk) = ||Yfc-e^Gfc(xfc,/fc,/fc)||^
(4.46)
and using the method presented in the previous subsection for the feature extraction
of a single scatterer. However, when multiple scatterers are located in the same
range, the minimization of C${fa, x.k, fk, fk) has numerous ambiguous solutions that
may lead to the artifact problem in the high resolution SAR image formation.
SPAR attempts to avoid the ambiguity problem by isolating out the most
dominant scatterer in Yk by using a 2-D rectangular window, which is determined
from and applied to the 2-D FFT of Y*,. The isolation process has the following
steps:
Step (i): Obtain Yk, the 2-D FFT of Yk, without zero padding.
Step (ii): Determine the 2-D window w(n,n) from Vfc. We first locate the
peak location (n+,+) of the magnitude of Vfc. We then fix to + and search for


65
Radar
Figure 5.1: Broadside spotlight-mode SAR data collection geometry.
It has been shown in Appendix 3B that R can be approximated as
Rq x cos 9 cos 0 y sin 9 cos 0 z sin 0,
where
o o +
[(x2 z2) sin0o cos 0O 2xz cos2 0O](0 0o)
X X +
y = y +
2-o
-(y2 + z2) cos 0o + 2a;zsin 0O
2o :
xy cos 0o + yz sin 0O
o
and
z = z
(x1 + y2) sin
2 ft,
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
Note that the second terms of the right sides of (5.8), (5.9), and (5.10) are due to
the range and elevation curvature effects and can be neglected for large 0. Let r0
= Since r = , then from (5.4), we have
d(t, 9, 0) = SXty,z exp
j4(7t/q7TQ+7)
(o o x cos 9 cos 0 y sin 9 cos 0
zsin 0)],
(5.11)


39
there are ambiguity problems associated with the semi-parametric data model. In
this section, we first analyze the possible data model ambiguities and then illustrate
their effects on SAR target feature extraction and image formation. The discussions
below will motivate the introduction of the SPAR algorithm, which will be presented
in detail in Chapter 4.4.
4.3.1 Model Ambiguities
Due to the flexibility of the data model in (4.2), there are various types of
ambiguities that may impact the feature extraction of each scatterer. Below we list
several types of the ambiguities inherent in the data model.
Type 1: Single scatterer
From (4.1), we note that ambiguity exists between x{)e:i = a;()e-7^+7r^ = £()e^, (4-14)
where x(n) = x(n) and = ^ + 7r. Ambiguity also exists between / and x(n) since
x{)ej2*f = (-l)x{)ej2nV--5) = SfriW2*?"
x(ri)e3
(4.15)
where x{n) = (l)nx() and / = / 0.5. The above two types of ambiguities cannot
be resolved.
Type 2: Two identical scatterers located in the same range
Let fa and fb, respectively, denote the cross-range locations of two identical scatterers
and let 4>a and 4>b, respectively, denote their phases. Then
x(n)
e0a+t27r/an) _|_ e0,+j27T/i,n)
2x(n) cos [-7T(fa fb)n +
eMfa+h)n, (4T6)
which indicates that two identical corner reflectors (trihedrals or dihedrals) located
in the same range but different cross-range positions fa and fb, respectively, can be
modeled by (4.1) as one scatterer located at (fa + fb)/2 in cross-range with x{n)


CHAPTER 1
INTRODUCTION
1.1 Background
RADAR stands for RAdio Detection And Ranging. By now, radar has been
used not only to detect and locate targets, as was originally intended, but to gen
erate images of illuminated scenes without the restriction of weather and time as
well. Radar image is defined as a two-dimensional (2-D) mapping of the reflectivity
corresponding to the scene illuminated by a radar. More specifically, a radar image
is the reconstruction of the illuminated scene from the electromagnetic reflectivity,
which is usually treated as a 2-D function in range and cross-range. The direction
in which a radar beam propagates is referred to as the range direction, while the
one perpendicular to the line of sight is called the cross-range direction, or the az
imuth direction (see Figure 1.1 for an illustration). Radar imaging [2] has played a
crucial role in both military applications, including reconnaissance and surveillance
[21], and civilian ones including geological and topographic map generation, weather
forecasting, scene classification, and target recognition [60, 65].
The quality of radar images depends greatly on the image resolution, which
is nominally referred to as the ability to distinguish two closely-spaced elements of a
target [48]. The higher the radar resolution, the more details of target features are
available for detection, characterization, and identification of targets of interest. As
such, much effort has been made to improve 2-D imaging radar resolutions, which
include range and cross-range resolutions. High range resolution can be achieved
by using wide bandwidth signals, such as linear frequency modulated (FM) chirp
pulses [48], since range resolution is inversely proportional to the transmitted signal
1


APPENDIX A
A SINGLE SCATTERER FEATURE EXTRACTION VIA SPAR
The estimates of {, x, /, /} can be obtained by minimizing the following NLS
cost function:
£
Ci(fxJ,/) = ||Y.-e*G(x,/,/)||
JV-1
= Iys" ~ x()e27Tfn+ =0
After simple calculations, we can rewrite (A.l) as
-1 f r
C2 (, x, /, /) = ] lly^ll2 + 7V x() ~ Re (
r,(2 nfn+4>)yT^N{f)
(A.l)
Tl0
1
e 3'(2'/+*)ylu'M))
*1 2
(A.2)
Minimizing (A.2) with respect to x(n) yields
x(n) = He
v N
-j(2irfn+) T
y nv*N(f)
, n = 0,1, , N 1.
(A.3)
Hence
x = -Re{e-"[Yja.*w(/)]0^(/)}
= ^[e-2*y,(/)0u.-s(/)].
(A.4)
Inserting (A.3) into (A.2), we obtain the NLS estimates {(f>,/,/} of {0,/,/} by
equivalently maximizing the following cost function:
-1
n=0
= \ x {|y>(/)l2 + R* [(yf,^(/))2 e-J[2'(2,>+2*1]} (A.5)
98


9
(3) We have devised a novel, robust, and computationally simple Semi-PARametric
(SPAR) algorithm for SAR target feature extraction and image formation based
on a flexible data model;
(4) Experimental results using ERIM and MSTAR Slicy data show that the relaxation-
based parametric approaches including RELAX and RELAX-NLS and the semi-
parametric algorithm SPAR, can yield SAR images with higher resolution than
FFT based methods. Also SPAR is more robust and computationally simple
than RELAX-NLS.
(II) 3-D target feature extraction and motion compensation via CLSAR
(1) We have established a 3-D data model;
(2) We have performed Cramr-Rao Bound (CRB) analyses of parameter estima
tion using various flight trajectories, which provide more insight into the per
formances associated with different curvilinear apertures;
(3) The effects of aperture errors on target feature estimation have been investi
gated and a relaxation-based approach for autofocus and target feature extrac
tion using CLSAR has been introduced.
1.4 Dissertation Outline
The remainder of the dissertation is organized as follows. Chapter 2 gives a
literature survey of emphasis on the topics considered, including 2-D SAR imaging, 3-
D SAR target feature extraction, and motion compensation (autofocus). In Chapter
3, we describe how to form super resolution 2-D SAR images via the relaxation-
based parametric spectral estimation algorithms including RELAX and RELAX-
NLS. A novel semi-parametric (SPAR) method is introduced in Chapter 4 for SAR
image formation when an accurate parametric data model in cross-range is difficult


87
Cartesian grid. When the Cartesian grid is fine enough, the errors introduced by the
approximation are negligible. In this chapter, we use a coarse Cartesian grid and the
FFT method to obtain an initial estimate of the parameters and then minimize (6.36)
via a multidimensional search method, such as the FMINV function in PV-WAVE.
The latter approach uses less computer memory.
6.3 Experimental and Numerical Results
We first present an experimental example to show the performance of the
AUTORELAX algorithm. The field data was obtained by the Deployable Signature
Measurement System (DSMS), Carderock Division, Naval Surface Warfare Center,
Bethesda, Maryland. The radar was carried on board of a helicopter. The radar
carrier frequency is 9.449 GHz and the bandwidth 0.498 GHz. The data set was
obtained with a curved aperture not exactly known but is roughly the same as the
dash line shown in Figure 6.1(a), where there are 64 look angles and 64 samples per
look angle. The radar was about 300 meters away from the ground target. The ground
target consists of 13 corner reflectors on the ground plane and 7 corner reflectors
mounted on a wooden tripod that is about 2.65 meters tall. The true distribution of
the scatterers is shown in Figure 6.1(b), where the centers of the squares denote the
locations of the scatterers in 3-D space and the length of each square is proportional
to the modulus of the RCS of the corresponding scatterer. The triangles show the
projections of the scatterer locations onto the ground plane and their sizes are also
scaled to be proportional to the RCSs of the scatterers.
Figures 6.1(c) and (d), respectively, show the scatterer distribution obtained
with the RELAX algorithm from the initial and an manually adjusted apertures
shown in Figure 6.1(a). We note that as compared to Figure 6.1 (b), the results in
Figure 6.1(d) are obviously better than those in Figure 6.1(c).
We now first consider autofocusing only in the elevation direction by as
suming that no errors exist in both the azimuth angles {#m}m=i and the distances


(a)
(b)
(c) (d)
Figure 4.1: Ambiguity effect on the SAR image formation in the absence of range
estimation errors, (a) True windowed FFT SAR image, (b) Windowed FFT image
of the first scatterer. (c) Windowed FFT image of the second scatterer. (d)
Combined windowed FFT image of the two scatterers. (The vertical and horizontal
axes are for range and cross-range, respectively.)


REFERENCES
[1] D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, and Y. Chu.
XPATCH: a high-frequency electromagnetic scattering prediction code and en
vironment for complex three-dimensional objects. IEEE Antennas and Propa
gation Magazine, 36(1):6569, February 1994.
[2] D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio.
Developments in radar imaging. IEEE Transactions on Aerospace and Electronic
Systems, 20(4):363-400, July 1984.
[3] W. J. Bangs. Array processing with generalized beamformers. Ph.D. dissertation,
Yale University, New Haven, CT, 1971.
[4] Sergio Barbarossa, Luigi Marsili, and Gaetano Mungari. SAR super-resolution
imaging by signal subspace projection techniques. AEU International Journal
of Electronics and Communications, 50(2): 133138, March 1996.
[5] G. R. Benitz. Adaptive high-definition imaging. SPIE Proceedings on Optical
Engineering in Aerospace Sensing, Orlando, FL, pages 106-119, April 1994.
[6] G. R. Benitz. High definition vector imaging for synthetic aperture radar. Pro
ceedings of the 31st Asilomar Conference on Signals, Systems and Computers,
Pacific Grove, CA, November 1997.
[7] Zhaoqiang Bi, Jian Li, and Zheng-She Liu. Super resolution SAR image for
mation via parametric spectral estimation methods. IEEE Transactions on
Aerospace and Electronic Systems, 35(1) :267281, January 1999.
[8] B. D. Bunday. Basic Optimization Methods. Edward Arnold Ltd, London, 1984.
[9] J. P. Burg. Maximum entropy spectral analysis. Proceedings of the 31th Meeting
Society of Exploration Geophysicists, Oklahoma City, OK, Oct. 1967.
[10] J. Capon. High resolution frequency-wavenumber spectrum analysis. Proceedings
of the IEEE, 57:1408-1418, August 1969.
[11] Walter G. Carrara, Ron S. Goodman, and Ronald M. Majewski. Spotlight Syn
thetic Aperture Radar Signal Processing Algorithms. Artech House Inc., Nor
wood, MA, 1995.
[12] I. J. Clarke. Supervised Interpretation of Sampled Data Using Efficient Imple
mentations of Higher-Rank Spectrum Estimation, Vol. II. Prentice Hall, Engle
wood Cliff, NJ, 1991.
[13] S. R. DeGraaf. SAR imaging via modern 2-D spectral estimation methods. SPIE
Proceedings on Optical Engineering in Aerospace Sensing, 2230:36-47, Orlando,
FL, April 1994.
104


8
Motion error compensation also plays a crucial role in accurately extracting
target features using CLSAR. Motion errors in 2-D SAR systems can be approximated
as phase errors in cross-range since the azimuth angle 6 is very small and phase errors
are mainly introduced by the uncertainty in the distance from the radar to the center
of the illuminated scene [29]. However, it is not the case for CLSAR since curvilinear
aperture determined by azimuth angle 9 and elevation angle is critical in obtaining
3-D information of the target. Hence motion errors in 6 and (j) may not be negligible.
Motion compensation (autofocus) in CLSAR turns out to be more complicated than
that in 2-D SAR systems. In Chapter 6, we investigate the effects of aperture errors
on 3-D target feature extraction and present a relaxation-based autofocus algorithm,
referred to as AUTORELAX, to extract target features in the presence of aperture
errors using CLSAR.
1.3 Contributions
The significance of this research lies in that it examines the possibility of super
resolution image formation and autofocus of small targets via spotlight-mode SAR
system by using relaxation-based parametric and semi-parametric spectral estimation
methods. Super resolution SAR images obtained via the proposed methods can
be further used for SAR image segmentation and automatic target detection and
recognition. This research makes contributions in the following two areas:
(I) 2-D super resolution SAR image formation
(1) We have extended RELAX to form enhanced SAR images of targets consisting
of point scatterers;
(2) We have extended RELAX-NLS for super resolution SAR image formation of
targets consisting of both trihedrals and dihedrals;


103
(x,y,z) = (4.9991,3.9346,2.0477). Note that in both examples, (x,y,z) are very
close to (x,y,z) and hence can be used to approximate (x,y,z).



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4.6 Cramr-Rao Bound (CRB) of the Semi-Parametric Data Model
Let Y be an N x matrix with its (n, )th element being y(n,fi) with
52
K
y(n, n) = ^ xk{n)eik + e(n, n), n 0,1, , N 1, n = 0,1, , N 1,
k=1
(4.51)
where K denotes the number of scatterers; {xk()}%=Q denotes the real-valued am
plitude function of for the /cth scatterer; q*)k and {fk,fk}, respectively, are the
constant phase and the frequency pair of the £;th scatterer; finally, {e(n, )} denotes
the unknown 2-D noise and clutter sequence. Then we can rewrite (4.51) as:
K
Y = £e*Wjv(/*)x£D(/fc) + E,
(4.52)
k=1
where E denotes an N x N matrix with e(n, n) being its (n, n)th element,
n t
>N{fk)
wv(/fc)
l e^k . e^¡k{N-\)
1 gj27r/fc . ej2irfh(N-l)
(4.53)
1 T
(4.54)
with (-)r denoting the transpose, D(fk) denoting the following diagonal matrix:
D(/*)=diag| l, ..., e^U-i) \
and
X/c =
*1 T
xk(0) xk(l) xk(N 1)
(4.55)
(4.56)
Let y = vec(Y) = [ yf y?
yL ]T with yn{n 1,2, , N) being the nth
column of Y. Then the data model can be rewritten as follows:
y = A/i. + e,
(4.57)
where
A -
[w(/l) OXl] ww(/i) (w/i(/jt)XK]WW(/i-)
(4.58)


CHAPTER 5
3-D TARGET FEATURE EXTRACTION USING CLSAR
5.1 CLSAR Data Model
We describe how to obtain 3-D target features via the full synthetic aperture
shown in Figure 1.2(a), which makes a preparation for the discussion on CLSAR. We
start by establishing one-dimensional (1-D) point scatterer data model of high range
resolution radar as a preparation for the establishment of 3-D data model herein.
5.1.1 1-D Data Model for High Range Resolution Radar
The range resolution of a radar is determined by the radar bandwidth. To
achieve high resolution in range, the radar must transmit wide band pulses, which
are often linear FM chirp pulses [29, 49]. A normalized chirp pulse can be written as
s{t) = e-A2*/ot+7t2), |t| < To/2, (5.1)
where f0 denotes the carrier frequency, 2q denotes the FM rate, and T0 denotes the
width of the pulse. We assume that /0, 7, and T0 are known. The signal returned by
a scatterer of a target has the form
r(t) = (5.2)
where ST is determined by the RCS of the scatterer and t denotes the round-trip
time delay. The demodulated signal d(t) is obtained by mixing r(f) with s*(t r0)
for some given r0 = 2R0/c, where ()* denotes the complex conjugate, c denotes the
propagation speed of the transmitted electromagnetic wave, and R0 denotes the range
distance from radar to the center of the illuminated patch.
_ (5Tet2(7r/o-7To)(T-To)e-j7(T-To)2ej27(r-ro)i (5-3)
63


CHAPTER 7
CONCLUSION
7.1 Summary
In this study, we have extended two parametric relaxation-based algorithms,
including RELAX and RELAX-NLS, and presented a novel semi-parametric ap
proach SPAR to form enhanced 2-D SAR images in different scenarios. An effective
relaxation-based parametric method, referred to as AUTORELAX, has also been pro
posed for both 3-D target feature extraction and motion compensation via CLSAR.
The RELAX and RELAX-NLS algorithms are parametric spectral estima
tion methods and provide the potential for significant resolution improvement of the
formed SAR images provided that the assumed data model is valid. RELAX is based
on the point scatterer (trihedral) data model and RELAX-NLS is devised according
to a mixed dihedral and trihedral data model, which describes a dihedral corner re
flector as a 2-D complex sinusoid with amplitude described as a sine function and
a constant phase in cross-range and with constant amplitude and phase in range.
Super resolution SAR images can be generated by first extracting the target features
via RELAX or RELAX-NLS based on the corresponding data models, and then ap
plying FFT to the synthesized phase history data with larger dimensions in both
range and cross-range by using the extracted target features. Experimental examples
with ERIM and MSTAR data in Chapter 3 have shown that RELAX and RELAX-
NLS can be used to obtain either higher resolution images from the same size data,
or, equivalently, equal-quality images from significantly fewer number of data sam
ples. Of the two methods, RELAX is more suitable for point scatterer targets and
94


86
shown in the previous chapter, the estimates {ak, fk, fk, fk}k=1 of {afc, fk, fk, fk}k=1
can be obtained with the RELAX algorithm by equivalently minimizing the following
NLS criterion:
/ K N M N-1
f {fe, fk,fk, a} j = ^2 m)~ m)i2 > (6-32)
' m=1 n=0
where y(n,m) = y(n,m)e~:iflm and z3(n,m) has the same form as z(n,m) in (6.5)
except that and iZm(n) are replaced by tXm(n), tym(n), and iZm(n),
respectively, which are determined by {9m, instead of {9m, 4>m}^=1. Let
M N-l
Fk (ak, fk, fk, /*.) = X] S |^(nm)
J 2k If Am W+fkiym (n)+fkizm Ml
| (6.33)
m=1 n=0
where tXm(n), tym(n), and tZm(n) are the same as tXm(n), tym(n), and tZm(n), respec
tively, except that {9m,(j)m}^=1 are replaced by {9mAm}m=i, and
K
yk(n,m) = y(n,m) Q,eAr[/Lm(n)+/Lm(ri)+/Lm(n)]^ (6.34)
il,i^k
Then minimizing Fk in (6.33) with respect to ak yields the estimate otk of ak:
Yfm-l En-0 yk(n> m)e~j2K[fkLmn+fkiym(n)+fkizm(n)]
Oik =
and
{fk, h, fk) = ar§ max
fk ,fk,fk
MN
M N-l
fk=fk,fk=fk,fk=fk
^ y ) yk(n, m)e f2n^kixm(n)+fkiym(n)+fkiZm(n)] (g_36)
m=1 n=0
Herein, we omit the detailed steps used for the 3-D target feature extraction
since they are the same as those in the previous chapter except that yk in (5.25)
is replaced by {yk(n,m)} in (6.34). The practical convergence in the iterations
of the RELAX algorithm now is determined by checking the relative change of the
cost function F fk, fk, fk}kz=^j in (6.32) between two consecutive iterations. To
speed up the RELAX algorithm via utilizing FFT, a Cartesian grid mapping approach
is introduced in Chapter 5 to approximately map {iXm(n), iyrn(n), iZrn(n)}m=i on a


This work is dedicated to my wife, Jing Wang, for her love and sharing my
happiness and depression in the past years; to my parents, who continue supporting
and encouraging me for my education.


