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Focus of attention for millimeter and ultra wideband synthetic aperture radar imagery

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Focus of attention for millimeter and ultra wideband synthetic aperture radar imagery
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Yen, Li-Kang, 1967-
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vii, 142 leaves : ill. ; 29 cm.

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Datasets ( jstor )
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False alarms ( jstor )
Mathematical vectors ( jstor )
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Radar ( jstor )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 137-141).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Li-Kang Yen.

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Full Text










FOCUS OF ATTENTION FOR MILLIMETER AND ULTRA
WIDEBAND SYNTHETIC APERTURE RADAR IMAGERY















By

LI-KANG YEN











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

1998













ACKNOWLEDGEMENTS



There are many people I would like to acknowledge for their help in the genesis of this

manuscript. I would begin with my parents, Chi-Meng Yen and Ai-Jui Sun, for their end-

less encouragement and support over the last six years. My little brother Li-Chiang Yen

has also constantly been providing constructive advice. Without their standing strong

behind me, it would be impossible for me to finish this thesis.

I would like to acknowledge my advisor, Dr. Jose Principe, for providing me with an

invaluable environment for the study of target detection and excellent guidance through-

out the development of this thesis. His influence will leave a lasting impression on me.

I would also like to thank the students, past and present, of the Computational Neu-

roEngineering Laboratory. The list includes, but is not limited to, Chuan Wang, Doxing

Xu and Quin Zhao for useful discussions on signal processing theory, and Albert Hsiao

and C. Pu for providing much needed recreational opportunities. There are certainly others

and I am grateful to all.

Finally, I would like to thank my girl friend, Bernice, for sharing my joys and tears in the

last two years of my Ph. D. journey. This memory is the thing I will endear forever.









ii














TABLE OF CONTENTS


Page
ACKNOWLEDGEMENTS ........................................... ii
ABSTRACT ...................................................... vi
CHAPTERS

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

1.1 M otivation for SAR ....................................... 1
1.2 SAR Imaging M odel ...................................... 2
1.3 MMW SAR Image Focusing........................................... 5
1.4 Motivation of UWB SAR: Resonance Effects.................... 7
1.5 UWB SAR Image Focusing................................ 10
1.6 SAR Image Data Sets ..................................... 13

2 BACKGROUND ........................................... 17

2.1 Representation of Signals In Gray-Scale Images ................ 18
2.2 Representation of Bipolar Transient Signals .................... 21
2.2.1 Gabor Bases .................. ...................... 22
2.2.2 Laguerre Bases ..................................... 23
2.3 The Neyman-Pearson Tests ................................. 29
2.4 Conclusion .................. ............................ 32

3 LOCAL INTENSITY TESTS FOR OPTIMAL DETECTABILITY...... 33

3.1 Formulation Of Optimal Intensity Detectors .................... 39
3.2 Statistical Properties Associated with the Intensity Detector ....... 43
3.2.1 Unbiased Estimators ................................. 43
3.2.2 The Detector Preserves the CFAR property ................ 44
3.3 Intensity Modeling of Targets ................. ........... . 45
3.3.1 Mathematical Background ............................ 46
3.3.2 The Modeling Procedure and Experiment Results ........... 48
3.4 Individual Target Size Estimation .......... .............. .. 53
3.4.1 Individual Target Size Estimation Procedure ............... 54


iii












Page
3.4.2 Experiment Results ................ .................. 56
3.5 Conclusion ............................................. 60

4 SUBSPACE DETECTION OF TARGETS IN UWB SAR IMAGES ..... 62

4.1 Target Detection Using Laguerre Networks .................... 63
4.1.1 The 1D Resonance Response Model ..................... 64
4.1.2 Formulation of the Proposed ID CFAR Detector............. 66
4.1.3 Detection Scheme and Simulation Results ................. 68
4.2 Target Detection Utilizing A Spatial Template .................. 79
4.2.1 The Spatial Template Model and the Detector Derivation ..... 80
4.2.2 The Detection Scheme and Simulation Results ............. 82
4.3 Subspace Detectors Extended with Data-Driven Templates ........ 86
4.3.1 Formulation of the Weighted Subspace Detector ............ 86
4.3.2 Subspace Parameter Tuning and Template Finding .......... 88
4.3.3 Simulation Results ................................... 91
4.4 Conclusion .................. ............................ 94

5 NEURAL NETWORK APPROACHES TO TARGET DETECTION .... 96

5.1 Target Detection Using Temporal Information .................. 96
5.2 The Target's Temporal Template Model ....................... 97
5.3 Sequential Detection Fusion Using A Neural Network............. 99
5.4 Training for the Fusion Neural Networks ...................... 100
5.5 Simulation results ........................................ 101
5.6 Conclusion ............................................. 105

6 QUADRATIC LAGUERRE DISCRIMINATOR..................... 106

6.1 Two Stage Detection Scheme ............................... 108
6.1.1 Prescreener: 1D Gamma-CFAR Detector.................. 108
6.1.2 Quadratic Laguerre Discriminator (QLD) ................. 112
6.1.3 Temporal Training for the Discriminator .................. 115
6.2 Discrimination Using Damped Sinusoidal Subspace ............. 117
6.3 Simulation Results ................. ....................... 119
6.4 Conclusion ............................................. 121

7 CONCLUSION.................. ............................ 124

iv













Page
7.1 Summary ............................................. 124
7.2 Future Work............................................. 129
APPENDICES
A THE STATISTICAL PROPERTY ASSOCIATED WITH THE LOCAL
INTENISTY KERNEL TEST .................................... 131
B THE STATISTICAL PROPERTY OF THE SUBSPACE DETECTOR ... 134
REFERENCES ................ ................................... 137
BIOGRAPHY SKETCH............................................. 142






































v












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


FOCUS OF ATTENTION FOR MILLIMETER AND ULTRA
WIDEBAND SYNTHETIC APERTURE RADAR IMAGERY


By

Li-Kang Yen

Aug 1998

Chairman: Dr. Jos6 C. Principe

Major Department: Electrical and Computer Engineering

The major goal of this research is to develop efficient detectors for Synthetic Aperture

Radar (SAR) images, exploiting the reflectivity characteristics of targets in different radar

types. Target detection is a signal processing problem whereby one attempts to detect a

stationary target embedded in background clutter while minimizing the false alarm proba-

bility. In radar signal processing, the better resolution provided by the Millimeter Wave

(MMW) SAR enhances the detectability of small targets. As radar technology evolves, the

newly developed Ultra Wideband (UWB) SAR provides better penetration capabilities to

locate concealed targets in foliage.

In this thesis we demonstrate that local intensity kernel tests can be formulated based on

the generalized likelihood ratio test (GLRT), while preserving constant false alarm rate

(CFAR) characteristics. Both the widely used two-parameter CFAR and the y -CFAR can

be viewed as special cases of the local intensity tests with different intensity kernels. It is

vi








demonstrated that the first-order Gamma kernel is a good approximation for the principal

eigenvector of the projected radial intensity of targets, which provides the optimal match-

ing intensity kernel. This also explains the better performance of the y -CFAR detector

over the two parameter CFAR detector.

We also developed different CFAR subspace detectors for UWB images, utilizing a

Laguerre function subspace. The driven response produced by natural clutter degrades the

performance of these subspace detectors. In addition to the driven response, the distin-

guishing feature of metallic targets in UWB is the resonance response. Therefore, we fur-

ther propose a two-stage detection scheme: y -CFAR detector followed by the quadratic

Laguerre discriminator (QLD). We evaluate every detector and discriminator using ROC

curves in a large area (about 2 km2) of imagery. The combined y-CFAR and quadratic

Laguerre discriminator improve the simple Laguerre subspace detector more than one

hundred fold for a perfect detection rate (Pd = 1),.
























vii













CHAPTER 1
INTRODUCTION

This project grew out of research conducted in the Computational Neuroengineering

Lab under Dr. Jose Principe on target detection in Synthetic Aperture Radar (SAR)

images. Target detection is a signal processing problem whereby one attempts to detect a

stationary target embedded in background clutter while minimizing the false alarm proba-

bility. In radar signal processing, the better resolution provided by the Millimeter Wave

(MMW) SAR enhances the detectability of small targets. As radar technology evolves, the

newly developed Ultra Wideband (UWB) SAR provides better penetration capabilities to

locate concealed targets under foliage. In this project, the different reflection features of

the targets in the images generated by MMW and UWB SAR are utilized to attack the

problem of target detection in the noisy background.


1.1 Motivation for SAR

Range resolution in real aperture radar system is generally defined in terms of system

bandwidth Af and propagation velocity, c, by


8 _c
range 2Af

while azimuth and elevation resolutions are defined by the operating wavelength ,, aper-

ture dimension in the orientation of interest, L, and range to target, R,


hR
6azimuth -


1





2

Automated target-recognition (ATR) systems generally require a large number of resolu-

tion cells on the target to achieve a specific performance level. Thus, the desire for

improving ATR capabilities has motivated increases in system bandwidths, operating

wavelength, and aperture sizes. To improve 8range, the system bandwidth Af can be

increased by the use of a chirp signal with high carrier frequency in MMW radar, or

directly by the use of an impulsive signal with large bandwidth in UWB radar.

Shorter wavelength X suffers from increased scattering and atmospheric attenuation.

Therefore wavelength must be operated in some desirable range. For a fixed operating

wavelength k of the radar, one can improve azimuth resolution by decreasing the range to

the target or increasing the aperture size in the azimuth dimension. Since target range is

usually dictated by operational constraints, and thus not subject to alteration, aperture

modification remains the only option for improving azimuth resolution azimuth .

In the real-aperture case, it is not practical to achieve high resolution by simply increas-

ing the physical aperture; for example, at k = 1 ft and R=1000 ft. (short range in most

applications), a 1000-ft. antenna would be required to achieve an azimuth resolution of 1

ft. Aperture sizes such as these are often impractical at fixed sites, and entirely unreason-

able in mobile applications. Thus, the ability to synthesize a large aperture by modeling it

as a coherent, linear array of smaller antennas is critical to achieving reasonably high azi-

muth resolutions.


1.2 SAR Imaging Model

The two-dimensional SAR imaging model, as shown in Figure 1, is considered. The

vectors (x, y) and (kx, k,) are used to identify the spatial coordinate and the spatial fre-

quency, where x and y -coordinate corresponds to the slant range and cross range, respec-





3

tively, The transmitting/receiving airborne radar moves along the line x = Xl, and it

makes a transmission and the corresponding reception at each position specified by

(X1, Y1 + u) for u e [-L, L], where the slant range X1 and cross range Y1 are constants.

Here, we assume that the radar stops, transmits a signal and receives the corresponding

reflection, and then moves to the next position. The assumption is unrealistic due to the

fact that radar is continuously moving. However, it is still a valid one since the aircraft's

moving speed is much smaller than the speed of light. The induced distance difference

between transmitting and receiving positions can be ignored.

While the radar illuminates the target area with a time dependent signal p(t), the round

trip delay T of the reflected signal received by the radar due to a point scatter at (x, y) is


t(x,y) = 2R(x,y)/C = 2J(Xl-x)2 + (Y+u-y)2/C

where R(x, y) is the distance between the radar and the point scatter. Thus the total ech-
oed reflection received s(u, t) at each moving position can be written as


s(u,t) = f(x,y)pt- (XI -x)2 + ( + u-y)2dxdy
(1)

where f(x, y) is the target area's reflectivity function, and the integral is taken all over the
target region illuminated by the radar in the (x, y) domain.





4



Radar Moving Path

(X Y, +L)

Synthetic Aperture

s(u, t) rY





(X, YI -L) p(t-R(x,y)) Z

height slant range

(x,y)
range- X



cross range/ iU minated area







Figure 1. Pictorial View of SAR image formation

Taking the Fourier transform of both sides of (1) with respect to t results in



s(u, Co)/P(o) = fJf(x,y)exp[-j2k (Xi -x)2 + (Y + u-y)2]dxdy (2)

The system model in (2) represents a 2D filtering process illustrated in Figure 2. The input

signal is f(x, y) the original reflectivity function, which passes through the 2D filter with

the impulse response H(x,y) = exp[-j2k/x +2] to generate the output signal

s(u, co) = s(u, co)/P(o) measured at (x,y) = (X1, Y1 + u).





5





f(x,y) --H(x, y) -(u, )l(x,y) = (X, Y, +u)


H(x,y) = exp[-j2kj y+2]


Figure 2. SAR image formation model: 2D filtering

Based on the system model in (2), SAR image focusing algorithms can be regards as to

restore the original reflectivity function f(x, y) from the received signal s(u, 0o), which

is the deconvolution with the system function H(x, y) = exp [-j2k i+ y2]. Apparently,

the difficulty of analyzing the deconvolution is that the transfer function H(kx, ky) (Bessel

function of the first kind) is too complicated. Thus, in different cases, different focusing

algorithms approximate the spherical wave function with different and simpler wave func-

tions so that the analysis of the deconvolution becomes easier.


1.3 MMW SAR Image Focusing

In this section, the principles of focusing the stripmap mode MMW SAR for surveil-

lance purpose is discussed, which is applied to form the images like ADTS data set of

DARPA.

In MMW SAR, the waveform p(t) in (1) to be transmitted is a chirp signal

exp(j0/ot +jKt2), where co is the carrier frequency and K is the modulation constant.

With a high carrier frequency o,, the chirp signal can be assumed to be operated with nar-

rowband characteristics, while its bandwidth is still large enough to improve the range res-





6

olution. Discarding the carrier frequency components of the chirp signal, s(u, co) in the

system model is given by


s(u, o) = s(u, co)/P(o)) = s(u, co) P*(o)

In other words, s(u, co) is formed by matched filtering the raw data with the chirp signal
in the frequency domain. With the approximation /i +q 1 + q/2 for q << 1, the Fresnel
approximation used in the system model for the spherical wave function H(x, y) in the far
field is given by


2k (X1-x)2 + (u -y)2_ 2k(X -x)+ (u-) (3)
Y(X -x)

The minus sign in (2) can be absorbed into k without loss of generality, and then the sys-
tem model with Y1 = 0 becomes


s(u, CO) f(x,y)exp[i2k(X -x)] exp (U -y)2 dxdy
Ii (XI X)

With direct algebraic simplification, one can get the following relationship


s(u, o) exp[j2kX] exp(j'u2/X1) Fx(2k,u) (4)

where Fx(2k, u) = Jf(x, y) exp [-2jkx] dx and denotes convolution in the u domain.
Notice that (4) suggests s(u, co) is the signal generated by blurring the signal Fx(2k, u)
with the chirp signal exp [ju2/X ] and its carrier frequency exp [-j2kX1 ]. Taking a sim-
ilar matched filtering approach to form s(u, co), we can match filter the signal s(u, co)
with (exp(ju2/Xl))* as


Fx(2k,u), exp[-j2kX1] exp(-ju2/X) s(u,co) (5)





7

(5) is the well known Fresnel approximation-based inversion for strip-mode SAR imag-

ing, and the imaging algorithm is computationally efficient since only the 2D Fourier

transform is involved [Ausherman et al, 1984].


1.4 Motivation ofUWB SAR: Resonance Effects

A fully polarimetric Ultra WideBand UWB foliage-penetrating (FOPEN) SAR was

recently designed mainly for the purpose of exploiting the capabilities to locate and recog-

nize targets embedded in foliage. The UWB system is considered to be ultra wideband

because of its high bandwidth occupancy. Let's define the relative bandwidth as the ratio

of bandwidth to center frequency


Q- f A (6)
fc (fhigh +flow)/2

A sensor is categorically wideband if its relative bandwidth is 0.1 < Q 0.5, and ultra-

wideband if its relative bandwidth is 0.5 < Q < 2.

In the narrowband case, a target's echo is typically modeled as a scalar number C, the

radar cross section (RCS) of the target. In general, however, RCS is a function of wave-

length, phase, aspect angle and polarization state. Wideband analysis thus motivates a

revised definition of a as a complex quantity, with both magnitude and phase compo-

nents. In the time domain, the complex RCS can be represented as a ringing or resonant

response of the target [Morgan and Larison, 1991]. If this resonant signature is sufficiently

unique, the target can be modeled and recognized by analysis of its resonant response. The

typical 1D responses of a dihedral and a tree are shown in Figure 3 and Figure 4, respec-

tively.






8



2 x 1 drivei resonance
response response
1 x10


Magnitude 0

-1 x106

-2x106
2 x 1 0 6 . . . . . ._._._._._. .. . .
0 50 100 150 200 250
n (sample index)


Figure 3. Typical response of a metallic object (a dihedral) in UWB.



2 0 0. . . .
2x10




-2x1 06
Magnitude
-4x10
driven
-6x106 response
0 50 100 150 200 250
n (sample index)


Figure 4. Typical response of a non-metallic object (a tree) in UWB.

The response of a resonant scatter to an incident wideband pulse will generally be com-

posed of two temporally distinct parts, referred to as the early-time or driven response and

the late-time or resonance response. The driven response is the echo of the incident pulse,

caused by local currents driven on the surface of the object; alone, it does not convey a

great deal of information about the scatter. The resonance response is a ring down of the

natural frequencies of the target excited by the incident pulse. These natural frequencies

are a function of the electrical dimensions of the metallic object.





9

The resonance response phenomenon is best illustrated with a canonical example. The

spatio-temporal distribution of current along a thin-wire dipole of length L is described by


--at 2nx .
i(x, t) = le sin --sin((ot)


where I is the value of the current at a current antinode, and o = 27c/X. The boundary

condition requires i(x, t) = 0 at x = 0 and x = L, which leads to the condition
2tnL
k7 k = 1, 2,.... For x = L/2,we get


--at
i(L/2, t) = le-atsin(7tk/2)sin(ot) k = 1, 2, ... (7)

The meaningful solutions to (7) exist for k odd. These are the so called radiating frequen-

cies. If we use the variable n = (k+ 1)/2 ,



i(L/2, t) = lne a"sin(ont) n = 1, 2,... (8)
n

Unless the forcing function that initiated the current flow is reapplied, the factor in e

indicates that the current decays with time at a rate proportional to an, which is also

known as the damping factor. Thus, if a dipole in empty space is illuminated by a incident

pulse, resonance at the odd frequencies within the band of the illuminating pulse will be

excited, and will decay exponentially once the forcing function has been removed.





10

For example, consider a resonator illuminated by a wideband pulse occupying a band

from flo and fhigh If the resonator has fundamental frequency and radiating harmonics

at on, then the resonator's response can be described by



-a\ t f flow ~Jn~ -high
R(t) = ZA(fn)Iene sin(o)nt)u(t) A(fn) = (9)
n

From the above equation, we can see that only metallic man-made objects in UWB radar

produce a damped sinusoidal response due to the wideband excitation. This phenomenol-

ogy of the reflection of the target is quite different from that in the conventional SAR. As

we can see in Figure 3 and Figure 4, the resonance response is a key feature for metallic

objects. It seems that a very promising approach for concealed target detection in UWB

SAR images is to exploit the resonance response of metallic objects.


1.5 UWB SAR Image Focusing

In this section, the principles of focusing the SAR images with some pulse p(t) trans-

mitted is discussed. In UWB radar, the transmitted pulse p(t) is a wideband signal with an

impulsive waveform, and the range resolution requirement can be met with the large band-

width. Now, the p(o) in (2) is of large bandwidth and no longer can be assumed to be nar-

rowband. Thus, to focus the UWB SAR image, the first step is to produce

s(u, co) = s(u, o)/P(co). Besides, the area illuminated by UWB is only near-field, so

the simpler Fresnel approximation is no longer appropriate. Instead, the more complicated





11

approximation, plane wave decomposition, for the spherical wave is used, and it can be

given by



exp [-j2kJ(X -x)2 + (Y, + u-y)2]
-2k (10)
Jk exp[j2 (2k)2-k (X,-x)+jku(Y, +u-y)] dkU
-2k

Substituting (10) into (2), one can get the following relationship


S(ku, c) = exp[ ,/4k2-k2k X +jkuY,]F( 4k2-k, k)

To be more clear, the above equation can be written as


exp [jkxX +jkyY1] F(kx, ky) = S(ku, o) (11)

where


kx = 4k2-k2 k = ku

Again, here (11) can be interpreted as a 2D filtering/deconvolution in the 2D spatial fre-

quency domain, where only S(ku, o) needs to be manipulated by changes of variables in

the continuous frequency domain. The corresponding manipulation in the discrete fre-

quency domain can be done by interpolation, and all the processes involved are illustrated

in Figure 5. Note that, to restore the signal f(x, y) from the observed signal s(u, t), one

only needs to compute


F(kx, ky) = S(kx, k,)/H(kx, ky)

The algorithm actually was first proposed by Stolt [Stolt and Weglein, 1985] for migrating

seismic data so that the reflection, with a hyperbola shape, to a point scatter would be





12

deconvolved to the single point, where the exciting waveform is assumed to be an impulse
function. Later it was reformulated and applied to SAR image processing by Soumekh
[Soumekh, 1990]. The flowchart of the algorithm is shown in Figure 6. Since this algo-
rithm uses more complicated plane wave approximation than Fresnel approximation, it
also can be applied for the SAR imaging operating in the stripmap mode at the cost of
more computations.




f(x, y) N H(x, y) ---(u, t)

2DFFT

2D FFT I 2D FFT S(k, o)
S' Interpolation


F(kx, ky) H(kx, ky) = S(k, ky)


Figure 5. 2D filtering/deconvolution for UWB SAR image focusing



exp[j4k2-uX1 +jk Y1]

S S(ku, co)
s(u,t) 2D DFT O




F(kx, ky) Interpolation
f(x,y).-- 2D IDFT
kue 6. Fw C t o k k


Figure 6. Flow Chart of the FFT Based SAR Focusing Algorithm





13

1.6 SAR Image Data Sets

In the experiments through this thesis, there are two different types of SAR image data

sets for simulation: the MMW Moving and Stationary Target Acquisition and Recognition

(MSTAR) data set and UWB SAR image data set.

The MSTAR public release data consists of X-band SAR images with 1 foot by 1 foot

resolution in one foot resolution spotlight mode. The target images contain one of three

T72 Main Battle Tanks (MBTs), one of three BMP2 Armored Personnel Carriers (APCs),

or a BTR70 APC. There are images of a test object (Slicy) available also. The target

images are 128 by 128 pixels and were collected on the ground near Huntsville, Alabama

by Sandia National Laboratory using a STARLOS sensor. There are 140 images of each

target at different poses (50 increments) In the latter simulation, all the target images of

T72s, BMP2s, and a BTR70s in 15 depression angle are used.

The UWB SAR raw data is collected by a moving UWB radar transmitting a impulsive

waveform with the bandwidth of 1 GHertz. Then the focused UWB image with 0.1 m by

0.3 m resolution is formed for a 2 (km)2 area, where natural clutter like trees or foliage,

man-made clutter like power lines, and 15 different types of vehicles are present. The

focused images are composed of 7 consecutive frames, each with 5376 by 2048 pixels are

used for the simulation of the proposed detectors. There are a total of 88 million pixels in

the images corresponding to the 2 (km)2 area.

For simulation, there are two image data runs of the same scene focused at different time

with the same 25 vehicle targets, but their locations changed on the ground. Besides, there

also exists man-made clutter like power lines. All the proposed detectors are tested on the

same single data run of 88 million pixel images.





14

For the neural network based detectors, it is necessary to prepare the training and testing

data set, where target and clutter examplars have to be included. To prepare the clutter

training and testing data set, one single data run is prescreened by 1D y -CFAR detector to

generate about 8000 false alarm detection points. These raw detections are clustered

within a user specified 5m radius, and only the detection with the largest statistics is pre-

sented for those clustered detections. Therefore, there are 2588 clustered false alarm

detections being reported. Among these false alarm points, the 300 points corresponding

to the largest 300 detection statistics are chosen to make the training clutter chips. As

described earlier, there are 50 vehicles in the UWB SAR data set. The locations corre-

sponding to the least 15 detection statistics are chosen for making training target chip sets.

Each image chip is of size 250 x 30. All the false alarm points and target locations are

used for generating the testing data set.





15




88M UWB SAR Images

15 4
1 3
0 -- -
-2048 I -|4

S2048 D yiCFAR I--- radial clustering
5376 (r=5m)





Clutter Testing Data Chips



Clutter Training Data Chips 2588 fe
I I2588 false alarm
S_ locally peak detections
first largest 300 *
false alarm detections -



S\- 30A
250





Figure 7. The formation of the clutter chip sets for training and testing





16




88M UWB SAR Images




S0
2048 Target Testing Data Chips

5376


S50 target chips
25 target chips



25 target chips

S4-_ 1330\--
1 2 15 target chips -_ 250
S0

_2048
5376 Target Training Data Chips



88M UWB SAR Images


Figure 8. The formation of the target chip sets for training and testing













CHAPTER 2
BACKGROUND

The design of a target detector can be divided into two phases: signal representation and

detector formulation. Through signal representation, targets are described more promi-

nently than clutter in terms of the extracted features, and the detector can be formulated to

detect these features. The two phases are so intertwined, that it's unavoidable to go back

and forth between these two phases before any powerful detector is derived.

The approaches to target detection in SAR images is to utilize the characteristic reflec-

tions of metallic objects. There have been many algorithms developed for stationary target

detection in various other applications, such as infra-red (IR) sensors and radar surveil-

lance [Reed et al, 1974][Reed and Yu, 1990]. In most of the scenes, the signals or targets

are dim or partially obscured by the varying noises in the background, and, hence, detect-

ability is severely degraded. To improve detectability, some of the developed algorithms

utilize the apriori information of targets, as well as linear mapping to enhance the target

features. These enhancement can be thought as a kind of signal representation, and then

the detector can be formulated to maximize the detection probability while minimizing the

false alarm rate. We would introduce the philosophy behind these developed detectors by

reviewing the related background knowledge: signal representation and detector formula-

tion.






17






18

2.1 Representation of Signals in Gray-Scale Images

In MMW SAR grey-scale images, one of the prominent features of targets is intensity.

To model the intensity templates of targets in the grey-scale images, some all positive ker-

nel can be used, since the intensity itself is always positive. Another merit of the all posi-

tive kernel is its interpretation as a weighting function, or window, for local statistics

estimation.

It is well known that the Gamma sequences are always positive and constitute a com-

plete set in 12. The k-th order 1D gamma kernel in the discrete domain is given by




gk, (n)= k(1 )n-kU(n-k). (12)
kC 1

where IA is the parameter that controls the scale of the kernel. The waveforms of gk [n]

for different order k = 0, 1, 2, 3 in the discrete time domain are shown in Figure 9.





k=0
0 >0 C------'--------------'--------I---------'---------'----------k -_
.-------- k=1
- k=2
.k=3





0 / ".- ---




Figure 9. The Gamma kernels of different order k=0, 1, 2, 3 in the discrete time
domain





19

The Z transform of the k-th order ID Gamma kernel can be written as

Gk(z) = z-


Therefore, the Z transform of the data projection into the k-th order ID Gamma kernel

can be written as


Xk(z) = G(z)Xk-_(z) G(z) = _
z-(1-)

That means the projection into each kernel can be simply obtained by passing the data

through the cascade of first-order kernels. That means for any causal signal x[n] # 0 for

0 < n < oo with finite energy, we can uniquely represent it by

00
ak, = x[m] 'gk,[m]
m=0

where ak, in fact is the projection of x[n] to the k-th Gamma sequence gk, 1[n]. We can

get ak, by convolving x[-n], with the k-th Gamma kernel:


ak, a = x[-n] gk, [n]

The complexity of the above recursive formula is only O(k), where k is the highest order

of Gamma kernels. From these arguments, the lD gamma kernel is likely appropriate for

ID intensity pattern modeling. Nonetheless, the more important case is to use 2D kernels

to extract the spatial information for detection.





20


Gt(z)


x(n) o- o Z- G,(z G (z)


xl(n) x2(n)/ xk(n)

Figure 10. The recursive ID Gamma filter structure

The extended 2D Gamma kernels is a circularly symmetric version of the 1D continuous
Gamma kernels, and the k-th order Gamma kernel gk[i,j] is given by [Principe et al,
1998b]

k+ 1
gk i,j] = ()kexp (-pt i2)
2 7k!
where gt is the parameter to control the scale of the kernel. The waveforms of different
order 2D circularly symmetric Gamma kernels are shown in Figure 11.














Figure 11. the 2D Circularly Symmetric Gamma kernels of different order k= 1, 4,
11, 21 in the discrete time domain for g = 0.8.





21

2.2 Representation of Bipolar Transient Signals

It has been pointed out that resonance response is an important feature for target detec-

tion in bipolar UWB radar signals [Chen et al, 1995]. To begin developing detection algo-

rithms for the resonance response, the first task is to provide a signal model to describe the

damped resonance response. The usual processing of transient signals like the resonance

response starts by transforming the time-domain signal to get a transform domain repre-

sentation. The usual linearly transformed representation used are Time-Frequency Repre-

sentation (TFR) and Time-Scale Representation (TSR), such as short time Fourier

transform (STFT) and Gabor transform [Gabor, 1946].

However, for any on-line detector, the computation complexity is always a big concern.

It's important for the chosen transform to have an efficient implementation, so that the

realization of on-line transform becomes feasible. Furthermore, there should be good rea-

sonings in choosing the transform so that a better representation could be expected.

Suppose we have the transient signal denoted by N x 1 vector x, and the signal sub-

space denoted by Nx M matrix S = [sI, s2,..., S*M], the signal model for x can be rep-

resented as x = Sa + n, where a is the projection vector, and n is the noise vector. A

good representation bases provides the projection space S where the energy is condensed

into a few bases. The limit of just one non-zero basis is called an eigendecomposition

which provides the best possible basis to detect the signal. The more components in a are

near zero, the better the representation is. We would like to find a signal subspace S which

meets the efficiency and "accuracy" to model the resonance response composed of the

damped components.





22

2.2.1 Gabor Bases

Gabor function subspace is a well-know damped sinusoidal subspace [Gabor, 1946]. It is

an intuitively reasonable choice for the resonance response composed of damped sinusoi-

dais by our previous argument. Suppose we have the damped sinusoidal subspace denoted

by Nx M matrix S(X,f):

S(X,f) = [s, s2 ..., SM]
si = [1, e(- i +j2nf )l,,. e(- i +j2nfi)(NN- )]T

where ki = [1l, 2, *..., XM] is the parameter vector of M-damping factors, and

f = [ff2, .,f **M] is the M-frequency parameter vector. S(X,f) can also be explicitly
expressed by



1 1 1
e (- ?, +J27tf) 1 e(- '2 +j2n7f2)1 e(- XM +j2tfM) 1
S(,f) = e +j2)1 (13)

e(-X, +j2tfi)(N- 1) e(-2 +j2rf2)(N- 1) e(- +j27f)(N- 1)


For the signal model y = S(,,f)a + n with the parameter (k,f) fixed, then the ML

estimate of a can be obtained as



a = [SH(i,)S(,j )f]-SH(,f)y (14)

Obviously, the complexity of (14) is O(N3), and it's a overwhelming computation burden

for a detection algorithm involved with the computation of a of Gabor bases, let alone the
overhead involved in estimating the parameter (),f).





23

2.2.2 Laguerre Bases

It is well known that, applying the Gram-Schmidt orthogonalization process to the fol-

kn
lowing sequences fi(n, u) = n u [Gottlieb, 1938], we obtain the k-th discrete Laguerre

sequence rk, j[n] given by



rk,[n] = exp(tn)Ak [(exp(-n)] (15)


where A is the forward difference operator, is the binomial coefficient, and pi is the
feedback parameter. Performing the k-th order difference operation it can be shown that

rk, [n] has the explicit form given by


00
rk, [] = pk z m)1( m) (3= exp(-L) (16)
m=0

The polynomials defined in (13) can be normalized to give the orthonomal Laguerre
sequence [Gottlieb, 1938]


k [n] = (-1)k rk,[n]

The waveforms of Ik, j[n] for different order k = 0, 1, 2, 3 in the discrete time domain
and frequency domain are shown in Figure 12 and Figure 13, respectively.







24







k=O
\D --------- k=1
k=2
---k=3


Magnitude -. ,


C D -. -. \ .,
-' \ \ / ./
I I






0 1 0 20 30 40 SO










k=0



Magnituden (same
Figure 12. the Laguerre kernels of different order k=0, 1, 2, 3 in the discrete time







domain
k=0









I[ Nn = I I
"-------- k=1





I "/ I
C .- .I D
m~,z / Fr e -ec
-I - k i=
II 0 i I .' ; 1 I

S\ i
~- I / \/ \j /
0 0 \ \V I 11 ". 1
010 0.5 1.0 1.1 2-0

Normalized Frequency



Figure 13. the Laguerre kernels of different order k=0, 1, 2, 3 in the frequency
domain



The Laguerre sequence lk, [n] satisfies the relation


oo

l1,[n]l1,,[n] =

n=O






25

where 65 is the Kronecker delta, i.e., Laguerre sequences constitute an orthogonal com-

plete set in 12, and it can be shown that the Laguerre sequences orthogonalized the

Gamma bases presented earlier [Silva, 1994]. Hence, for any causal signal x[n] # 0 for

0 < n < oo with finite energ, we can uniquely represent it by


00oo
x[n] = ak, l kk[n]
k=0
00
ak, 1= x[n]l k,t[n]
n=0

where ak, i is the projection of x[n] to the k-th Laguerre sequence 1k, [n]. We can get

ak, g by convolving x[-n] with the k-th Laguerre kernel:


ak, = x[-n] Ik, [n]

The Z transform of the Laguerre sequence is given by

Go

Li, (z) = li(n, u)z-n = (u2 (- i 0 (17)
n=oUZ-
So, with



Lo,(z) = (Ji'- )/(1-Pz-1) 0<.<1, (18)

we have


Lk+ 1, (z) = LZ(z) Lk, (z)

z-1 0 LI(z) 0

This shows that the projection into Laguerre space can be implemented by a cascade of

identical all-pass filters with transfer function L,(z) preceded by a low-pass filter with





26

transfer function Lo, (z). Let the response of each stage k be denoted by xk(n). The dif-

ference equations for computing the response of each stage recursively are as follows



xk+I(n) = Pkik+l(n- 1) +xk(n-1)-pxk(n)
(20)
Xo(n) = p.xo(n 1) + tx(n) 0 < < 1

The complexity of the above recursive formula is only O(K), where K is the highest

order of Laguerre kernels. Compared to the complexity of Gabor bases, O(N3), the com-

putation load is largely reduced. Due to the simplicity of recursive implementation and

good modeling for the damped signals, we will focus on using Laguerre recurrent net-

works to extract the projection information of the original resonance response and develop

our detection algorithms based on it.

In the following, we will see the general subspace representation of a target's response

using Laguerre bases, while the signal representation by the Gabor functions is also shown

for comparison. Suppose we have the Nx 1 signal vector

x(n) = [x(n), ..., x(n + N- 1)] T, and it belongs to a general N x M subspace S, where

the column vectors of S are the modeled kernels like Laguerre bases or Gabor bases. That

means x(n) = Sa, where a is the Mx 1 projection vector, and it can be obtained by


a = (STS)-ITx(n).





27

For the orthogonal subspace like Laguerre subspace denoted as L, it would reduce to

a = LTx(n). To have an indication of how much energy is captured by the modeled sig-

nal subspace, we propose to use the criteria given as



JS(n) = (21)
x(n)Tx(n)

which is a ratio of the energy in the projection subspace, S, to the energy in the original

signal, x(n). For the target signal in Figure 14-(a), the corresponding JL(n) for the

Laguerre subspace, L, and JG(n) for the Gabor subspace, G, are shown in Figure 14-(b)

and Figure 14-(c), respectively. For both cases, we use 25 kernels with the length of the

region of support equal to 128. Thus, x(n) is a 100 x 1 vector, and L or G is a 100 x 25

subspace matrix. In this case, it can be observed that for this target, the Laguerre subspace

representation is at least as good as the Gabor representation, since the JL(n) is always

larger than JG(n). Moreover, the computation complexity for the Laguerre subspace rep-

resentation is much less.





28



I ....1 ...



Magnitude




n (sample index)

(a) the original signal





JL(n)



n (sample index)
(b) JL(n) with Laguerre subspace




JG(n)--




n (sample index)
(c) JG(n) with Gabor subspace


Figure 14. The subspace representation of the target response using different
bases (a) the target's original response (b) the Laguerre subspace
representation with the first 25 order Laguerre kernels and p.=0.7 (c)
the Gabor representation with 25 damped sinusoidal bases, where the
damping factor X =0.95





29

2.3 The Nevman-Pearson Tests

Most of the target detection algorithms have been developed using the generalized like-

lihood ratio test (GLRT) approach which is in fact a Neyman-Pearson test to maximize the

detectability while given a fixed false alarm probability. In the following, we will present

a brief review of the theory [Scharf, 1988].


say Hi


S : Observation space
So



say Ho

Figure 15. Decision Regions


Let X = [X1 ,X2,...,XN] be a N-dimensional random vector of observations with joint

probability density function (pdf) fx(x10) where 0 is a parameter of the density func-
N
tion. Any specific realization x = [x1 ,z,... XN] of X will be a point in R where R is

the set of all real numbers. Detection problems can be viewed as two hypothesis testing

problems, in which we have to decide between one of two hypotheses, which we will label
N
Ho and H about the pdf fx(x ), given an observation vector x in R Let 0 be the
set of all possible values of 0. We usually identify Ho with one subset 00 of 0 values

and H1 with a disjoint 01, so that = 00 u 01. This may be normally expressed as



H1 : x has pdf fx(xlH1) = fx(xl0, 0 e 01)
(22)
Ho : x has pdf fx(xlHo) = fx(xIO, 0 E 0o)





30

In the signal detection literature, hypothesis Ho is usually called the null hypothesis, and
H1 is called the alternative hypothesis.

A test for hypothesis H against Ho may be specified as a partition of the sample space
S = Rn of observations into disjoint subsets S1 and So, so that x falling in S1 leads to
acceptance of H with Ho accepted otherwise. This may be expressed by a test function
t(x), which is defined to have value t(x) = 1 for x e S1 and value e t(x) = 0 for
x e SO. Let's use the following denotations:


S(0) = Ist(x)fx(xlHo)dx = s, t(x)fx(x|9, e 0o)dx

P(0) = st(x)fx(xIHl)dx = fst(x)fx(x 9, e o,)dx

, where PF is called the probability of a false alarm; PD is the probability of a detection.
Furthermore, we would use PM = 1 PD as the probability of a miss. The false alarm
and detection probability may be conveniently combined as the power function of the
detector as follows:

PD(0) if 0 e 01
P(O) = PD e o
PF(0) if 0 E 00

Without loss of generality, let's assume, for each subset 01 and 00, there is only one
single parameter 01 and 00, respectively. Our goal is to design a test, such that PD is
maximized (or PM is minimized), under the constraint that PF = a' a. Using
Lagrange multipliers, the cost function can be defined as:


J = PM+X[PF-a]

= I f(x Hi)dx+ [j f(xHo)dx-a] (23)





31

Obviously, if PF = a, J is minimized when PM is minimized. (23) can also be writ-
ten as

J= [1-a]+ J [f(xl|H)-Xf(x|H0)]dx
So

For any positive value k, the above cost function can be minimized by minimizing the
second term in the right hand side. Therefore, J is minimized by minimizing the likeli-
hood ratio


f(xlH1) f(xl01)
t(x)
f(xIHo) A(x100)

Then the desired test is the likelihood ratio test (LRT), which can be written as


H1
f(xl01) H
t(x) = ) H (24)
f(x|00) Ho

Generally, for real-world signal processing, the parameters 01 and 00 have to be esti-
mated from the observed data. If the most likely estimates (called the maximum likelihood
-ML estimate) 61 and 02 are substituted into the LRT for the parameters 01 and 00, then
the LRT becomes the so-called generalized likelihood ratio test (GLRT). To satisfy the
constraint, we have to choose X, so that PF = a. Then we require


PF = f(t0)dt = a

More importantly, under Ho, if the test is invariant to 0o, or, in other words, the f(t) is
independent of 00, that means PF = Jf(t)dt = a. Then the test t is a constant false
alarm rate (CFAR) detector. It's equivalent to say that if the f(t) is independent of 00, the
threshold X used for detection can be uniquely calculated in terms of a in advance with-
out resorting to 0o. Therefore, any derived detector would be more practical with the
CFAR characteristics.





32

2.4 Conclusion

In this chapter, we introduced the Gamma bases for the signal representation in intensity

images like MMW SAR images. Moreover, Laguerre bases and Gabor bases are intro-

duced for representation of transient signals such as the resonance response in UWB. In

the following chapters, depending on the different reflectivity characteristics of targets,

different bases are utilized to represent the target reflections, and then we will take the

GLRT approach to formulate the detectors based on the representation. All the proposed

detection schemes are simulated on the extensive data set: the MMW Moving and Station-

ary Target Acquisition and Recognition (MSTAR) data set and UWB SAR image data set.














CHAPTER 3
LOCAL INTENSITY TESTS FOR OPTIMAL DETECTABILITY

Target detection, the first stage in a radar system, is an important problem in signal pro-

cessing. The approach to target detection depends on the reflectivity characteristics of tar-

gets. The most distinguishing features of target reflections in different radar systems

should be utilized for detection, so that the detection probability can be maximized while

the associated false alarm rate is minimized. In MMW SAR images, targets are known to

contain many point scatters with large reflections, due to the metallic corers. Thus the

straightforward approach to "target detection" in MMW SAR has been to detect point

scatters. Assuming the background clutter intensity has a locally Gaussian distribution, the

two parameter CFAR test [Goldstein, 1973] to detect point scatters with intensity x can be

written as:


t x- t
t-


where p. is the estimated local mean, and a is the estimated local variance. As a pre-

screener in MIT Lincoln Lab's ATR system [Novak et al, 1993], the CFAR detector uses

the pixels in the clutter and rectangular stencil as illustrated in Figure 17-(b) to estimate

these two parameters. The corresponding 3D windowing function with the amplitude of

the stencil of the two parameter CFAR is shown in Figure 16-(b). The CFAR detector is a

normalized contrast comparison detector, which computes the ratio of the energy of the

tested pixel to that of the local background clutter, and it is widely used because of its sim-

plicity [Novak et al, 1993].


33





34

Notice that the CFAR detector is formulated to test every single pixel in the SAR image

separately, apparently discarding all the spatial information about target's point scatters.

MMW SAR technology provide us with better azimuth resolution of small targets, so that

better 2D target signature information is contained in the SAR images. To utilize the spa-

tial information of targets, Kelly [Kelly, 1989] first proposed that each target in the image

can be described by a completely known template or signature with an unknown scalar

gain. In a similar approach, Li [Li and Zelino, 1996] proposed a detector with targets' tem-

plates for the SAR images, where the templates can be represented by binary matrices

with "1" elements to represent the point scatter locations. One of the problems with target

detection in SAR images is that targets would have different reflection patterns due to dif-

ferent poses. Theoretically, a detector can work pretty well by incorporating all the tem-

plate information for targets due to pose variation. Practically, the detection system would

become too complicated to be implemented.

The following notation is used throughout this chapter: a 2D image data matrix is

denoted by a boldface capital letter such as X, and a 1D data vector by a boldface lower-

case letter such as x. Since we are working with image chips, without loss of generality,

all the data matrices will be assumed of finite region of support defined by 0N =

{(ij) | -N< i,j
dimension (2N+ 1) x (2N+ 1) with the origin in its center.

The y-CFAR detector [Kim and Principe, 1996] was proposed to generalize the two

parameter CFAR detector utilizing circularly symmetric Gamma kernels Gk to extract the





35

spatial information in the neighborhood of tested pixels, where Gk is the k- th order

Gamma kernel, and each of its element gk[i,j] is given by


k+1
gkj L] ( )k-l 'exp (--L t+2) (i,j) E nN
27tk!

where pt is the parameter to control the region of support of the Gamma kernels. Suppose

we have the data image X with the pixel under test in the center and two different order

Gamma kernels Gt and Gc. The y -CFAR detector can be written as


gt x-gT x
t = (25)
g[(x0x)- (g x)2

where x = vect{X},gt = vect{Gt} and gc = vect{Gc}. Comparing the form of the

CFAR detector with that of the y -CFAR detector, we can see that the two parameter CFAR

detector is extended by replacing the term gtx for the single intensity of the tested pixel

x. Besides, the local mean estimation gcTx and variance estimation Jg(x x) (gC x)2

is substituted for CP and <, respectively.

Although the extension to y -CFAR detector was heuristic, the y -CFAR detector has

been shown to significantly outperform the CFAR detector [Principe et al, 1998a], with

the choice of gt = gl = vect{G1} and gc = g15 = vect{G15}. The 3D windowing

function and the corresponding 2D region of support of the y -CFAR detector are shown

Figure 16-(a) and Figure 17-(a), respectively. The extension is intuitive in two aspects: gl

is chosen for gt, since G1 has a peaky shape like 6[nx, ny], which is used as Gt in the

CFAR detector as illustrated in Figure 16. Besides, g15 is chosen for g, in that G15 has





36

the similar size of the box-like guard-band proposed by MIT lincoln Laboratory for an

intermediate value of t.













(a) y-CFAR stencil (b) CFAR stencil


Figure 16. The 3D windowing function used by y-CFAR detector and CFAR
detector.






guard area,





(a) y-CFAR region of support (b) CFAR region of support


Figure 17. The 2D regions of support associated with the stencils used by y -CFAR
detector and CFAR detector.