88
{R0(m)}M=1. The AUTORELAX algorithm converges after 6 iterations. Figure 6.2
shows the target scatterers extracted by AUTORELAX with the search interval for
{A0m}^f=1 being 0.34. Comparing Figures 6.2(b) and 6.1(d) with Figure 6.1(b),
we find that AUTORELAX works well and the AUTORELAX results are slightly
better than those obtained by using the manually adjusted aperture. Figure 6.3 shows
that the manually adjusted and autofocused apertures fit quite well after adding a
line to the former. Since the results in Figure 6.2(b) are better than those in Figure
6.1(d), it appears that a linear phase error (in (j)m as a function of 6m) was introduced
in the manually adjusted aperture. The linear phase error can cause shifts in /*, and
the amount of shift is different for different scatterer k, as can be seen from (6.14).
Consider next autofocusing in both the elevation and azimuth directions by as
suming that no errors exist in {R0(m)}^=1. The AUTORELAX algorithm converges
after 7 iterations. Figure 6.4 shows the results obtained with the search intervals for
{A0m}£f=1 and {A9m}^=1 being 0.34 and 0.006, respectively. Compared with
the results obtained by autofocusing only in the elevation direction, we find that aut
ofocusing in both directions provides little further improvement to the accuracy of
the estimated target parameters in this example. Hence for this example, it appears
that the aperture errors mainly occur in the elevation direction.
For this experimental example, we have also used AUTORELAX to extract
target features when we assumed that errors exist in both and
{Ro(m)}M=1. Again, as compared with Figure 6.2(b), we have noticed little change
in the extracted target parameters. It could be that DSMS has already done a good
job compensating for the errors in {Ro(m)}^=l with some traditional method so that
they are now negligible as compared with the errors in { Finally, we use a simulation example to show that the accuracy of target
feature extraction via CLSAR is very sensitive to the accuracy of the curvilinear
aperture. We assume that there are 20 scatterers with a similar distribution to that


62
(e) (f)
Figure 4.5: Comparison of SAR images obtained via different algorithms for the
Slicy data hbl5533.015 (0 azimuth and 30 elevation angles), (a) Windowed 2-D
FFT SAR image from the XPATCH data, (b) Windowed 2-D FFT SAR image from
the Slicy data, (c) Hybrid SAR image, (d) SPAR SAR image, (e) RELAX-NLS
SAR image, (f) Modified RELAX-NLS SAR image, (c) ~ (f) are all obtained from
the data used in (b). (The vertical and horizontal axes are for range and cross-range,
respectively.)


15
and parametric methods for feature extraction and image formation of SAR targets
has not been addressed in the literature before.
2.2 3-D SAR Feature Extraction
One relatively mature technology for 3-D SAR imaging and feature extrac
tion is IFSAR [24, 55, 69, 27, 70] which exploits two different apertures. A simple
implementation of a IFSAR is to use two antenna displaced in the cross-track plane
to obtain two coherent and parallel measurements. The relative height information
can be estimated from the phase difference between the two measurements. The use
of interferometric techniques with SAR imagery to provide height information makes
IFSAR suitable for 3-D imaging since it inherits the finer resolutions (in range and
cross-range) associated with the spotlight-mode SAR and also has reasonable height
resolution. Many 2-D modern spectral estimation algorithms including Capon [10],
APES [38], and MUSIC [58] have been generalized for 3-D imaging and target feature
extraction using IFSAR [40, 47]. An NLS method that outperforms MUSIC in speed
and estimation accuracy was also reported in [40]. However, such a system cannot
resolve more than one target scatterer at the same projected range and cross-range
but at different heights [40].
Another relatively new technology for 3-D imaging is curvilinear SAR. CLSAR
traverses a curvilinear aperture path and avoids the height ambiguity problem inher
ent in IFSAR systems. CLSAR images are useless in practice due to high sidelobes.
However, CLSAR can be used in conjunction with modern spectral estimation meth
ods to accurately extract 3-D features of small targets, which consist of a finite number
of isolated point scatterers. CLSAR is still at its exploratory stage of development
and only a few studies were reported in the literature. In [33], a coherent CLEAN
algorithm was presented by Knaell to effectively eliminate sidelobes in CLSAR im
ages. The CLEAN algorithm was originally proposed in radio astronomy [28] and


64
The term e_i7^T_T0^2 in (5.3) is usually close to a constant for all rm¡n < r, r0 < rmax,
where rmax and rm¡n correspond to the maximum and minimum values, respectively,
of the round-trip time delays between the scatterers of a target and the radar. This
term can also be partially removed [49]. Let D(u) denote the Fourier transform of
~ ~ u2
d(t). Then the inverse Fourier transform of D(uj)eJi:i will have the term e_J7^T_To^
removed. Yet this removal can only be approximate since d(t) is not known for all
t and hence D(lo) is not known exactly. The closer e_J7(T-T)2 is to a constant for
rmin < t < rmax, the better its removal. With this removal, we have
d[t) = (5TeJ'2{7r/-7ro){T-7'o)ej27{T-To)i, (5.4)
which is a complex sinusoid with frequency 27(7-r0) and amplitude <5Te-?2(7r-f0-7T)(T~T0b
We know rmax and rmjn approximately since we assume that the altitude, antenna
beamwidth, and grazing angle of the radar are known. We also assume that (rmax
rmin) < To- Then for T0/2 + rmax < t < T0/2 + rmin, the scatterers of the target at
different ranges correspond to different frequencies of the signal d(t), while the RCSs
of the scatterers are proportional to the amplitudes of the corresponding sinusoids.
The ranges and RCSs of the target scatterers are the 1-D target features.
5.1.2 Full Synthetic Aperture Radar
A broadside data collection geometry in a spotlight-mode SAR is shown in
Figure 5.1 [29], The XY Z coordinate system is centered on a small patch of ground,
where a target is located. The ground is illuminated by a narrow radio frequency
(RF) beam from the moving radar that rotates (with radius Rq) around the coordinate
origin. In Figure 5.1, R denotes the distance between the radar and a scatterer at
the position (x,y,z), and 9 and (j) are the azimuth and elevation angles of the radar
relative to the XYZ coordinate system. We assume that 9, 4>, and R0 are known.
The range R of the scatterer located at (x, y, z) can be written as
R = [(Ro cos 9 cos 4> x)2 + {Ro sin 9 cos y)2 + (Rq sin

75
where Ir denotes an x (N) matrix with each column and row containing
only one non-zero unit element corresponding to the locations of the available data
samples in the rectangular grid. Note that the larger the dimensions of the rectangular
grid, the more accurate the mapping approximation. For large enough dimensions,
however, the noise will become the dominant source of error. For our examples, we
chose iV = N = = 128. As compared to Polar-to-Cartesian interpolation, the grid
mapping approach avoids the interpolation step needed by the former, which could
be complicated for an arbitrary curvilinear aperture. The former approach, however,
is computationally more efficient due to its smaller data dimensions.
The steps of applying RELAX to yr are similar to those for the case of full
aperture except that Equations (5.26) and (5.27) above are replaced by
[Ir(afc <8> a* afc)]'^ yrk
Oik =
N
and
Uk =Uk ,Uk =Uk ,Wk =Uk
{c5fc, ujk} = arg max [Ir(afc afc a*)] yTk
(5.40)
(5.41)
respectively, where
Yrk = Yr
K
Y <*dr
(5.42)
Let yric be similar to y except that the elements in y that are missing in yTk are
replaced with zeros in yTk. Then the right hand side of (5.41) can also be computed
by applying 3-D FFT to yTk. Since the dimensions of the rectangular grid are larger
than the dimensions of the original data due to zero-filling, the 3-D FFT of yTk is
approximately periodic and we should limit our attentions to only one period. For our
examples, the former is four times as large as the latter and hence the peak searching
in (5.41) is limited to the frequency intervals [ 2n x (0.125) 2tt x 0.125 ] Note
that the computational advantages of FFT over DFT (discrete Fourier transform)
and backprojection diminish as the dimensions of the rectangular grid increases,
especially for curvilinear apertures.


APPENDIX C
CALCULATING (x,y,z) FROM (x,y,z)
Given R0, 0O, and (x,y,z), (x,y,z) can be determined from (5.8), (5.9), and
(5.10) as follows:
vlv+ 2ViRo [vi (l + r&) (R0 + 2t]2z) + 7?iRo 2rj¡x] ye + R20 [2r% (l + 772) y2
+47i + 1 )2 2 + + 4R0???7?2 (5 + rfe) z 877^771 (l + rfe) xz
+4R0 (2?7i 3772) x + Rlrj[ (-8 + 772)] y4 877^ (771^0 + 2r)\x) yy3
+2772i?o [25 (1 + 77!) +Ro?]2 (3 + ril) + 2t7^77i] t/2t/2 87]lR40y3y +
= 0, (C.l)
z = {vivh4 + Ro [RoVi(2 + V2) + Zvir]2z(l + vl)] y2
-2yyxRly R^^^y2} / [i?0(^2/ + yv¡)y\ (C.2)
and
2 = \ {v?W + Ro?7i [277i(1 + 772) + 772(77!Ro 2772)] y2
-2rj2R20yy + Ro%y2(l + ^2)} / [-Ro(t7iS/ + VvDv] > (C.3)
where 771 = cos0o and 772 = sin0O- Note that the left side of (C.l) is an eighth-order
polynomial, and hence has eight zeros. We pick the root that is closest to y as the
solution for y.
We now present two examples. First, let Rq = 15, 00 = 7t/5, and (x,y,z) =
(0.5,0.4,0.3). We obtain (x,y,z) = (0.5005,0.3849,0.3078). (We remark that the
other seven possible solutions for y are either complex or far away from y.) Next, let
R0 300, 0O = 7t/4, and (x,y,z) = (5,4,2). We obtain
102


13
In recent years, many sophisticated and high resolution methods based on
point scatterer data model, such as eigendecomposition based methods [4], including
MUSIC (Multiple Signal Classification) [58] and ESPRIT (Estimation of Signal Pa
rameters via Rotational Invariance Technology) [56], and relaxation-based nonlinear
least squares (NLS) method such as RELAX [39], have also been applied to super
resolution SAR image formation. The eigendecomposition based methods are signal
or noise subspace based approaches and generally are not suitable for SAR image
formation since they require that the number of data samples be much larger than
that of the dominant point scatterers and the process of whitening the noise sub
space (used in MUSIC) will destroy the spatial inhomogeneities of the terrain clutter
[15]. The RELAX algorithm [39] was originally proposed for sinusoidal parameter
estimation in colored noise. RELAX obtains the parameter estimates by minimizing
the NLS cost function via a sequence of FFTs, which is conceptually and computa
tionally simple and hence robust. RELAX coincides with the maximum likelihood
(ML) method when the noise is white and is also asymptotically (for large numbers
of data samples) statistical efficient when the noise is an AR or autoregressive mov
ing average (ARMA) process that can be described by a finite number of unknowns
[63]. Due to the iterations performed in each step of RELAX, a set of more accurate
initial estimates are available, which make RELAX converge to the global minimum
with a high probability. RELAX can also be referred to as SUPER CLEAN when
compared with two other approximate relaxation methods including CLEAN [28] and
MCLEAN (more clean) [23]. CLEAN was first proposed in radio astronomy and later
used in microwave imaging [66]. If no iteration is performed in each step of RELAX,
then it is equivalent to CLEAN. RELAX becomes MCLEAN if only one iteration
is performed in each step of RELAX. Hence RELAX is a computationally efficient
and robust algorithm with good statistical performance and is suitable for super
resolution SAR image formation of targets consisting of point scatterers (trihedrals).


83
6.2 The Relaxation-Based Autofocus Algorithm (AUTORELAX)
The AUTORELAX algorithm obtains the estimates {9m, and
A A 'L o 0
{&k,fk,fkifk}k=v respectively, of the true values {9m, m, 7?m}£f=1 and
{ak, fk, fk, fk}k=i by minimizing the following NLS criterion:
/ K \ M N-l
Cs ^ ^ |?/(n,m) x(n,m)|2, (6.23)
m=1 n=0
where
x(
(6.24)
;(n, m) = (X akej2*\Jktxm(n)+fktym(n)+fktZm(n)] ^ gj'r/m ,
with tXm(n), tym(n), and tZm(n) defined in (6.2), (6.3), and (6.4), respectively. Note
that as shown below, we will determine 9m and cj)m via a search method. Hence there
is no need to use the approximations such as those in (6.11), (6.12), and (6.13) for
txm (n)) y (n) > and tzm (n) j respectively. We use an alternating optimization approach
to minimize C5 in (6.23) and the AUTORELAX algorithm is outlined as follows:
Step 1: Extract the target features {ak, fk, fk, fk}k=i with the RELAX algorithm
from an initial curvilinear aperture by assuming that {r¡m 0}£f=1.
Step 2: Update the curvilinear aperture {0m,0m}=1 and the phase error {r7m}m=i
with Steps (1) (4) (to be discussed in Section 6.2.1). (If {r]m}^=1 is known to be
negligible, then only update {9m,(/)m}^=1 with Step (1).)
Step 3: Redetermine the target parameters with the RELAX algorithm by using
the curvilinear aperture {9m, (f>m}^=l and {vm}^=1 obtained in Step 2.
Step 4: Repeat Steps 2 and 3 until practical convergence.
The practical convergence of AUTORELAX is determined similarly to that
of the RELAX algorithm by checking the relative change of the cost function C5 in
(6.23) between two consecutive iterations.
6.2.1 Aperture Parameter Estimation
Assume that the target feature estimates {&k, fk, f fk}k=i are given- Then
6m, 4>m, and f}m can be determined by minimizing the following Cm with respect to


29
without the background clutter are illustrated in Figures 3.1 (c) ~ (f) with K = 59.
Note that the RELAX images have a higher resolution than the FFT images for the
dominant scatterers. Utilizing only 25% of the 40 x 40 phase history data, i.e., using
a 20 x 20 phase history data, we form the 2-D FFT SAR images shown in Figures
3.2(a) and (b). Figures 3.2(c) ~ (f) show the RELAX images with K 41. Compar
ing Figures 3.2(f) and 3.2(b), we note that the two images are quite similar although
the former uses only 25% of the data used by the latter.
Consider next an example of the MSTAR Slicy data consisting of both trihe
dral and dihedral corner reflectors collected by the SNL using the STARLOS sensor.
The field data was collected by a spotlight-mode SAR with a carrier frequency 9.559
GHz and bandwidth 0.591 GHz. The radar was about 5 kilometers away from the
ground target shown in Figure 3.3. The SAR images are obtained when the tar
get is illuminated by the radar from the azimuth angle 0 and elevation angle 30.
The range and cross-range resolutions of the original data (54 x 54) are 0.3 meters
and 0.32 meters, respectively, and the windowed 2-D FFT SAR image (obtained
via Taylor window) is shown in Figure 3.4(a). To demonstrate the super resolution
property of RELAX and RELAX-NLS, we use part of the original data, whose di
mension is 32 x 32. Figure 3.4(b) shows the windowed 2-D FFT images with the
spoiled resolution 0.51 x 0.54 meters obtained from this reduced size data. RELAX
and RELAX-NLS are also applied to this 32 x 32 phase history data matrix. Figures
3.4(c) and (d), respectively, show the windowed RELAX SAR images with and with
out estimated background clutter with K = 36. We note that RELAX images have a
better resolution than the FFT images for the dominant trihedrals and resemble the
FFT images for dihedrals even though the data model in the cross-range dimension
used by RELAX is not correct for this example. The windowed RELAX-NLS SAR
images with and without estimated background clutter with K 9 are illustrated
in Figures 3.4(e) and (f), respectively. We note again that the RELAX-NLS images


Elevation Angle (degrees)
93
i¡t.ra>in*
(c) (d)
Figure 6.5: (a) Curvilinear aperture for the simulation example, (b) True scatterer
distribution for the simulation example, (c) Scatterer distribution obtained from the
initial aperture (dashed line) in (a) by using RELAX with K = 20. (d) Scatterer dis
tribution obtained via AUORELAX with K 20 and using the autofocused aperture
(dotted line) shown in (a).


101
2xy sin 9 cos <¡>\cos 0O (sin 0O)0] + 2yz sin 9 cos 0[sin 0O + (cos 0O)ip\
+2xz cos 9 cos 0[sin 0O + (cos 0O)ip]
2(.in/ cos 0O + yz sin 0o) sin 9 cos 0 + (2xz sin 0O) cos 9 cos 0
+ (2a:zcos2 0o)0-
Hence R is simplified to (5.6).
(B.5)


31
(e) (f)
Figure 3.1: RELAX SAR Images by using the 40 x 40 ERIM data with K = 59.
(a) Unwindowed 2-D FFT image, (b) Windowed 2-D FFT image, (c) Unwindowed
RELAX image without background clutter, (d) Windowed RELAX image without
background clutter, (e) Unwindowed RELAX image with background clutter, (f)
Windowed RELAX image with background clutter.


21
We consider using GAIC (see [39] and the references therein for details) to
determine K, the number of sinusoids, by assuming the unknown noise and clutter
being white. The estimate K of K can be determined as an integer that minimizes
the following GAIC cost function:
/ Nl 1 \
GAIC^ = N\n EE |e(n, n)|2 j + j\n[ln(NN)](4:K + l), (3.8)
where
K
^ 710 71=0
(n, n) = y(n, n) ^ Ajte-7^'cn+Wfcn\ n = 0,1, , N 1, n = 0,1, , N 1,
k=1
(3.9)
4K + 1 denotes the total number of unknown real-valued parameters (of which 4K
are for the sinusoids and 1 is for the white noise), and 7 is a parameter of user choice.
3.1.2 Image Formation
We can form SAR images by using the point scatterer data model and the
target features extracted via RELAX. SAR images obtained via RELAX are referred
to as RELAX SAR images or simply RELAX images. The steps of using RELAX
for SAR image formation are outlined as follows.
Step 1: Obtain the parameter estimates of {o^, uk, 4}^ yia RELAX by using the
measured phase history data (see Section 3.1.1 for details).
Step 2: Generate ys{ns,s), the simulated phase history data of large dimensions,
from the estimated parameters and based on the data model in (3.1):
K
, ns = 0,1, , CAT 1, s = 0,1, , ( 1,(3.10)
y.
,{ns,ns) = ^ ake-
j(uikns+ujkns)
fc=1
where £ (£ > 1) denotes the extrapolation factor, which is a parameter of user
choice, and 0>fc, and u>k, respectively, denote the estimates of ak, u)k, and wfc,
k = 1,2, , i£.
Step 3: Form RELAX SAR images containing only the dominant target features by


10
to establish. In Chapter 4, we also present a modified RELAX-NLS algorithm by
using the initial conditions provided by SPAR to save the amount of computations
needed by the original RELAX-NLS algorithm. In Chapter 5, a CLSAR data model
is established and RELAX is introduced for 3-D target feature extraction via CLSAR.
Chapter 6 investigates the ambiguity problems inherent in the proposed CLSAR data
model when aperture errors are present. A relaxation-based autofocus algorithm for
3-D target feature extraction in the presence of aperture errors is also presented in
Chapter 6. Finally, Chapter 7 give the summary and conclusions of this research and
outlines the future work.