Although y -CFAR has a decent performance, the performance is affected by the param-

eters I for gl and g15. The parameter values for maximal detectability are found through

training on the collected target and clutter chips. The performance surface of the y -CFAR





37

detector with respect to the parameters i for gl orgl5 for a training data set is shown in

Figure 18 [Principe et al, 1998b]. It can be recognized that parameter values of p for gl

or g15 is very critical to the detector's performance. Forseeably, each target would require

a different value of p1 or i15, but this was not done in the previous work, where the i

values were optimized over the training set. That argument is related to the two unsolved

problems for y -CFAR detector: First, how close to the optimum is the Gamma kernel gl ?

Secondly, is the learned size of the guard-band for gc good enough for the local statistics

estimation?

The first question must be answered by formulating the detector using the GLRT to

investigate what the functionality of g, is, and then we can further pursue the best possi-

ble candidate for g The guard-band size is a trade-off between two extremes. In one

side, the guard-band size needs to be as small as possible so that the estimated statistics

would be "local" to the target. On the other hand, the guard-band size should be chosen

large enough so that the estimation in the guard-band would be "independent" of the tar-

gets. In the previous CFAR detection approaches proposed by MIT Lincoln Lab, the

guard-band size is fixed at 80x80, which is about a 20Mx20M area to include the largest

size target supposed to be detected. However, that fixed guard-band size is not suitable for

all the targets. As shown in Figure 19, the guard-band size is much larger than these two

targets near the noisy clutter environment, and the estimated mean or variance would be

increased because many unnecessary clutter pixels are included. This larger guard-band

size would leads to a smaller y -CFAR testing statistics, and a probable missed detection.






38


Actually, these two target chips fail to be detected by the two parameter CFAR detector on


the results reported by Kim [Kim and Principe,1996]. Although the performance of the y -


CFAR detector can be improved by finding the better guard-band size through training, it

is still fixed in operation and sub-optimal. Observing these facts, we would like to come

up with a scheme to estimate the target size on-line for maximum detectability.







600

500,

400 -

300-

200,

100>.
40
30 40

10 10
0 0



Figure 18. The performance surface (false alarm surface) of the y -CFAR detector
with respect to the parameters p. for gl org15.


















Figure 19. Two embedded targets of TABLIS24 data set with their CFAR stencils.





39

3.1 Formulation Of Optimal Intensity Detectors

Observing targets in MMW SAR images, we can see that, in addition to the bright point

scatters, the whole target exhibits brighter reflection compared to the background because

of its metallic material. Although the intensity is not as large as that of the point scatters,

it's reasonable to assume that there exists higher intensity around the point scatters of tar-

gets, and target detection in the intensity image can be regarded as an intensity detection

problem. As a result, for the (2N+ 1) x (2N + 1) intensity image X, target detection can

be formulated as the two hypothesis testing problem:


H1 : x = aht+w w N(mi, c2i)
Ho : x= w w N(mi, 2i)


where ht is the intensity kernel, a is the unknown amplitude, x = vect{X}, i is the
(2N+ 1)2 x 1 vector with all its component equal to 1, and w is assumed to be white
Gaussian noise, with mean vector mi and variance a2i. Both m and G2 are unknown sca-
lar parameters to be estimated. Note that ht is assumed to be a known intensity kernel that
provides maximum detectability, and we will show later how to model the kernel from the
data set.

Without loss of generality, we can impose the constraint hTi = 1. So, the likelihood

ratio test can be written as


Pr(H) exp- x ah- mil 2)
t Pr(xlHl) 22
t -
Pr(xI|Ho) 1 l 2
exp- x-mi\ |





40

Taking the log of both sides of the above equation, we get


t = ah-mi x-mill 2)

Expansion of the right hand side of the above equation leads to


a
t = 2~(2hfx-ahrh,-2hTmi)
(26)
= a [h(x-mi)-a (26)


where l = IIhtIJ2. Here, we would take the generalized likelihood ratio test (GLRT)

approach. That means we would maximize (26) by using the maximum likelihood (ML)

estimate of a:



t = argmax{ a hf(x -mi)- "a1 (27)


Straightforwardly, the ML estimate of a is given by

-1
'1

Then, the estimated a can be plugged into (26) to yield



t = -a (28)
T2I22 202





41

Without affecting the monotonic increase of the testing statistics, we can multiply both

sides of (28) by 2r1, yielding



t "a2 h(x-mi)2




h[x-m
since hTi = 1. In practice, the statistics -- < 0 can be discarded, so taking the

square root on the both sides of (29) yields


h x-m
t = (30)


Suppose we have a spatial weighting vector he with a region of support disjoint from that

of ht's. That means, without loss of generality, with N = (2N+ 1) x (2N+ 1), he and

ht can be assumed to be


ht = [hto, ..., ht(K-), 0, ....., O]T
h, = [0, ......,0, h .., hc(_l)1]

In terms of he, we propose to estimate the parameters m and a by the local statistical

information as follows:


m = hcx
(31)
2 = chT[X{i ] x = x-mi





42

where c = 1/(1 hchc) is the coefficient needed for unbiased estimation of a2, Substi-

tuting the above estimators of m and a2 into (29) and absorbing the constant Jc in the

denominator, the detector becomes


hTx-h T
t = (32)


The detector is essentially an intensity detector, which functions as a normalized correlator

between the kernel and the testing image with the background clutter mean taken out.

Therefore, the success of the intensity detector is highly dependent on the shape of the ker-

nel ht. For optimality, ht should match the target intensity signature and it must be appro-

priately designed so that the maximal detectability is obtained.

Both the y -CFAR detector and CFAR detector can be framed into a kernel matching

detector. To see this, the variance estimator can be expanded as



hT[Xi@x] = C hci (xi-m)2
i=0...N-1
(33)
N-1 N-l N-i
= hci -x -2m hi -xi + m2 E hci
i=0 i=0 i=0

After direct simplification, (33) can be written in vector form as


a2 = c[hc(X x)-(h TX)2] (34)

Plugging (34) into (32) yields



h Tx h x)
h (x Q x) (hhx)2





43

The main difference between the two-parameter CFAR and the y -CFAR detector is the

intensity kernel of the point scatters to be matched: In the two-parameter CFAR detector

case, the intensity kernel ht is simply h,[i,j] = 6[i,j], while ht = gt in the y-CFAR

detector's case.


3.2 Statistical Properties Associated with the Intensity Detector

3.2.1 Unbiased Estimators

The estimator 0 would be called unbiased for 0 if E{(} = 0. Let

x = [x0, ..., x. ] T. We assume that the elements of x, x0, ..., x._ are independent

Gaussian random variables with mean m and variance 2 That means E{x} = mi and

E{x2} = &2i. In the following, we show that the two estimators m and a2 given in (31)

are unbiased.

First, we show that m is unbiased:


E{m} = E{hrx} = hcE{x}
= mhCi = m

Secondly, we show that 72 is unbiased. The first term in the right hand side of(34) can be

written as


E{hcx2} = hcE{x2}
= (m2 + a2)hci (36)
= m2+ a2

Then it can be shown (see Appendix A) that the second term in the right hand side of (34)

can be written as


E{(hcx)2} = m2 + (hhc)o2 (37)





44

Combining (36) and (37), then we get


E{a2} = E{c[hCTX2-(hCx)2]}
= [(m+ 2)- (2 + (h+hc2)]
= c[l-h Thc]C2
= C2

3.2.2 The Detector Preserves the CFAR Property

To say that a detector has the property of a CFAR test requires showing that, under H0,

the associated probability function of the testing statistics is independent of the parameters

for the assumed noise probability density function (PDF). That is the same approach used

by Robey [Robey et al, 1993] to show that the AMF detector preserves the CFAR statis-

tics. In our case, the testing statistics t in (32) can be shown as a CFAR test, if, under Ho,

its probability function f(t) is independent of the parameters, mean m and variance a2,

of the assumed Gaussian noise.

Notice that (32) can be written as


[(h- x-hrx)/]
t = (38)
Jhc[j ]/a2

It can be observed that the numerator of t, (htx hcx)/o, can be written as



(hfx-hcx)/a = ht J-h- J

= hTv-hcv

where v = (x-mi)/o, denoted as v = [vo, ..., v. ]T, and its elements vo, ..., v

are normalized independent variables with zero mean and variance 1. The numerator can





45

be written as a liner combination of the independent Gaussian random variable
vo, ..., v_ 1 Since ht and hc have exclusive regions of support, the numerator is still a
Gaussian random variable with zero mean and variance J hth + h[rhc. Furthermore, it

can be shown (See Appendix A) that the denominator term can be written as


hCT[ ]/G2 = h C[v v] -(hCv)2 (40)

Substituting (39) and (40) into (38), then the detection statistics now can be rewritten as


h[v-hr[v
t = (41)
hC[v v] -(hcv)2

The second term in the denominator has a quadratic form in terms of hTjv, which is a

Gaussian random variable with zero mean and variance Jh~hc, so it is 2. As to the first

term in the denominator, when h[ = [1/K, ..., 1/K, 0, ..., 0], the first term become j ,
and the denominator is ~_ The testing statistics will have a t-distribution. For other
cases where h T belongs to some kernel, the term h T[v v] does not fit any particular
probability function, and it is difficult to derive a close form of the PDF. However, since

the first term is still in terms of v, which is a normalized Gaussian random variable with
zero mean, its PDF definitely is independent of m and 0. By the same argument, the
PDFs of the denominator and the testing statistics are also independent of m and 0,
respectively under Ho. Therefore the intensity detector is a CFAR detector.

3.3 Intensity Modeling of Targets

In the previous section, we show that both y -CFAR and CFAR detector can be cast into

intensity detectors with unspecified intensity kernel ht. Then the question arises: what is

the best intensity kernel ht to match targets. To answer this question, we seek to design a

circularly symmetric kernel ht for the intensity detector. The added constraint of cir-





46

curlarly symmetry is imposed so that the change of reflectivity due to pose changes can be

avoided, and the analysis is simplified.


3.3.1 Mathematical Background

Suppose we have a 2D continuous image f(x, y) with background clutter mean removed.
The statistics of correlating f(x, y) with 2D circularly symmetric kernel h(x, y) can be
written as


t = h(x,y) f(x,y) dxdy
(42)
= f h(r, 0) f(r, ) rdrdO

where h(r, 0) and f(r, 0) are the corresponding image and kernel function in the polar

coordinate domain, respectively. Since h(r, 0) is circularly symmetric, it is independent
of the variable 0, then (42) can be written as


t = rh(r) fR(r) dr = h(r) fR(r) dr
(43)

where


h(r) = rh(r) fR(r) = f0f(r, 0) dO (44)

Note that h(r) is still a symmetric kernel. Suppose the kernel fR(r) can be modeled by
some deterministic signal p(r) and noisy mismatch signal s(r) as fR(r) = p(r) + E(r).
Then (43) can be written as


t = f h(r) (p(r) + (r)) dr (45)





47

SThen, the problem of designing a 2D symmetric kernel h(x, y) to maximize the correla-

tion in (42) is simplified to the problem of designing 1D radial kernel h(r). If we assume

that e(r) is white Gaussian noise, from the matching filter point of view, the SNR is max-
imized when



h(r) = p(r) (46)


Note that, for some specific ro, fR(ro) in (44) can be written as



1 f"f(ro, 6) ro dO
fR(r ) = 2nro0 (47)


It means that, for some specific ro, we can get -F(ro) by integrating the image's inten-
2Tn
sity along the points of the circle with radius ro, and then normalize it with respect to

2nro. So, TlfR(r) can be viewed as a 1D radial representation of the 2D image data

f(r, 0) or f(x, y). Since the SAR data image is discrete, we will formulate the above cor-
responding procedures in the discrete spatial domain. For the discrete image chip F with
finite region of support fIN, the corresponding part of (47) can be approximated by



fR[r] = f J[i,j]/NR(r) (48)
i,j 3 i2 +j2 = r2

where r ON = {r2 = i2 +j2 I -N< i,j N i,j e integer}, and NR(r) is the num-

ber of the points with the same radius distance r in the discrete image f[i,j]. We can

think that fR[r] is the equivalent of (47) in the discrete domain. The mapping relationship

between r and (i,j) is illustrated in Figure 20, where we can easily see that r E {0, 1,

1.414, 2.236, 2.828,...}, and it's not difficult to figure out that there are
N 1 = (N+ 1)N/2 elements in ON. Since there is only 1 point with image index

(0,0) having the radial distance r = 0, therefore we have NR(O) = 1. There are four






48

points whose indices are (0,1), (0,1), (0,-1) and (-1,0), respectively, with radial distance

r = 1, Thus we have NR(1) = 4. In the same manner, we can get NR(1.414) = 4,

NR(2.236) = 8, and etc.




0 *

(-2,2) (-1,2) (0,2) (1,2) (2,2) 2.828 2.236 2 2.236 2.828

(-2,-1) (-1,1) (0,1) (1,1) (2,1) 2.236 1.414 1 1.414 2.236

** (-2,0) (-1,0) (0,0) (1,0) (2,0) ** ** 2 1 0 1 2 **

(-2,1) (-1,-1) (0,-1) (1,-1) (2,-1) 2.236 1.414 1 1.414 2.236

(-2,-2) (-1,-2 (0,-2) (1,-2) (2,-2) 2.828 2.236 2 2.236 2.828

S *
S *



Figure 20. The usual image indices with (0,0) in the image center (left), and the
corresponding radial indices (right).


3.3.2 The Modeling Procedure and Experiment Results

The problem left out is how to build a good intensity kernel model for p[r] using the

information fR[r] computed from the target data set we had. As we mentioned earlier, the

features of targets which we are interested in for detection are around "bright" point scat-

ters. Therefore, we propose the following 3-step procedure to build the intensity kernel

model p(r)

(1) align each target chips so that the brightest point is in the center

(2) get the ID radial projection of each target chips






49

(3) find the intensity kernel with maximal projection for the data set by means of princi-

pal component analysis (PCA).

The essence of the intensity detector is correlation between signal fR(r) and intensity

kernel p(r), and, equivalently, the projection of signal fR(r) into p(r). To implement the

maximal projection, PCA can be applied to find the eigenvector corresponding to the larg-

est eigenvector of the data correlation matrix of the collected data chips. The largest eigen-

vector can be used as the intensity kernel p(r). Then the intensity detector can be

regarded as performing maximal eigenfiltering [Haykin, 1991].




2N--

JN PCA and Gamma
Kernel Matching
---" o ff N --






Figure 21. The 3-step intensity kernel modeling procedure


Let the elements in DN be sorted in the ascending order and form the radius vector

r = [ro, ..., r(N+1)N/2-1]. That means r = [0 1.414 2.236 2.828, ...]T. Notice that the

first element of r is 0. Without affecting generality, we can just use a smaller number like

0.7 instead. Suppose we have the 2D aligned data chips X0, ..., XN_ I, and the corre-

sponding 1D radial projection vectors are denoted as xo, ..., xN_ 1, where the i-th com-





50

ponent of xn is the radial projection along radial distance ri, the corresponding element in

r. The criteria used for principal component analysis (PCA) is given by




J = argmax llXi_-uTx 2

(49)

where u is the eigenvector with the largest eigenvalue of the data correlation matrix

zxi(xi)T. For the three types of the targets BMP2, BTR70, and T72 in MSTAR data set,
the largest eigenvectors are shown in the most right hand side of Figure 22, Figure 23, and

Figure 24, respectively. To implement the intensity detector in (42) by means of DFT fil-

tering, h(r) rather than h(r) = rh(r) in (43) is needed, and the corresponding signatures

in the discrete spatial domain are shown in the Figure 25.

The previous results show that the intensity detector with the Gamma kernels works

well, so we rather prefer to use the Gamma kernels to model the intensity pattern since it is

a computationally simpler operation. The projection modeling can be written as




J = argmax Z xi--ai k a, = xT gk,
k,t i=o
(50)

where gk, = r gk, p. The components in gi, can be specifically written as

k+1
gk, (r) = r gk,(r) = r. 2 (rY)k-exp(-I-r).
2ntk!

To implement the intensity detector in (42) by means ofDFT filtering, it's necessary to

get h(r) rather than h(r) = rh(r) in (43). That's the reason why gk, is used. The best

gk, is found through parameter searching with respect to both k and gt. In our experi-






51

ment, we exhaustively substitute the parameter value k { 1, 2,..., 20} and

ue {0.01, 0.02, 0.03, ..., 0.99} into (50) to find the best gk, For the three types of the

targets BMP2, BTR70, and T72 in MSTAR data set, the best kernel order k are all 1, and

the parameter p is 0.47, 0.51, and 0.53. Since in the real application, we have to fix the

parameter value for the detector. The modeled radial representation gk, p, with k = 1 and

p = 0.5, are shown in the right of Figure 22, Figure 23, and Figure 24, respectively.

while the corresponding Gamma kernels gk, in the discrete spatial domain is shown in

the Figure 25. From this analysis, it is clear why the y -CFAR detector work so well. The

first order circularly symmetric Gamma kernel is approximately the maximal eigenvector

for radial target intensity, which is the best kernel for the GLRT test.



-- 1 st order Gamma kernel
] - 1st Principal component


F u 2 T M o Ioo B








Figure 22. The Modeling of the Radial Projection of BMP2






52



e -- 1st order Gamma kernel
0. - 1st Principal component




.... -1



__ ooo *o B



Figure 23. The Modeling of the Radial Projection of BTR70




-- 1st order Gamma kernel
1 - 1st Principal component

I
In oo










Figure 24. The Modeling of the Radial Projection of T72






53



S1st order Gamma kernel
- 1st Principal component


0.20 020 0.20







BMP2 BTR70 T72



Figure 25. The actual 1D slice of the modeled intensity pattern in the radial
direction for the three types of targets in the MSTAR data set


3.4 Individual Target Size Estimation

First, we will present an example to see why the y-CFAR or the CFAR detector's perfor-

mance is closely related to the guard-band size. The previous two targets near noisy clutter

with their radial intensity projection are shown in Figure 26. The guard-band size used by

MIT Lincoln Lab's ATR system is fixed at 42 x 42. Notice that in both cases the clutter

within sample 42 has higher local mean or variance. It would lead to a contaminated test-

ing statistics resulting in the miss detection of those two targets. Suppose the guard-band

size can be set around 30, where most of the target energy are included. Then the esti-

mated mean and variance of the local clutter would be smaller, and the testing statistics

would be higher so that the two targets would more probably be detected. Our purpose in

the following is to dynamically determine the guard-band size for the stencil for each tar-

get, instead of finding a single guard-band size through off-line training. This goal means

that all quantities must be estimated for each image chip.






54















Radial Projection

'7o j 2oo[ l | l



7SO0




o 20 o eo o 2 o eo



Figure 26. The two targets of Tablis24 data embedded in the clutter of Mission 90
data set. The vertical line shows the MIT stencil to estimate the local
statistics.


3.4.1 Individual Target Size Estimation Procedure

Essentially, the intensity detector match the intensity pattern with the radial projection of

data image above the estimated noise mean. In that sense, the "target extent" meaningful

to the intensity detector is where most of the intensity of targets's radial projection lies.

We select this radius as the mean intensity of the local clutter level. Inspired by the results

in the last section, the following 4-step scheme is proposed to estimate the target size by

determining where target's radial projection falls off the estimated noise mean.





55

(1) iT = gcx. The local mean p. is estimated by gcx, where the guard-band size of the

Gamma kernel g, includes the largest target to be detected.

(2) Xg, = Gt X, where means 2D circular convolution. Get enhanced image chip
Xg, by filtering the original image chip X with the first order Gamma kernel Gt, the

approximate principal eigenvector. This processing basically can be viewed as spatial

eigenfiltering.

(3) xR[r] = V x[i,j]/N(r). Get the radial projection representation xR[r] of
i,j 3 i2 +j2 = r2
the filtered image data chip Xg .

(4) Find the smallest r, such that xR[r] < p..

The above scheme can be illustrated by




Target Enhancement: radial projection
convolved with 1st
order gamma kernel





Image chips radius finding





Local Mean Estimation:
convolved with 15th the mean of the noise
order gamma kernel



Figure 27. Target size estimation scheme in MMW SAR images.






56

3.4.2 Experiment Results

The enhanced image chips, their radial projections and the estimated target size for the

two targets BMP2 and T72 in MSTAR data set are shown in Figure 28, respectively, while

those for the two embedded targets in TABLIS data set are shown in Figure 29. From the

radial representations, it can be immediately observed that the SNR is improved due to the

spatial filtering. The SNR for the different data are listed in Table 1 where the average

SNR is improved approximately by 4.6 db. Form Table 3 the estimated size for each data

set is obtained with NMSE around 15%. That provides us with enough information for the

setting of the guard-band size, since the guard-band size can be set 15% larger. More

importantly, most of the information in our scheme exists already to compute the y -CFAR

detector. There is not too much overhead involved.





57

Image Enhancement






SRadial Projection





4 4









Figure 28. The associated processed data involved in target size estimation for the
two targets BMP2 and T72 in MSTAR data set.





58


Image Enhancement









I Radial Projection




















Figure 29. The associated processed data involved in target size estimation for the
two embedded targets in TABLIS24 data set.






59







Target type Series No. Smallest SNR Largest SNR Avg SNR
improvement improvement improvement
(db) (db) (db)
BMP sn_9563 3.7 6.8 5.1
BMP sn_9566 4.0 6.4 5.0
BMP sn_c21 4.0 7.0 5.0
BTR70 soc71 3.0 6.6 4.5
T72 sn_132 3.3 6.3 4.6
T72 sn_812 3.2 6.5 4.6
T72 sns7 3.2 7.0 4.8
Tablis24 Mission 90 1.2 10.8 5.3


Table 1. The SNR improvement for different data set






Target type Series No. Smallest Detection Largest Detection Avg Detection
Statistics improved Statistics improved Statistics improved
BMP sn_9563 3.1 8.2 4.9
BMP sn_9566 3.0 6.8 4.4
BMP sn_c21 3.1 9.0 4.6
BTR70 sn_c71 1.9 6.6 3.4
T72 sn_132 2.7 6.6 4.3
T72 sn_812 2.6 8.1 4.3
T72 sn_s7 2.5 8.3 4.6
Tablis24 Mission 90 1.5 10.8 7.8


Table 2. The detection statistics improvement for different data set






60



Target type
Series No. Actual Size Mean Est. Size Mean Est. Size NMSE
BMP sn_9563 8.5 8.72627 1.30892 15.0%
BMP sn_9566 8.5 9.00103 1.34748 15.0%
BMP sn_c21 8.5 8.80738 1.09854 12.5%
BTR70 sn_c71 9.0 9.24721 0.879623 9.5%
T72 sn_132 10.0 9.76500 1.56374 16.0%
T72 sn_812 10.0 9.99818 1.54033 15.4%
T72 sn_s7 10.0 10.05790 1.70741 17.0%
Tablis24 Mission 90 8-12 12.0%


Table 3. The estimated target size and the associated statistics

3.5 Conclusion

This chapter formulates the CFAR detector as an intensity detector by applying the

GLRT formulation developed in Chapter 2. We were able to show using the ML approach

that the CFAR stencil is intrinsically linked with the optimality of the test. The stencil can

be thought as the correlation template, and as such for optimality, it should match the

intensity profile of the targets. We then analyze the radial intensity profiles of targets from

MSTAR through principal component analysis (PCA). It turns out that the first eigenfunc-

tion of the radial intensity profile of targets can be well approximated by the first gamma

kernel. Hence this explains why previous results [Kim and Principe, 1996] have shown

that the y -CFAR outperformed the delta funcion stencil proposed by MIT/Lincoln Labo-

ratories.