LIST OF FIGURES
1.1 Illustration of 2-D radar imaging 2
1.2 Illustration of SAR imaging modes, (a) Spotlight-mode SAR. (b)
Strip-map SAR 3
1.3 Possible apertures for a 3-D SAR system, a) Full aperture, b) Parabolic
aperture, (c) Orthogonal subaperture. ( 9 and denote the azimuth
and elevation angles, respectively.) 6
1.4 Mesh plots of the modulus of the RCS obtained by using 2-D FFT with
different apertures (range information suppressed for the illustration
purpose only), (a) Full aperture as shown in Figure protect 1.2(a). (b)
Curvilinear aperture as shown in Figure 1.2(c) 7
3.1 RELAX SAR Images by using the 40 x 40 ERIM data with K 59. (a)
Unwindowed 2-D FFT image, (b) Windowed 2-D FFT image, (c) Un
windowed RELAX image without background clutter, (d) Windowed
RELAX image without background clutter, (e) Unwindowed RELAX
image with background clutter, (f) Windowed RELAX image with
background clutter 31
3.2 RELAX Images by using the 20 x 20 ERIM data with K 41. (a)
Unwindowed 2-D FFT image, (b) Windowed 2-D FFT image, (c) Un
windowed RELAX image without background clutter, (d) Windowed
RELAX image without background clutter, (e) Unwindowed RELAX
image with background clutter, (f) Windowed RELAX image with
background clutter 32
3.3 Target photo taken at 45 azimuth angle 33
3.4 SAR images of the Slicy data (0 azimuth and 30 elevation angles). (a)
Original windowed 2-D FFT image, (b) Windoed 2-D FFT image with
spoiled resolution, (c) ~ (d): Windowed RELAX image without and
with background clutter with K = 36. (e) ~ (f): Windowed RELAX-
NLS image without and with background clutter with K = 9. (c) ~
(f) are obatined using the data in (b). (The vertical and horizontal
axes are for range and cross-range, respectively.)
vii
34


12
[37]. Empirically, it was observed in [38] that Capon appears to give biased spec
tral estimates whereas APES is unbiased. This observation was later proved in [37]
by using the second-order Taylor expansion technique. Also the statistical perfor
mance of the Capon estimator degrades rapidly as the filter length increases, while
that of APES remains unaffected. The study in [37] gives a compelling reason for
preferring APES to Capon in most spectral estimation applications including SAR
imaging. In cases where an appropriate parametric data model of targets is not
available, APES is a good candidate for SAR imaging which provides much lower
sidelobes and narrower spectral peaks than Capon and the conventional FFT based
methods. However, significant resolution improvement cannot be obtained by using
non-parametric methods since they do not fully exploit the characteristics of targets,
especially when such information is available.
Parametric Spectral Estimators
When parametric data models can be used to describe radar targets, paramet
ric spectral estimators provide attractive alternatives for super resolution SAR image
formation. The Burgs maximum entropy (ME) method [9] was applied to radar im
age enhancement by Mancill in 1977 [44]. Unlike FFT methods which assume that
the signal beyond the observations is zero, ME methods make linear prediction of the
signal beyond the observation window so that they can achieve better resolution than
FFT methods. Identical to ME methods in principles, the autoregressive (AR) model
based methods [15, 13, 20, 19, 26] have attracted a lot of research attention. However,
the high resolution property of the AR based methods are highly dependent on the
presence of observation noise [31]. As the signal-to-noise ratio (SNR) decreases, so
does the resolution. On the other hand, the AR based methods are more appropriate
for SAR image formation of distributed scatterers rather than point scatterers since
the AR spectral estimator by nature is a continuous spectral estimator.


6
(a)
0
$
(c)
e
Figure 1.3: Possible apertures for a 3-D SAR system, a) Full aperture, b) Parabolic
aperture, (c) Orthogonal subaperture. ( 9 and <¡> denote the azimuth and elevation
angles, respectively.)
no accurate parametric data model is available in cross-range. Based on the flexible
data model, we present a robust and computationally simple semi-parametric (SPAR)
algorithm for target feature extraction and SAR image formation by exploiting a
hybrid of non-parametric and parametric spectral estimation methodes.
It has been shown in [62, 16] that three-dimensional (3-D) target features
including the height information as well as the 2-D target features provide very useful
information in certain applications, such as automatic target recognition (ATR). 3-D
SAR image formation and feature extraction are becoming active research areas. It
is possible to obtain 3-D information of an illuminated scene with Spotlight-mode
SAR when a non-straight line synthetic aperture is exploited, which is determined by
the azimuth angle 9 and elevation angle . In our work, we will also study 3-D target
feature extraction and motion compensation (autofocus) using curvilinear SAR, in
which the trajectory of a spotlight-mode SAR is a suitable curvilinear aperture.
Among existing 3-D SAR imaging techniques, interferometric SAR (IFSAR)
is now a well-developed technique that plays an important role in 3-D (as well as 2-D)
imaging. However, IFSAR suffers from ambiguity problems [11, 40]. Theoretically,
3-D target features and images can also be obtained when the trajectory of SAR is a
full aperture as shown in Figure 1.2(a) even though such an aperture is not feasible in


108
[61] T. Sderstrm and P. Stoica. System Identification. Prentice-Hall International,
London, U.K., 1989.
[62] B. D. Steinberg and B. Kang. Radar detection sensitivity as a function of target
dimensionality. IEEE International Radar Conference, pages 106-110, 1990.
[63] P. Stoica and R. L. Moses. Introduction to Spectral Analysis. Prentice-Hall,
Englewood Cliffs, NJ, 1997.
[64] P. Stoica and A. Nehorai. Performance study of conditional and unconditional
direction-of-arrival estimation. IEEE Transactions on Acoustics, Speech, and
Signal Processing, ASSP-38(10):1783-1795, October 1990.
[65] Kiyo Tomiyasu. Tutorial review of synthetic-aperture radar (SAR) with appli
cations to imaging of the ocean surface. Proceedings of the IEEE, 66(5):563-583,
May 1978.
[66] J. Tsao and B. D. Steinberg. Reduction of sidelobe and speckle artifacts in
microwave imaging: The CLEAN technique. IEEE Transactions on Antennas
and Propagation, 36:543-556, April 1988.
[67] D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and C. V. Jakowatz, Jr. Phase gradi
ent autofocus a robust tool for high resolution SAR phase correction. IEEE
Transactions on Aerospace and Electronic Systems, 30:827-835, July 1994.
[68] G. N. Yoji. Phase Difference Auto Focusing for Synthetic Aperture Radar Imag
ing. U.S. Patent 4,999,635, 1991.
[69] H. Zebker and R. M. Goldstein. Topographic mapping from interferometric
synthetic aperture radar observations. Journal of Geophysical Research, 91:4993-
4999, April 1986.
[70] H. A. Zebker, S. N. Madsen, J. Martin, K. B. Wheeler, T. Miller, Y. Lou, G. Al
berti, S. Vetrella, and A. Cucci. The TOPSAR interferometric radar topographic
mapping instrument. IEEE Transactions on Geoscience and Remote Sensing,
30(5):933-940, September 1992.


71
respectively, where
yCJb = y c
K
i
= l,ijk
{Ic( )} 0
(5.30)
Let yCfc be similar to y*, except that the elements in y*. that are missing in yCk are
replaced with zeros in yCk. Then the right hand side of (5.29) can also be computed
by applying 3-D FFT to yCk.
5.3 Performance Analysis of Parameter Estimation via CRB
Consider first the case of the full aperture. Let Q = Ejee^} be the noise
covariance matrix, which is arbitrary and unknown. The extended Slepian-Bangs
formula for the ijth element of the Fisher information matrix (FIM) has the form
[3, 64]:
{FIM},, = tr (cr^Q-'q;) + 2Re [(<*" A")'. Q~' (Aa)'] (5.31)
where X) denotes the derivative of X with respect to the zth unknown parameter,
tr(X) denotes the trace of X, and Re(X) denotes the real part of X. Note that FIM
is block diagonal since Q does not depend on the parameters in (Aa), and (Aa)
does not depend on the elements of Q. Hence the CRB matrix for the target features
of interest can be calculated from the second term on the right side of (5.31). Let
V =
ReJ (a) InF (a) a/r ojt u>T
where
a;
-i T
CO l 0J2 LOfc
CO =
lT
CO i O2 Ll>k
and
co
iT
CO 1 C2 COk
(5.32)
(5.33)
(5.34)
(5.35)


84
{Om,(pm,Vm}, where
Nl
Cm(Om,(l)m,Vm) = '^2\y{n,m) x^n^m)]2, n = 0, , N 1, m = 1,2, , M,
71=0
(6.25)
where x\(n, m) has the same form as x(n, m) in (6.24) except that {ak, fk, fk, fk}k=i
are replaced by {fc, fk, fk, fk}k=1. To simplify the optimization of Cm, we determine
{0m, Step (1): Obtain \f9jj1, > by minimizing the following Cm\ with respect to {$777, 5
where
JV-l
cmi (9m,fm) = l^771) ~ i(n>m)|2 n = 0, ,N- 1, m = 1, M,
n=0
(6.26)
where i(n, m) has the same form as z(n,m) in (6.5) except that {ak, fk, fk, fk}k=i
are replaced by {k, fk, /*,, /jtlitLi- When there are errors in both the elevation and
O o
azimuth directions, we can estimate 0m and Qm by the alternating minimization
^ o
approach, i.e., by iteratively fixing the estimate 0m of 0m and minimizing Cmi with
/V o
respect to 9m, and then fixing the estimate 9m of 0m and minimizing Cmi with respect
to 0m until practical convergence. The practical convergence in the alternating
minimization approach is determined by the relative change of the cost function Cmi.
In the numerical examples, we terminate the iteration when the relative change of the
cost function Cmi between two consecutive iterations is less than 10-3. When there
are errors in only one angular direction, i.e., either 6m or 4>m is to be determined,
then the minimization of Cmi is a simple one-dimensional search problem.
Step (2): Determine phase errors {fjm}m=1 by minimizing the following cost func
tions Cm2:
JV-l
Cm2 {Vm) = \y(nim) z2(n,rn)e:,11m\2 (6.27)
n=0


70
Step (2): Assume K = 2. Compute y2 with (5.25) by using {cok,Cjk,cuk,ak}k=i
obtained in Step (1). Obtain {bk, £ok, u>k, dfe}fe=2 from y2. Next, compute yi by using
{h,k,u>k,k}k=2 and then redetermine {£>fc, 4, 5fc, dfc}fc=1 from yi.
Iterate the previous two substeps until practical convergence is achieved (to
be discussed later on).
Step (3): Assume K = 3. Compute y3 by using {cjfe, 4, 4, ak}2k==1 obtained in Step
(2). Obtain {04, Cok,u)k, &k}k=3 from y3. Next, compute yi by using {4, 4, k, k}k=2
and redetermine {)k,Cok,Cok,ak}k=i from y^ Then compute y2 by using
{k,L)k,u)k,k}k=i and redetermine {uk, wk, u>k, aik}k=2 from y2.
Iterate the previous three substeps until practical convergence.
Remaining Steps: Continue similarly until K is equal to the desired or estimated
number of sinusoids. (Whenever K is unknown, it can be estimated from the available
data, for instance, by using the generalized Akaike information criterion (GAIC) rules
which are particularly tailored to the RELAX method of parameter estimation. See,
e.g., [39].)
The practical convergence in the iterations of the above RELAX method
may be determined by checking the relative change e of the cost function
C\ ^{¡)¡fc, u)fc, )k, k}£=1^J in (5.24) between two consecutive iterations. Our numerical
examples show that the iterations usually converge in a few steps.
5.2.2 Curvilinear Aperture
The RELAX algorithm for the curvilinear aperture is similar to that of the
full aperture except that Equations (5.26) and (5.27) above are replaced by
[{Ic(afc <8>fc)} <8>afc]HyCfc
Oik
MN
and
Uk Uk !Uk =Uk ,Uk=Uk
{u)k,k,ujk} = arg max [{Ic(a* afe)} a.k]H yCk
Uk,Uk,Uk
(5.28)
(5.29)


APPENDIX B
APPROXIMATION OF THE RANGE R
The range R in (5.5) can be approximated as follows. Under the conditions
k ^ Rft < l' and ~k < we have
R
Rq
Rq
2 i 2 i 2~\ 1/2
x y z x y ~r z *
1 2 cos 9 cos (ft 2^- sin 9 cos (ft 2 sin (ft H ~
Ro Rq Rq Rft
i x a j. y a j. z x2 + y2 + z2
1 cos 9 cos (ft - sin 9 cos (ft - sin (ft d 5
i?o No /t0 2jRq
{x cos 9 cos (ft + y sin 9 cos (ft + z sin (ft)'
2/§
(B.l)
Let ip = (ft 0o, where is the average of all (ft used to form the synthetic aperture.
For very small ip, we have
cos (ft fa cos (fto (sin (B.2)
sin (ft fa sin (fto + (cos (fto)ip.
(B.3)
For very small 9, we have cos9 fa 1, sin# ~ 9. Then keeping first-order terms yields
x2 sin2 (ft + x2 sin2 9 cos2 (ft + y2 sin2 (ft + y2 cos2 9 cos2 (ft + z2 cos2 (ft
fa x2 sin (ft[sm (ft0 + (cos (ft0)ip\ + y2 sin (ft[sin (ft0 + (cos (fto)ip]
+y2 cos 0[cos (ft0 (sin (ft0)ip] + z2 cos 0[cos (ft0 (sin (ft0)ip]
fa [(x2 + y2) sin (ft0] sin (ft + [(y2 + z2) cos (ft0\ cos 9 cos +[(x2 z2) sinocos(fto\ip. (B.4)
We also have
2xy cos 9 sin 9 cos2 (ft + 2yz sin 9 cos 0 sin 0 -p 2xz cos 9 cos (ft sin (ft
100


4.1 Ambiguity effect on the SAR image formation in the absence of range
estimation errors, (a) True windowed FFT SAR image, (b) Windowed
FFT image of the first scatterer. (c) Windowed FFT image of the
second scatterer. (d) Combined windowed FFT image of the two
scatterers. (The vertical and horizontal axes are for range and cross
range, respectively.) 58
4.2 Ambiguity effect on the SAR image formation in the presence of range
estimation errors, (a) Windowed FFT image of the first scatterer.
(b) Windowed FFT image of the second scatterer. (c) Combined
windowed FFT image of the two scatterers with £ = 1 (without
extrapolation), (d) Combined windowed FFT image with £ = 2.
(The vertical and horizontal axes are for range and cross-range, re
spectively.) 59
4.3 Comparison of SAR images formed using different algorithms for sim
ulated data at high SNR (a2 = 0.6). (a) True SAR image, (b) Win
dowed 2-D FFT SAR image, (c) Hybrid SAR image, (d) SPAR SAR
image, (e) RELAX-NLS SAR image, (f) Modified RELAX-NLS SAR
image. (The vertical and horizontal axes are for range and cross-range,
respectively.) 60
4.4 Comparison of SAR images formed using different algorithms for sim
ulated data at low SNR (a2 = 6). (a) Windowed 2-D FFT SAR image.
(b) Hybrid SAR image, (c) SPAR SAR image, (d) RELAX-NLS SAR
image, (e) Modified RELAX-NLS SAR image. (The vertical and hor
izontal axes are for range and cross-range, respectively.) 61
4.5 Comparison of SAR images obtained via different algorithms for the
Slicy data hbl5533.015 (0 azimuth and 30 elevation angles), (a)
Windowed 2-D FFT SAR image from the XPATCH data, (b) Win
dowed 2-D FFT SAR image from the Slicy data, (c) Hybrid SAR
image, (d) SPAR SAR image, (e) RELAX-NLS SAR image, (f) Mod
ified RELAX-NLS SAR image, (c) ~ (f) are all obtained from the
data used in (b). (The vertical and horizontal axes are for range and
cross-range, respectively.) 62
5.1 Broadside spotlight-mode SAR data collection geometry 65
5.2 Curvilinear apertures, a) The circular aperture, b) The Arc-1 aper
ture. c) The Arc-2 aperture 73
5.3 The CRBs of the target parameters as a function of the circular aper
ture radius when K = 1, a = 1, u) = = = 0, and o2 = 40. a) The
CRB of the complex amplitude a. b) The CRB of u. c) The CRB of
Co. d) The CRB of Go 77
5.4 3-D plots of K = 8 scatterers extracted by using RELAX with the
indoor experimental data, (a) Obtained with full aperture as shown in
Figure 1.2(a). (b) Obtained with curved aperture as shown in Figure
1.2(c)
viii
78


92
in
CD
45.4
1 11 1 1 1 1 1 1 1 T r ' 1 1 11 i
(degre
45.2
45.0
:
- j
\
s . \
0
44.8
\
. \
cn
c
<
44.6
_ \
\
: Initial aperture \ SN
c
o
44.4
_ .. : Aperture obtained by AUTORELAX after 7 iterations **. _
\
\
-+-1
o
>
0
44.2
44.0
\
LlJ
J i-L i.i. i-i i-l i i i i i i I i i i i
-3-2-10 1 2 3
Azimuth Angle (degrees)
(a)
v iuertions
Figure 6.4: Autofocused curvilinear aperture and scatterer distribution obtained with
AUTORELAX by autofocusing in both the elevation and azimuth directions and
using K = 20 for the experimental example, (a) Autofocused curvilinear aperture
(dotted line), (b) Scatterer distribution.


99
Let ys{n, f) denote the nth element of ys(/), n = 0,1, , N 1. Then cj) is given by
(A.6)
Inserting (A.6) into (A.5) and ignoring the scaling factor, we can simplify (A.5) to:
i = jargj^1 [yJ>M/)] .
N-1
CMJ) = +
71=0
Nl
E [yl"N(f)]2 e-'
71=0
llys(/)ir +
N-1
-j2ir(2f)n
n=0
(A.7)
Then the NLS estimates {/, /} of {/, /} are determined by
{/, /} = argmax C4(/, /). (A.8)
/,/
Note that ys(/) in (4.37) can be obtained by applying 1-D FFT to each column of Ys
and the second term in (A.7) can also be readily obtained by applying 1-D FFT to
the sequence {y2s{fi, f)}n=o w^h 2/ as the frequency variable. Hence {/,/} can be
obtained via a 2-D search for the location corresponding to the peak of Cif, /), which
can be computed efficiently via 1-D FFTs. Note also that padding with zeros for the
1-D FFTs is necessary to achieve high accuracy for the frequency estimates. An
alternative approach is to find an approximate location corresponding to the global
maximum with 1-D FFT without much zero-padding and then use the approximate
location as the initial condition to find a more accurate position via, for example,
alternatively using the FMIN function in MATLAB.


37
4.2 Review of the APES Algorithm
The APES [38] algorithm belongs to the adaptive filter-bank class, in which
the adaptive finite impulse response (FIR) filter is designed based on not only the data
to be processed, but also the interference at the frequency of interest. APES mimics
the maximum likelihood estimator of a complex sinusoid in circularly symmetric
zero-mean Gaussian noise. It has been shown in [37] that APES has better statistical
performance than Capon [10], another popularly used adaptive filter-bank approach
for SAR imagery.
Let {zn}^I{g1 denote the available 1-D data sequence and have the form:
zn = a(co)ex-p{jnu>} + n = 0,1, , N 1, (4.3)
where a(u) denotes the complex amplitude of a sinusoid with frequence lo and en(u)
denotes the unmodeled noise and interference at u. Then APES can be used to obtain
the complex amplitude estimate a(co) of a(u) from {zu}^Zq. It has been shown in
[38] that the least-squares estimate of a{u) using the forward and backward FIR
filter output has the form:
&(lu)
h//(a;)Z(w) -I- /3(co)ZH (uj)h(uj) ,
(4.4)
where h(u>) denotes the impulse response of an M-tap FIR filter and has the following
form:
h(w) =
l T
hi(u) h2(co)
hM(uj)
(4.5)
Z(u>) and Z(cu), respectively, denote the normalized Fourier transforms of {zi}^M
and {zi.e., the Fourier transforms of {zj}^QM and {z}^qM divided by (N
M + 1), where z and zrespectively, denote the overlapping subvectors of the data
vector z =
z0 z i
data sequence z =
zn-i
ZN-1 ZN-2
and the complex conjugated and backward-ordered
and have the form
Z=
T
Zi+1
Zi+M1
i = 0,1, , N M,
(4.6)


25
and
{t,tt} = argmax
u%(ujt)Ytv*(ujt)\2
N
(3.21)
Thus {ujt,u>t} corresponds to the peak location of |aj^(a>)Ytuj*(jt)\2, which can be
calculated efficiently via 2-D FFT, and at corresponds to the complex height of the
normalized 2-D FFT of Yt at {jt,Ct}.
Feature Extraction of a Single Dihedral Corner Reflector
Let Yd denote the data of a single dihedral in the presence of unknown noise
Erf. Then
Yrf = adUJN(u>d)gr -I- Erf, (3.22)
where {uid, ud ad, b, r) are the target features of a dihedral corner reflector, and
r -\T
g =
9(0) 0(1)
g(-1)
(3.23)
with g() sinc[f)7r(r )]eJ'w {rf,c¡)rf,rf, b, f} of {ad,tod, Cob, r} are determined by minimizing the following cost
function:
Crf (ad, ud, id, 6, r) = || Yrf adu>N(ud)gr\\2F (3.24)
It can be shown that ad is determined by
. ^N^d)Yrfg*
ad Y||g||2
and {uod,u)d,b, f} can be obtained by maximizing the cost function
r, l \ \^N^d)Ydg*\2
c3 (ujd,ud,b,T) ^||g||2 (3.26)
RELAX-NLS maximizes C3 in (3.26) by utilizing an alternating maximization ap
proach [8], i.e., alternatively updating {ujd, u>d, b, f} while fixing other parameter esti
mates at their recently determined values. Note that four-dimensional search over the
parameter space is performed to estimate the dihedral corner reflector parameters.