With this understanding we proposed a method to adapt the guardband for each individ-

ual target, which is much better than the average guardband size proposed in [Principe et






61

al, 1998b]. This procedure to determine the guardband will make the CFAR test more

robust and should improve performance even more.














CHAPTER 4
SUBSPACE DETECTION OF TARGETS IN UWB SAR IMAGES

Early radar systems detected targets in the air. Illuminated targets would return a large

energy reflection which is contaminated by the thermal noise generated by the radar sys-

tem itself. To detect the large energy reflection of targets, the so-called one parameter

Constant False Alarm Rate (CFAR) detector compares the amplitude of the testing cell

with the noise amplitude, the "adaptive threshold," estimated from the neighboring cells.



cell under test


I ll *LI li *** G***1I I 1 +1
o *** M/2-1 M/2 *** M-1
,, threshold



1/M


Figure 30. One parameter CFAR detector


Many other target detection algorithms have been developed based on the observation that

the reflection of targets has larger energy compared to that of natural clutter [Goldstein,

1973]. As technology improved, SAR has also been applied to the surveillance of ground

targets. In MMW SAR images, background clutter can no longer be assumed to have a

global Gaussian distribution due to the different reflections of the ground textures. How-

ever, it is still valid to assume that clutter has a local Gaussian distribution, since texture



62






63

changes slowly. In this case, "target detection" becomes detection of point scatters as

introduced earlier.

UWB was developed to take advantage of the better penetration capability of the low

frequency components in the transmitted waveform with large bandwidth. Consequently,

metallic objects reflect the resonance response following the driven response due to the

wideband excitation. Several algorithms have been developed to directly detect the driven

response of targets in the UWB SAR images [Kapoor et al, 1997]. However, some natural

clutter would also produce large energy returns, resulting in poor performance. Instead,

the target resonance responses may be utilized for discrimination from natural clutter

[Sabio, 1994][Chen et al, 1995]. Resonance responses are composed of damped sinusoi-

dals, so it is reasonable to assume they can be modeled by some "matched" subspace, in

the sense that the energy of resonance response would concentrate only on a few bases,

where natural clutter contains little energy. Then an approach similar to detecting transient

signals [Friedlander and Porat, 1989] can be used to develop a resonance detector.


4.1 Target Detection Using Laguerre Networks

There have been many algorithms developed for target detection in UWB SAR. Basi-

cally, as described earlier, UWB SAR provides foliage penetration capabilities. Moreover,

man-made metallic objects in UWB radar produce a damped sinusoidal response in addi-

tion to the large energy reflections, which also could be generated by non-metallic objects

such as trees. Therefore, it is important to explore the 1D information of the resonance

response of metallic objects to develop improved target detection algorithms [Yen and

Principe, 1997a]. This is the approach taken in this work.





64

4.1.1 The ID Resonance Response Model

The resonance response, denoted by x, is assumed to be a length N vector belonging to

a 12 finite-dimensional linear space of discrete functions. The space is the so-called signal

space, and it can be represented by H, where H is a N x M matrix. That means the col-

umn vectors { h,, 1 < m < M} of H are the basis vectors of the signal space. A specific

instance of x can be expressed as


x = Ha

where a is the M x 1 projection vector. The signal subspace is application dependent.

Wavelets, STFT or Gabor transforms are all possible choices. If the bases are chosen

according to the resonance response shape, we usually only need K out of the M bases to

constitute our signals, so only K components in a would be nonzero. Ideally, we would

like a sparse implementation, in the sense that K will be much smaller than M (K < M).

That means the bases should be matched to the signal.

This model can be generalized by letting the signal space be dependent upon some

parameter vector 0, such as the scale parameter p in Gamma or Laguerre kernels. The

vector may contain the waveform-shape parameters. Real-life signals will seldom obey

this model, so some deviation from it should be included. Therefore, we will modify the

above model as


x(O) = Ha(O) + e(0)

where e(0) is the Nx 1 mismatch signal vector. By definition, this mismatch signal is

orthogonal to the signal subspace spanned by H. It is assumed that the energy contained

in e(0) is small with respect to the total signal energy. In addition, the received transient






65

signal is also assumed to be corrupted by additive white Gaussian noise. Hence, after sam-

pling, the final model could be described as:


y(O) = Ha() + e() + w

where w is the N x 1 noise vector, whose elements are assumed to be mutually uncorre-

lated Gaussian random variables. If we apply any linear transformation to the resonance

response signal y by a Mx N matrix L, then


z = Ly = LHa+Le+Lw

When L is chosen as the left inverse of H so that L H = IM, where I is a M x M iden-

tity matrix, then we get


z = a+Le+Lw
(51)
= a+Lw

where ws = e + w is still assumed to be white Gaussian noise, with zero mean and

unknown variance a2. We assume that there are only K nonzero projections for the repre-

sentation vector a, and that those K component locations are known, but their values are

unknown. Let the nonzero representation vector be denoted by aNz. Then it can be com-

puted by


aNz = Sa (52)

where S is a Nx N diagonal matrix of rank R describing the locations of known com-

ponents. That means


(1 if i = j = known component location
[] = elsewher(53)
v 0 elsewhere





66

Then the measurement model is given by

y = HaNZ + i
= HSa + v

Consequently, the transformed vector z can be rewritten as



z = Ly = LHSa+L (54)
= Sa+v

where v = Lw is the colored Gaussian noise vector with covariance a2Q, where

Q = LLT. (54) is a well known subspace model described by [Scharf, 1988].


4.1.2 Formulation of the Proposed 1D CFAR Detector

Based on the previous signal model, we have the following two hypothesis testing prob-

lem


H0 : z =v v N(O, 2Q)

H1 : z = Sa+v v- N(O, 2 Q)


The GLRT statistic is defined by

t = max{21ogfi(z)}-21ogfo(z)

where f,(z) and fo(z) are the corresponding probability functions under Ho and H1,

respectively. Then, we have


21ogf,(z) = -log(27)-logl e-(z-as)7 Q-(z-aS)






67

Following [Friedlander and Porat, 1989], we can show that the GLRT statistics for this

problem becomes



z = z zcT Q -l
t 2 (55)


where


c = ( IN-S)z and QC = (IS)Q

IN is a N x N identity matrix. In our UWB SAR scenario, we assume 02 unknown. It can
be estimated by


a2 = (1/p)uTU

from the neighboring p x 1 sample vector u, which are assumed to be independent of the

testing sample. So, our GLRT test statistics is given by


T lz T -1 Ho
S- > threshold (56)
u u H1

It can be shown (see Appendix A.2) that the numerator in (56) is a chi-square distributed

random variable with K degrees of freedom, and that the testing statistics t is a random

variable with F-distribution. The numerator in (56) is the energy contained in the whole

space subtracted by that contained in the null space. Thus, it can be viewed as the energy

in the signal space. So, the testing statistics uses the ratio of the energy contained in the

signal space to the estimated background noise energy to decide if the return is a target or

not.





68

4.1.3 Detection Scheme and Simulation Results

Laguerre recurrent networks are used to capture the projection information of the UWB

radar signals, due to the simplicity of recursive implementation and a reasonable subspace

for representing the damped signals as shown in Chapter 2. Based on the test statistics

(56), we can implement our detection scheme as shown in Figure 31. The projection of the

signal to the Laguerre bases is simply extracted form each tap of the recurrent network,

and the projection energy can be computed by squaring the tap values. The GLRT can be

computed simply by summing up all the projection energy.




L(z) (
xo(n) I x(n) x2(n) xk(n)
<+ Z-1 4 -<+l L(z) FL(Z)] 10

SL (n) L2(n) Lk(n)

( )2 )2 ( )2 ( )2




GLRT


Figure 31. GLRT with recursive Laguerre Networks

First, we will show that the proposed detection scheme works for the signal composed of

Laguerre functions embedded in nonstationary noise. And then, we will show that our

algorithm works well for a UWB SAR down range profile. We defer the complete testing

of the detector for the final implementation.







69








10



-5

-5

-- 1 0 -- o--.- . o'o '.' 4 5'o--'-'-
0 500 1 000 1 500 2000
(a) The three Laguerre kernels
10

5



--5

-1 0
0 500 1 000 1 500 2000
(b) The three Laguerre kernels embedded in non-Gaussian noise

1 OO


60
60
4-O

20 -

O 500 1 000 1 500 2000

(c) The detection statistics




Figure 32. The signals of the three kernels embedded in the noise and the
corresponding detection statistics



Figure 32-(a) shows the first three Laguerre kernels with the parameter [ = 0.7. The


generated Laguerre function signal embedded in the noise are shown in Figure 32-(b),


where the non-stationary Gaussian white noise has variance 1 in the first 1000 samples


and variance 4 in the following 1000 samples. It is impossible by eye to see where the sig-


nal is. The corresponding detection statistics using (56) are shown in Figure 32-(c). It is






70


obvious that our statistics can differentiate the three embedded Laguerre transients even

when occurring in nonstationary noise. However, notice that the signals belong to the

basis set which represent the best performance situation.

Next, we will deal with the ID UWB SAR down range profile shown in Figure 33,

where three types of objects exist. The samples from 550 to 750 correspond to a power

line, the samples from 3150 to 3350 correspond to a vehicle target, and the samples from

samples from 4000 to 4200 correspond to tree reflections. Note that in terms of SNR, this

situation is more benign than the synthetic example since most of the returns are of lower

amplitudes. However, now the target is only approximately modeled by Laguerre func-

tions as shown in Chapter 2, and more importantly there are high returns of trees which do

not belong to targets.



1.5x106
1.0x10

0
-5.0 x 10
-1.0x10
-1 O>c 10
-1 .5 x10 6
0 1000 2000 3000 4000 5000 6000



Figure 33. The ID UWB SAR signal


To design a signal subspace constructed for Laguerre bases, we will first investigate how

the UWB transient signal is represented by the parametric Laguerre space. There are two

basic questions: what are the bases to be used and what is best value of Pt ? Intuitively, the


JL(n) in (21) may be examined along the ID UWB signal samples for the optimization of






71

p. However, the goal of designing a signal space for detection is to get a peaky statistics

around the target neighborhood. That means we would optimize the values of p such that

the projection is as large as possible over a small window 1 around the largest target

reflection. Therefore, instead of using JL(n) for a signal y(n), 0 I n < N1 we propose

the following criteria for the optimization of gp



no+I-1
"peak = argmax E JL(k)
P-9 k= n,
[ (57)

where 1 is empirically determined to be 20. We use as many as 40 Laguerre bases to con-

struct our Laguerre signal subspace, where the parameter pt can be optimized using the

criteria (57). For the vehicle in Figure 33, the relationship between the average Jpeak and

p. is shown in Figure 34. This figure clearly shows that for pt = 0.7 the criterion is maxi-

mized, i.e., the subspace is best matched to the target response.






Jpeak



0.0 0.2 0.'4. 0.0 O.0 1.0



Figure 34. The Jpeak with respect to the parameter p for the targets in Figure 33.


A 40th order Laguerre space with Ip = 0.7 are used to construct the signal subspace to

capture the signal energy. The results for the vehicle and the power lines are shown in Fig-





72

ure 35-(a) and Figure 36-(a), while the resulting JL(n) for the foliage is shown in Figure

37-(a) for comparison. The Jpeak for the three objects closest to 1 occur at n = 564,

n = 3256, and n = 4096. Theoretically speaking, the power lines act like a metallic

object also producing a strong resonance response. It can be observed that the tree pro-

duces a response which has an amplitude comparable but smaller than the other two

objects. However, we would like to investigate if the energy is concentrated in the same

bases as the target.

The corresponding distribution of the projection energy are shown in Figure 35-(b), Fig-

ure 36-(b) and Figure 37-(b), respectively. We can see that the projections of the three

objects have a large overlap in the bases, but there are some differences. The power line

has a dominant component in the first Laguerre basis, with a second emphasis on bases

around 7. The target has a clean peak around order 7 and a second concentration around

order 15, while the tree has the largest projection around order 15. So, there are significant

differences in the energy pattern in the subspace that can be further explored to improve

the discrimination of the targets from the clutter.






73







-JL(n>

n=654


n
(a) The JL(n) with Laguerre subspace


at n=654







order k
(b) The normalized projection energy in the Laguerre bases


Figure 35. The Laguerre subspace representation for the power lines in Figure 33
(a) the JL(n) around the sample from 550 to 750 (b) the distribution of
the normalized projection energy in the Laguerre bases for the sample
n=654






74








n=3256



n
(a) The JL(n) with Laguerre subspace



L at n=3256







order k
(b) The normalized projection energy in the Laguerre bases


Figure 36. The Laguerre subspace representation for the vehicle in Figure 33 (a)
the JL(n) around the sample from 3150 to 3350 (b) the distribution of
the normalized projection energy in the Laguerre bases for the sample
n=3256






75








. nn=4096



n -
(a) The JL(n) with Laguerre subspace



at n=4096






order k
(b) The normalized projection energy in the Laguerre bases


Figure 37. The Laguerre subspace representation for the tree in Figure 33 (a) the
JL(n) around the sample from 4000 to 4200 (b) the distribution of the
normalized projection energy in the Laguerre bases for the sample
n=4096


We would like to find the size of the space that contains most of the energy of the

objects. So, we order the projection energies and plot them. The corresponding results for

n = 654, n = 3256, and n = 4096 are shown in Figure 38. We also can see from Fig-

ure 35, Figure 36 and Figure 37 that the signal energy is concentrated in the first 15

Laguerre bases.





76




atn=654





(a)




atn=3256




(b)


0.0 at n=4096






(c)


Figure 38. The effect of changing the number of bases to the subspace energy for
the three objects (a) the power lines (b) the vehicle (c) the tree

From these results, we can assume that the resonance responses of the power lines and

the vehicle are within the subspace expanded by first 15 order Laguerre kernels. Thus, we

can use the Laguerre recurrent networks of order 15 and parameter P equal to 0.7, since it

was observed that the Laguerre subspace constructed by the first 15 kernels captures 90%






77

energy of the resonance response. The detection statistics of the proposed GLRT using the

full 15 bases is shown in Figure 39. This preliminary experiment shows that with the cho-

sen parameters, the algorithm is able to discriminate the vehicle and power lines from nat-

ural clutters with a large margin, but of course this is a single case of each object.

400 -



200



0 1000 2000 3000 4000 5000 6000


Figure 39. The detection statistics corresponding to the ID UWB SAR down
range profile in Figure 33


Next, we will try to demonstrate the effect of using the appropriate signal subspace for

detection. Taking the distribution of the foliage in Figure 37-(b) into consideration, we can

intuitively argue that the signal space for the three object types can be constructed from

three different combination of the Laguerre bases. The Laguerre kernels of the first four

Laguerre bases can be used for the man-made clutter, power lines, since the subspace

energy in these bases is dominant. It is equivalent to partitioning the full space into the sig-

nal subspace and non-signal subspace, i.e., the 15 x 15 partition matrix S in (53) can be

set as


[S] = if i =j = 1,2,3, 4
S= (0 if i =j =5,...,15


Based on the same reasoning, the Laguerre kernels of order k = 5, 6, 7, 8, 9 can be

used to construct the signal subspace for the vehicle while the next five order Laguerre






78


kernels can be used for the tree. Depending on the different matching signal subspace, the

corresponding detection statistics are shown in Figure 40.







;I 00---00
1234 5 15
order

ll..,.ti lal. ,danl, .lllilalJ .ll L,,,Jdl ln illdlll IllU.IIIId,lih ll lllll l llIl, ,,IJ ...
- -3 -? -D - - - - - -3 -3 -3 -3 -3 -3 -
(a) The detection statistics with signal subspace constructed by Laguerre bases
of order 1, 2, 3 and 4





- o----o - o---o
1 ---45 6---910---15
-= II order

C 0 1C oB o C3o -- 000 0000 0000
(b) The detection statistics with signal subspace constructed by Laguerre bases
of order 5, 6, 7, 9 and 9


00 0 0


:0--- 00
:1 --- 89 10 11---15:
0 order
-1 -3 = -3 - -S -3 - -O 0 - 3 3 - = 3 -
(b) The detection statistics with signal subspace constructed by Laguerre bases
of order 10, 11, 12, 13, 14 and 15



Figure 40. The detection statistics corresponding to the 1D UWB SAR down
range profile in Figure 33 using different combination of bases for
signal subspace





79

From the results in Figure 40, it can be observed that the use of appropriately designed

signal subspace will improve the detectability of the corresponding objects. Comparing

Figure 37-(b) with Figure 38-(b), we can also expect that there is an overlap between tar-

gets and trees. The design of a signal subspace for improved detectability of a single

object may be straightforward, but for all the targets, it will be much tougher since the

clutter response needs to be minimized at the same time.


4.2 Target Detection Utilizing A Spatial Template

In the previous section, only the resonance response information along one down range

profile is explored. One should be reminded that, in addition to the unique wideband exci-

tation, UWB SAR also provides us with improved azimuth resolution of the small targets,

so that abundant spatial information is carried into the images like the MMW SAR.

In MMW SAR images, target reflection signatures vary as their poses change, and it is

impractical to have different reflection templates with respect to different poses. Although

the same problem occurs for modeling the driven response signatures of targets in the

UWB SAR images, the resonance response of each target is at least theoretically indepen-

dent of its pose. Thus it seems reasonable that under the assumption of a template for the

target's resonance response, a unique model is applicable to detect the resonance response

of the target even with different poses. We assumed that, for the target's cross range

extent, the resonance responses along different down range profiles concentrates on a sub-

set of bases in the "matched" signal subspace. Therefore, each target of interest is modeled

with a signal subspace template, and a more robust test statistics preserving constant false

alarm rate (CFAR) can be formulated [Yen and Principe, 1997b].





80

4.2.1 The Spatial Template Model and the Detector Derivation

The resonance response of a target in UWB is spatially contained in J neighboring

down range profiles Yl,Y2, **.*J, where each down range profile yj is a Nj x 1 vector.

So, we can collect these J different down range profiles to constitute a N x 1 target tem-

plate vector


y = [yy T ...,JT]T

where N is given by N = N1 + ... + N Notice that the components yj of y do not need

to be adjacent to each other, and Nj doesn't need to be equal to Ni when i #j. Then the

resonance response template vector y can be written as


y = x+ (58)

where x = [x, x2, ...jJ] and w = [wT, wf, ...,wT]T are the corresponding ideal

signal vector and the noise vector respectively. Both are of dimension N. Usually, wj is

assumed to be white and Gaussian, with zero mean and unknown variance 02, and wi is

independent of wj, when i #j.

Each ideal signal component xj of x is assumed to belong to a known Mj -dimensional

signal subspace represented by a M x Nj matrix, L If we apply a linear transform LT

to the y then we get



zj = LTyj = Sja +vj (59)

where aj is the representation vector, and Sj a N x Nj matrix of rank Kj describing the

locations of known components. At most one element in each row and each column would





81

be equal to one, and the remaining of the elements are zero. vj = LTfw is the colored
Gaussian noise vector with covariance a2LfLj.

Let's denote the temporal representation vector by a = [aT, aT, ...,a]T. L is the

M x N template signal subspace matrix, which is a block diagonal matrix given by


L1 0

L= L2

0 Lj

Then, from (59), the transformed temporal vector Z = [zT, zT, ...,T]T can be given by


z = LTy = Sa+v (60)

where S a Nx N component selection given by


S1 0

S = s2

0 S,

v is a Gaussian noise vector with covariance matrix r2Q, where Q = LTL. Based on

the previous signal model, we thus have the following two hypothesis testing problem


Ho : z = V V N(O, 2Q)
H1 : z = Sa+v v N(O, A 2Q) (61)





82

Following the approach leading to (56), it can be shown that the GLRT statistics for this

problem is given by


T -1TQ 1 H
z Q z zcT -clzc H0
t = > threshold (62)
H1
a H,
where


zc = (INV-S)z and Qc = (I-S)Q

,and IN is a Nx N identity matrix.


4.2.2 The Detection Scheme and Simulation Results

The implementation of the spatial detection scheme shown in Figure 41 is implemented

in the recursive Laguerre projection structure shown in Figure 42. The projection of the

signal to each Laguerre basis can be extracted form each tap of the recurrent network.

With the spatial extent information of targets, the GLRT can be computed simply by sum-

ming up the projection energy needed, since Q in (62) is an identity matrix for the orthog-

onal Laguerre bases.




F* 4.T spileeincm



ID down range profile U_ Transform




Figure 41. The spatial detection scheme






83



x(n) xyi-(n) xy.ik(n)
LO(z L(z (z)
()2 ()2 ()2
L(z) L(z L(z
()2 () -() -


L(z) (z) L(z




GLRT



Figure 42. GLRT implementation with recursive Laguerre Networks

We will show that the detection statistics of the proposed spatial template scheme out-

perform by 3 db over that of the ID scheme, which may reduce the false alarm rate. There

is another variable that needs to be determined which is the target extent in cross range.

Since the resolution in the cross range is 0.3 m, we would use 10 down range profiles cor-

responding to a 3 m extent in the cross range, which is reasonable for most targets. The 2D

UWB SAR image is shown in Figure 43. There is only one target in this 2D image, which

corresponds to the high intensities located around sample 3000 along the down range. The

high intensities along the down range from sample 4000 to sample 5000 are due to the

reflections of the foliage.






84




5000



4000




2000


1000


0
0 50 100
cross range


Figure 43. The original UWB SAR image


Previous work shows that the resonance responses of the two targets are within the sub-


space expanded by the first fifteen Laguerre kernels with up = 0.7. Therefore, we use a 15

dimension projection space implemented by the Laguerre networks with 15 taps and


u. = 0.7. The detection statistics of the usual GLRT based on the 1D resonance model is

shown in Figure 44.1 and Figure 44.2. The algorithm is able to determine the targets

around sample 3000 along the down range from clutter. However, there is a false alarm

with detection statistics as high as the target.