96
for 3-D target feature extraction via CLSAR and motion compensation (autofocus)
for CLSAR in Chapters 5 and 6, respectively. A CLSAR data model has been first
established and the ambiguity problems inherent in the data model have been in
vestigated when aperture errors are present. To provide a guide for the choice of
CLSAR trajectory, the performance of different curvilinear apertures is analyzed via
Cramr-Rao Bound, which is the lowest bound achievable by any unbiased estima
tors. Autofocusing for CLSAR becomes more complicated since the aperture errors
in CLSAR, unlike in the 2-D SAR systems, can no longer be approximated as phase
errors only in cross-range due to the non-negligible errors introduced in the azimuth
and elevation angles of the trajectory. We have presented a relaxation-based autofo
cus algorithm, referred to as AUTORELAX, for 3-D target feature extraction using
CLSAR in the presence of aperture errors. AUTORELAX can be used to simulta
neously extracts 3-D target features and compensates motion errors including phase
errors and aperture errors by minimizing an NLS cost function via an alternating
optimization approach. Experimental examples have been used to demonstrate the
effectiveness of AUTORELAX for 3-D target feature extraction using CLSAR data
in the presence of curvilinear aperture errors.
7.2 Future Works
In this study, we have concentrated on the study of SAR target feature extrac
tion, super resolution image formation, and autofocus via relaxation-based paramet
ric and semi-parametric methods. However, the problems involved in inverse SAR
(ISAR) become more complicated and difficult. Compared with the stationary targets
of SAR, targets being imaged in ISAR are often engaged in complicated maneuvers
that combine translational and rotational motions and furthermore, the motions be
tween the radar and the moving targets in ISAR are non-cooperative. Hence, the


89
Estimation Error
Magnitude
Range (/)
Cross-range (/)
Height (/)
AUTORELAX
0.015
0.00288
0.00179
0.00348
RELAX
0.296
0.31544
0.01858
0.24444
Table 6.1: Estimation errors for the parameters of the highest scatterer in the simu
lation example before and after autofocusing.
in the experimental example above. The true aperture is the dot line shown in Figure
6.5(a) and the scatterer distribution is shown in Figure 6.5(b). Here we simulate the
case where the aperture errors exist only in the elevation direction. Figure 6.5(c)
shows the scatterer distribution obtained with the RELAX algorithm from the initial
aperture (dash line) shown in Figure 6.5(a). We note that the scatterer distribution
in Figure 6.5(c) is quite different from the true one in Figure 6.5(b). When used
with the simulation data, AUTORELAX converges after four iterations. The auto-
focused aperture is the dot line shown in Figure 6.5(a) and the scatterer distribution
obtained with AUTORELAX is illustrated in Figure 6.5(d). We see that the scat
terer distribution obtained by AUTORELAX is almost the same as the true one. We
also notice that the shape of the autofocused aperture is closer to the true one than
the initial aperture. The constant difference between the autofocused and the true
apertures will cause phase errors in {ck}k=n which cannot be eliminated due to the
ambiguity problems discussed in Section 6.1.To quantitatively illustrate the accuracy
improvement of the parameter estimates via AUTORELAX, we compare the esti
mation errors of the highest scatterer in Figure 6.5(b) before and after autofocusing.
Table 6.3 shows the differences between the true values and their estimates obtained
via RELAX and AUTORELAX, respectively.


ACKNOWLEDGEMENTS
I am sincerely grateful to my advisor and committee chairman, Dr. Jian
Li, not only for her academic guidance and insightful suggestions throughout the
development of this dissertation and providing an invaluable environment for the
research, but also for her financial support in the past years.
I would like to thank my graduate committee members, Dr. Jos C. Principe,
Dr. Fred J. Taylor, Dr. William W. Edmonson, and Dr. David C. Wilson, for their
time and interest in serving on my supervisory committee.
I would also like to thank Dr. Zheng-She Liu, Dr. Renbiao Wu, and the fellow
students, past and present, of the Spectral Analysis Laboratory (SAL) for their great
help.
Finally, I would like to gratefully acknowledge all the people who helped me
during my Ph.D program.


7
(a) (b)
Figure 1.4: Mesh plots of the modulus of the RCS obtained by using 2-D FFT with
different apertures (range information suppressed for the illustration purpose only),
(a) Full aperture as shown in Figure protect 1.2(a). (b) Curvilinear aperture as shown
in Figure 1.2(c).
practice. Nevertheless, a suitable subset of the full aperture can be used as a trajectory
of SAR to obtain 3-D information. This technique is known as curvilinear SAR
(CLSAR). CLSAR traverses a curvilinear aperture path, which is a subset of the full
aperture (see, for example, Figures 1.2(b) and (c)), and collects a sparsely sampled
dataset, from which 3-D information can be obtained. CLSAR avoids ambiguity
problems inherent in IFSAR systems, but suffers from severe high sidelobes when
used with FFT to form SAR images due to the limited size of measurements. This
observation is illustrated in Figure 1.2. (Figures 1.2(a) and (b) show the modulus
of the RCS of a single point scatterer obtained by applying 2-D FFT to the data
collected via the full aperture as shown in Figure 1.2(a) and the curvilinear aperture
as shown in Figure 1.2(c).) Due to its high sidelobes, CLSAR is of little interest in
SAR imaging. Yet CLSAR can be used with parametric spectral estimation methods
to extract 3-D target features of small targets consisting of a finite number of isolated
point scatterers. In Chapter 5, we will establish a CLSAR data model and describe
how the RELAX algorithm [39] can be used for 3-D target feature extraction using
CLSAR with different curvilinear apertures.


54
j2neJ(t>k {[uft(fk) Xfc] (uN(fk) dN)}
(4.64)
with
and
1N
i T
0 1 TV- 1
j2-ire^k {[w^(/jfc) Xfc Ody^] wjv(/fc)}
(4.65)
(4.66)
with
lJV
n T
0 1 TV 1 (4.67)
Let F4 denote an TVTV x K matrix whose [(k 1)TV + ]th column is determined
by
eJ(^k {[wjj d>yv(/fc)] u?yv(/fe)} ) k = 1,2,... ,K, = 1, 2, , TV, (4.68)
where wn is an TV x 1 vector with the th element being unit and all other elements
being zeros. Let
F =
Fi F2 F3 F4
(4.69)
Then the CRB matrix for the parameter vector 77 is given by:
CRB(ij) = [2Re(FffQ-1F)]_1.
(4.70)
4.7 Numerical and Experimental Results
We demonstrate and compare the SAR image formation performances of
SPAR, and the modified RELAX-NLS algorithm with both simulated and experi
mental examples. The algorithms are also compared with Hybrid and RELAX-NLS.
In the following examples, the dimensions of the original SAR phase history data
matrix are TV = TV = 32 and GAIC with 7 = 5.5 is used to determine K for the
relaxation-based feature extraction algorithms of SPAR and Hybrid. The threshold
Ti used in the isolation process of SPAR is 10% of the peak value. The maximization
of C4(/,/) in (4.41) is done in two steps. First, initial frequency estimates / and /


73
(a) (b) (c)
Figure 5.2: Curvilinear apertures, a) The circular aperture, b) The Arc-1 aperture,
c) The Arc-2 aperture.
Aperture
CRB(a)
CRB(o;)
CRB(cj) CRB(o>)
Full
-5.9875
-51.4511
-51.4511 -51.4511
Circular
5.3663
-39.3415
-40.8469 -40.8788
Parabolic
5.8693
-39.3415
-40.4915 -39.6521
L-shaped
4.8584
-39.3415
-38.2414 -38.2414
Arc-1
11.7482
-39.3415
-31.7272 -31.7272
Acr-2
17.2473
-39.3415
-26.8015 -27.4636
Table 5.1: Comparison of the CRBs (in dB) of the target features for the cases of the
full, circular, parabolic, L-shaped, Arc-1, and Arc-2 apertures when M 63, K 1,
cu = (2> = ) = 0, a = 1, and cr2 = 40.
circle whose center is at the upper right corner of the full aperture shown in Figure
1.2(a) and whose radius is 31, as shown in Figure 5.2(b), and the Arc-2 aperture
is one half of a parabolic aperture whose vertex is at the lower right corner of the
full aperture and who starts at the upper left corner of the full aperture shown in
Figure 1.2(a), as shown in Figure 5.2(c). These curvilinear apertures are subsets
of the full aperture in Figure 1.2(a) and are made to be as large as possible. For
example, the radius of the circular aperture is 15.5 in this example. Table 1 shows
the CRBs of the target parameters. As expected, the CRB for lu is the same for


RELAXATION-BASED METHODS FOR SAR TARGET
FEATURE EXTRACTION AND IMAGE FORMATION
By
ZHAOQIANG BI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY.
UNIVERSITY OF FLORIDA


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
May 1999
Winfred M. Phillips
Dean, College of Engineering
Mihran J Ohanian
Dean, Graduate School


80
exactly due to atmospheric turbulence and platform position uncertainty. In 2-D SAR
imaging, it is generally assumed that the errors in {9m, 0m}W_1 are negligible and the
errors in {R0(m)}^=1 cause phase errors along the synthetic aperture. In CLSAR,
however, the errors in {9m, 0m}£f=1 may no longer be negligible since the aperture
shape is critical for 3-D target feature extraction. Our problem of interest herein is to
compensate for the curvilinear aperture errors in {R0(m)}^=1 and {9m, (f)m}^=1 and
extract the 3-D target parameters {ok, /*,, fa, fk}k=i from {y(n, m)},n = 0,1, , N
1, m = 1, 2, , M. We first analyze aperture errors in CLSAR and then present an
autofocus algorithm, referred to as AUTORELAX, for 3-D target feature extraction
in the presence of aperture errors.
6.1 Aperture Error Effects
We will consider the approximations and the ambiguity problems in our data
model in the presence of aperture errors. We omit the dependence on n in the
following analysis For the sake of notation convenience. For the broadside data
collection geometry shown in Figure 5.1, 9m is very small. For very small 9m, we have
sin(0m) 9m and cos(9m) 1. These approximations also hold for the true look
angles of the radar. Then according to (6.2), (6.3), and (6.4), respectively, we have
xm~ (cos 0m),
(COS 0J 9m,
and
(sin 0m),
o o
where {9m, 0m}m=i denote the true look angles of the radar and
4(tt/o 7r0 + 71)
(6.6)
(6.7)
(6.8)
t =
Consider first the errors in {cj)m}^=1 when {9m =^m,i?0(n^) = Ro}m=i- Let
0m 0m +A0m-
(6.9)


76
We now present an experimental example to demonstrate the performances
of the RELAX algorithm for 3-D target feature extraction with a CLSAR. We use
e = 0.01 to test the convergence of the RELAX algorithm in this example. The
indoor data used herein was obtained by the Radar Signature Branch, Naval Air
Warfare Center, Mugu, California. The radar carrier frequency is 9.968 GHz and the
bandwidth 1.524 GHz. The 32 x 32 x 32 data set was obtained with the full aperture
shown in Figure 1.2(a) with 32 samples in each dimension and the angular increments
were 0.28. The target consists of K = 8 corner reflectors with a cubic configuration
and was about 15 meters away from the radar. The scatterers were about 0.5 meters
apart.
Figure 5.4.2 shows the extracted scatterers when K = 8 is used with RELAX.
Figure 5.4.2(a) is obtained when the full aperture shown in Figure 1.2(a) is used to
extract the 3-D target features and Figure 5.4.2(b) is obtained when the L-shaped
aperture shown in Figure 1.2(c) is used. The centers of the circles denote the locations
of the extracted scatterers in 3-D space and the radius of each circle is proportional
to the modulus of the RCS of the corresponding scatterer. The triangles show the
projections of the scatterer locations onto the horizontal plane and their sizes are also
scaled according to the RCSs of the scatterers. Note that Figure 5.4.2(b) is similar
to Figure 5.4.2(a) even though the former is obtained by using only 6.15% amount
of data used by the latter.


6.1 (a) Curvilinear apertures for the experimental example, (b) True scat-
terer distribution, (c) 3-D plot of K = 20 scatterers extracted from
the initial aperture (dashed line) in (a) with RELAX, (d) 3-D plot
of K = 20 scatterers extracted from the manually adjusted aperture
(solid line) in (a) with RELAX 90
6.2 Autofocused curvilinear aperture and scatterer distribution obtained
with AUTORELAX by autofocusing only in the elevation direction
and using K = 20 for the experimental example, (a) Autofocused
curvilinear aperture (dotted line), (b) Scatterer distribution 91
6.3 (a) Manually adjusted (solid line) and autofocused (dotted line) curvi
linear apertures for the experimental example, (b) Fitting the manu
ally adjusted aperture to the autofocused aperture by adding a line to
the former 91
6.4 Autofocused curvilinear aperture and scatterer distribution obtained
with AUTORELAX by autofocusing in both the elevation and azimuth
directions and using K 20 for the experimental example, (a) Auto
focused curvilinear aperture (dotted line), (b) Scatterer distribution. 92
6.5 (a) Curvilinear aperture for the simulation example, (b) True scatterer
distribution for the simulation example, (c) Scatterer distribution ob
tained from the initial aperture (dashed line) in (a) by using RELAX
with K = 20. (d) Scatterer distribution obtained via AUORELAX
with K = 20 and using the autofocused aperture (dotted line) shown
in (a) 93
IX


3
Figure 1.2: Illustration of SAR imaging modes, (a) Spotlight-mode SAR. (b) Strip-
map SAR.
processing radar returns from a sequence of locations when a nominally-sized antenna,
which is carried on board of an aircraft, sequentially moves along the flight path
(cross-range direction). Consequently, a synthetic antenna with an effective antenna
length proportional to the synthetic aperture can be realized even with a small real
antenna. The longer the synthetic aperture, the finer the cross-range resolution
achieved. It has been demonstrated in [29] that the cross-range resolution of SAR
depends only on the wavelength A and the synthetic aperture interval. To achieve
such a desirable cross-range resolution, many techniques have been developed to
record data, compensate motion errors, and construct high quality SAR images ever
since the 1960s. (Detailed descriptions of these techniques are well documented in
many books, e.g. [29, 11]). SAR has grown so dramatically through advances in
digital signal processing, computing power, and microwave technology that it is one
of the most important techniques for high resolution radar image formation.
Spotlight-mode and strip-map SAR are two commonly used SAR imaging
modes. In a spotlight-mode SAR, antenna beams continuously steer to the same
terrain patch (scene) illuminated by a radar when the aircraft traverses the flight
path, while the antenna steering of a strip-map SAR is fixed relative to the flight
path when the aircraft moves along the flight path (see Figure 1.2 for the difference).


42
where £ denotes an extrapolation factor (£ > 1) and is a parameter of user choice.
Note that the super resolution property of the so-formed SAR images is determined
by the feature extraction algorithm and £ > 1 is only used to demonstrate the super
resolution property of the target feature extraction algorithm. The estimated noise
and clutter data matrix is
(n, n) = y(n, ) ss(n, ), n = 0,1, , N 1, = 0,1, , 1, (4.24)
which is also important in many SAR applications since, for example, important
target information such as the target shadow information is contained in (n, ). We
cannot extrapolate (n, ) in either range or cross-range since no parametric data
model is available for (n, ).
To obtain SAR images with low sidelobes in range via 1-D FFT, we apply 1-D
windows to ss(n, ) and (n, ) in range. We obtain a new (£Y) x data matrix Y
as follows:
{y(n, ) = ss(n, n)ws(n) + £(n, )we(n), n = 0,1, , N 1, = 0,1, , 1,
y(n, ) = ss(n, )ws(n), n = N, N + 1, , £Y 1, = 0,1, , 1,
(4.25)
where y(n, fi) denotes the (n, )th element of Y and ws{n) and we(n) are 1-D windows
of lengths £Y and N, respectively, satisfying
Civ-i
53 Ws(n) = C-W. (4.26)
n=0
and
N-1
^2we(n) = N. (4.27)
n0
Note that scaling (n,) in y(n,) by a factor of £ is necessary since the range
dimension of ss(n,) is £ times of that of (n, ). The steps needed for SAR image
formation are as follows:
Step (1): Form Y from y(n, ) by using (4.23), (4.24), and (4.25).


44
k
Oik
4>k
fk
Ik
bk
Tk
k=l
9.6
0
0.1
-0.3
0.3
18.6
k=2
6.4
0
0.1
0.1
0.2
18.6
Table 4.1: True parameters of the two dihedrals used in Figures 1 and 2.
to simulate the dihedrals, where and brespectively, are proportional to the
maximal RCS and the length of the kth dihedral corner reflector and rdenotes the
peak location of the data sequence and is determined by the orientation of the kth.
dihedral. The size of the simulated data matrix is 32 x 32 (i.e., N = = 32).
The parameters for the two dihedrals are given in Table 1. An ambiguous set of
target features can be obtained by choosing fi /2 = 0.1, f1 = f2 = f = 0.2,
4>i = 0, and (¡>2 7t/2 in (4.19) and (4.20). The windowed FFT SAR images of
the two scatterers are shown in Figures 4.1(b) and (c), respectively, which differ
considerably from the two dihedral scatterers in Figure 4.1(a). The combined SAR
image of the two scatterers is given in Figure 4.1(d), which is exactly the same as
the true image shown in Figure 4.1(a). However, due to the presence of noise and
clutter, parameter estimation errors are inevitable. The errors in range are the main
cause of the artifact problem in the high resolution SAR image formation. In Figure
4.2, we assume that all parameters are accurate except that /2 = /2 + 0.01 = 0.11.
Figures 4.2(a) and (b), respectively, show the windowed FFT images of the two
aforementioned scatterers in the presence of estimation errors and Figure 4.2(c)
shows the combined SAR image. By comparing Figures 4.1(a) and 4.2(c), we note
that an extra line (artifact) shows up next to the short dihedral. The reason is that
due to the estimation errors, ^ f2. Hence the two scatterers in Figures 4.2(a)
and (b) are not exactly in the same range and cannot be combined perfectly to
obtain the two dihedral lines in Figure 4.1(a). This problem becomes even worse
when super resolution SAR images are formed via data extrapolation in range. The


105
[14] S. R. DeGraaf. Sidelobe reduction via adaptive FIR filtering in SAR imagery.
IEEE Transactions on Image Processing, 3(3) :292301, May 1994.
[15] S. R. DeGraaf. SAR imaging via modern 2-D spectral estimation methods. IEEE
Transactions on Image Processing, 7(5):729-761, May 1998.
[16] D. E. Dudgeon and R. T. Lacoss. An overview of automatic target recognition.
The Lincoln Laboratory Journal, 6(1):310, 1993.
[17] P. H. Eichel, D. C. Ghiglia, and C. V. Jakowatz, Jr. Speckle processing methods
for synthetic-aperture -radar phase correction. Optics Letters, 14:1-3, January
1989.
[18] P. H. Eichel and C. V. Jakowatz, Jr. Phase-gradient algorithm as an optimal
estimator of the phase derivative. Optics Letters, 14:1101-1103, October 1989.
[19] A. Farina, A. Forte, F. Prodi, and F. Vinelli. Superresolution capabilities in
2D direct and inverse SAR processing. International Symposium on Noise and
Clutter Rejection in Radars and Imaging Sensors, pages 151-156, Kanagawa
Science Park, Japan, November 1994.
[20] A. Farina, F. Prodi, and F. Vinelli. Application of superresolution techinques to
radar imaging. Chinese Journal of Systems Engineering and Electronics, 5(1):1-
14, January 1994.
[21] M. T. Fennell and R. P. Wishner. Battlefield awareness via synergistic SAR and
MTI exploitation. IEEE Aerospace and Electronic Systems Magazine, 13(2):39-
45, February 1998.
[22] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Uni
versity Press, Baltimore, MD, 1984.
[23] P. T. Gough. A fast spectral estimation algorithm based on the FFT. IEEE
Transactions on Signal Processing, 42:1317-1322, June 1994.
[24] L. C. Graham. Synthetic interferometer radar for topographic mapping. Pro
ceedings of the IEEE, 62:763-768, June 1974.
[25] R. O. Harger. Synthetic Aperture Radar System: Theory and Design. Chapter
VI, Phase and Motion Errors. Academic Press Inc., New York, 1970.
[26] G. R. Heidbreder. Maximum entropy methods in coherent radar imaging. In-
ternationl Journal of Imaging Systems and Technology, 2:239-247, 1990.
[27] D. N. Held and J. D. OBrien. Norden systems 2-dimensional interferometric
synthetic aperture radar flying testbed. Proceedings of the 1992 IEEE Inter
national Geoscience and Remote Sensing Synoisum (IGARSS), Houston, TX,
2:646-648, May 1992.
[28] J. A. Hbgbom. Aperture synthesis with a non-regular distribution of interfer
ometer baselines. Astronomy and Astrophysics Supplements, 15:417-426, 1974.
[29] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thomp
son. Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach.
Kluwer Academic Publishers, Norwell, MA, 1996.


66
where Sx,y,z is proportional to the RCS of the scatterer located at (x,y,z). For
4(tt/o 7Tq + 7)(-o Ro)/c d[t, 9, 0) ~ Sx,y,z
pjiitx+yty+ztz)
c )
(5.12)
where
ty
4(tt/0 7T0 + 7) cos 9 cos 0
5
c
4(tt/o 7T0 + 7) sin 0 cos 0
and
t, = -
4(tt/0 7^0 + 7) sin0
(5.13)
(5.14)
(5.15)
Note that d(t, 9,0) is a 3-D complex sinusoid. The frequencies of the 3-D sinusoid
correspond to the 3-D location (x, y, z) of the scatterer, while the amplitude is propor
tional to its RCS. Note that (x, y, z) is not the true location (a;, y, z) of the scatterer,
but is close to (x, y, z) for large R0. (See the Appendix 3C on how to calculate (x, y, z)
from (x, y, z).) When a target has multiple scatterers, d(t, 9, 0) in (5.12) will be a sum
of sinusoids. The 3-D locations and RCSs of the target scatterers are the 3-D target
features. Since usually the samples on the t, 9, and 0 axes are uniformly spaced, the
samples of tx, ty, and tz occur at the points of a polar grid. Hence Polar-to-Cartesian
interpolation may be needed for the data samples to occur at rectangular grid points.
(See Section 5.4.2 for an alternative approach.)
After Polar-to-Cartesian interpolation and sampling, the signal obtained by
the 3-D full aperture SAR in the presence of noise can be written as:
K
y(n, , n) = ^ akeJ
(wfcn+Ofc+w^Ti)
e(n, , n),
(5.16)
fc=l
where n = 0,1, , N 1, = 0,1, , 1, and = 0,1, , 1, with
N, N, and denoting the numbers of available data samples in the three dimen
sions; K denotes the number of sinusoids; a-jt and {wjt, 4, 4} denote the unknown


CHAPTER 3
PARAMETRIC METHODS FOR SAR IMAGING
When an appropriate data model is available for radar targets, parametric
spectral estimation algorithms, which are less sensitive to the model mismatch, offer
the potential to significantly improve the resolution of the formed SAR images. In
this chapter, we consider forming enhanced SAR images with robust relaxation-based
methods including RELAX and RELAX-NLS.
3.1 RELAX
When the dominant features of targets can be approximated as point scatter
ed, enhanced SAR images can be formed by using the target features extracted via
RELAX and the point scatterer data model. We first briefly introduce the RELAX
algorithm [39] and then discuss how to form SAR images using RELAX.
3.1.1 Review of the RELAX Algorithm
Assume that there are K dominant point scatterers or trihedral corner re
flectors in a target. An ideal trihedral acts as a triple-bounce reflector and can be
modeled as a strong point scatterer [53] since its radar cross section (RCS) can be
considered as a constant due to the very small angle variation of the radar beam in
SAR. Then the target data model in the presence of unknown noise has the form:
K
y(n, ) ake^ulkn+Jkn'> + e(n, ), n = 0,1, , N 1, fi = 0,1, , 1,
k=1
(3.1)
where N and denote the numbers of data samples in range and cross-range, respec
tively; ok and {ujk, 4}, respectively, denote the unknown complex amplitude and 2-D
18


40
modulated by 2cos[7r(/a fb)n + (a 4>b)/2]. Thus the data model in (4.1) cannot
be used to describe each of the two corner reflectors in this case.
Type 3: Two different scatterers located in the same range
Assume that two different corner reflectors with parameters {^()}^1 f, /}?=1
are located in the same range. Then the target model in the absence of noise has the
form:
r(n, n) = Xi{n)e^ieP2ir^n+^in\ n = 0,1, , N 1, n = 0,1, , N 1.
=1
(4.17)
With straightforward calculations, we can rewrite (4.17) as
2
r(n,) = Xj () e3^' eJ'27r(A n+/"); (4.18)
2=1
where 2 = i + 7t/2 with denoting an arbitrary phase, f1=f2 = f,f1 = f2 = f
with / denoting an arbitrary cross-range location,
and
i(n) = ^2xi(n) cos[2vr(/i f)n + (<^ 1)],
=i
2(n) = sin[27r(/ /)+(& ~ 0i)]-
2=1
(4.19)
(4.20)
Note from (4.18) that {0, {(n)}^r=01,/,/,}=1 are the ambiguous features of
{cj)U {xi()}%~,f,fi}2i=v
Type 4: Multiple scatterers located in the same range
When more than two scatterers are located in the same range, the data model in the
absence of noise can still be written as (4.18) except that
L
i() = 5^z()cos[27r(/j f)+ ( &)], (4.21)
2=1
and
X2(n) = Y^xi{n) sin[27r(/ f)n+ (<& 0i)],
2 = 1
(4.22)


33
Figure 3.3: Target photo taken at 45 azimuth angle.


38
and
zNil zN-i-2
, i = 0,1, , N M,
(4.7)
"N-i-M
respectively; finally, (3(u) exp{j(N l)u>}. It has been shown in Appendix C
of [38] that by using a certain conjugate symmetric condition, a(u) in (4.4) can be
reduced to
a(uj) = hH (lo)Z(uj). (4.8)
The adaptive FIR filter h.APEs{u) corresponding to APES is derived in Ap
pendix A of [38] and has the form:
Q 1(c<;)a(a;)
¡APES
M =
where the steering vector a(o>) is given by
(4.9)
a(u) =
1 T
l gjw . ej(Ml)t
(4.10)
and Q(oi) denotes the estimate of the covariance matrix of the noise and is given by:
Q(w) = R Z(w)Zff(u>),
(4.11)
with
ZM = ^=[ Z(w) ZH] (4.12)
and R = (R + R)/2 denoting the average of the forward and backward sample
covariance matrices R and R. The forward sample covariance matrix R is obtained
from Zi as follow:
NM
R =
N
- y
M + l z
i=0
Zjzf,
(4.13)
and the backward sample covariance matrix R can be obtained similarly from z
4.3 Data Model Ambiguities and Their Effects on SAR Image Formation
Target feature extraction methods devised based on the data model in (4.2)
are robust against data mismodeling errors due to the model flexibility. However,


85
where
K
2(12, m) = ^ ^keRAfkixm{n)+'ikiym(,n)+JkiZm{n)}^ n = 0,1, - -, TV 1, m = 1, 2, , M,
k=1
(6.28)
and tXm(n), iym(n), and iZm(n) are the same as iIm(n), tyrri(n), and tZm(n), respec
tively, except that {9m, (¡)m}!^=1 are replaced by obtained in Step (1).
This step is similar to [43] and we have
Vm = angle(z^ym), m = 1, 2, , M,
(6.29)
where
and
ym
-\ T
2/(0, m) 2/(1, m)
y(N 1, m)
Zo
.n
1 T
(6.30)
(6.31)
2(0, m) z2(l,m) z2(N-l,m)
Note from (6.29) that we do not need the search over parameter space to determine
{Vm}m=1 and hence the errors in {R0(m)}^=l are easier to deal with than those in
{emAm}M
m= 1
Step (3): Repeat Step (1) by replacing y(n,m) with y(n,m) = y(n,m)e m,
m = 1,2, , M, where {r)m}^f=1 are determined in Step (2).
Step (4): Repeat Steps (2) and (3) until practical convergence, which is deter
mined by the relative change of the cost function Cm in (6.25) between two consec
utive iterations. In the numerical examples, we terminate the repetition of Steps (2)
and (3) when the relative change of Cm is less than 10~3 between two consecutive
iterations.
Note that if the errors in {R0(m)}^=1 are known to be negligible, then Step
(1) alone is sufficient for the autofocusing.
6.2.2 Target Feature Extraction
Assume aperture parameter estimates {9m, problem becomes the target feature extraction problem considered in Chapter 5. As


78
(a)
(b)
Figure 5.4: 3-D plots of K = 8 scatterers extracted by using RELAX with the indoor
experimental data, (a) Obtained with full aperture as shown in Figure 1.2(a). (b)
Obtained with curved aperture as shown in Figure 1.2(c).


32
(a)
(b)
# #

(e) (f)
Figure 3.2: RELAX Images by using the 20 x 20 ERIM data with K = 41. (a)
Unwindowed 2-D FFT image, (b) Windowed 2-D FFT image, (c) Unwindowed
RELAX image without background clutter, (d) Windowed RELAX image without
background clutter, (e) Unwindowed RELAX image with background clutter, (f)
Windowed RELAX image with background clutter.


43
Step (2): Apply the normalized 1-D FFT to each column of Y to obtain
an intermediate matrix and then apply 1-D APES to each row of the intermediate
matrix. (See [41] for the efficient implementation of APES.) Note that the normalized
1-D FFT has the form
rN-1
(4.28)
71=0
4.3.3 Model Ambiguity Effects on SAR Image Formation
All of the aforementioned types of ambiguities will have no effect on the SAR
image formation if no parameter estimation errors exist since the scatterers will be
perfectly reconstructed by using any of the possible ambiguous data models. For
example, when there are two identical scatterers located in the same range, the data
model in (4.2) can still be used for SAR image formation since the two scatterers are
now described as one scatterer with (4.16), which still fits the data model of (4.2)
with K = 1. Hence the original SAR image can still be reconstructed by using the
parameters of the one scatterer described by the right side of (4.16).
In the presence of parameter estimation errors due to the presence of noise
and clutter, however, Types 1 and 2 ambiguities discussed in Section 3.1 will have
little effect on SAR image formation, whereas Types 3 and 4 ambiguities can result in
artifact problems for the high resolution SAR image formation. Generally speaking,
the higher the SNR, the more accurate the parameter estimates and hence the less
significant the artifact problem. The effect of Type 3 ambiguity on SAR image
formation in the presence of estimation errors is illustrated in Figures 4.1 and 4.2.
(The effects are similar for Type 4 ambiguity.) Figure 4.1 is obtained by assuming
no parameter estimation errors. Figure 4.1(a) shows the FFT image of a target
consisting of two dihedrals of different lengths located in the same range. We use
xk(n) = aksmc[bkn(n rk)], k = 1,2, n = 0,1, 31,
(4.29)


81
Let
tn
-4tt/o
(6.10)
It can be shown that for very small A0m and 7r/0 7(t r0), where \t ro| < To/2,
we have
and
icos 0m -to(sin 00)A0m,
(6.11)
w 0 O
~ to (COS 0Q) 0m,
(6.12)
O O
tsin 0m +i0(cos 0o)A0m,
(6.13)
where 0O is the average of all 0m, m = 1, 2, , M, and hence is a constant. Then
^ exp {j2tt(fktXm + fktym + AO} exp j j2?r
/fct COS (f)rn + fk^O COS (¡>q9ti
+fkt sin 0
exp < j27r
-fkh sin o +/fcio cos 0O
A0
(6.14)
Equation (6.14) shows that if A0m is a constant, then the phase error due to A0m
and the phase of are ambiguous. Hence the phase of can never be determined
O ~
exactly in the presence of A0m. If A0m is a linear function of 9m, then fk and the
linear phase error due to A0m are ambiguous and cannot be determined exactly.
o
Similarly, we can analyze the errors in {0m}m=i when {0m =0m, i?0(m) =
Ro}m=v Let
=#m +A0m. (6.15)
o
For very small 9m and A9m and for 7r/0 7(i r0), where |i r0| < T0/2, we have
O
tXm ~ t COS 0m,
o
~ to COS 0o (9m TA9m) 1
(6.16)
(6.17)
and
o
i sin 0m .
(6.18)


53
with and (g>, respectively, denoting the element-wise product and Kronecker prod
uct of two vectors.
H =
(4.59)
and e = vec(E).
Next, we derive the CRB matrix for the data model in (4.57) where the noise
covariance matrix is assumed to be arbitrary and unknown. Let Q E{eeH} be
the noise covariance matrix with (-)H denoting the conjugate transpose. Then the
extended Slepian-Bangs formula for the ijth element of the Fisher information matrix
(FIM) has the form [3, 64]:
{FIM},, = tr (q-'Q'CT'Q;) + 2Re [(/A1)! Q_1 (A/i)j] (4.60)
where X) denotes the derivative of X with respect to the th unknown parameter,
tr(X) denotes the trace of X, and Re(X) denotes the real part of X. Note that FIM
is block diagonal since Q does not depend on the parameters in (A/x), and (A/x)
does not depend on the elements of Q. Hence the CRB matrix for the target features
of interest can be calculated from the second term on the right side of (5.31). Let
(4.61)
where
and
V =
fT
fT xe
(f) =
4> 2
f =
fi
2
K
f -
h
h
Ik
xe =
T
T
X2
T
XK
1 t
-i T
(4.62)
Let Fi, F2, and F3 denote NN x K matrices whose A;th columns are determined,
respectively, by
j*k {[u(/i)0xJ8m(/i)),
(4.63)


14
Recently, RELAX-NLS [42] was introduced for the feature extraction of targets
consisting of both trihedrals and dihedrals based on a mixed data model rather than
the point scatterer data model. An ideal trihedral acts as a triple-bounce reflector
and can be modeled as a strong point scatterer [53]. The point scatterer data model,
however, is no longer valid in cross-range for an ideal dihedral corner reflector, which
acts as a double-bounce reflector, since its radar cross section (RCS) is described by
a sine function, i.e., sinc(0) = sin(0)/0, where 0 denotes the angle between the radar
beam and the line perpendicular to the dihedral.
Super resolution SAR images obtained via modern spectral estimation meth
ods provide more details required for the detection and automatic target recognition
(ATR) than the conventional FFT based methods. Quantitative results showing the
advantages of using these sophisticated spectral estimation methods in SAR appli
cations begin to appear in the literature. For example, it was shown in [51, 50] that
a significant improvement in ATR performance can be achieved by using SAR im
ages with enhanced resolution obtained via the reduced-rank variations of the Capon
method.
Compared with non-parametric spectral estimators, high resolution paramet
ric spectral estimators provide great potential for SAR imaging with better resolution
and lower sidelobes when the assumed data model is valid. On the other hand, para
metric methods are more sensitive to data model mismatch than non-parametric
methods. Hence, practically only robust super resolution parametric methods can be
used for SAR image formation. We will use robust relaxation-based algorithms in
cluding RELAX and RELAX-NLS to form SAR images of different man-made targets
in our work. In cases where an appropriate data model of man-made targets is not
available in cross-range, robust non-parametric spectral estimators with good perfor
mances such as APES and high resolution parametric methods may work together to
obtain satisfactory results. To the best of our knowledge, a hybrid of non-parametric


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.
RELAXATION-BASED METHODS FOR SAR TARGET
FEATURE EXTRACTION AND IMAGE FORMATION
By
Zhaoqiang Bi
May 1999
Chairman: Dr. Jian Li
Major Department: Electrical and Computer Engineering
Synthetic aperture radar (SAR) has been a mature but actively researched
technology due to its day-and-night and all-weather capability of offering high res
olution imaging for both military and civilian applications. As the foundation of
automatic target detection and recognition, SAR imaging and autofocusing continue
to attract more research interest. Non-parametric spectral estimation methods are
robust methods for SAR image formation. However, non-parametric methods cannot
be used to significantly improve the resolution of the formed SAR images since they
generally do not fully exploit the characteristics of radar targets of interests even
when such information is available.
In this dissertation, efforts have been made to form super resolution two-
dimensional (2-D) SAR images via relaxation-based parametric methods. The
relaxation-based optimization methods have been proved to be quite useful in several
x


97
moving target images obtained via conventional SAR processing algorithms are typi
cally mislocated and smeared in cross-range due to phase errors and range migrations
induced by the motions. The amount of smearing generally depends on the charac
ter and magnitude of the target motion during the data coherent collection interval
(CTI). The future work to be conducted includes the establishment of an effective
ISAR data model considering range migrations and phase errors and how to apply
the relaxation-based optimization approach to simultaneous ISAR autofocusing and
image formation.


107
[45] C. E. Mandil and J. M. Swiger. A map drift autofocus technique for correcting
higher order sar phase errors. 27th Annual Tri-Service Radar Symposium Record,
Monterey, CA, pages 391-400, June 1981.
[46] S. L. Marple, Jr. Digital Spectral Analysis with Applications. Upper Saddle
River, Prentice Hall NJ, 1987.
[47] R. P. Marzban. Interferometric Synthetic Aperture Radar Image Formation and
Segmentation. M.S Thesis, University of Florida, Ganiesville, FL, 1996.
[48] Dean L. Mensa. High Resolution RADAR Imaging. Artech House, Inc., 610
Washington St., Dedham, MA, 1981.
[49] D. C. Munson, Jr., J. D. OBrien, and W. K. Jenkins. A tomographic formulation
of spotlight-mode synthetic aperture radar. Proceedings of the IEEE, 71:917 -
925, August 1983.
[50] L. M. Novak, G. R. Benitz, and G. J. Owirka. Classifier performance using
enhanced resolution SAR data. Proceedings of the IEE International Conference
on Radar, pages 634-638, Edinburgh, UK, October 1997.
[51] L. M. Novak, G. R. Benitz, G. J. Owirka, and L. A. Bessette. ATR performance
using enhanced resolution SAR. SPIE Proceedings on Optical Engineering in
Aerospace Sensing, Orlando, FL, pages 332-337, April 1996.
[52] F. Prodi, F. Vinelli, and F. Dente. An autofocus techniques of SAR data based
on superresolution. International Geoscience and Remote Sensing Symposium,
pages 1586-1588, August 1993.
[53] August W. Rihaczek and Hershkowitz Steven J. Radar Resolution and Complex-
Image Analysis. Artech House Inc., Norwood, MA, 1996.
[54] E. A. Robinson. A historical perspective of spectrum estimation. Proceedings of
the IEEE, 70:885-907, September 1982.
[55] E. Rodriguez and J. M. Martin. Theory and design of interferometric synthetic
aperture radars. IEE Proceedings F, 139(2):147-159, April 1992.
[56] R. Roy and T. Kailath. ESPRIT Estimation of signal parameters via ro
tational invariance techniques. IEEE Transactions on Acoustics, Speech, and
Signal Processing, ASSP-37(7):984-995, July 1989.
[57] G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum. Radar Cross
Section Hand Book. Plenum Press, New York, 1970.
[58] R. O. Schmidt. Multiple emitter location and signal parameter estimation. IEEE
Transactions on Antennas and Propagation, AP-34(3):276-280, March 1986.
[59] C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe. Some early developments
in synthetic aperture radar system. IRE Transactions on Military Electronics,
MIL-6(2):111-115, April 1962.
[60] Paul L. Smith, Jr., Kenneth R. Hardy, and Kenneth M. Glover. Applications
of radar to meteorological operations and research. Proceedings of the IEEE,
62(6):724-745, June 1974.