The detection statistics of the usual GLRT based on the spatial resonance template

model, with the Laguerre recurrent networks, is shown in Figure 45.1 and Figure 45.2

Comparing the detection statistics of target in Figure 45.1 with that of the foliage in Figure

45.2, we can see that the target's peak detection statistics is 3-db higher than the peak sta-

tistics due to the foliage.






85













:?10 0 o10







Figure 44.1 The detection statistics based on 1D resonance Figure 44.2 The detection statistics based on 1D resonanc
model along down range 1801-3600 model along down range 3601-5400



Figure 44. The detection statistics based on ID resonance model





x~o .l01S 1.o*11'



1 00










Figure 45.1 The detection statistics based on 2D resonance Figure 45.2 The detection statistics based on 2D resonance
model along down range 1801-3600 model along down range 3601-5400



Figure 45. The detection statistics based on spatial resonance template model





86

4.3 Subspace Detectors Extended with Data-Driven Templates

In the previous two sections, the ID and 2D subspace detectors are presented. We

showed that there is a chance to improve discriminability of targets if the right signal sub-

space is chosen appropriately. The question is how to find the most appropriate signal sub-

space for a collection of targets. In the proposed subspace detectors, there exist unknown

template matrices such as S in (56) to describe the energy distribution in the subspace for

each target in the data set. It's implied by the previous transient signal detection algo-

rithms that the we can use the "1" component in the matrix S to indicate if the correspond-

ing axis extracts the target's projection energy or not. It is a rough representation since a

hard-limit decision instead of soft decision is used. But most importantly, a procedure to

find the relevant basis is not easy due to the variability of responses among the targets.

This section demonstrates how the subspace detector can be derived by incorporating a

"soft decision" template matrix, where the elements are rational numbers between 0 and 1,

and that corresponds to the target subspace. A data-driven approach is proposed to build

the parameters and template matrix S. In the end, we will compare the performance of all

the proposed detectors in terms of their receiver operational characteristics (ROC) curves.


4.3.1 Formulation of the Weighted Subspace Detector

Let's assume that the ideal resonance response y belongs to a known M-dimensional

orthogonal signal subspace represented by a N x M matrix, L, i.e. y = La + n, where a

is the representation vector. Suppose y concentrates on only m out of the M bases. Let

the nonzero representation vector be denoted by az. Then it can be given by aNz = Sa,






87

where S is a Nx N diagonal matrix of rank m indicating the locations of known com-

ponents. Then the measurement model can be written as


Yr = LaNZ+n
= LSa+n

If we apply a linear transform L T to y, then we get


z = LTy = a+v v = LTn

Comparing with the derivation of (56), the testing statistics for this problem can be simpli-

fied as


TSZ H
zt Z >l
t ^ < T (63)
CT Ho


where T is the threshold for tr, and a can be estimated from the neighboring Nu x 1 sam-

ple vector u by


2 = (uru)/Nu.

Basically, (63) utilizes the information in S, and we don't assume any apriori informa-

tion about how targets are represented in the signal subspace. To enhance the performance

of the detector, we propose to apply a deterministic M x M diagonal weighting matrix

W = diag([wo, ..., WM_ ]) with I WIJ = 1 and wo, ..., WM_1 > 0 to the transform rep-

resentation q, so that now the new representation can be written as


l1/2Z = WI/2LTy = W1/2a + Wv1/2






88

Then the GLRT becomes


T HI
t z Wz >
t- 2- < T (64)
H0
SH2

The new test is still a CFAR test with F distribution. In practice, a can be absorbed into

the threshold and the test just becomes tr = Z Wz. We can view the proposed W as an

energy template in the subspace, which "best" matches the distribution of the energy of

the different targets' resonance response in the transform domain.


4.3.2 Subspace Parameter Tuning and Template Finding

All the subspace methods like Laguerre functions, Gabor Transforms or Short Time

Fourier Transforms, have some parameters to control the bases. For a single exemplar we

showed in 4.1.3 how this can be accomplished. However, for all the targets of interests,

one has to take a more systematic approach to find a single "signal" subspace. As sug-

gested in (64), the subspace detector may be enhanced by imposing a weighting template

for targets to provide good discriminant capabilities from clutter. To incorporate the apri-

ori target information, a data driven approach is naturally chosen to find the parameters for

the signal subspace and the weighting template. Since the Laguerre functions are used to

model the resonance response, we will focus on reliably estimating the feedback parame-

ter t for the Laguerre subspace and the associated template matrix W adaptively through

the data set.

Suppose we have K different target down range profiles y(1), ...,y(K), and the Jpeak


in (57) for each profile is denoted as Jpeak, y(k) The subspace detector compares the

energy captured by the modeled subspace to that in the null space, so the goal is to build






89


the Laguerre signal subspace parameterized by pt to provide the projection as large as pos-

sible for the target data set. Therefore, we propose the following criteria to compare the

detectability of targets:




Jo = argmin Jpeaky(k)/K (65)
I k=0


Then the feedback parameter Ip is searched to maximize the proposed criteria. For 450

down range profiles from 15 targets in the training data set (See 6.1.3), the Jo is mini-

mized by using pt in the 15 Laguerre kernels of the Laguerre function space. The result is

shown in Figure 46, and one can say that the value of 0.7 for p can lead to the most satis-

factory results.









o. C CI
.0






0o.o o.z 0..-- o0.0 0. 1 .o
p.


Figure 46. The performance indices Jo with respect to the feedback parameter p.


After a reasonable value of p is obtained, we still need to select the corresponding sub-

space representation z(1), ..., z(K). The goal of the template matrix W is to choose the

subspace where most of the energy of the training targets lies. Here we utilized again the

ideas of PCA. We would like to find the best direction of the target clusters in the sub-






90

space, which is given by the largest eigenvector. Therefore the template vector

w = [wo, ...- WM_ ]T is again proposed to be the eigenvector for the data set

z( 1), ..., z(K) The above process can be illustrated by Figure 47.


Laguerre Subspace *
Projection
y(2) I z(2)
y() z(l)






PCA analysis for
finding the template


maximizing Jo for tuning
the feedback parameter


Figure 47. Laguerre subspace parameter tuning and template finding scheme

Finally, we would come up with a weighted Laguerre subspace detector scheme shown

in Fig.48.





91



qo L(z) f 92 gK






WO Wi W W2
w0i








Figure 48. The Weighted Subspace Detector


4.3.3 Simulation Results

We will show the performance of three proposed detection schemes: ID subspace detec-

tor, 2D subspace detector, and the weighted 1D subspace detector in terms of the receiver

operation characteristics (ROC) curves. The UWB SAR images (about 2 (km)2) for the

simulation contains 50 targets embedded in the clutter. For all the schemes, the Laguerre

space with the first 15 order kernels and feedback parameter i = 0.7 is used as the signal

subspace to capture the resonance response. For the 1D subspace detector, only a single

down range file is used, while 10 down range profiles are used at the same time as the tem-

plate for the 2D subspace detector. The results of the three detection schemes are shown in

Fig.49.






92



1.0
Weighted Laguerre
.0. Subspace Detector
S2D Laguerre
.id Subspace Detector
Pd D Laguerre
Subspace Detector
0.4

0.2 -2



FA/km2


Figure 49. The ROC curves of subspace detectors


The weighted Laguerre subspace detector has the best performance (5010 false alarms

for Pd = 1), while the ID Laguerre subspace detector performs worst (25368 false


alarms for Pd = 1). Although the 2D detection scheme utilizes more information in the


cross range than the ID scheme, the performance enhancement is marginal. The resonance

response theoretically exists along several down range profiles but apparently the driven

response of trees also enhances comparably the clutter detection statistics. The results sug-

gest that the more down range profiles in the 2D scheme only provide the merit of statis-

tics averaging. However, the detection performance is largely improved by using the

weighting mask in the subspace energy domain, especially in the operating range where

Pd is close to 1 as shown by the ROC curve of the weighted Laguerre subspace detector.

This result suggests us that the better discriminant power between targets and clutter, espe-

cially the clutter with severe interference, comes from the weighting in the Laguerre

power spectrum.






93


The weighted subspace detector can be viewed as a match filtering of the signal power

spectrum in the Laguerre space. To get a better insight of the functionality of the scheme,

we alternatively analyze the weighed Laguerre subspace detector in the Fourier domain.

Since the focus is the subspace energy, the Fourier power spectrum of the "match filter" is

investigated. The frequency response of the weighed Laguerre subspace detector is shown

in Figure 50-(a), and the power spectrum of the 256 data samples of the vehicle beginning

at 654 in Figure 33 is shown in Figure 50-(b). Figure 50 suggests that the weighted sub-

space detector matches the energy in the target resonance response as we would expect

from the largest eigenfilter response.






,. o. x 8
1 S0 1 0O








(a) the frequency response of the weighted Laguerre subspace detector






=. Ox,1




0 S =1 00 1 SO

(b) the frequency response of the data samples of a vehicle


Figure 50. The frequency response of the weighted Laguerre subspace detector
and the signal




Full Text
7
(5) is the well known Fresnel approximation-based inversion for strip-mode SAR imag
ing, and the imaging algorithm is computationally efficient since only the 2D Fourier
transform is involved [Ausherman et al, 1984].
1.4 Motivation of UWB SAR: Resonance Effects
A fully polarimetric Ultra WideBand UWB foliage-penetrating (FOPEN) SAR was
recently designed mainly for the purpose of exploiting the capabilities to locate and recog
nize targets embedded in foliage. The UWB system is considered to be ultra wideband
because of its high bandwidth occupancy. Lets define the relative bandwidth as the ratio
of bandwidth to center frequency
A/
(fhigh +floWV2
(6)
A sensor is categorically wideband if its relative bandwidth is 0.1 < Q < 0.5, and ultra-
wideband if its relative bandwidth is 0.5 < Q <2.
In the narrowband case, a targets echo is typically modeled as a scalar number a, the
radar cross section (RCS) of the target. In general, however, RCS is a function of wave
length, phase, aspect angle and polarization state. Wideband analysis thus motivates a
revised definition of a as a complex quantity, with both magnitude and phase compo
nents. In the time domain, the complex RCS can be represented as a ringing or resonant
response of the target [Morgan and Larison, 1991]. If this resonant signature is sufficiently
unique, the target can be modeled and recognized by analysis of its resonant response. The
typical ID responses of a dihedral and a tree are shown in Figure 3 and Figure 4, respec
tively.


12
deconvolved to the single point, where the exciting waveform is assumed to be an impulse
function. Later it was reformulated and applied to SAR image processing by Soumekh
[Soumekh, 1990]. The flowchart of the algorithm is shown in Figure 6. Since this algo
rithm uses more complicated plane wave approximation than Fresnel approximation, it
also can be applied for the SAR imaging operating in the stripmap mode at the cost of
more computations.
f(x,y)
1
H(x, y)
s(u, t)
1
1
, 2D FFT
2D FFT
1
1 2D FFT kku,(o)
i
1
t
^ Interpolation
F(kx, ky) .
H(kx,ky)
S(kx, ky)
Figure 5. 2D filtering/deconvolution for UWB SAR image focusing
Figure 6. Flow Chart of the FFT Based SAR Focusing Algorithm


demonstrated that the first-order Gamma kernel is a good approximation for the principal
eigenvector of the projected radial intensity of targets, which provides the optimal match
ing intensity kernel. This also explains the better performance of the y-CFAR detector
over the two parameter CFAR detector.
We also developed different CFAR subspace detectors for UWB images, utilizing a
Laguerre function subspace. The driven response produced by natural clutter degrades the
performance of these subspace detectors. In addition to the driven response, the distin
guishing feature of metallic targets in UWB is the resonance response. Therefore, we fur
ther propose a two-stage detection scheme: y-CFAR detector followed by the quadratic
Laguerre discriminator (QLD). We evaluate every detector and discriminator using ROC
curves in a large area (about 2 km1) of imagery. The combined y-CFAR and quadratic
Laguerre discriminator improve the simple Laguerre subspace detector more than one
hundred fold for a perfect detection rate (Pd = 1),.
Vll


Mutiplying 1/a2 on both sides of (85) yields
*-i rx-m^2 N-i
i = 0
2 M ^
i = 0
JC W'v2
At-1
i = 0
Writing (86) in the vector form leads to
/icr[*Jt]/a2 = /icr[vv]-(/icrv)


77
energy of the resonance response. The detection statistics of the proposed GLRT using the
full 15 bases is shown in Figure 39. This preliminary experiment shows that with the cho
sen parameters, the algorithm is able to discriminate the vehicle and power lines from nat
ural clutters with a large margin, but of course this is a single case of each object.
O 1 OOO 2000 3000 4-000 5000 6000
Figure 39. The detection statistics corresponding to the ID UWB SAR down
range profile in Figure 33
Next, we will try to demonstrate the effect of using the appropriate signal subspace for
detection. Taking the distribution of the foliage in Figure 37-(b) into consideration, we can
intuitively argue that the signal space for the three object types can be constructed from
three different combination of the Laguerre bases. The Laguerre kernels of the first four
Laguerre bases can be used for the man-made clutter, power lines, since the subspace
energy in these bases is dominant. It is equivalent to partitioning the full space into the sig
nal subspace and non-signal subspace, i.e., the 15 x 15 partition matrix S in (53) can be
set as
if i = j = 1,2, 3,4
if i =j = 5,15
Based on the same reasoning, the Laguerre kernels of order k = 5, 6, 7, 8, 9 can be
used to construct the signal subspace for the vehicle while the next five order Laguerre


114
Figure 63. The optimal discriminant functions for the two class data samples with
Gaussian distribution (a) The two class data samples have the same
Gaussian distribution (b) The two class data samples have Gaussian
distributions with unequal covariance matrices
In the simpler case that the target and clutter data samples have unequal diagonal covari
ance matrices £, = of I and Ec = ofl, where of of, the optimal discriminant func
tion can be simplified as [Duda and Hart, 1973]
M M
1 = Z qmu2m + X WmUm + c (74)
m = 1 m = 1
where wm and qm are the associated weights that need to be adjusted. Inspired by these
observations and the excellent results of applying quadratic discriminant functions in
[Principe et al, 1998b], we propose to use (74) as the quadratic Laguerre discriminator
(QLD) where um for 1 < m < M is the component of the Laguerre subspace energy vec
tor u = [uu ..., uM]7 as described earlier.
Notice that the simplicity of implementation of any proposed detection scheme is a
important concern. That is the reason why we choose (74) over (73) to be the discrimina
tor. QLD also has the same form of the so-called Quadratic detector [Picinbono and
Duvaut, 1988], where qm and wm have different solutions under different optimization


126
by imposing a spatial template. For a simple example, the detector outputs where higher
and more distinguished from clutter.
The unsolved issue with the Laguerre subspace detector is to find the signal subspace for
best detectability. I propose a weighted Laguerre subspace detector which includes an
energy pattern template to enhance the discriminability for the signal subspace. The ideas
of PCA are utilized again to find the largest eigenvector of targets in the subspace, which
is then used as the weighting template to determine the signal subspace. Hence, the
weighted Laguerre subspace detector can be interpreted as a maximum eigenfilter for the
target subspace, and an improvement in performance is expected.
The proposed three subspace detectors were tested in the UWB data covering 2 {km)2.
Although the weighted Laguerre subspace detector has the best performance as expected,
it is far from appropriate because it still creates too many false alarms (around 5000 false
alarms for P 1). The performance of all the subspace detectors are degraded due to
the large energy in the driven response of natural clutter. To improve detectability, the
detection scheme has to avoid the impulsive response, while exploiting the resonance
response of metallic objects.
The large energy contained in the driven response always precedes the resonance for tar
gets. Therefore it is reasonable to detect the driven response first by applying an intensity
based detector such as y-CFAR test which has been shown excellent performance for
MMW SAR. Then a discriminator for the resonance response is applied to reduce the false
alarms.


34
Notice that the CFAR detector is formulated to test every single pixel in the SAR image
separately, apparently discarding all the spatial information about targets point scatters.
MMW SAR technology provide us with better azimuth resolution of small targets, so that
better 2D target signature information is contained in the SAR images. To utilize the spa
tial information of targets, Kelly [Kelly, 1989] first proposed that each target in the image
can be described by a completely known template or signature with an unknown scalar
gain. In a similar approach, Li [Li and Zelino, 1996] proposed a detector with targets tem
plates for the SAR images, where the templates can be represented by binary matrices
with 1 elements to represent the point scatter locations. One of the problems with target
detection in SAR images is that targets would have different reflection patterns due to dif
ferent poses. Theoretically, a detector can work pretty well by incorporating all the tem
plate information for targets due to pose variation. Practically, the detection system would
become too complicated to be implemented.
The following notation is used throughout this chapter: a 2D image data matrix is
denoted by a boldface capital letter such as X, and a ID data vector by a boldface lower
case letter such as x. Since we are working with image chips, without loss of generality,
all the data matrices will be assumed of finite region of support defined by C1N =
{(i,j) | N< i,j dimension (2N + 1) x (IN + 1) with the origin in its center.
The y-CFAR detector [Kim and Principe, 1996] was proposed to generalize the two
parameter CFAR detector utilizing circularly symmetric Gamma kernels Gk to extract the


59
Target type
Series No.
Smallest SNR
improvement
(db)
Largest SNR
improvement
(db)
Avg SNR
improvement
(db)
BMP
sn_9563
3.7
6.8
5.1
BMP
sn_9566
4.0
6.4
5.0
BMP
sn_c21
4.0
7.0
5.0
BTR70
sn_c71
3.0
6.6
4.5
T72
sn_132
3.3
6.3
4.6
T72
sn_812
3.2
6.5
4.6
T72
sn_s7
3.2
7.0
4.8
Tablis24
Mission 90
1.2
10.8
5.3
Table 1. The SNR improvement for different data set
Target type
Series No.
Smallest Detection
Statistics improved
Largest Detection
Statistics improved
Avg Detection
Statistics improved
BMP
sn_9563
3.1
8.2
4.9
BMP
sn_9566
3.0
6.8
4.4
BMP
sn_c21
3.1
9.0
4.6
BTR70
sn_c71
1.9
6.6
3.4
T72
sn_l 32
2.7
6.6
4.3
T72
sn_812
2.6
8.1
4.3
T72
sn_s7
2.5
8.3
4.6
Tablis24
Mission 90
1.5
10.8
7.8
Table 2. The detection statistics improvement for different data set


123
We were expecting an improvement, but not as much improvement as experimentally
verified. In fact the false alarms for a Pd=l were reduced from 3,000 to 100 (one order
magnitude reduction). This is our best false alarm result for target detection in UWB, and
although higher, it is in the same range of performance of MMW SAR discrimination
algorithms. This gives us hope that our methodology can lead to realistic ATR systems
using UWB SAR.


118
(1)forming the residual
k-1 M
= y xj1^ ^ xj'-1) xj') = aj'Tsjfi (77)
7=1 7=*+i
We can think of this step as a kind of deflation of y to zft, which is only composed of
44-
(2)To find the ML estimate of (A.,/), the following criteria is maximized
(A,/) = ax^max yS[SHS]~l SHy = av^max ^s(X,j)
Accordingly, the parameters (A¡,f¡) can be found now by two ID searches of DFT in the
(A¡,fi) plane of E^ ^, which now can be considered to contain only single component
aksk and can be written as:
N~' p(.~h +jW\)np(-X+j2nJ)n
N-\
e;:
-2 Xn
(78)
For a fixed value of A, say A = X, E^ is only a function of and thus E^ ^ has
a ridge that runs parallel to the A axis at / = f¡ in the (A,f) plane. Hence we can find the
value of f so that L is maximized, and, Then, the value of A can be used in (79) to find the
N-1
E,:
,(-X.| +X)n
^S(X¡,f)
N-1
e;:
-2 Xn
(79)
(3)estimate a and update the subspace
1 = [4,4 4,a¡j-]T
(80)


30
In the signal detection literature, hypothesis H0 is usually called the null hypothesis, and
H, is called the alternative hypothesis.
A test for hypothesis Hj against H0 may be specified as a partition of the sample space
S = Rn of observations into disjoint subsets Sj and S(), so that x falling in 5, leads to
acceptance of Hj, with H0 accepted otherwise. This may be expressed by a test function
t(x), which is defined to have value t(x) = 1 for x g 5j and value e t(x) = 0 for
x e S0. Lets use the following denotations:
^(0) = j/(x)/x(x|H0)i/x = Js/(x)/x(x|0,0 g Q0)dx
PD(V = j5'(x)/x(xlHiMx = s/(x)/x(x|0,0G 0])i/x
, where PF is called the probability of a false alarm; PD is the probability of a detection.
Furthermore, we would use PM = 1 PD as the probability of a miss. The false alarm
and detection probability may be conveniently combined as the power function of the
detector as follows:
Pd(Q) if 0 g 0,
P(Q) = \
lPF(0) if 0 G 0O
Without loss of generality, lets assume, for each subset 0, and 0O, there is only one
single parameter 0, and 0O, respectively. Our goal is to design a test, such that PD is
maximized (or PM is minimized), under the constraint that PF = a Lagrange multipliers, the cost function can be defined as:
J P m+^[Pf ct]
= f /(xIHj^x + aIJ /(x|H0)x a
L's,
(23)


37
detector with respect to the parameters p for g¡ org15 for a training data set is shown in
Figure 18 [Principe et al, 1998b], It can be recognized that parameter values of p for g{
or £]5 is very critical to the detectors performance. Forseeably, each target would require
a different value of pj or p]5, but this was not done in the previous work, where the p
values were optimized over the training set. That argument is related to the two unsolved
problems for y-CFAR detector: First, how close to the optimum is the Gamma kernel g, ?
Secondly, is the learned size of the guard-band for gc good enough for the local statistics
estimation?
The first question must be answered by formulating the detector using the GLRT to
investigate what the functionality of gl is, and then we can further pursue the best possi
ble candidate for g,. The guard-band size is a trade-off between two extremes. In one
side, the guard-band size needs to be as small as possible so that the estimated statistics
would be local to the target. On the other hand, the guard-band size should be chosen
large enough so that the estimation in the guard-band would be independent of the tar
gets. In the previous CFAR detection approaches proposed by MIT Lincoln Lab, the
guard-band size is fixed at 80x80, which is about a 20Mx20M area to include the largest
size target supposed to be detected. However, that fixed guard-band size is not suitable for
all the targets. As shown in Figure 19, the guard-band size is much larger than these two
targets near the noisy clutter environment, and the estimated mean or variance would be
increased because many unnecessary clutter pixels are included. This larger guard-band
size would leads to a smaller y -CFAR testing statistics, and a probable missed detection.