CHAPTER 6
AUTOFOCUS IN CLSAR
To analyze the effect of CLSAR aperture errors, we first rewrite the CLSAR
data model in (5.12) with respect to radar viewing angles. Assume that a curvilin
ear aperture consists of M different viewing angles and let denote the
elevation and azimuth angle pairs of the M look angles of the radar. Let y(n,m),
n = 0,1, , AT 1, denote the one-dimensional (1-D) data samples obtained after
dechirping from the mth viewing angle of the radar. Let
4(7t/o 7To + 7tn)
t"n.
n = 0,1, , N 1,
(6.1)
where tn denotes the time samples. Let tXjn (n), tVm (n), and tZm (n),respectively, denote
the time samples of the mth look angle, where
and
tXrn (n) = tncos(9m) cos(0m),
(6.2)
tym{n) = in sin(9m) COS(0m),
(6.3)
(^) n sin(0m),
(6.4)
with m = 1, 2, , M and n = 0,1, , AT 1. Then y(n, m) has the form
K
y(n, m) = ^ akej2AfktxmW+fktym(n)+fktzm(n)\ A + e(n, m), m 1, 2, , M,
Jb=l
(6.5)
where e(n, m) denotes the noise and clutter and K is the number of scatterers.
We have assumed that {0m,0m}m=i and Ro{m) = Ro, m = 1,2, , M, are
known exactly in the CLSAR data (6.5). For a practical curvilinear SAR system, how
ever, the radar positions relative to the XYZ coordinate system may not be known
79


23
3.2 RELAX-NLS
When targets consist of both trihedral (point scatterers) and dihedral corner
reflectors, we can form enhanced SAR images by using the RELAX-NLS algorithm
[42], We first give a brief review of the RELAX-NLS algorithm and then discuss SAR
image formation via RELAX-NLS.
3.2.1 Review of the RELAX-NLS Algorithm
RELAX-NLS [42] is devised to identify and obtain target features of multiple
corner reflectors (trihedral and dihedral) based on a mixed parametric data model.
An ideal dihedral corner reflector acts as a double-bounce reflector and the RCS
achieves the maximum when the radar beam is perpendicular to the dihedral and
falls off as a function of sinc(0) = sin(0)/9 when the angle between the radar beam
and the line perpendicular to the dihedral increases [49, 57]. Then dihedrals can be
modeled as
Kd
sd(n,n) = ^ adksmc{nbk(n-Tk)]e3{Udkn+Uikn\n = 0,1, , N-l, = 0,1, , N-1,
k=1
(3.14)
where N and denote the numbers of the available data samples in range and
cross-range, respectively; adk, {ujdk,dk}, and bk, k = 1, 2, , Kd, are, respectively,
proportional to the maximal RCS, the central location, and the length of the A:th
dihedral corner reflector; rk, k 1, 2, ,Kd, denotes the peak location of the data
sequence in cross-range and is determined by the orientation of the fcth dihedral
corner reflector; finally, Kd is the number of the dihedral corners. Assume that a
target consists of Kd dihedrals and Kt trihedrals and K = Kd + Kt- Then the target
data model in the presence of noise and clutter has the form:
y(n, ft) = sd(n, ) + s(n, ) + e(n, ), n = 0,1, , N 1, = 0,1, , 1,
(3.15)


36
The received signal reflected from a target scatterer herein is modeled as:
s(n, ) = x()e?2*Un+fn\ n = 0,1, , N 1, = 0,1, , 1, (4.1)
where N and denote the dimensions of the available data samples in range and
cross-range, respectively; x() is an arbitrary unknown real-valued function of
determined by the radar cross section (RCS) of the scatterer; 0 is a constant phase;
finally, {/, /} is the frequency pair proportional to the 2-D location (range and cross
range) of the scatterer. This data model is essentially semi-parametric since little
parameterization is done in cross-range.
Assume that a target consists of K scatterers. Then the target data model in
the presence of noise has the form:
K
y(n, ) = ^ Xk{)e^k2R^knJrk1^ + e(n, ), n = 0,1, , N 1, = 0,1, , 1,
k=1
(4.2)
where {^()}^1 denotes the real-valued amplitude function of for the kth scat
terer; cf)k and {.fk,fk}, respectively, are the constant phase and the frequency pair
of the A;th scatterer; finally, {e(n,)} denotes the unknown 2-D noise and clutter
sequence.
Since SAR images are often used in SAR applications, our problem of interest
herein is to estimate the target parameters {fik, {xk()}%~Q, fk, fk}k=1 from the 2-
D data sequence {y(n, )} and then to form high resolution SAR images with the
estimated target parameters. Before presenting the effects of the semi-parametric
data model in (4.2) on SAR image formation and the SPAR (Semi-PARametric)
algorithm for feature extraction and image formation, we first briefly introduce the
APES algorithm.


45
larger the extrapolation factor £, the more significant the artifact problem since the
difference between fx and f2 is exaggerated £ times. Figures 4.2(d) shows the SAR
image obtained with £ = 2. By comparing Figures 4.2(d) and (c) (here £ = 1,
without extrapolation), we note that the artifact next to the short dihedral becomes
more significant. Severe artifacts may exist at low SNR since the accuracy of the
parameter estimates is poor. The SPAR algorithm we present below attempts to
avoid this problem by using windows to isolate the multiple scatterers located in the
same range.
4.4 The SPAR Algorithm
SPAR can be summarized by the following two steps:
Step 1: Scatterer Isolation based Target Feature Extraction: See Chapter
4.4.1 below for details.
Step 2: SAR Image Formation: See Chapter 4.3.2 for details, where the
estimated target features are obtained by using Step 1.
4.4.1 Target Feature Extraction
The basic idea behind SPAR is to extract the features of each scatterer sepa
rately. Before we present the target feature extraction algorithm, we first summarize
the steps needed for the feature extraction of a single scatterer as a preparation. The
generalized Akaike information criterion (GAIC) is also introduced to estimate K,
the number of scatterers, at the end of this subsection.
Feature Extraction of a Single Scatterer
The data model of a single scatterer in the presence of noise has the form:
ys(n, ) = s(n, ) + es(n, ), n = 0,1, , IV 1, = 0,1, , 1, (4.30)


106
[30] S. M. Kay. Modern Spectral Estimation: Theory and Application. Prentice-Hall,
Englewood Cliffs, N.J., 1988.
[31] S. M. Kay and S. L. Marple, Jr. Spectrum analysis a modern perspective.
Proceedings of IEEE, 69(11) :13801418, November 1981.
[32] Klauder, Jr., A.C. Price, S. Darlington, and W. J. Albersheim. The theory and
design of chirp radars. The Bell System Technical Journal, 39(4):745-808, July
1960.
[33] Kenneth Knaell. Three-dimensional SAR from curvilinear apertures. SPIE Pro
ceedings on Optical Engineering in Aerospace Sensing, 2230:120-134, Orlando,
FL, April 1994.
[34] Kenneth Knaell. Three-dimensional SAR from practical apertures. SPIE Pro
ceedings on Optical Engineering in Aerospace Sensing, San Diego, 2562:31-41,
1995.
[35] Kenneth Knaell. Three-dimensional SAR from curvilinear apertures. Proceedings
of the 1996 IEEE National Radar Conference, pages 220-225, Ann Arbor, MI,
May 1996.
[36] V. Larson and L. Novak. Polarimetric subspace target detector for SAR data
based on the Huynen dihedral model. Proceedings of SPIE, 2487:235-250, Or
lando, Florida, April 1995.
[37] Hongbin Li, Jian Li, and Petre Stoica. Performance analysis of forward-backward
matched-filterbank spectral estimators. IEEE Transactions on Signal Processing,
46(7): 1954-1966, July 1998.
[38] J. Li and P. Stoica. An adaptive filtering approach to spectral estimation and
SAR imaging. IEEE Transactions on Signal Processing, 44(6):1469-1484, June
1996.
[39] J. Li and P. Stoica. Efficient mixed-spectrum estimation with applications to
target feature extraction. IEEE Transactions on Signal Processing, 44:281-295,
February 1996.
[40] Jian Li, Zheng-She Liu, and Petre Stoica. 3-D target feature extraction via
interferometric SAR. IEE Proceedings on Radar, Sonar and Navigation, 144(2),
April 1997.
[41] Zheng-She Liu, Hongbin Li, and Jian Li. Efficient implementation of Capon and
APES for spectral estimation. IEEE Transactions on Aerospace and Electronic
Systems, 34(4):1314-1319, October 1998.
[42] Zheng-She Liu and Jian Li. Feature extraction of SAR targets consisting of
trihedral and dihedral corner reflectors. IEE Proceedings on Radar, Sonar and
Navigation, 145:161-172, March 1998.
[43] Zheng-She Liu and Jian Li. SAR motion compensation and feature extraction
via MCRELAX. Journal of the Optical Society of America A, 15(3):599-610,
March 1998.
[44] C. E. Mancill. Enhancement of radar imagery by maximum entropy processing.
Proceeding of the Human Factors Society 21 st Annual Meeting, pages 241-243,
1977.


72
Let
F= A j'A Du Do Do
(5.36)
where A is defined in (5.20), the Ath columns of D^, D, and Dq are d[a.k
afc]/3wfc, 3[afc <8> fc (g> afc]/<9d)fc, and d[afc afc afe]/du)fc, respectively, and
(5.37)
Then the CRB matrix for the parameter vector rj is given by:
CRB(rj) = [2Re(FHQ_1F)]_1.
(5.38)
For the case of curvilinear aperture, the CRB matrix for the target parameters
is similar to the one in (5.38) except that the A in (5.38) is now replaced by Ac.
5.4 Numerical and Experimental Results
We first use CRBs to investigate the performances of different curved apertures
for target feature extraction and then demonstrate with an experimental example the
performance of the RELAX algorithm for 3-D target feature extraction using CLSAR.
5.4.1 Performance Analysis of Different Curvilinear Apertures via CRBs
Since CRBs are the best unbiased performance any asymptotic estimator can
achieve, we not only compare CRBs of target features when different curved apertures
are used for 3-D target feature extraction, but compare them with the CRB of the
full aperture. Without loss of generality, let us consider the case of a single scatter
with a. 1 and u Co = = 0. The additive noise is assumed to be zero-mean
white Gaussian with variance a2 = 40. The full aperture data is generated according
to (5.16) with N = = = 32.
Consider first the example of M = 63. The curved apertures we consider
include the parabolic one in Figure 1.2(b), the L-shaped one in Figure 1.2(c), and
the ones in Figure 5.2. For the arc apertures, the Arc-1 aperture is a quarter of a


59
(a)
(b)
(c) (d)
Figure 4.2: Ambiguity effect on the SAR image formation in the presence of range
estimation errors, (a) Windowed FFT image of the first scatterer. (b) Windowed
FFT image of the second scatterer. (c) Combined windowed FFT image of the
two scatterers with C = 1 (without extrapolation), (d) Combined windowed FFT
image with £ = 2. (The vertical and horizontal axes are for range and cross-range,
respectively.)


Devotion (meters')
90
Figure 6.1: (a) Curvilinear apertures for the experimental example, (b) True scatterer
distribution, (c) 3-D plot of K = 20 scatterers extracted from the initial aperture
(dashed line) in (a) with RELAX, (d) 3-D plot of K = 20 scatterers extracted from
the manually adjusted aperture (solid line) in (a) with RELAX.


26
Identification of Crner Reflector
It has been shown in [42] that Cd is much less than Ct when the true corner
reflector is a dihedral and Ct is almost the same as Cd when the true corner reflector
is a trihedral. Also b 26f when a trihedral is mistakenly considered as a dihedral,
where Sf is the smallest positive solution to
sm(N5fir)
N
~2~'
(3.27)
sin(5/7r)
Then a corner reflector can be identified by checking the estimate b and the cost
functions Ct in (3.19) (with {at,u)t, ojt} replaced by {at,u)t,u)t}) and Cd in (3.24)
(with {cOdi Cbd, Od-, b, r} replaced by {j, uid, b,f}. A corner reflector is identified as
a dihedral if
cd
>0.1 and b > 25f.
Steps of RELAX-NLS are outlined as follows:
Step (1): Assume K 1. Identify and obtain the parameters of a single
corner reflector.
Step (2): Assume K = 2. Subtract the signal obtained in Step (1) from
{y(n,)} and then identify and obtain parameters of the second corner reflector.
Subtract the second signal from {y(n,n)} and re-identify and redetermine the pa
rameters of the first signal.
Iterate the previous two substeps until practical convergence.
Step (3): Assume K = 3. Subtract the two signals obtained in Step (2) from
{y(n,)} and then identify and obtain the parameters of the third signal. Subtract
the second and third signals from {y(n, )} and re-identify and redetermine the first
signal. Subtract the first signal (most recently determined) and the third signal from
{y(n,n)} and then re-identify and redetermine the second signal.
Iterate the previous three substeps until practical convergence.
Remaining Steps: Continue similarly until K K (K can be determined
by using GAIC to be discussed later on).


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jian Li Chairman
Associate Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Jos C. Principe
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Frederick J. Taylor
Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
William W. Edmonson
Assistant Professor of Electrical and
Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David C. Wilson
Professor of Mathematics


CHAPTER 2
LITERATURE SURVEY
In this chapter, we give a brief review of the topics related to the work, namely
super resolution 2-D SAR image formation, 3-D SAR target feature extraction, and
SAR motion compensation.
2.1 2-D Super Resolution SAR Image Formation
Since SAR image formation can be considered as a spectral estimation prob
lem, modern spectral estimation methods find extensive applications in improving
the resolution and accuracy of the formed SAR images. Excellent texts on spectral
estimation are available [63, 30, 46] and historical and modern perspectives on the
general topics of spectral analysis can be found in [54, 31]. A brief review of super res
olution SAR image formation via non-parametric and parametric spectral estimation
methods is given below.
Non-parametric Spectral Estimators
An important class of non-parametric spectral estimators are the adaptive
filterbank approaches which can be used to form SAR images with better resolu
tion and lower sidelobes than the FFT based methods [15]. The adaptive sidelobe
reduction (ASR) approach [14] and the widely used Capon method [10] as well as
its reduced rank variations [5, 6] are members of this class. A recently introduced
matched-filter bank based complex spectral estimator that is of interest is the APES
(Amplitude and Phase Estimation of a Sinusoid) method [38]. Despite their different
derivations, it was found that both Capon and APES are members of the general
class of the matched-filterbank (MAFI) approaches to complex spectral estimation
11


51
scatterers. However, the Hybrid SAR images may suffer from more severe artifact
problem than SPAR, especially at low SNR. When no multiple scatterers are located
in the same range, SPAR and Hybrid perform similarly.
4.5 Modified RELAX-NLS Algorithm
RELAX-NLS [42] is a parametric approach for the feature extraction of targets
consisting of both trihedrals and dihedrals. It is based on a mixed data model in
which x(n) is modeled as a real-valued constant for a trihedral or a sine function of
for a dihedral. Like SPAR, RELAX-NLS extracts the target features by minimizing
an NLS cost function via a relaxation-based approach. However, RELAX-NLS is
computationally expensive since a 4-D search over the parameter space is required
for dihedral corner reflectors. Since SPAR is more robust and computationally more
efficient than RELAX-NLS, the former can be used to provide the initial conditions
needed by the latter.
Let {fa, fa, fk, fk}k=i denote the parameter estimates obtained via SPAR
according to the data model in (4.2), where K is the estimated number of scatterers
obtained via GAIC. The SPAR estimates {fa, fa, fk, fk}k=1 cannot be used directly
as initial conditions for RELAX-NLS. The initial conditions are obtained by applying
the first step of RELAX-NLS [42] to each Ufc, k 1,2, , K, where
k
Ufc = Y £ e**G ifrjiji), k = 1,2, , K. (4.50)
=l^zk
Once we have the initial conditions, we can use the last step of RELAX-NLS [42] to
obtain the dihedral and trihedral parameter estimates, which are then used for SAR
image formation [7]. We refer to this approach as the modified RELAX-NLS.


69
parameter estimates for the first K 1 scatterers can be used to initialize the search
for the parameters of the Kth one. We first briefly make a preparation for the RELAX
approach. Let
R
y/c = y a(i <8> ^), (5.25)
A A A
where , , and are formed, similarly to those in Equation (5.21), from
and respectively, and {)j,¡)j,u;j, are assumed to have been estimated.
Then minimizing C4 in (5.24) with respect to ak yields the estimate k of ak:
Oik
[a* <8) afc (g) afc]H yfc
NNN
Uk Uk t^k-Uk !<*>k =Uk
(5.26)
and
{A, O)fc, CUfc}
arg max
la* at a*]Hyfc
(5.27)
Hence {0fc, Dfc, } can be obtained as the location of the dominant peak of the 3-D
periodogram,
h 2
[a* afe 0 a*] yk
N
which can be computed efficiently with 3-D FFT. Note that padding with zeros
for the 3-D FFT is necessary to achieve high accuracy. An alternative approach is
to find an approximate location corresponding to the global maximum with the 3-D
FFT without much zero-padding and then use the approximate location as the initial
condition to find a more accurate position via a multidimensional search method, such
as the FMINV function in PV-WAVE. We used the latter approach in our examples
presented in Section 5.4.2. Note that ak is easily computed from the complex height
of the peak of [a*, kk ak]H yk/(N).
With the above preparations, we now proceed to present the steps of the
RELAX algorithm for 3-D target feature extraction with the full aperture SAR.
Step (1): Assume K = 1. Obtain {(2>k, k, G)k, ak}k=i from y by using (5.27) and
(5.26).


CHAPTER 4
SEMI-PARAMETRIC METHODS FOR SAR IMAGING
4.1 Problem Formulation and Data Model
Compared with parametric spectral estimation methods, non-parametric spec
tral estimation methods are robust, but cannot be used to significantly improve the
resolution of the formed SAR images. As can be seen from the previous chapter that
relaxation-based parametric methods can be used to generate super resolution SAR
images when the data model for targets of interest is appropriate. However, the super
resolution property of parametric methods will be compromised when the appropriate
data model is not available. In this chapter, we consider the case where it is difficult
to establish a good parametric data model in cross-range for target scatterers due
to a variety of physical phenomena including glints, resonances, and motion-induced
phase errors, etc. Instead of modeling the target of interest as 2-D sinusoid (point
scatterer) in both range and cross-range, we apply herein a semi-parametric model
and simply assume that the data model in cross-range due to one or more corner re
flectors is a complex sinusoid with an arbitrary amplitude and a constant phase. Due
to the flexibility of this data model, feature extraction and image formation meth
ods based on this data model should be more robust compared with those based on
the approximated dihedral and trihedral data models in Chaper 3. The robustness,
however, comes at the cost of small resolution improvement in cross-range over the
conventional fast Fourier transform methods due to assuming little in cross-range. To
preserve the robustness, we use a refined non-parametric spectral estimation method
such as APES (Amplitude and Phase Estimation of a Sinusoid) algorithm [38] to
improve the cross-range resolution.
35


(a)
(b)
(c)
(e) (f)
Figure 4.3: Comparison of SAR images formed using different algorithms for simu
lated data at high SNR (a% = 0.6). (a) True SAR image, (b) Windowed 2-D FFT
SAR image, (c) Hybrid SAR image, (d) SPAR SAR image, (e) RELAX-NLS SAR
image, (f) Modified RELAX-NLS SAR image. (The vertical and horizontal axes are
for range and cross-range, respectively.)


30
not only have a better resolution than the 2-D FFT images, but also more closely
reflects the characteristics of the photo shown in Figure 3.3.