36
the similar size of the box-like guard-band proposed by MIT lincoln Laboratory for an
intermediate value of p.
(a) y-CFAR stencil (b) CFAR stencil
Figure 16. The 3D windowing function used by y-CFAR detector and CFAR
detector.
(a) y-CFAR region of support (b) CFAR region of support
Figure 17. The 2D regions of support associated with the stencils used by y -CFAR
detector and CFAR detector.
Although y -CFAR has a decent performance, the performance is affected by the param
eters p for g] and g]5. The parameter values for maximal detectability are found through
training on the collected target and clutter chips. The performance surface of the y -CFAR


BIOGRAPHY SKETCH
Mr. Li-Kang Yen was bom September 20,1967. He earned his bachelors degree in elec
trical engineering from the National Sun-Yet-Sen University, Taiwan, in 1989. Since 1994,
he pursued his Ph. D. degree in the Computational NeuroEngineering Laboratory at the
University of Florida, during which time he is focused (Ph.D. topic) on applying adaptive
signal processing to target detection in mm-wave SAR imagery. He was also a graduate
research assistant in the Electronic Communications Laboratory at the University of Flor
ida from 1996 until 1998, during which time he conducted researches in the areas of SAR
image migration and adaptive target detection.
142


20
GJz)
x{ri) o
\G^z)
xx(n) x2{n)/
t
xk()
Figure 10. The recursive ID Gamma filter structure
The extended 2D Gamma kernels is a circularly symmetric version of the ID continuous
Gamma kernels, and the A>th order Gamma kernel gk[i,j] is given by [Principe et al,
1998b]
gk[Uj] ~ 7(aA2+./2/ exp(pV/2+72)
2nk\
where p is the parameter to control the scale of the kernel. The waveforms of different
order 2D circularly symmetric Gamma kernels are shown in Figure 11.
Figure 11. the 2D Circularly Symmetric Gamma kernels of different order k=l, 4,
11,21 in the discrete time domain for p = 0.8.


70
obvious that our statistics can differentiate the three embedded Laguerre transients even
when occurring in nonstationary noise. However, notice that the signals belong to the
basis set which represent the best performance situation.
Next, we will deal with the ID UWB SAR down range profile shown in Figure 33,
where three types of objects exist. The samples from 550 to 750 correspond to a power
line, the samples from 3150 to 3350 correspond to a vehicle target, and the samples from
samples from 4000 to 4200 correspond to tree reflections. Note that in terms of SNR, this
situation is more benign than the synthetic example since most of the returns are of lower
amplitudes. However, now the target is only approximately modeled by Laguerre func
tions as shown in Chapter 2, and more importantly there are high returns of trees which do
not belong to targets.
Figure 33. The ID UWB SAR signal
To design a signal subspace constructed for Laguerre bases, we will first investigate how
the UWB transient signal is represented by the parametric Laguerre space. There are two
basic questions: what are the bases to be used and what is best value of p? Intuitively, the
JL(n) in (21) may be examined along the ID UWB signal samples for the optimization of


104
ral detection scheme has better discriminant power and reduced the false alarms to 875.
However, from the overall detection system performance point of view, this result is still
unsatisfactory.
Number of delays
Figure 55. The number of false alarms with response to the number of delays in
the temporal detection scheme
Pd o s -
YMkrn
the discriminant power of
temporal detection scheme
1D y CFAR detector
Figure 56. The ROC curve of the temporal detection scheme


125
well approximated by the first order Gamma kernel. Therefore, I conclude that what the
y -CFAR detector actually does is maximum eigenfiltering, resulting in better performance
when compared with the delta function stencil proposed by MIT/Lincoln Laboratories.
This fact was experimentally verified by [Kim and Principe, 1996], but no justification was
available.
Modeling the radial intensity of targets with the first Gamma kernel leads to a new very
efficient way of computing the guard band size for each target. Although this improve
ment was not thoroughly tested in large data sets, it has the potential to improve the false
alarm rate of the y -CFAR detector. So, my contribution to the y -CFAR detector was both
a better understanding of the detector and modification to improve its performance.
Inspired by the success of applying the GLRT approach to design the detector in MMW
SAR, we continue to follow this theoretical approach to design detectors for the transient
resonance response, one of the key features for targets in UWB. Due to the preliminary
evidence established in Chapter 2 that Laguerre basis represent well the resonance
response of targets, our detection scheme is centered around Laguerre kernels, with the
added advantage of fast computation. This is also a contribution of my work. I compared
the Laguerre bases with the Gabor decompositions and found that the Laguerre bases lead
to better performance.
A ID Laguerre subspace detector is first formulated, and it works for nonstationary
noise. Through the analysis of the responses to natural clutter, man-made clutter and a tar
get, we found that subspace projection did enhance discriminability of targets from man
made clutter and natural clutter. To integrate the cross-range information provided by SAR
for better detectability, the Laguerre subspace detector is also extended to the 2D scheme


65
signal is also assumed to be corrupted by additive white Gaussian noise. Hence, after sam
pling, the final model could be described as:
y(Q) = Ha(Q) + e(0) + w
where w is the N x 1 noise vector, whose elements are assumed to be mutually uncorre
lated Gaussian random variables. If we apply any linear transformation to the resonance
response signal y by a M x N matrix L, then
z = Ly = LHa + Le + Lw
When L is chosen as the left inverse of H so that L H \M where I is a Mx M iden
tity matrix, then we get
z = a + Le + Lw
(51)
= a + Lw
where w = e + w is still assumed to be white Gaussian noise, with zero mean and
unknown variance a2. We assume that there are only K nonzero projections for the repre
sentation vector a, and that those K component locations are known, but their values are
unknown. Let the nonzero representation vector be denoted by aAZ. Then it can be com
puted by
a = Sa (52)
where S isa NxN diagonal matrix of rank R describing the locations of known com
ponents. That means
if i j known component location
elsewhere
(53)


4
Figure 1. Pictorial View of SAR image formation
Taking the Fourier transform of both sides of (1) with respect to t results in
5(u, oo)/P(co) = |J/(x,y)exp[-j2kJ(Xl-x)2 + (7, +u-y)2]dxdy (2)
The system model in (2) represents a 2D filtering process illustrated in Figure 2. The input
signal is f(x, y) the original reflectivity function, which passes through the 2D filter with
the impulse response H(x,y) = exp[j2kjx2 + y2] to generate the output signal
s(u, co) = s(u, co)/P(a) measured at (x,y) = (X¡, Y{ + u).


Ill
An alternative worthwhile exploring for simplicity is to implement a 1D y -CFAR test on
every down range profile. We propose the ID y-CFAR detector, which can be explicitly
given by
tyCFAR ~
(g,*x-gc*x)
Jgc x2- (gc x)2
(71)
where means inner product, x is the N x 1 ID down range data vector, and gt as well
as gc belongs to the ID Gamma kernels gk = [gy.(O),..., gk(N 1)]. Here the local sta
tistics is also estimated by the ID Gamma kernel gc so that we can take advantage of the
recursive Gamma filtering shown in Figure 61 to reduce the computation complexity. Like
the 2D case, the ID gc can be put ahead of the ID gt to implement the front guard-band
stencil as shown in Figure 61.
¡GQQ
-x?
&*/ 8*'X
ID recursive Gamma filtering
1D y-CFAR stencil
Figure 61. ID recursive Gamma filtering and 1D y-CFAR front guard-band kernel
The ROC curves in Figure 62 show that the ID y -CFAR detector with front guard-band
kernel has the better performance in our simulation. Furthermore, compared with Figure
60, Figure 62 also suggests that the ID y-CFAR detector with the ID front guard-band
stencil is more powerful than 2D y -CFAR detector with 2D front guard-band stencil.


83
xdn)
) | Xy_l()|
Lo(z:
()2
L(z)
?
02
L(z)
Lh
02
L(z
1

02
1
'
L(z)
Hj-n)I

L(z)
()2
LoO|
j[7)2|n
L(z)
... THUHi
|l(z:
0i
GLRT
Figure 42. GLRT implementation with recursive Laguerre Networks
We will show that the detection statistics of the proposed spatial template scheme out
perform by 3 db over that of the ID scheme, which may reduce the false alarm rate. There
is another variable that needs to be determined which is the target extent in cross range.
Since the resolution in the cross range is 0.3 m, we would use 10 down range profiles cor
responding to a 3 m extent in the cross range, which is reasonable for most targets. The 2D
UWB SAR image is shown in Figure 43. There is only one target in this 2D image, which
corresponds to the high intensities located around sample 3000 along the down range. The
high intensities along the down range from sample 4000 to sample 5000 are due to the
reflections of the foliage.


103
tion is totally suppressed, and that there are more detections corresponding to the accurate
target locations.
target
(b) The detection statistics of clutter
Figure 54. The detection statistics based on the fused temporal detectors d (a) The
detection statistics around the target along down range 1801 ~ 3600 (b)
the detection statistics of clutter along down range
Next, we apply the proposed temporal detection scheme as a discriminator following the
ID y-CFAR detector. The ID y-CFAR detector will be discussed in detail in 6.1.1, and
here we just show the ROC curve of ID y-CFAR detector in Figure 56. The same training
data set described in 6.1.3 is used for our proposed temporal detection scheme, where the
number of the taps has to be first determined.
Under 100% detection rate for the training data set, we can get the number of false
alarms with respect to the number of delays in the temporal detection scheme as shown in
Figure 55. Using 30 delay taps resulted in the least false alarms. Fixing the number of
delays at 30, the ROC curve of the temporal detection scheme following the ID y-CFAR
detector is shown in Figure 56. It can be observed that for the training data set the tempo-


129
7.2 Future Work
In this dissertation, we have explored different stencils for local intensity tests, and com
pared both linear and quadratic discriminant functions to implement detection tests. There
are many possible ways of improving the state of the art in this area. Here I will only pro
vide a few directions that are directly coupled with my work.
The issue of stencil design is critical for local intensity tests, and the best way to analyze
it is to think in terms of projection spaces. The Gamma and Laguerre spaces are an inter
esting possibility due to the fast computation, but multi-resolution decompositions are an
appealing alternative. They should be compared with the present detectors.
Moreover, we restricted our study to only two stencil possibilities: 1-D stencils and radi
ally symmetric 2D stencils. Information about texture in the image is therefore largely lost
in both of these stencils. It is well known that texture conveys information about the back
ground, so it should be utilized at the front end detection. The challenge is that texture is
difficult (and time consuming) to quantify, so designing stencils that will be able to help in
the analysis is a worthwhile pursuit. But with texture information it may be possible to
configure the detector with different thresholds for targets in the open or in trees, which is
a big problem in practical SAR applications.
The second aspect that should be further investigated is the comparison of discriminant
functions. Results show that quadratic discriminant functions perform better than linear
discriminants, but in reality the quadratic is optimal only for Gaussian distributed data. In
general the local intensity of targets and clutter are only approximately Gaussian distrib
uted, so more alternatives should be investigated. Here I propose the use of artificial neu
ral networks (ANNs) because they are able to create arbitrary discriminant functions.


67
Following [Friedlander and Porat, 1989], we can show that the GLRT statistics for this
problem becomes
T 1 T -1
z Q z zc Qc zc
* 2
a
(55)
where
zc = ( l^)z and Qc = (I^fi
1^ is a N x N identity matrix. In our UWB SAR scenario, we assume a2 unknown. It can
be estimated by
a2 = (1 /p)uT u
from the neighboring p x 1 sample vector u, which are assumed to be independent of the
testing sample. So, our GLRT test statistics is given by
t =
z Q z zc Qc
T
u u
H0
> threshold
H,
(56)
It can be shown (see Appendix A.2) that the numerator in (56) is a chi-square distributed
random variable with K degrees of freedom, and that the testing statistics tis a random
variable with F-distribution. The numerator in (56) is the energy contained in the whole
space subtracted by that contained in the null space. Thus, it can be viewed as the energy
in the signal space. So, the testing statistics uses the ratio of the energy contained in the
signal space to the estimated background noise energy to decide if the return is a target or
not.


116
where / = argmax{ tycfar(x(W }>and /j, /2, /3 and /4 are user defined ranges.
In our experiments /] = 10, /2 = 40, /3 = 20 and l2 = 50are used. As to the desired
signal for the data vectors x(i) corresponding to i < ipeak and i k + / interpolate the desire signal, so that we would come up with the whole desired signal, for
example, shown in Figure 64.
Figure 64. ID down-range target profile and its corresponding desired signal
Next, we would discuss how to find the weight qm and wm of the quadratic Laguerre
discriminator through temporal processing. If we denote the data vector by
y(i) = [y(0, yU+ l),..., y(i + N1 )] 7 at time index i, the corresponding desired signal by
d(i), and the corresponding Laguerre subspace energy vector by
u(i) = [i(i'), ..., uM(i)]r. Using the LMS approach [Haykin, 1991], the instantaneous
weight qm and wm can be updated by
0'+ 0 = 9m(0 + Tl m(0(^(0-(0)
(76)
wm(i'+l) = wm(i) + r\ um(i)(d(i)-t(i))
In our experiments, with r\ = 0.05, the training converges within 50 batch iterations.


64
4.1.1 The 1D Resonance Response Model
The resonance response, denoted by x, is assumed to be a length N vector belonging to
a l2 finite-dimensional linear space of discrete functions. The space is the so-called signal
space, and it can be represented by H, where H is a TV x M matrix. That means the col
umn vectors {hm, 1 < m < M} of H are the basis vectors of the signal space. A specific
instance of x can be expressed as
x = Ha
where a is the Mx 1 projection vector. The signal subspace is application dependent.
Wavelets, STFT or Gabor transforms are all possible choices. If the bases are chosen
according to the resonance response shape, we usually only need K out of the M bases to
constitute our signals, so only K components in a would be nonzero. Ideally, we would
like a sparse implementation, in the sense that K will be much smaller than M (KM).
That means the bases should be matched to the signal.
This model can be generalized by letting the signal space be dependent upon some
parameter vector 0, such as the scale parameter p in Gamma or Laguerre kernels. The
vector may contain the waveform-shape parameters. Real-life signals will seldom obey
this model, so some deviation from it should be included. Therefore, we will modify the
above model as
x(0) = Ha(Q) + e(0)
where e(0) is the Ax 1 mismatch signal vector. By definition, this mismatch signal is
orthogonal to the signal subspace spanned by H. It is assumed that the energy contained
in e(0) is small with respect to the total signal energy. In addition, the received transient


112
Pd
1 O 1 CD 1 O 1 O 1 O
FA/£m2
ID yCFAR with front guard-band kernel
ID yCFAR with rear guard-band kernel
Figure 62. ROC comparison of 1-D g-CFAR processing with front and
following clutter kernels (hilbert envelope data).
We do not fully understand this result since the 2D stencil should have increased the
robustness of the test. The shape of the 2D stencil may have to be further fine tuned.
Instead of covering 180 angle, the stencil should be a wedge of 120 degrees.
6.1.2 Quadratic Laguerre Discriminator (OLD)
We would try to extend the weighted Laguerre subspace detector to the discriminator
needed for UWB SAR images. First, let us briefly review the difference between the
GLRT and a quadratic discriminantor. Suppose we have a TV x 1 data vector denoted by y.
Applying a M-dimensional Laguerre signal subspace matrix represented by a N x M
matrix, L, we can get the corresponding Laguerre representation vector by
z = [zu .zM]r. Then weighted Laguerre subspace detector in (64) be written as


82
Following the approach leading to (56), it can be shown that the GLRT statistics for this
problem is given by
where
t =
Tr\~x
z Q z
ztQc
a2
H0
> threshold
H,
(62)
Zc = (IArS)Z and QC = (IyyS)Q
, and IN is a N x N identity matrix.
4.2.2 The Detection Scheme and Simulation Results
The implementation of the spatial detection scheme shown in Figure 41 is implemented
in the recursive Laguerre projection structure shown in Figure 42. The projection of the
signal to each Laguerre basis can be extracted form each tap of the recurrent network.
With the spatial extent information of targets, the GLRT can be computed simply by sum
ming up the projection energy needed, since Q in (62) is an identity matrix for the orthog
onal Laguerre bases.
L,
-
11
ID down range profile
Figure 41. The spatial detection scheme


90
space, which is given by the largest eigenvector. Therefore the template vector
w = [w0, ...,wM_]]r is again proposed to be the eigenvector for the data set
z{ 1), ..z{K). The above process can be illustrated by Figure 47.
Laguerre Subspace
Projection
T(2)
J(l)
I
T
T
maximizing J0 for tuning
the feedback parameter
*( 2)
2(1)
- 2(0)
PCA analysis for
finding the template
Figure 47. Laguerre subspace parameter tuning and template finding scheme
Finally, we would come up with a weighted Laguerre subspace detector scheme shown
in Fig.48.


13
1.6 SAR Image Data Sets
In the experiments through this thesis, there are two different types of SAR image data
sets for simulation: the MMW Moving and Stationary Target Acquisition and Recognition
(MSTAR) data set and UWB SAR image data set.
The MSTAR public release data consists of X-band SAR images with 1 foot by 1 foot
resolution in one foot resolution spotlight mode. The target images contain one of three
T72 Main Battle Tanks (MBTs), one of three BMP2 Armored Personnel Carriers (APCs),
or a BTR70 APC. There are images of a test object (Slicy) available also. The target
images are 128 by 128 pixels and were collected on the ground near Huntsville, Alabama
by Sandia National Laboratory using a STARLOS sensor. There are 140 images of each
target at different poses (5 increments) In the latter simulation, all the target images of
T72s, BMP2s, and a BTR70s in 15 depression angle are used.
The UWB SAR raw data is collected by a moving UWB radar transmitting a impulsive
waveform with the bandwidth of 1 GHertz. Then the focused UWB image with 0.1 m by
0.3 m resolution is formed for a 2 {km)2 area, where natural clutter like trees or foliage,
man-made clutter like power lines, and 15 different types of vehicles are present. The
focused images are composed of 7 consecutive frames, each with 5376 by 2048 pixels are
used for the simulation of the proposed detectors. There are a total of 88 million pixels in
the images corresponding to the 2 {km)2 area.
For simulation, there are two image data runs of the same scene focused at different time
with the same 25 vehicle targets, but their locations changed on the ground. Besides, there
also exists man-made clutter like power lines. All the proposed detectors are tested on the
same single data run of 88 million pixel images.


78
kernels can be used for the tree. Depending on the different matching signal subspace, the
corresponding detection statistics are shown in Figure 40.
(a) The detection statistics with signal subspace constructed by Laguerre bases
of order 1,2,3 and 4
(b) The detection statistics with signal subspace constructed by Laguerre bases
of order 5, 6, 7, 9 and 9
(b) The detection statistics with signal subspace constructed by Laguerre bases
of order 10, 11, 12, 13, 14 and 15
Figure 40. The detection statistics corresponding to the ID UWB SAR down
range profile in Figure 33 using different combination of bases for
signal subspace


115
criteria. In our methodology, the data-driven approach will be used again to find the
weights qm and wm to provide better discriminant capability.
6.1.3 Temporal Training for the Discriminator
The discriminator is applied on the points of interests produced by the detector. In our
case, the QLD is directly applied on the detection points generated form the 1D y -CFAR
detector. As shown by the ROC curve in Figure 62, about 2357 false alarm points for
100% detection rate is obtained after clustering. Among these false alarm points, the 300
points corresponding to the largest 300 detection statistics are chosen to make the training
clutter chips. As described earlier, there are 50 vehicles in the UWB SAR data set. The 15
target locations corresponding to the least 15 detection statistics are chosen for making
training target chips. Each image chips is of size 250 x 100. All the false alarm points and
target locations are used for the testing data set.
Next, we would discuss how to find the weight wm and qm of the quadratic Laguerre
discriminator through temporal training. In order to catch the temporal information to
achieve discrimination, our desired signal for the ID down range profiles of targets has to
be carefully designed. In our methodology, the desired signal is designed to guide the dis
criminator to catch information in the resonance response after the driven response which
is indicated by the peak detection statistics of the y -CFAR detector. Thus, the desired sig
nal for the target down range profile of target chips is designed as
1 for i e [ipeak-li,ipeak + l2\
0 for i £ Upeak~^3 ipeak + ^
d(i) =
(75)


91
Figure 48. The Weighted Subspace Detector
4.3.3 Simulation Results
We will show the performance of three proposed detection schemes: ID subspace detec
tor, 2D subspace detector, and the weighted 1D subspace detector in terms of the receiver
operation characteristics (ROC) curves. The UWB SAR images (about 2 {km)2) for the
simulation contains 50 targets embedded in the clutter. For all the schemes, the Laguerre
space with the first 15 order kernels and feedback parameter p= 0.7 is used as the signal
subspace to capture the resonance response. For the ID subspace detector, only a single
down range file is used, while 10 down range profiles are used at the same time as the tem
plate for the 2D subspace detector. The results of the three detection schemes are shown in
Fig.49.


100
with the 2-2-1 structure, as shown in Figure 51, to extend the previous fusion rule in (69)
can be written as
/( Mj, uf) = tanh(a0+fljMj-i-a22)
(70)
u- = tanh(a0 + antr + ai2td) for i = 1,2
where a0\ af and af are the weights for the second layer, and ai0, an and ai2 are the
associated weights for the hidden nodes uf and u2\ Note that (70) is definitely a general
ization of (69), and has the advantage that all the weights and the thresholds can be
adapted by using the training data to give better performance.
Figure 51. Temporal Detection Fusion with recursive Laguerre Networks
5.4 Training for the Fusion Neural Networks
In our methodology, the desired signal is designed to train the networks to catch the tem
poral information, our desired signal along the down range has to be carefully designed.
Suppose we have the ID target profile vector y, the corresponding desire signal vector by
d. Since the total response denoted as y can be divided temporally into two regions such


84
5000
4000
QJ>
^ 3 00 0
ca
i
o
2000
1 OOO
O
O 50 1OO
cross range
Figure 43. The original UWB SAR image
Previous work shows that the resonance responses of the two targets are within the sub
space expanded by the first fifteen Laguerre kernels with p = 0.7. Therefore, we use a 15
dimension projection space implemented by the Laguerre networks with 15 taps and
p = 0.7. The detection statistics of the usual GLRT based on the ID resonance model is
shown in Figure 44.1 and Figure 44.2. The algorithm is able to determine the targets
around sample 3000 along the down range from clutter. However, there is a false alarm
with detection statistics as high as the target.
The detection statistics of the usual GLRT based on the spatial resonance template
model, with the Laguerre recurrent networks, is shown in Figure 45.1 and Figure 45.2
Comparing the detection statistics of target in Figure 45.1 with that of the foliage in Figure
45.2, we can see that the targets peak detection statistics is 3-db higher than the peak sta
tistics due to the foliage.