50
{^fc, 5c*¡, /fc, /fc}*=i,3, calculate Y2 from Y2, and then redetermine {0fc, xfe, fk, /fc}fc=2
from Y2.
Iterate the previous three substeps until practical convergence.
Remaining Steps: Continue similarly until K is equal to the desired or
estimated number of scatterers.
The practical convergence in the iterations of the above relaxation-based
NLS algorithm may be determined by checking the relative change e of the cost
function C5 f{0fc,Xfc,/fc,/fc}in (4.44) between two consecutive iterations. Our
numerical and experimental examples show that the algorithm usually converges in
a few iterations.
We can determine K, the number of scatterers in (4.2), by extending the
generalized Akaike information criterion (see [61] for details). By assuming that the
noise is white, the estimate iY of iY is determined as an integer that minimizes the
following extended GAIC cost function:
(N-1-1 \
EE \pc(n,n)\2\ + y\n[\ny(NN)][K(N + 3) + 1], (4.48)
n=0 =0 /
where 7 is a constant of user choice,
k
fcin, ) = y(n, )Xk{n)e^ke:2'K^kn+^kn\ n = 0,1, , TV1, = 0,1, , TV1,
fc=i
(4.49)
and Tf(TV + 3) + 1 is the total number of real-valued unknown parameters (of which
K(TV + 3) is for the scatterers and 1 is for the white noise variance).
Note that the NLS estimates of {(j)k, xfc, fk, fk}k=i can also be estimated from
determined in (4.45), rather than {Yfc}^, via the above relaxation-based
optimization algorithm. We refer to this approach as the Hybrid method. When
multiple scatterers are located in the same range, Hybrid may be computationally
more efficient than SPAR since Hybrid does not isolate the scatterers so that multiple
scatterers located in the same range can be more efficiently described as at most two


28
where ns = 0,1, , £N 1 and fis = 0,1, , ( 1 with £ being an extrapolation
factor (£ > 1); dk, Codk, u>dk, h, and fk, respectively, denote the estimates of adk,
Ujdk, Qdk, bk, and Tk, k = 1, 2, , Kd\ finally, Jfe, wifc, and utk, respectively, denote
the estimates of atk, cotk, and Qtk, k = 1, 2, , Kt. The clutter estimate (n, )
used in Step 3 is now determined by (3.29) with Kt and Kd replaced by Kt and Kd,
respectively. Note that scaling the simulated data for the dihedrals by a factor of £
in (3.30) is needed since the sine function goes to zero as ns increases or decreases
away from rk. Note also that £ is a parameter of user choice and £ > 1 is just
used to demonstrate the super resolution property of RELAX-NLS. Finally, since the
Fourier transforms of the sine functions of sufficient lengths do not result in sidelobes,
when ys(ns,s) in (3.30) is windowed, the second term of ys(jis,fis) is multiplied by
ws(ns,s) and the first term is multiplied by wi(ns), where w\ins) is a 1-D window
sequence satisfying
CAT-1
w{na) = (N. (3.31)
7ls=0
3.3 Experimental Results
In this section, we demonstrate the image formation performances of RELAX
and RELAX-NLS with the experimental data. In the following examples, the extrap
olation factor £ = 2 is used and GAIC with 7 = 4 and 7 = 18 are used to determine
K for RELAX and RELAX-NLS, respectively. Kaiser windows with shape parameter
(3 6 are used whenever needed.
First consider SAR image formation via 2-D FFT and RELAX by using a
portion of the 2-D data corresponding to some roof rims collected by one of the two
apertures of the ERIMs (Environment Research Institute of Michigans) DCS inter
ferometric synthetic aperture radar (IFSAR). Figures 3.1(a) and (b), respectively,
show the unwindowed and windowed 2-D FFT images obtained by zero-padding the
40 x 40 phase history data. The unwindowed and windowed RELAX images with and


46
where s(n,n) is given in (4.1) and {es(n,n)} denotes the unknown 2-D noise and
clutter sequence. Let
UnU)
1 ej2irf
T
ej2nf(N-l)
(4.31)
and
Wff(/) =
-\T
1 ej2nf . ej2Trf(N-l)
(4.32)
where (-)T denotes the transpose. Let D(/) denote the following diagonal matrix:
D(/) = diag|l, ..., eK*W-i)J.
(4.33)
Define
T 1T
x x(0) x(l) x( 1) (4-34)
Let Ys be an N x matrix with its (n, )th element being ys(n,). Then we can
rewrite (4.30) as:
Ya = e*G(x,/,/)+Es, (4.35)
where
G(x,/,/) = w(/)xTD(/), (4.36)
and Es denotes an N x TV matrix with es(n,n) being its (n,)th element. Let yS,
= 0, 1, ,iV 1, denote the th column of Ys and define
Mf) = Y WM), (4.37)
where ()* denotes the complex conjugate. Then the NLS estimates jx, , /,/j of
{x, , f, /} are (see Appendix A for the detailed derivations):
x = ^Re [eJ0ys(/) 0w^(/)] (4.38)
where Re (a:) denotes the real part of x and denotes the Hadamard matrix product
or the element-wise product of two matrices;
i = arg ( [y]>M/)]V22,2/,4 (4.39)


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4
The cross-range resolution of strip-map SAR is limited by the azimuth beam-width,
while that of spotlight-mode SAR is independent of the azimuth beam-width since the
antenna steers its beam to continuously illuminate the same scene. Hence Spotlight
mode SAR is more suitable for high resolution SAR image formation of a small scene
than strip-map SAR.
1.2 Scope of the Work
Spotlight-mode SAR has been extensively applied to 2-D high resolution im
age formation and target feature extraction of small scenes due to its finer resolutions
and readily controllable flight path, which is a straight line. The 2-D target features
include the radar cross section (RCS) and the 2-D location (range and cross-range) of
each scatterer. One topic of our work is super resolution 2-D image formation from
phase history data collected by spotlight-mode SAR. The super resolution herein is
defined as the resolution that is better than the Rayleigh limits determined by the
transmission wavelength and the length of the synthetic aperture of SAR. A conven
tional approach for 2-D image formation from SAR phase history data is the fast
Fourier transform (FFT), which is computationally efficient and robust. However,
SAR images obtained via FFT suffer from poor resolution and high sidelobes. SAR
image formation can be considered as a spectral estimation problem since SAR images
are the reconstruction of scene reflectivity intensities versus space locations. Mod
ern spectral estimation methods including parametric and non-parametric methods
have found extensive applications in improving the resolution and accuracy of the
formed SAR images. In particular, parametric spectral estimation methods are more
suitable to form super resolution SAR images when parametric data models can be
used to model the target. The super resolution of the formed SAR images can be
achieved by extrapolating the phase history data beyond observations using an appro
priate data model and the parameter estimates. We will discuss how to form super


other applications, such as radio astronomy, microwave imaging, and spectral estima
tion. The relaxation-based methods are extended for super resolution SAR imaging
of radar targets consisting of only trihedrals or both trihedrals and dihedrals. We
have also devised a robust and computationally simple SPAR (Semi-PARametric) al
gorithm for 2-D SAR imaging based on a flexible semi-parametric data model when
it is difficult to establish an accurate target data model in cross-range. Hence SPAR
takes advantages of both parametric and non-parametric spectral estimation meth
ods to form enhanced SAR images. Numerical and experimental examples have been
used to demonstrate the performances of the proposed algorithms. We have observed
that the relaxation-based parametric methods provide super resolution SAR images
when the assumed data model is valid; otherwise SPAR performs better.
Three-dimensional (3-D) target features, including the height information,
radar cross section (RCS), and 2-D location (range and cross-range), provide quite
useful information for such applications as automatic target recognition. Thus, ef
forts have also been made in this dissertation to devise an effective relaxation-based
algorithm, referred to as AUTORELAX, for both 3-D target feature extraction and
motion compensation via curvilinear SAR (CLSAR), a novel technology which is
still at its developing stage. The proposed AUTORELAX algorithm is shown to be
promising when evaluated by using both the experimental and simulated examples.
xi


24
where
Kt
st(n, n) = y^jatke^utkn+mkn\ n = 0,1, , N 1, = 0,1, , 1, (3.16)
k=1
with and {aHk,tk}k=v respectively, denoting the unknown complex am
plitudes and 2-D unknown frequencies of the Kt trihedrals. The RELAX-NLS al
gorithm proposed in [42] can be used to identify and effectively estimate the tar
get features{adk, bk, Lodk, Odk, Tk}kii and {atk, utk, tk by minimizing the following
NLS criterion:
C2 (^{atk > fc=l> iadk > bk,Udk dk rA:}fc=i^ || Y St Sd\\F (3-17)
where [Y]y = y(i,j), [St]*j = st(i,j), and [Sd]ij = sd(i,j). RELAX-NLS converts
the problem of multiple corner reflector identification and feature extraction to that
of a sequence of single reflector identification and feature extraction by using the
relaxation based optimization approach. The identification and feature extraction of
a single corner reflector in RELAX-NLS is described as follows:
Feature Extraction of a Single Trihedral Corner Reflector
Let Yt denote the data of a single trihedral in the presence of unknown noise
Et. Then
Y t = cx.tu n + E, (3.18)
where at and respectively, are proportional to the RCS and 2-D location
of the trihedral. The NLS estimates {at,ujt,tit} of {at, ut, Cot} can be obtained by
minimizing the following cost function
Ct {auuuQt) = ||Yj atuN(y and respectively, are given by
v%(ut)Ytu;*()t)
NN
(3.20)


BIOGRAPHICAL SKETCH
Zhaoqiang Bi was born in Heilongjiang Province, China on October 15, 1968.
He attended Huazhong University of Science and Technology, Wuhan, China in
September 1986, where he earned his Bachelor of Engineering degree in electrical
engineering in July 1990. He earned his Masters degree in communication and elec
tronic systems from Communication, Telemetering, and Telecontrol Research Insti
tute (CTI) in Shijiazhuang, China in March 1993. He worked as an IC designer for
digital communication systems with advanced EDA (electronic design aided) tools in
CTI from 1993 to 1996. He has been a graduate research assistant in the Spectral
Analysis Lab (SAL) of the Department of Electrical and Computer Engineering at
the University of Florida since August 1996, during which time he has conducted
research in the area of radar signal processing, including target feature extraction
and synthetic aperture radar image formation.
109


82
Then
ah exp {j2v(fktXm + fktVm + /fctZm)} ~ ak exp l j2ir
o o
fkt cos 0m +fkt0COS 00Or
+fkt sin 0
exp j2-Kfkt0 cos 0o A6,,
(6.19)
which shows that a constant A9m also results in the ambiguity between the phase
error due to A9m and the phase of ak. Hence the phase of ak can never be determined
O ~
exactly in the presence of A9m. Also, if A9m is a linear function of 9m, then fk and
the linear phase error due to A9m are ambiguous and cannot be determined exactly.
o O
Finally, consider the errors in Ro(m) when 9m 9m and 0m =0m, m
1,2, , Af. Let
R0(m) R0 + AR0(m). (6.20)
Replacing R0 in (5.7) with Ro(m) and for small Ai?0(m), large R0(m), and 7r/0
7(i r0), where |i t0| < T0/2, we have from (5.11)
9, 0)
ux,y,zt
j {xtx +ytv +ztz) ejt0 ARq (m)
(6.21)
Hence the errors in {R0(m)}!^=1 result in phase errors along the synthetic aperture,
which is consistent with the analysis in [29, 43]. Then the z(n,m) in (6.5) should,
for this case, be replaced by
x(n, m)
K
E
. k=\
Oik &
j2ir[fktxm(n)+fktym(,n)+fktzmM} l p3Vm
(6.22)
where t0AR0(m) is the phase error caused by AR0(m). Note that this phase
error differs from the phase errors in (6.14) and (6.19) since it does not depend on the
parameters of the A;th scatterer and is easier to deal with. If AR0(m) is a constant,
then the phase error due to AR0(m) and the phase of ak are ambiguous. Hence
the phase of ak can never be determined exactly in the presence of ARo(m). If
o
AR0(m) is a linear function of 9m, then fk and the linear phase error due to AR0(m)
is ambiguous and cannot be determined exactly.


68
row containing only one non-zero unit element corresponding to the locations of the
available data samples. Then
yc = Aca + e, (5.22)
where
Ac = {Ic(i i)} ai {Ic(aK a^)} aK
The unknown sinusoidal parameters {cok, k, k, ctk, }(L1 are the features of our
interest and are to be estimated from the yc collected by CLSAR. In the following
sections, we will consider parametric 3-D target feature extraction using CLSAR.
5.2 The RELAX Algorithm
The RELAX algorithm [39] can be extended to extract 3-D target features. We
first consider using RELAX with the full aperture for 3-D target feature extraction
and then extend this approach to the case of curvilinear apertures.
5.2.1 Full Aperture
The RELAX algorithm [39] minimizes the following NLS cost function:
C4 = ||y- Ac*||\ (5.24)
where ||-|| denotes the Euclidean norm. When the noise e(n, , ) is the zero-mean
white Gaussian random process, the NLS estimates obtained with RELAX coincide
with the ML estimates of the target features. When the noise is colored, the NLS
estimates are no longer the ML estimates, but they still possess excellent statistical
performance [39].
We present below the relaxation-based minimization approach that leads to a
conceptually and computationally simple method. For each fixed K, the intermediate
number of scatterers, we perform a complete relaxation-based search by letting only
the parameters of one scatterer vary and freezing all others at their most recently
determined values. In this way, we will also take advantage of the fact that the
(5.23)


27
The practical convergence in the iterations can also be determined by check
ing the relative change of the cost function C2 in (3.17) between two consecutive
iterations.
We can also determine K, the total number of trihedrals and dihedrals, by
extending the GAIC discussed in Section 3.1 and assuming white noise and clutter.
The estimate K of K can be determined as an integer that minimizes the following
extended GAIC cost function:
(N-1-1 \
EE Ie(n, n)|2 I + 7ln[ln(AGV)](4iG + 6Kd + 1), (3.28)
TiO =0 /
where
Kd
e(n, n) = y(n, n) dd,sinc[7r6fe(n fk)\e
j{^dkn+ihdkn)
Kt
J2tkej{Cjt*n+£lt*), (3.29)
k=l
k=1
with n = 0,1, , N 1 and = 0,1, , 1; 7, as before, is a parameter of user
choice; K = Kt + K¡i with Kt and Kd denoting the numbers of trihedral and dihedral
corners, respectively, determined by RELAX-NLS given K\ finally, 4Kt + 6Kd + 1
is the total number of unknown real-valued parameters (of which 4Kt and 6Kdi
respectively, are for the trihedral and dihedral corners and 1 is for the white noise).
3.2.2 Image Formation
We can form enhanced SAR images of targets consisting of both trihedrals
and dihedrals by using the extracted target features via RELAX-NLS and the mixed
data model in (3.15). The SAR images obtained via RELAX-NLS are referred to
as RELAX-NLS SAR images or simply RELAX-NLS images. To form RELAX-NLS
images, the dominant target features are first extracted by using the RELAX-NLS
algorithm (see Section 3.2.1 for details) and then Steps 2 and 3 discussed in Section
3.1.2 are used except that the simulated phase history data of large dimensions used
in Step 2 is now determined by
Kd Kt
ys(ns, s) = CX] dfcsincMfc(s fk)}eUdkns+dknA + ^ (3 30)
k=1 fc=l


74
all of the curvilinear apertures since u is in the range direction, which is normal
to the plane of the curvilinear apertures. Comparing the CRBs for the L-shaped,
Arc-1, and Arc-2 apertures, we note that as expected, the more curved the aperture
is, the lower the CRBs for the target features. Since the Arc-2 aperture is the least
curved, it has the largest CRBs. When the aperture becomes one straight line and
hence no longer curved, the CRBs go to infinity since we can no longer extract 3-D
target features. Note also that from the top to bottom of Table 5.4.1, the CRBs
for J and Co increase since the aperture length decreases. The CRB for a, however,
does not always increase. Consider next the circular aperture in Figure 5.2(a) when
ui = Q = 0. Figure 5.4.2 shows the CRBs of the target parameters as a function
of the radius of the circular aperture for different M. Note that as expected, the
larger the radius and/or M, the lower CRBs of the target parameters. Note also
that, as expected, the CRB for co does not change with the radius for the fixed
M since again to is in the range direction, which is orthogonal to the plane of the
curvilinear aperture.
5.4.2 Experimental Examples
Before we present an experimental example to demonstrate the performance
of the RELAX algorithm for 3-D target feature extraction with a CLSAR, we first
introduce an alternative to the Polar-to-Cartesian interpolation. Instead of Polar-
to-Cartesian interpolation, we created an N x N x rectangular grid and mapped
the data sample at each (tx,ty,tz) to the nearest grid point, where tx, ty, and tz are
functions of t, 9, and 4> (see Equations (5.13) to (5.15)). Let N denote the number of
available data samples. Then for both full and curvilinear apertures, the data model
can be written as
yr = Ir Aa + e,
(5.39)


LIST OF TABLES
4.1 True parameters of the two dihedrals used in Figures 1 and 2 44
5.1 Comparison of the CRBs (in dB) of the target features for the cases
of the full, circular, parabolic, L-shaped, Arc-1, and Arc-2 apertures
when M = 63, K = 1, uj = Q = Q = 0, a = 1, and a2 = 40 73
6.1 Estimation errors for the parameters of the highest scatterer in the
simulation example before and after autofocusing 89
vi


19
frequencies of the fcth sinusoid or point scatterer, k = 1,2, K] {e(n, )} denotes
the unknown noise sequence. Note that {cok,cok} corresponds to the 2-D location of
the kth scatterer and ak is determined by the RCS and range of the fcth scatterer.
The NLS estimates {ak,ujk,LOk}k=1 f {ak, ^k)k=i can be obtained by minimizing
the following NLS criterion [39]:
Ci ({ak,ujk,u>k}^=1)
K
Y Y cvN(uk)uj:fr(ujl.)ak
k=1
(3.2)
where ()T denotes the transpose; || \\p denotes the Frobenius norm; Y is an N x N
matrix with y(n, ) being its (n)th element;
^N^k)
1 ejuJk
T
ejuk(N-l)
(3.3)
n T
^N^k) 1 e^k e^k^N b (3-4)
RELAX minimizes the NLS cost function C\ in (3.2) by using a complete relaxation-
based search, i.e., for each K, the intermediate number of scatterers, only the param
eters of one scatterer vary while all others are fixed at their most recently determined
values. Let
K
~ (3.5)
Yt = Y- ^2 Q¡W/(w¡)w5(iij),
iz=\,izjLk
where {, >, are assumed to have been estimated. Then {o;*, ¡>*, are
obtained by
Oik =
NN
\Uk=Uk,Uk=Uk
(3.6)
where (-)H and ()*, respectively, denote the conjugate transpose and complex con
jugate, and
{ujk,uk} = argmax \u>%(ujk)YkuJ(k) |2 (3.7)
Uk ,Uk
which indicates that {tuk,uk} can be obtained as the location of the dominant peak
of the periodogram |a;^(w*;)Yfea;^.(;fc)|2 /N, which can be efficiently calculated
via FFT. Then, ak can be calculated from the complex amplitude of the peak of


20
\uj%(u>k)Ykuj*()k)^ /N. The RELAX algorithm is summarized as follows:
Step (1): Assume K 1. Obtain {Ck,u>k,ak}k=i from Y by using (3.7) and (3.6).
Step (2): Assume K = 2. Compute Y2 with (3.5) by using {uok, uk, k}k=i ob
tained in Step (1). Obtain {uk,)k,k}k~2 from Y2. Next, compute Yi by using
{vk,u>k, ak}k=2 and then redetermine {ujk, uk, k}k=\ from YL.
Iterate the previous two substeps until practical convergence is achieved.
Step (3): Assume K = 3. Compute Y3 by using {k,k,k}k=i obtained in Step
(2). Obtain {u)k,k,k}k=3 from Y3. Next, compute Yi by using {A, k, k}\=2 and
redetermine {k,u)k, k}k=i from Yx. Then compute Y2 by using {tbk,uk, k}k=1
and redetermine {u)k, Ok, o¡k}k=2 from Y2.
Iterate the previous three substeps until practical convergence.
Remaining Steps: Continue similarly until K is equal to K, which can be deter
mined by using generalized Akaike information criterion (GAIC) to be discussed later
on.
The practical convergence in the iterations may be determined by checking
the relative change e of the cost function C\ ({)k,u)k,&k}k=i) in (3.2) between two
consecutive iterations.
It has been demonstrated in [39] that RELAX is related to CLEAN [28], which
is only an approximate relaxation-based method that minimizes the NLS criterion.
RELAX turns out to be CLEAN when no iteration is performed in each step of
RELAX. Due to the iterations performed in each step of RELAX, a more accurate
initial can be obtained, which makes RELAX converge to the global minimum with a
high probability. Meanwhile, RELAX is asymptotically (for large number of samples)
efficient, while CLEAN gives biased estimates, especially when two scatterers are
closely spaced. The RELAX algorithm, thereby, can also be referred to as SUPER
CLEAN.