93
The weighted subspace detector can be viewed as a match filtering of the signal power
spectrum in the Laguerre space. To get a better insight of the functionality of the scheme,
we alternatively analyze the weighed Laguerre subspace detector in the Fourier domain.
Since the focus is the subspace energy, the Fourier power spectrum of the match filter is
investigated. The frequency response of the weighed Laguerre subspace detector is shown
in Figure 50-(a), and the power spectrum of the 256 data samples of the vehicle beginning
at 654 in Figure 33 is shown in Figure 50-(b). Figure 50 suggests that the weighted sub
space detector matches the energy in the target resonance response as we would expect
from the largest eigenfilter response.
O SO 1 OO 1 so
(a) the frequency response of the weighted Laguerre subspace detector
(b) the frequency response of the data samples of a vehicle
Figure 50. The frequency response of the weighted Laguerre subspace detector
and the signal


101
as yd and yr in (66), our N x 1 temporal signal d can also be equivalently divided as
d- [dj¡, dJ]T, where dd = [0, 0,..., 0]r and dr [1, 1, ..., l]r. Figure 52 shows an
example of how the desired signal is designed. Notice that here the two regions are heuris-
tically divided. Latter in 6.1.3, we will see how it can be divided more reasonably with the
help of ID y-CFAR detection results.
Once the desired signal has been designed, if we have the data vector
y(i) = [y(0,y0'+1), ...,y(i + N-1)] at time index i, the optimal weights aiQ, an and
ai2 for i = 1,2 in (70) can be trained by back-propagation approach [Haykin, 1994].
Magnitude
Figure 52. ID down-range profile data of a target, a dihedral, and its
corresponding desire signal
5.5 Simulation results
First, we will show that the performance of the subspace detector can be improved using
the sequential detection for one single target. Then we will apply the temporal fusion
detection scheme on a finite data set to investigate its discriminant power.


136
The denominator in is still the same chi-square distributed random variable as in the case
under H0, so the statistics Ms a non-central F-distributed random variable with the non
centrality parameter given by
r = aTSTPP HSa/o2 .


23
2.2.2 Laguerre Bases
It is well known that, applying the Gram-Schmidt orthogonalization process to the fol-
k YX
lowing sequences f¡(n, u) n u [Gottlieb, 1938], we obtain the k-th discrete Laguerre
sequence rk ^[rc] given by
r^n] = exp(pw)A*
exp(pw)
(15)
where A is the forward difference operator, (^J is the binomial coefficient, and p is the
feedback parameter. Performing the k-th order difference operation it can be shown that
rk u[] has the explicit form given by
W"1 = L (V)t)C) p = exp<"M) ,16)
m = 0
The polynomials defined in (13) can be normalized to give the orthonomal Laguerre
sequence [Gottlieb, 1938]
/,[] =(-D*Jp- rUn]
The waveforms of L ,,[] for different order k 0, 1, 2, 3 in the discrete time domain
and frequency domain are shown in Figure 12 and Figure 13, respectively.


128
achieving a discrimination power at 1:24 ratio. This is our best false alarm result for target
detection in UWB, and although higher, it is in the same range of performance of MMW
SAR discrimination algorithms. This gives us hope that this methodology can lead to real
istic ATR systems using UWB SAR.
To summarize, the extensive testing results of all the proposed detection schemes for
UWB SAR images are listed in Table 4 in terms of false alarms for the three important
operation points Pd = 0.8, Pd = 0.9 and Pd = 1.0. It is impressive that the QLD
improves significantly the prescreener performance, and at the same time uses much less
computation than the Gabor Discriminator. Based on the evidence in [Principe et al,
1998b] and the performance of QLD in UWB SAR, we can argue that the quadratic dis
criminant function is preferable to the linear discriminant function should be first applied
for reducing false alarms in MMW and UWB SAR imagery.
Pd=0.8
Pd=0.9
Pd=1.0
ID Laguerre Detector
18519
20413
25368
2D Laguerre Detector
14038
19862
21124
Weighted Subspace Detector
4023
4792
5010
2D y CFAR Detector
2114
2891
4367
ID y CFAR Detector
1885
2037
2588
Temporal Discriminator
1678
1742
1803
Gabor Discriminator
787
941
1034
Temporal Discriminator
694
752
801
QLD Discriminator
74
89
126
Table 4. The performance of the proposed detection scheme for UWB SAR images


109
ric stencil is used, the estimated mean and variance of the local background noise will be
increased due to the inclusion of the target resonance response appearing after the driven
response. To avoid this problem, the previous circularly symmetric guard-band kernel
must be modified. We propose to divide the 2D circularly symmetric stencil into two
parts: the front detection and the rear detection stencils. The front detection stencil utilizes
only the front half plane ahead of the target in the down range profile. For the sake of com
parison, the full stencil will also be used. The two 3D windowing kernels with the front
guard-band stencil and rear guard-band stencil are shown in Figure 59-(a) and Figure 59-
(b), respectively.
rear guard-band stencil front guard-band stencil
Figure 59. 2-D modified kernels for resonance experiments.
We will experimentally show that the proposed front guard-band stencil has better per
formance than the full guard-band stencil. The UWB SAR data for the simulation contain
8 consecutive frames, and each frame has 5376 x 2048 pixels. The data corresponds to an
area of the size 2 (km)2, where 50 different vehicles are embedded. The y-CFAR detec
tors in (25) with the modified front guard-band kernel is applied to the UWB SAR data to
investigate the performance. Notice that here we still use the fifteenth order circularly


22
2.2.1 Gabor Bases
Gabor function subspace is a well-know damped sinusoidal subspace [Gabor, 1946], It is
an intuitively reasonable choice for the resonance response composed of damped sinusoi-
dals by our previous argument. Suppose we have the damped sinusoidal subspace denoted
by TV x M matrix S(k,f):
~ [si> *2 sm\
s: = [ 19 e(- A- +jWi) 1 s ? g(- A +j2nfi)(N- 1)j T
where Af = [A,,, A2, ...,kM]T is the parameter vector of M-damping factors, and
/ = is the M-frequency parameter vector. S(X,f) can also be explicitly
expressed by
1
1
1
S(KJ) =
e(-X{+j2nf{)\
e(-X2 +j2nf2)\
+j2nfu)\
(13)
e{-Xx+j2nf{N-1) e(-X2+j2nf2)(N-1) e(~^M+j2nfM)(N-1)
For the signal model y = S{X,f)a + n with the parameter (A,/) fixed, then the ML
estimate of a can be obtained as
= [SH(X,f)S(i.,f)YxSH(i.,f)y (14)
Obviously, the complexity of (14) is 0(N^), and its a overwhelming computation burden
for a detection algorithm involved with the computation of a of Gabor bases, let alone the
overhead involved in estimating the parameter (A,f).


21
2.2 Representation of Bipolar Transient Signals
It has been pointed out that resonance response is an important feature for target detec
tion in bipolar UWB radar signals [Chen et al, 1995]. To begin developing detection algo
rithms for the resonance response, the first task is to provide a signal model to describe the
damped resonance response. The usual processing of transient signals like the resonance
response starts by transforming the time-domain signal to get a transform domain repre
sentation. The usual linearly transformed representation used are Time-Frequency Repre
sentation (TFR) and Time-Scale Representation (TSR), such as short time Fourier
transform (STFT) and Gabor transform [Gabor, 1946].
However, for any on-line detector, the computation complexity is always a big concern.
Its important for the chosen transform to have an efficient implementation, so that the
realization of on-line transform becomes feasible. Furthermore, there should be good rea
sonings in choosing the transform so that a better representation could be expected.
Suppose we have the transient signal denoted by TV x 1 vector x, and the signal sub
space denoted by N x M matrix S = [s,, s2,..., sM], the signal model for x can be rep
resented as jc = Sa + n, where a is the projection vector, and n is the noise vector. A
good representation bases provides the projection space S where the energy is condensed
into a few bases. The limit of just one non-zero basis is called an eigendecomposition
which provides the best possible basis to detect the signal. The more components in a are
near zero, the better the representation is. We would like to find a signal subspace S which
meets the efficiency and accuracy to model the resonance response composed of the
damped components.


CHAPTER 1
INTRODUCTION
This project grew out of research conducted in the Computational Neuroengineering
Lab under Dr. Jose Principe on target detection in Synthetic Aperture Radar (SAR)
images. Target detection is a signal processing problem whereby one attempts to detect a
stationary target embedded in background clutter while minimizing the false alarm proba
bility. In radar signal processing, the better resolution provided by the Millimeter Wave
(MMW) SAR enhances the detectability of small targets. As radar technology evolves, the
newly developed Ultra Wideband (UWB) SAR provides better penetration capabilities to
locate concealed targets under foliage. In this project, the different reflection features of
the targets in the images generated by MMW and UWB SAR are utilized to attack the
problem of target detection in the noisy background.
1.1 Motivation for SAR
Range resolution in real aperture radar system is generally defined in terms of system
bandwidth A/ and propagation velocity, c, by
8
range
2A/
while azimuth and elevation resolutions are defined by the operating wavelength X, aper
ture dimension in the orientation of interest, L, and range to target, R,
8
azimuth
IR
L
1


127
I designed and tested several stencils for intensity detectors to take advantage of the dif
ferent phenomenology of UWB SAR (the driven response). Unexpectedly, the ID y-
CFAR works better than all the subspace detectors in our data set, producing less than
3,000 false alarms for a Pd=l. We can consider the y-CFAR as a special case of a GLRT
with a subspace defined by gamma bases. The Laguerre bases are an orthogonalization of
the Gamma space, so they are closely related. The big difference is that we choose a priori
which are the bases to represent the signal and the background (gl and gl5 kernels),
instead of using all of them or a weighted version using eigendecompositions. This result
suggests that the picking the guard band as in the ID y-CFAR produces better immunity
to the severe noisy environment than maximizing projection as done in the subspace
detectors. It also points out that discriminant power is always a trade-off between robust
ness and sensitivity. The Gabor detector provides a performance slightly inferior to the
weighted Laguerre detector. Therefore we would prefer to further exploit the Laguerre
detector as a discriminator due to its computational simplicity and better performance.
Finally I tested the GLRT subspace detector against quadratic discriminant functions to
investigate their relative performance. The GLRT subspace detector like the proposed
weighted Laguerre detector can be considered a matched filter for the signal subspace
energy, so it produces a linear discriminant function from the classification point of view.
Matched filters are known to be optimal discriminants only for the case of signal and clut
ter data samples with equal covariance matrices. The real data does not share this property
so quadratic discriminant function should perform better.
The experiments on the testing data set show that the QLD reduced the 2588 false
alarms triggered by the ID yCFAR detector to 107 false alarms at 100% detection rate,


28
Magnitude
n (sample index)
(b) JL(n) with Laguerre subspace
(c) JG(n) with Gabor subspace
Figure 14. The subspace representation of the target response using different
bases (a) the targets original response (b) the Laguerre subspace
representation with the first 25 order Laguerre kernels and p=0.7 (c)
the Gabor representation with 25 damped sinusoidal bases, where the
damping factor A, =0.95


27
For the orthogonal subspace like Laguerre subspace denoted as L, it would reduce to
a = L rx( n ). To have an indication of how much energy is captured by the modeled sig
nal subspace, we propose to use the criteria given as
Js(n)
aTSSTa
x(n)Tx(n)
(21)
which is a ratio of the energy in the projection subspace, S, to the energy in the original
signal, For the target signal in Figure 14-(a), the corresponding JL(n) for the
Laguerre subspace, L, and JG(n) for the Gabor subspace, G, are shown in Figure 14-(b)
and Figure 14-(c), respectively. For both cases, we use 25 kernels with the length of the
region of support equal to 128. Thus, x(n) is a 100 x 1 vector, and L or G is a 100 x 25
subspace matrix. In this case, it can be observed that for this target, the Laguerre subspace
representation is at least as good as the Gabor representation, since the JL{n) is always
larger than JG(n). Moreover, the computation complexity for the Laguerre subspace rep
resentation is much less.


18
2.1 Representation of Signals in Grav-Scale Images
In MMW SAR grey-scale images, one of the prominent features of targets is intensity.
To model the intensity templates of targets in the grey-scale images, some all positive ker
nel can be used, since the intensity itself is always positive. Another merit of the all posi
tive kernel is its interpretation as a weighting function, or window, for local statistics
estimation.
It is well known that the Gamma sequences are always positive and constitute a com
plete set in /2. The k-th order ID gamma kernel in the discrete domain is given by
Sk,^n) = (12)
where p is the parameter that controls the scale of the kernel. The waveforms of gk ^[n]
for different order k 0, 1, 2, 3 in the discrete time domain are shown in Figure 9.
Figure 9. The Gamma kernels of different order k=0, 1, 2, 3 in the discrete time
domain


71
p. However, the goal of designing a signal space for detection is to get a peaky statistics
around the target neighborhood. That means we would optimize the values of p such that
the projection is as large as possible over a small window / around the largest target
reflection. Therefore, instead of using JL{n) for a signal y(n), 0 the following criteria for the optimization of p
n0 + l-\
I -4W
k=no ,
[ (57)
where / is empirically determined to be 20. We use as many as 40 Laguerre bases to con
struct our Laguerre signal subspace, where the parameter p can be optimized using the
criteria (57). For the vehicle in Figure 33, the relationship between the average Jpeak and
p is shown in Figure 34. This figure clearly shows that for p = 0.7 the criterion is maxi
mized, i.e., the subspace is best matched to the target response.
Jpeak = argmax
P
J,
peak
Figure 34. The Jpeak with respect to the parameter p for the targets in Figure 33.
A 40th order Laguerre space with p = 0.7 are used to construct the signal subspace to
capture the signal energy. The results for the vehicle and the power lines are shown in Fig-


29
2,3 The Neyman-Pearson Tests
Most of the target detection algorithms have been developed using the generalized like
lihood ratio test (GLRT) approach which is in fact a Neyman-Pearson test to maximize the
detectability while given a fixed false alarm probability. In the following, we will present
a brief review of the theory [Scharf, 1988].
say H,
S : Observation space
sayH04/
Figure 15. Decision Regions
Let X = be a N-dimensional random vector of observations with joint
probability density function (pdf) /x(x|0), where 0 is a parameter of the density func-
N
tion. Any specific realization x = [xj An] f X will be a point in R where R is
the set of all real numbers. Detection problems can be viewed as two hypothesis testing
problems, in which we have to decide between one of two hypotheses, which we will label
N
H0 and H,, about the pdf /x(x|0), given an observation vector x in R Let 0 be the
set of all possible values of 0. We usually identify H0 with one subset 0O of 0 values
and H| with a disjoint 0j, so that 0 = 0O u 0j. This may be normally expressed as
Hj : x has pdf /x(x|Hj) = /x(x|0, 0 e 0,)
H0 : x has pdf /x(x|H0) = /x(x|0, 0 e 0O)
(22)


APPENDIX A
THE STATISTICAL PROPERTY ASSOCIATED WITH THE LOCAL
INTENISTY KERNEL TEST
Suppose x = [x0, ..xN_, ]T with its elements, x0, ..xN_¡ being indpendent Gauss
ian random variables with mean m and variance a2. It can be shown that, for a weighting
vector hc = [hc0,hc^N_i>]r with the constraint h?ci 1,
E{(hTcx)2} = m2 + (hTchc)a2.
First, expanding E{(hTcx)2} directly yields
E{{hlx)2} = E\
Z haxi Z h"iX:
CJ J
\i = 0...N 1 y\j = 0...N 1 J
Z hM Z hcjx
(82)
i = 0...N 1
v = 0...N i y
The expectation term in the above equation can be further written as
£*( Z Vy) =A(E{*?>+ I A^{*,>E{*,}
l j = 0...N 1 J y = 0, l,f + 1 AT1
= hci(m2 + a2) + ^ hc-m2
j = o, i v-i
= *Ci<*2 + Z hcjm2
j 0...N 1
= /?Ci-a2 + m2
(83)
131


68
4.1.3 Detection Scheme and Simulation Results
Laguerre recurrent networks are used to capture the projection information of the UWB
radar signals, due to the simplicity of recursive implementation and a reasonable subspace
for representing the damped signals as shown in Chapter 2. Based on the test statistics
(56), we can implement our detection scheme as shown in Figure 31. The projection of the
signal to the Laguerre bases is simply extracted form each tap of the recurrent network,
and the projection energy can be computed by squaring the tap values. The GLRT can be
computed simply by summing up all the projection energy.
Figure 31. GLRT with recursive Laguerre Networks
First, we will show that the proposed detection scheme works for the signal composed of
Laguerre functions embedded in nonstationary noise. And then, we will show that our
algorithm works well for a UWB SAR down range profile. We defer the complete testing
of the detector for the final implementation.


113
(72)
m = 1
where wm is the weight to be adjusted and um = z2 for 1 < m < M. Therefore, the vector
u = [uu ..., uM]7 can be viewed as a Laguerre subspace energy vector. From the classi
fication point of view, the detection function in (72) is a linear discriminant function for
the two class data samples: clutter and targets. Let the covariance matrix of the target and
clutter samples be denoted as I, and Zc, respectively. The linear discriminant function is
optimal for the classification problem under the assumption that the data samples from
clutter and targets have the same Gaussian distribution with an equal diagonal covariance
matrix, i.e., T.t = Ec = a2/, where I is an identity matrix. [Duda and Hart, 1973], For
the case where 'Lt*'Lc the optimal discriminant function becomes [Duda and Hart,
1973]
MM M
M
1 = Z X ?pqUpUq + X WmUm + c
(73)
p = 1 q = 1 m = 1
m = 1
where wm and qm are the weights that need to be adjusted. The difference between the
linear and quadratic discriminant functions can be also illustrated as Figure 63.


11
approximation, plane wave decomposition, for the spherical wave is used, and it can be
given by
exp[~j2kJ(Xl -x)2 + (7, +u-y)2]
2k (10)
exp\j2j(2k)2-kl(Xx -x) +jku(Yl +u-y)] dku
-2k
Substituting (10) into (2), one can get the following relationship
o) = exp \j J4 k¡X, +jkuY,]F(Ju2 *)
To be more clear, the above equation can be written as
expt/'^A", +jkyYl] F(kx, ky) = S(ku, of) (11)
where
K = JAk2~kl ky = ku
Again, here (11) can be interpreted as a 2D filtering/deconvolution in the 2D spatial fre
quency domain, where only S(ku, co) needs to be manipulated by changes of variables in
the continuous frequency domain. The corresponding manipulation in the discrete fre
quency domain can be done by interpolation, and all the processes involved are illustrated
in Figure 5. Note that, to restore the signal f(x,y) from the observed signal s(u, t), one
only needs to compute
F{kx,ky) = S{kx,ky)/H{kx,ky)
The algorithm actually was first proposed by Stolt [Stolt and Weglein, 1985] for migrating
seismic data so that the reflection, with a hyperbola shape, to a point scatter would be


108
Detection Stage
Figure 57. Two Stage Detector Scheme
6.1 Two Stage Detection Scheme
Following a similar two stage approach for UWB, we will first apply an intensity based
detector to prescreen the driven response of both foliage and targets. Then the area flagged
by the prescreener would be further examined by the discriminator utilizing the temporal
information in the resonance response to reduce false alarms. So, basically we have a two
stage detection system like the one proposed by [Novak et al, 1993]. The only difference is
the nature of the tests that are particular to UWB phnomenology.
Driven Response
Prescreener
Resonance Response
Discriminator
Figure 58. Two stage detection scheme for UWB
6.1.1 Prescreener: ID Gamma-CFAR Detector
The y-CFAR detector with circularly symmetric 2D kernels is very effective to detect
point scatters in MMW SAR images. However, in UWB if the original circularly symmet-


19
The Z transform of the A>th order ID Gamma kernel can be written as
Gfc) =
^ (1 |J.)
Therefore, the Z transform of the data projection into the A>th order ID Gamma kernel
can be written as
Xk(z) = G(z)Xk_ j (z) G(z) =
That means the projection into each kernel can be simply obtained by passing the data
through the cascade of first-order kernels. That means for any causal signal x[n] 0 for
0 < n < oo with finite energy, we can uniquely represent it by
ak,v=
m = 0
where ak in fact is the projection of x[n] to the k-th Gamma sequence gk We can
get ak ^ by convolving x[n], with the k-th Gamma kernel:
ak,VL = x\~n]
The complexity of the above recursive formula is only O(k), where k is the highest order
of Gamma kernels. From these arguments, the ID gamma kernel is likely appropriate for
ID intensity pattern modeling. Nonetheless, the more important case is to use 2D kernels
to extract the spatial information for detection.


69
1 O
1 O L . . i . J
O 6 0 0 1 O O O 1 600 2 00 0
(b) The three Laguerre kernels embedded in non-Gaussian noise
1 O O
SO
60
-4-0
20
O
O 600 1 OOO 1 600 2000
(c) The detection statistics
-
L
r i
w :
Li
Figure 32. The signals of the three kernels embedded in the noise and the
corresponding detection statistics
Figure 32-(a) shows the first three Laguerre kernels with the parameter p = 0.7. The
generated Laguerre function signal embedded in the noise are shown in Figure 32-(b),
where the non-stationary Gaussian white noise has variance 1 in the first 1000 samples
and variance 4 in the following 1000 samples. It is impossible by eye to see where the sig
nal is. The corresponding detection statistics using (56) are shown in Figure 32-(c). It is


31
Obviously, if PF = a, J is minimized when P^ is minimized. (23) can also be writ
ten as
y=Ml-a] + f [/(xlH,) V(x|H0)]x
S0
For any positive value X, the above cost function can be minimized by minimizing the
second term in the right hand side. Therefore, J is minimized by minimizing the likeli
hood ratio
_/[xlH,) _y(x|eI)
'w Ax|H) y(x|e0)
Then the desired test is the likelihood ratio test (LRT), which can be written as
t{x)
Ax|0,) <'
Ai|0o) H0
(24)
Generally, for real-world signal processing, the parameters 0, and 0O have to be esti
mated from the observed data. If the most likely estimates (called the maximum likelihood
-ML estimate) 0i and 02 are substituted into the LRT for the parameters 0j and 0O, then
the LRT becomes the so-called generalized likelihood ratio test (GLRT). To satisfy the
constraint, we have to choose X, so that PF a. Then we require
PF | J[t\Qo)dt a
More importantly, under H0, if the test is invariant to 0O, or, in other words, the f(t) is
|*00
independent of 0O, that means PF = \ f(t)dt = a. Then the test t is a constant false
alarm rate (CFAR) detector. Its equivalent to say that if the f(t) is independent of 0O, the
threshold X used for detection can be uniquely calculated in terms of a in advance with
out resorting to 0O. Therefore, any derived detector would be more practical with the
CFAR characteristics.