56
same range. In this example, Hybrid and SPAR have the same estimated number of
scatterers, K 7. Figure 4.3(b) shows the windowed 2-D FFT SAR image obtained
by applying the normalized 2-D FFT to the windowed data matrix. SAR images
formed via Hybrid, SPAR, RELAX-NLS and the modified RELAX-NLS algorithms
are shown in Figures 4.3(c) through 4.3(f), respectively. We note that at high SNR,
the Hybrid image is similar to the SPAR image. Both of the parametric RELAX-NLS
and the modified RELAX-NLS algorithms outperform their semi-parametric coun
terparts SPAR and Hybrid since the data model used by the parametric methods is
correct rather than approximate. For this example, our simulations show that the
ratios between the MATLAB flops needed by Hybrid, SPAR, the modified RELAX-
NLS, and RELAX-NLS over the flops needed by the windowed FFT method are 27.4,
28.4, 50.1, and 70.8, respectively. Note that both SPAR and Hybrid are computa
tionally more efficient than RELAX-NLS and the modified RELAX-NLS, with the
modified RELAX-NLS being more efficient than RELAX-NLS.
Consider next a simulated example with low SNR. The SAR phase history
data is the same as in the above example except that the noise variance is increased
to = 6. SAR images obtained by using the windowed 2-D FFT, Hybrid, SPAR,
RELAX-NLS, and the modified RELAX-NLS are shown in Figures 4.4(a) through
4.4(e), respectively. Note that the artifact problem starts to show up in the Hybrid
image in Figure 4.4(b) due to large parameter estimation errors. By comparing
Figures 4.4(c) with (b), it can be seen that SPAR can effectively mitigate the artifact
problem.
Finally, consider an experimental example of SAR image formation by using
the MSTAR Slicy data collected by imaging an object consisting of both trihedral
and dihedral corner reflectors, which is shown in Figure 3.3. The data was collected
by the Sandia National Laboratory (SNL) using the STARLOS sensor. The field
data was collected by a spotlight-mode SAR with a carrier frequency of 9.559 GHz


49
the interval n\ < n+ < n2 so that the magnitude of Vfe is above a certain threshold,
say within the interval. Similarly, we can fix n to n+ and search for the interval
i < + < 2. Then the N x rectangular window w(n,fi) has unit value for
ni < n < n2 and \ < < 2 and zero value elsewhere. The threshold T we use in
our numerical and experimental examples is 10% of the peak value of the magnitude
of Vfc.
Step (iii): Determine Yk by applying 2-D inverse FFT (IFFT) to Vfc W,
where the (n, )th element of W is w(n,n).
Instead of minimizing Ce{4>k, Xfc, fk, fk), we now minimize
2
5
F
C7((j)k, xjt, A, A) == Y*, eJ^Gfc(xfe, A, /fe)
(4.47)
where Yk is used to replace Yk in C'6(^fc,Xfe, fk, fk), by using the method presented
in Section 4.1.1.
With the above preparations, now we provide the steps of the scatterer isola
tion and relaxation based NLS algorithm, which are the substeps of Step 1 of SPAR.
Step I: Assume K = 1. Calculate Y from Y by using the isolation process.
Obtain {/>fc, xfc, fk, fk}k=i from Y.
Step II: Assume K = 2. Compute Y2 with (4.45) by using {(frki'X-k-, fk, f k}k=\
obtained in Step I. Calculate Y2 from Y2. Obtain {(¡>k,xk, fk, fk}k=2 from Y2. Next
compute Yx with (4.45) by using {k,xk, fk, fk}k=2, calculate Y\ from Yi, and then
redetermine { Iterate the previous two substeps until practical convergence is achieved (to
be discussed later on).
Step III: Assume K = 3. Compute Y3 with (4.45) by using {(/fc, x.k, fk, fk}k=i
obtained in Step II. Calculate Y3 from Y3. Obtain { compute Yi with (4.45) by using {(j>k, xfc> fk, Jk}\=2^ calculate Yx from Y3, and then
redetermine {(j)k,x.k, fk, fk}k=1 from Yx. Then compute Y2 with (4.45) by using


55
are obtained via 1-D FFT with zero-padding to a total length of 128 in range and to a
total length of 64 in cross-range. Next, these initial estimates are refined by using the
FMIN function in MATLAB alternately, i.e., by updating / while fixing / at its most
recently determined value and vice versa, until practical convergence, which is de
termined by checking the relative change of the cost function C^(f,f). We have used
10-3 to determine the convergence of this fine search as well as the relaxation-based
algorithm. The extrapolation factor £ = 8 is used in range for SPAR and Hybrid
and in both range and cross-range for RELAX-NLS and the modified RELAX-NLS.
Both 1-D and 2-D Kaiser windows with shape parameter ¡3 = 6 are used whenever
needed.
First consider a simulated example with high SNR. The SAR phase history
data matrix is simulated by assuming that there are four trihedrals and three di
hedrals in the presence of zero-mean white complex Gaussian noise with variance
o2n = 0.6. The amplitude functions for the four trihedrals are generated as follows:
xk[n) = 1, A: = 1,2,3, = 0,1, , 1, (4.71)
and
x4() = 2, = 0,1, , JV 1. (4.72)
The amplitude functions for the three dihedrals are
xb{n) = 9.6sinc[0.37r( 18.6)], ft = 0,1, , 1, (4-73)
and
xk() = 6.4sinc[0.27r( 18.6)], k = 6,7, ft = 0,1, , 1, (4-74)
where sinc(x) = sin(x)/x. Figure 4.3(a) shows the modulus of the true SAR image.
Note that two of the dihedrals are located in the same cross-range and are closely
spaced in range and two of them are located in the same range. Of the four trihe
drals, two of them are closely spaced in range and the other two are located in the


16
later was used in microwave imaging [66]. Encouraged by the initial results in [33],
Knaell later presented in [34] a correlation technique which works in conjunction with
the coherent CLEAN algorithm to reduce sidelobe interference. Furthermore, Knaell
in [35] compared the performances of the coherent CLEAN method and two other
methods including a maximum likelihood (ML) method and a sidelobe leakage reduc
tion algorithm [12] in extracting 3-D target features by using CLSAR datasets. We
will discuss how RELAX (SUPER CLEAN) can be extended for 3-D target feature
extraction using CLSAR in Chapter 5.
2.3 SAR Motion Compensation
Aperture errors exist due to atmospheric turbulence and platform uncertainty.
These undesirable errors may significantly degrade the quality of SAR images in sev
eral ways. For example, aperture errors may cause image geometric distortion, spu
rious targets, loss of resolution, and decrease in image contrast [32, 25] depending on
their natures. Therefore, effective motion compensation methods are indispensable
for SAR imaging and target feature extraction in practice. In 2-D SAR systems, mo
tion errors mainly cause phase errors in cross-range [29]. Autofocus techniques that
compensate for phase errors using radar phase history data have been extensively
exploited by 2-D SAR systems. Of the existing autofocus algorithms, mapdrift [45]
and phase difference [68] algorithms are used to compensate for phase errors asso
ciated with certain order polynomials, whereas the phase-gradient autofocus (PGA)
algorithm [17, 18, 67] is independent of the phase error model, thereby is more ro
bust. Autofocus algorithms based on super resolution spectral estimation methods
also begin to appear in the literature [43, 52], All the aforementioned 2-D autofocus
algorithms assume that phase errors change in cross-range only and are constant for
all range bins. This assumption is generally valid for the 2-D SAR systems where


57
and bandwidth of 0.591 GHz. The radar was about 5 kilometers away from the
ground object. The data was collected when the object was illuminated by the radar
from approximately the azimuth angle 0 and elevation angle 30. To cross-validate
the experimental results given below, XPATCH [1], a high frequency electromagnetic
scattering prediction code for complex 3-D objects, was used to generate very high
resolution phase history data for the object shown in Figure 3.3. The data generated
by XPATCH has a resolution of 0.038 meters in both range and cross-range, and the
corresponding windowed FFT SAR image is shown in Figure 4.5(a). (We have used
the log scale for all of the images shown in Figure 4.5.) The original experimental
Slicy data has a resolution of 0.3 meters in range and 0.32 meters in cross-range.
The 32 x 32 data matrix we used to demonstrate the performance of our algorithms
has a spoiled resolution of 0.51 meters in range and 0.54 meters in cross-range. The
windowed 2-D FFT SAR image of this data matrix is shown in Figure 4.5(b). Figure
4.5(c) shows the Hybrid image with K = 7 (obtained via GAIC). Figure 4.5(d)
shows the SPAR image with K 7 (obtained via GAIC). Figures 4.5(e) and (f)
show the SAR images obtained via RELAX-NLS and the modified RELAX-NLS
with K = 7, respectively. Note that Hybrid has a more severe artifact problem
than SPAR. Note also that the SPAR image shown in Figure 4.5(d) appears to fit
Figure 4.5(a) and the characteristics of the object in Figure 3.3 well. However, the
parametric algorithms are not as robust as SPAR since the parametric images shown
in Figures 4.5(e) and (f) do not fit Figure 4.5(a) as well with one of the scatterers
mis-identified and mis-located. For this experimental example, the ratios between the
MATLAB flops needed by Hybrid, SPAR, the modified RELAX-NLS, and RELAX-
NLS over the flops needed by the windowed FFT method are 29.2, 16.7, 32.8, and
43.1, respectively. Note that SPAR can sometimes be faster than Hybrid!


61
(e)
Figure 4.4: Comparison of SAR images formed using different algorithms for simu
lated data at low SNR ( SAR image, (c) SPAR SAR image, (d) RELAX-NLS SAR image, (e) Modified
RELAX-NLS SAR image. (The vertical and horizontal axes are for range and cross
range, respectively.)


5
resolution SAR images by using the target features extracted via relaxation-based
methods including RELAX [39] and RELAX-NLS [42] in Chapter 3. The relaxation-
based optimization approach is one of the effective ways to minimize a nonlinear least
squares (NLS) cost function and has been proved to be quite useful in many applica
tions, such as radio astronomy [28], microwave imaging [66], spectral estimation [23],
and sinusoidal parameter estimation and target feature extraction [39, 42], RELAX
was originally proposed for the parameter estimation of complex sinusoids in colored
noise. Compared with other NLS algorithms, RELAX is relatively robust against
data mismodeling errors [39]. We will show that RELAX can be used to enhance
SAR images of targets consisting of point scatterers. However, the point scatterer
data model may not be valid in practice since, for example, for many man-made
targets, such as vehicles and buildings, much of the returned energy is caused not
only by trihedral corner reflectors, but also by dihedral corner reflectors of a target
[36]. We also describe how to form super resolution SAR images of targets consisting
of both trihedrals and dihedrals using RELAX-NLS [42]. RELAX-NLS is based on
a mixed data model, which models a trihedral as a point scatterer and a dihedral as
a complex sinusoid with a constant phase and a variable amplitude described by a
sine function in cross-range. We will demonstrate the super resolution property of
the RELAX and RELAX-NLS algorithms with experimental data including MSTAR
and ERIM (Environmental Research Institute of Michigan) data.
Parametric methods may outperform non-parametric methods provided that
the assumed data models are valid. In the more likely cases where it is difficult to
establish a good parametric data model in cross-range due to a variety of physical
phenomena including glints, resonances, and motion-induced phase errors [15], a more
flexible data model is desired and non-parametric and parametric methods may be
combined together for 2-D SAR target feature extraction and image formation. In
our work, we also establish a more flexible data model for man-made targets when


34
(a)
(b)
v.
"1. W
(C)
(d)
(e) (f)
Figure 3.4: SAR images of the Slicy data (0 azimuth and 30 elevation angles),
(a) Original windowed 2-D FFT image, (b) Windoed 2-D FFT image with spoiled
resolution, (c) ~ (d): Windowed RELAX image without and with background clutter
with K = 36. (e) ~ (f): Windowed RELAX-NLS image without and with background
clutter with K = 9. (c) ~ (f) are obatined using the data in (b). (The vertical and
horizontal axes are for range and cross-range, respectively.)


17
significant phase errors are caused by motion measurement errors, such as along-track
velocity errors or line-of-sight acceleration errors [11].
As in 2-D SAR systems, motion error compensation is also indispensable for
3-D SAR systems. Autofocus in CLSAR, however, becomes more complicated since
the aperture errors can no longer be approximated as phase errors only in cross-range
since the errors introduced in 6 (azimuth angle) and <¡> (elevation angle) may not be
negligible and the extracted 3-D target features are sensitive to the shape of the
curvilinear aperture (see the discussion in Chapter 6). To the best of our knowledge,
autofocus for CLSAR has not been reported in the literature, partly because CLSAR
is a relatively new technology.


77
1 1 1
1
Full Aperture
M=32
_
M=64
* *% *..
-- M=80
*
-
1 1 1
1
10
Radius
(a)
<3,
m -40
oc
O
"T"
T
Full Aperture
M=32
* M=64
M=80
10
Radius
-36
1 1 1
i r
_
-38
-
-40
-
-42
-
-44
Full Aperture
M=32
-
-46
- -M=64
-- M=80
-
-48

-50
-
-52
1 1 1
1 L_
10
Radius
(b)
T"
T
Full Aperture
M=32
M=64
-- M=80
10
Radius
(c) (d)
Figure 5.3: The CRBs of the target parameters as a function of the circular aperture
radius when K = 1, a = 1, to = 0, and cr2 = 40. a) The CRB of the
complex amplitude a. b) The CRB of to. c) The CRB of Co. d) The CRB of Co.


4.4.1 Target Feature Extraction 45
4.5 Modified RELAX-NLS Algorithm 51
4.6 Cramr-Rao Bound (CRB) of the Semi-Parametric Data Model 52
4.7 Numerical and Experimental Results 54
5 3-D TARGET FEATURE EXTRACTION USING CLSAR 63
5.1 CLSAR Data Model 63
5.1.1 1-D Data Model for High Range Resolution Radar .... 63
5.1.2 Full Synthetic Aperture Radar 64
5.1.3 Curvilinear SAR 67
5.2 The RELAX Algorithm 68
5.2.1 Full Aperture 68
5.2.2 Curvilinear Aperture 70
5.3 Performance Analysis of Parameter Estimation via CRB 71
5.4 Numerical and Experimental Results 72
5.4.1 Performance Analysis of Different Curvilinear Apertures
via CRBs 72
5.4.2 Experimental Examples 74
6 AUTOFOCUS IN CLSAR 79
6.1 Aperture Error Effects 80
6.2 The Relaxation-Based Autofocus Algorithm (AUTORELAX) . 83
6.2.1 Aperture Parameter Estimation 83
6.2.2 Target Feature Extraction 85
6.3 Experimental and Numerical Results 87
7 CONCLUSION 94
7.1 Summary 94
7.2 Future Works 96
APPENDIXES
A A SINGLE SCATTERER FEATURE EXTRACTION VIA SPAR . 98
B APPROXIMATION OF THE RANGE R 100
C CALCULATING (x,y,z) FROM (x,y,z) 102
REFERENCES 104
BIOGRAPHICAL SKETCH l0g
v


47
where arg(a;) denotes the argument of a complex variable x; finally,
{/,/} = argmax C4(/, /),
fif
(4.40)
where
The steps needed to obtain the NLS estimates of a single scatterer are summarized
as follows:
Step (I): Use (4.37) to obtain ys(/) and obtain the cost function C^f, f)
according to (4.41). Determine {/, /} by maximizing C4(/, /) using the method given
in Appendix A.
Step (II): Calculate 4> according to (4.39) with {/,/} replaced by {/,/}
obtained in Step (I).
Step (III): Calculate x via (4.38) with {0,/,/} replaced by {0,/,/} deter
mined in Steps (I) and (II), respectively.
Feature Extraction of Multiple Scatterers
When a target consists of multiple scatterers, we can obtain the NLS esti
mates of the target features based on (4.2) by using a relaxation-based optimization
approach. Let
T
Xfc Xfc(0) xk(l) xk(N 1)
(4.42)
and let Y and E be N x N matrices with their (n,n)th elements being y(n,n) and
e(n, ), respectively. Then we can rewrite (4.2) as
K
(4.43)
k=1
where Gk(xk, fk, fk) has the same form as the G(x,/,/) in (4.36) except that x, /,
and / are replaced by xfc, fk, and fk, respectively. Let y, = 0,1, , 1, be the


41
where L > 2 denotes the number of scatterers located in the same range. Thus the
L scatterers located in the same range are considered as two scatterers when using
the data model in (4.2).
Before we discuss the impact of the model ambiguities on SAR image for
mation, we first describe how the image formation is done if we have the estimated
model parameters.
4.3.2 Image Formation
Assume for now that we have extracted the target features based on any of
the ambiguous data models. For notational convenience, we will use the notation
used in (4.2). Since the target data model in range is a sum of several complex
sinusoids with constant amplitudes and phases, we can use the estimated sinusoidal
parameters to simulate a data matrix with a larger dimension in range and then use
FFT to demonstrate the super resolution property of the feature extraction algorithm
we shall present. Yet we cannot extrapolate the estimate {()} of {^()} since it is
assumed to be an arbitrary unknown real-valued function of and hence FFT cannot
be used to obtain SAR images with enhanced resolution in cross-range. Instead, we
use 1-D APES [37, 38, 41] in cross-range when forming SAR images since APES
belongs to the class of matched interbank spectral estimators and provides lower
sidelobes, narrower spectral peaks, and more accurate spectral estimates than FFT.
Let {ss(n, )} denote the simulated data sequence with a larger dimension in
range based on the estimated target features {(j)k, {xk()}^~Q fk, fk}^=l of
{(/)k, {xk()}^ZQ, /fc, fk}k=i where K denotes the estimate of the scatterer number
K. Then
K
ss(n, n) Xk()e^kkU\ n = 0,1, , £N 1, = 0,1, , 1,
fc=i
(4.23)


Elevation Angle (degrees) Elevation Angle (degrees)
91
Figure 6.2: Autofocused curvilinear aperture and scatterer distribution obtained with
AUTORELAX by autofocusing only in the elevation direction and using K = 20 for
the experimental example, (a) Autofocused curvilinear aperture (dotted line), (b)
Scatterer distribution.
Azimuth Angle (degrees) Azimuth Angle (degrees)
(a) (b)
Figure 6.3: (a) Manually adjusted (solid line) and autofocused (dotted line) curvi
linear apertures for the experimental example, (b) Fitting the manually adjusted
aperture to the autofocused aperture by adding a line to the former.


67
complex amplitude and 3-D frequencies of the kth sinusoid, k = 1,2, , K; fi
nally, {e(n, , )} denotes the unknown noise sequence. The sinusoidal frequencies
{ujk = 2nfk,uJk = 27Tfk,,d>k = 2irfk} correspond to the 3-D location of the kth
scatterer of a radar target and ctk is determined by its RCS.
Let
y =
2/(0,0,0) 2/(1,0,0) y(N- 1,0,0) 2/(0,1,0) y( 1,1,0)
y(N 1,1,0) 2/(0, 1, 1) 2/(1, 1, TV 1)
T
y(N 1, 1,7V 1)
(5.17)
Let e be defined similarly from e(n, , ) as y from y(n, , ). Then the data model
in (5.16) can be rewritten as:
y = Act + e,
(5.18)
where
a =
T
Ot\ Ot2 Otx )
and
A =
ai <8> i ai 3lK
ak )
where denotes the Kronecker product [22] and
1
i
1
afc =
e3^k
>
ejk
, a k
ejQk
ej(N-l)uik
ej(-l)£jk
ej(-l)ik
5.1.3 Curvilinear SAR
(5.19)
(5.20)
(5.21)
Assume that the curvilinear aperture used in CLSAR consists of M different
radar viewing angles. Then the received data vector yc in CLSAR, is an (MN) x 1
subvector of y in (5.18). Let Ic denote an M x () matrix with each column and


2
Flight Path

Cross-Range
Figure 1.1: Illustration of 2-D radar imaging.
bandwidth. The cross-range resolution 5C of a conventional radar or real aperture
radar (RAR) is range-dependent and is determined by
(1.1)
where R represents the range of the target; A and L, respectively, denote the wave
length and the size of the antenna aperture. To increase cross-range resolution (given
R), an obvious way is either to reduce the wavelength, or to increase the size of the
antenna. Using shorter wavelength requires more transmission power since electro
magnetic signals with smaller wavelengths are more likely absorbed by the atmo
sphere. In the meantime, it is not feasible to increase the antenna size too much,
especially when the radar is carried on board of an airplane or spacecraft. Hence
a conventional radar or RAR inherently has a limited cross-range resolution. It is
desired to achieve high cross-range resolution without reducing the wavelength and
exploiting large physical antennas. Synthetic aperture radar (SAR) is a solution and
can achieve high cross-range resolution independent of range.
The SAR technology was first proposed in the 1960s [59]. Instead of exploiting
large antennas and reducing wavelength, SAR makes use of a synthetic aperture to
achieve high cross-range resolution. A synthetic aperture is formed by coherently


95
RELAX-NLS is more effective for SAR image formation of targets consisting of both
trihedral and dihedral corner reflectors.
A more robust semi-parametric algorithm, referred to as SPAR, was proposed
in Chapter 4 based on a more flexible and robust data model for man-made targets
when it is difficult to establish an accurate parametric data model in cross-range
for targets of interest. Instead of using the aforementioned approximate dihedral
and trihedral data model, all scatterers of targets in SPAR are modeled as 2-D com
plex sinusoids with arbitrary amplitudes and constant phases in cross-range and with
constant amplitudes and phases in range. The SAR images with super resolution in
range obtained via SPAR can be achieved by applying FFT to the phase history data
with a large dimension in range synthesized by using the extracted features and the
parametric data model in range. The cross-range resolution of SAR images cannot
be significantly improved via SPAR over the conventional FFT method due to the
non-parametric data model in cross-range. Yet it can be improved slightly by using
APES, which is a matched filterbank based non-parametric method with better sta
tistical performance than Capon. It has been shown in Chapter 4 with numerical and
experimental examples that SPAR can not only form accurate and super resolution
SAR images by taking advantages of both parametric and non-parametric spectral
estimation methods, but also effectively mitigate the artifacts in SAR images due
to the flexible data model by first isolating out the most dominant scatterer via a
rectangular window. Another advantage of SPAR is that it can be used to provide
good initial conditions needed by other parametric target feature extraction methods
to reduce the total amount of computations needed. Numerical and experimental
examples have shown that SPAR is more robust and computationally simple than
RELAX-NLS and its variation discussed in Chapter 4.
CLSAR is a relatively new technology for 3-D target feature extraction and is
still at its developing stage. In the study, we have presented parametric algorithms