CHAPTER 2
BACKGROUND
The design of a target detector can be divided into two phases: signal representation and
detector formulation. Through signal representation, targets are described more promi
nently than clutter in terms of the extracted features, and the detector can be formulated to
detect these features. The two phases are so intertwined, that its unavoidable to go back
and forth between these two phases before any powerful detector is derived.
The approaches to target detection in SAR images is to utilize the characteristic reflec
tions of metallic objects. There have been many algorithms developed for stationary target
detection in various other applications, such as infra-red (IR) sensors and radar surveil
lance [Reed et al, 1974] [Reed and Yu, 1990], In most of the scenes, the signals or targets
are dim or partially obscured by the varying noises in the background, and, hence, detect
ability is severely degraded. To improve detectability, some of the developed algorithms
utilize the apriori information of targets, as well as linear mapping to enhance the target
features. These enhancement can be thought as a kind of signal representation, and then
the detector can be formulated to maximize the detection probability while minimizing the
false alarm rate. We would introduce the philosophy behind these developed detectors by
reviewing the related background knowledge: signal representation and detector formula
tion.
17


121
ison, the Gabor subspace detector with 15 damped sinusoidal bases are used as a discrimi
nator, while the associated parameters (A,,f) are found by (78).
Compared to the ROC curves of 1D y-CFAR, the Gabor discriminator slightly improved
the detection performance even with the parameters (A,,/) tuned. There are two possible
explanations: (1) The damped sinusoidal bases are not orthogonal. A non-orthogonal sub
space representation is redundant, and makes the classification more difficult. (2) The dis
crimination function provided by the damped sinusoidal subspace detectors is still linear,
and does not have enough differentiation capability. It is the ID y-CFAR detector com
bined with Quadratic Laguerre discriminator which yields the best performance (126 false
alarms for Pd = 1). This result confirms our reasoning that a general quadratic discrimi
nant function is better than the GLRT for the two class discrimination problem, where the
two class data, target and non-target data samples at least are of the Gaussian distribution
with different variance.
6.4 Conclusion
In this final chapter we explore a couple of other ideas to improve the performance of
target detection algorithms for UWB SAR. The first idea is to go back to the Gamma
CFAR test which provided excellent performance for MMW SAR. We can consider the y -
CFAR as a special case of a GLRT with a subspace defined by gamma bases. As we said
the Laguerre bases are an orthogonalization of the Gamma space, so they are closely
related. The big difference is that we choose a priori which are the bases to represent the
signal and the background (gl and gl 5 kernels), instead of using all of them or a weighted


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141


CHAPTER 3
LOCAL INTENSITY TESTS FOR OPTIMAL DETECTABILITY
Target detection, the first stage in a radar system, is an important problem in signal pro
cessing. The approach to target detection depends on the reflectivity characteristics of tar
gets. The most distinguishing features of target reflections in different radar systems
should be utilized for detection, so that the detection probability can be maximized while
the associated false alarm rate is minimized. In MMW SAR images, targets are known to
contain many point scatters with large reflections, due to the metallic corners. Thus the
straightforward approach to target detection in MMW SAR has been to detect point
scatters. Assuming the background clutter intensity has a locally Gaussian distribution, the
two parameter CFAR test [Goldstein, 1973] to detect point scatters with intensity x can be
written as:
= Z
a
where p is the estimated local mean, and a is the estimated local variance. As a pre-
screener in MIT Lincoln Labs ATR system [Novak et al, 1993], the CFAR detector uses
the pixels in the clutter and rectangular stencil as illustrated in Figure 17-(b) to estimate
these two parameters. The corresponding 3D windowing function with the amplitude of
the stencil of the two parameter CFAR is shown in Figure 16-(b). The CFAR detector is a
normalized contrast comparison detector, which computes the ratio of the energy of the
tested pixel to that of the local background clutter, and it is widely used because of its sim
plicity [Novak et al, 1993].
33


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, Chair
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.
John G Harris
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.
Jian Li
Associate Professor of Electrical
and Computer Engineering


CHAPTER 4
SUBSPACE DETECTION OF TARGETS IN UWB SAR IMAGES
Early radar systems detected targets in the air. Illuminated targets would return a large
energy reflection which is contaminated by the thermal noise generated by the radar sys
tem itself. To detect the large energy reflection of targets, the so-called one parameter
Constant False Alarm Rate (CFAR) detector compares the amplitude of the testing cell
with the noise amplitude, the adaptive threshold, estimated from the neighboring cells.
cell under test
l/M
Figure 30. One parameter CFAR detector
Many other target detection algorithms have been developed based on the observation that
the reflection of targets has larger energy compared to that of natural clutter [Goldstein,
1973]. As technology improved, SAR has also been applied to the surveillance of ground
targets. In MMW SAR images, background clutter can no longer be assumed to have a
global Gaussian distribution due to the different reflections of the ground textures. How
ever, it is still valid to assume that clutter has a local Gaussian distribution, since texture
62


25
where 5- is the Kronecker delta, i.e., Laguerre sequences constitute an orthogonal com
plete set in /2, and it can be shown that the Laguerre sequences orthogonalized the
Gamma bases presented earlier [Silva, 1994]. Hence, for any causal signal x[n] 0 for
0 < n < oo with finite energ, we can uniquely represent it by
*[] = X
k= 0
oo
= £*[] 4,^"]
n = 0
where ak ^ is the projection of x[n] to the k-th Laguerre sequence lk ^[w]. We can get
ak n by convolving jt[n] with the k-th Laguerre kernel:
ak, = x[-]
The Z transform of the Laguerre sequence is given by
So, with
LU ^ = Z ln> M)z
n = 0
(1 WZ_1)¡ 1
z > 0
(17)
L0,n(z) = (Vl-p2)/(l-^ ) 0 < p < 1 (18)
we have
^k+\,\S-z^ ^k, |i(z)
-1
zv'i =
Zw = P-TT 0 1 pz'
(19)
This shows that the projection into Laguerre space can be implemented by a cascade of
identical all-pass filters with transfer function Z (z) preceded by a low-pass filter with
r1


88
Then the GLRT becomes
t
zWz
' 2
a
2'
<
Hr,
(64)
A 2
The new test is still a CFAR test with F distribution. In practice, cf can be absorbed into
the threshold and the test just becomes tr = z Wz We can view the proposed W as an
energy template in the subspace, which best matches the distribution of the energy of
the different targets resonance response in the transform domain.
4.3.2 Subspace Parameter Tuning and Template Finding
All the subspace methods like Laguerre functions, Gabor Transforms or Short Time
Fourier Transforms, have some parameters to control the bases. For a single exemplar we
showed in 4.1.3 how this can be accomplished. However, for all the targets of interests,
one has to take a more systematic approach to find a single signal subspace. As sug
gested in (64), the subspace detector may be enhanced by imposing a weighting template
for targets to provide good discriminant capabilities from clutter. To incorporate the apri-
ori target information, a data driven approach is naturally chosen to find the parameters for
the signal subspace and the weighting template. Since the Laguerre functions are used to
model the resonance response, we will focus on reliably estimating the feedback parame
ter p for the Laguerre subspace and the associated template matrix W adaptively through
the data set.
Suppose we have K different target down range profiles j(l), ...,y(K), and the Jpeak
in (57) for each profile is denoted as Jpeak y^ky The subspace detector compares the
energy captured by the modeled subspace to that in the null space, so the goal is to build


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McCorkle, M., and Nguyen, L., Ultra wideband bandwidth synthetic aperture radar
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McLachlan, G. J., and Basford, K. E., Mixture Models: Inference and Applications to
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noise, IEEE Trans. Inform. Theory, vol. IT-18, pp.241-250, Mar. 1972
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ICASSP vol. 4, pp25-28, 1994
139


57
Image Enhancement
Figure 28. The associated processed data involved in target size estimation for the
two targets BMP2 and T72 in MSTAR data set.


89
the Laguerre signal subspace parameterized by p to provide the projection as large as pos
sible for the target data set. Therefore, we propose the following criteria to compare the
detectability of targets:
4
K- 1
\
= argmin
ft
'£JJpeak,y(k)/K
\k = 0 )
(65)
Then the feedback parameter p is searched to maximize the proposed criteria. For 450
down range profiles from 15 targets in the training data set (See 6.1.3), the J0 is mini
mized by using p in the 15 Laguerre kernels of the Laguerre function space. The result is
shown in Figure 46, and one can say that the value of 0.7 for p can lead to the most satis
factory results.
ft
Figure 46. The performance indices JQ with respect to the feedback parameter p
After a reasonable value of p is obtained, we still need to select the corresponding sub
space representation z( 1),..., z(K). The goal of the template matrix W is to choose the
subspace where most of the energy of the training targets lies. Here we utilized again the
ideas of PCA. We would like to find the best direction of the target clusters in the sub-


63
changes slowly. In this case, target detection becomes detection of point scatters as
introduced earlier.
UWB was developed to take advantage of the better penetration capability of the low
frequency components in the transmitted waveform with large bandwidth. Consequently,
metallic objects reflect the resonance response following the driven response due to the
wideband excitation. Several algorithms have been developed to directly detect the driven
response of targets in the UWB SAR images [Kapoor et al, 1997]. However, some natural
clutter would also produce large energy returns, resulting in poor performance. Instead,
the target resonance responses may be utilized for discrimination from natural clutter
[Sabio, 1994][Chen et al, 1995]. Resonance responses are composed of damped sinusoi-
dals, so it is reasonable to assume they can be modeled by some matched subspace, in
the sense that the energy of resonance response would concentrate only on a few bases,
where natural clutter contains little energy. Then an approach similar to detecting transient
signals [Friedlander and Porat, 1989] can be used to develop a resonance detector.
4.1 Target Detection Using Laguerre Networks
There have been many algorithms developed for target detection in UWB SAR. Basi
cally, as described earlier, UWB SAR provides foliage penetration capabilities. Moreover,
man-made metallic objects in UWB radar produce a damped sinusoidal response in addi
tion to the large energy reflections, which also could be generated by non-metallic objects
such as trees. Therefore, it is important to explore the ID information of the resonance
response of metallic objects to develop improved target detection algorithms [Yen and
Principe, 1997a]. This is the approach taken in this work.


41
Without affecting the monotonic increase of the testing statistics, we can multiply both
sides of (28) by 2r), yielding
t =
since hfi = 1. In practice, the statistics
square root on the both sides of (29) yields
hf^x mi)^2
c
h]x m's2
c
hjx m
(29)
< 0 can be discarded, so taking the
hlx m
t = (30)
a
Suppose we have a spatial weighting vector hc with a region of support disjoint from that
of ht s. That means, without loss of generality, with N = (2N + 1) x (2iV+ 1), hc and
ht can be assumed to be
~ [h/0 ht(K-iy 0]"
hc ~ [0j > 0, hcK, ..., hc(_^]T
In terms of hc, we propose to estimate the parameters m and a by the local statistical
information as follows:
m = hTcx
2 = c/icr[ii]
x = x mi
(31)


75
order k
(b) The normalized projection energy in the Laguerre bases
Figure 37. The Laguerre subspace representation for the tree in Figure 33 (a) the
JL(n) around the sample from 4000 to 4200 (b) the distribution of the
normalized projection energy in the Laguerre bases for the sample
n=4096
We would like to find the size of the space that contains most of the energy of the
objects. So, we order the projection energies and plot them. The corresponding results for
n = 654, n = 3256, and n = 4096 are shown in Figure 38. We also can see from Fig
ure 35, Figure 36 and Figure 37 that the signal energy is concentrated in the first 15
Laguerre bases.


85
Figure 44.1 The detection statistics based on ID resonance
model along down range 1801 ~3600
Figure 44.2 The detection statistics based on 1D resonanc
model along down range 3601-5400
Figure 44. The detection statistics based on ID resonance model
Figure 45.1 The detection statistics based on 2D resonance
model along down range 1801-3600
Figure 45.2 The detection statistics based on 2D resonance
model along down range 3601-5400
Figure 45. The detection statistics based on spatial resonance template model


38
Actually, these two target chips fail to be detected by the two parameter CFAR detector on
the results reported by Kim [Kim and Principe, 1996], Although the performance of the y-
CFAR detector can be improved by finding the better guard-band size through training, it
is still fixed in operation and sub-optimal. Observing these facts, we would like to come
up with a scheme to estimate the target size on-line for maximum detectability.
Figure 18. The performance surface (false alarm surface) of the y-CFAR detector
with respect to the parameters p forg, org15.
Figure 19. Two embedded targets of TABLIS24 data set with their CFAR stencils.


60
Target type
Series No.
Actual Size
Mean Est. Size
Mean Est. Size
NMSE
BMP
sn_9563
8.5
8.72627
1.30892
15.0%
BMP
sn_9566
8.5
9.00103
1.34748
15.0%
BMP
sn_c21
8.5
8.80738
1.09854
12.5%
BTR70
sn_c71
9.0
9.24721
0.879623
9.5%
T72
sn_132
10.0
9.76500
1.56374
16.0%
T72
sn_812
10.0
9.99818
1.54033
15.4%
T72
sn_s7
10.0
10.05790
1.70741
17.0%
Tablis24
Mission 90
8-12
*
*
12.0%
Table 3. The estimated target size and the associated statistics
3.5 Conclusion
This chapter formulates the CFAR detector as an intensity detector by applying the
GLRT formulation developed in Chapter 2. We were able to show using the ML approach
that the CFAR stencil is intrinsically linked with the optimality of the test. The stencil can
be thought as the correlation template, and as such for optimality, it should match the
intensity profile of the targets. We then analyze the radial intensity profiles of targets from
MSTAR through principal component analysis (PCA). It turns out that the first eigenfunc
tion of the radial intensity profile of targets can be well approximated by the first gamma
kernel. Hence this explains why previous results [Kim and Principe, 1996] have shown
that the y -CFAR outperformed the delta funcin stencil proposed by MIT/Lincoln Labo
ratories.
With this understanding we proposed a method to adapt the guardband for each individ
ual target, which is much better than the average guardband size proposed in [Principe et


51
ment, we exhaustively substitute the parameter value k e {1,2, 20} and
u e {0.01, 0.02, 0.03, ..., 0.99} into (50) to find the best gkt[l. For the three types of the
targets BMP2, BTR70, and T72 in MSTAR data set, the best kernel order k are all 1, and
the parameter p is 0.47, 0.51, and 0.53. Since in the real application, we have to fix the
parameter value for the detector. The modeled radial representation gki M, with k = 1 and
p = 0.5, are shown in the right of Figure 22, Figure 23, and Figure 24, respectively,
while the corresponding Gamma kernels gk M in the discrete spatial domain is shown in
the Figure 25. From this analysis, it is clear why the y-CFAR detector work so well. The
first order circularly symmetric Gamma kernel is approximately the maximal eigenvector
for radial target intensity, which is the best kernel for the GLRT test.
Figure 22. The Modeling of the Radial Projection of BMP2


47
. Then, the problem of designing a 2D symmetric kernel h(x,y) to maximize the correla
tion in (42) is simplified to the problem of designing ID radial kernel h(r). If we assume
that e(r) is white Gaussian noise, from the matching filter point of view, the SNR is max
imized when
h(r) = p(r) (46)
Note that, for some specific r0, fR(r0) in (44) can be written as
P/Oo, 0) r0 dQ
5 (47)
2717-0
It means that, for some specific r0, we can get ^-F(r0) by integrating the images inten-
271
sity along the points of the circle with radius r0, and then normalize it with respect to
2ti7-0 So, -^/(t-) can be viewed as a ID radial representation of the 2D image data
f{r, 0) or f(x,y). Since the SAR data image is discrete, we will formulate the above cor
responding procedures in the discrete spatial domain. For the discrete image chip F with
finite region of support Q v, the corresponding part of (47) can be approximated by
2>>
/['] = Y, AUVNR(r) (48)
i,j 3 i2 +j2 = r2
where r e N = {r2 = i2 + j2 | N< i,j < N i,j e integer} and NR(r) is the num
ber of the points with the same radius distance r in the discrete image f[i,j]. We can
think that fR[r] is the equivalent of (47) in the discrete domain. The mapping relationship
between r and (i,j) is illustrated in Figure 20, where we can easily see that re {0, 1,
1.414, 2.236, 2.828,...}, and its not difficult to figure out that there are
(N+ 1\
I 2 I = (N+ \)N/2 elements in On. Since there is only 1 point with image index
(0,0) having the radial distance r = 0, therefore we have NR(0) = 1. There are four


45
be written as a liner combination of the independent Gaussian random variable
v0, Since ht and hc have exclusive regions of support, the numerator is still a
Gaussian random variable with zero mean and variance Jhjht + hTchc. Furthermore, it
can be shown (See Appendix A) that the denominator term can be written as
hTc[xx]/a2 /ij[v v]-(/iJv)2 (40)
Substituting (39) and (40) into (38), then the detection statistics now can be rewritten as
hTtv-hlv
t = === (41)
Jh£[v <8> v] (h£v)2
The second term in the denominator has a quadratic form in terms of hTcv, which is a
Gaussian random variable with zero mean and variance Jh^hc, so it is %2. As to the first
term in the denominator, when hTc = [ 1 /K, ..., 1 /K, 0,..., 0], the first term become y2,,
and the denominator is x3 The testing statistics will have a t-distribution. For other
cases where hTc belongs to some kernel, the term hTc[v v] does not fit any particular
probability function, and it is difficult to derive a close form of the PDF. However, since
the first term is still in terms of v, which is a normalized Gaussian random variable with
zero mean, its PDF definitely is independent of m and 0. By the same argument, the
PDFs of the denominator and the testing statistics are also independent of m and 0,
respectively under H0. Therefore the intensity detector is a CFAR detector.
3.3 Intensity Modeling of Targets
In the previous section, we show that both y -CFAR and CFAR detector can be cast into
intensity detectors with unspecified intensity kernel ht. Then the question arises: what is
the best intensity kernel ht to match targets. To answer this question, we seek to design a
circularly symmetric kernel ht for the intensity detector. The added constraint of cir-


135
And our testing statistics t can be also written as
= yTppLHy/g2
uTu/o2
Under H0, we have y = w, so now the new numerator is given by
(87)
wTPpLHw/2 = Wpplhw/g\\2 (88)
(88) is a quadratic form in the Gaussian random variable Pphw/<3 with zero mean and
variance Ppjj So, its a central chi-square distributed random variable, with a number
of degrees of freedom is equal to the rank of PPljj The denominator in (87) is a qua
dratic form in the Gaussian random variable m/ct with zero mean and variance I. So, its
also a central chi-square distributed random variable, with a the number of degrees of free
dom is equal to its Denison. So, the statistics t is a central F-distributed random variable.
Under Hx, we have y = Sa + w, so now the new numerator of t can be given by
(Sa + w)TPpLH(Sa + w)/o2 = ^PpLH(Sa + H)/a||2 (89)
(89) is a quadratic form in the Gaussian random variable PP H(Sa + w)/a with mean
equal to Pp^Sa/o and variance PpLn So, its a non-central chi-square distributed
random variable, with a number of degrees of freedom is equal to the rank of PpLy¡ and
the non-centrality parameter is given by
r = aTSTPpHSa/a2


117
6.2 Discrimination Using Damped Sinusoidal Subspace
In the previous implementation of the subspace detector of (56), the Laguerre function
space is used as the projection subspace to capture the resonance response, but the
Laguerre function is not the only choice for the modeling subspace. Generally, the well-
know damped sinusoidal subspace like Gabor functions also seems to be a reasonable
choice, but the estimation of the parameters for the model is not a trivial problem. In the
following, one of the fast estimation methods for the parameters of the damped sinusoidal
functions is discussed, and the results of the detector with damped sinusoidal subspace is
provided for a comparison. Suppose we have the damped sinusoidal subspace denoted by
NxM matrix S(X,f) in (13). Then the signal model can be rewritten as
y = SCk,f)a + n. With the parameter set of (X,f) fixed, the damped sinusoidal signal
subspace can be used for implementing the testing statistic in (56) as a detector.
To have better discriminant power for (56), the parameter (Kf) can be estimated by
many algorithms [Li and Stoica, 1996]. It is shown that the data-driven iterative procedure
proposed in [Umesh and Tufts, 1996] is one of the most robust estimation methods, which
in fact can be considered as the iterative version of the matching pursuit using the dictio
nary formed by the damped sinusoids.
After the initialization, the iterative procedure update components ak of a, and sk of S
for k = 1, ..., M one by one at each iteration. In the i-th iteration, the k-th components
ajh.') of a, and of S are updated as the following:


FOCUS OF ATTENTION FOR MILLIMETER AND ULTRA
WIDEBAND SYNTHETIC APERTURE RADAR IMAGERY
By
LI-KANG YEN
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
1998


79
From the results in Figure 40, it can be observed that the use of appropriately designed
signal subspace will improve the detectability of the corresponding objects. Comparing
Figure 37-(b) with Figure 38-(b), we can also expect that there is an overlap between tar
gets and trees. The design of a signal subspace for improved detectability of a single
object may be straightforward, but for all the targets, it will be much tougher since the
clutter response needs to be minimized at the same time.
4.2 Target Detection Utilizing A Spatial Template
In the previous section, only the resonance response information along one down range
profile is explored. One should be reminded that, in addition to the unique wideband exci
tation, UWB SAR also provides us with improved azimuth resolution of the small targets,
so that abundant spatial information is carried into the images like the MMW SAR.
In MMW SAR images, target reflection signatures vary as their poses change, and it is
impractical to have different reflection templates with respect to different poses. Although
the same problem occurs for modeling the driven response signatures of targets in the
UWB SAR images, the resonance response of each target is at least theoretically indepen
dent of its pose. Thus it seems reasonable that under the assumption of a template for the
targets resonance response, a unique model is applicable to detect the resonance response
of the target even with different poses. We assumed that, for the targets cross range
extent, the resonance responses along different down range profiles concentrates on a sub
set of bases in the matched signal subspace. Therefore, each target of interest is modeled
with a signal subspace template, and a more robust test statistics preserving constant false
alarm rate (CFAR) can be formulated [Yen and Principe, 1997b].


122
version using eigendecompositions. Hence, the y-CFAR utilizes heuristic knowledge in
its construction, but it is a GLRT test.
We can not utilize directly the circularly symmetric kernels of the MMW SAR due to the
different phenomenology of UWB SAR (the driven response). We therefore propose a
new 2D stencil for the g-CFAR tuned for UWB SAR. We also use a simplistic ID stencil
for UWB SAR.
Up to now we have not compared the performance of our systems with alternate sub
spaces. So in this chapter we develop a subspace using the Gabor functions and compare
its performance with the weighted Laguerre subspace and the 2D and ID gCFAR detec
tors. The comparisons show that the ID y-CFAR works better than all of the others in our
data set, producing less than 3,000 false alarms for a Pd=l. This was not expected,
because we thought that the weighted Laguerre detector, which chooses optimally the sub
space from the training set data should outperform this simple detector. The Gabor detec
tor provides a performance slightly inferior to the weighted Laguerre detector. Therefore
we would prefer the Laguerre detector due to its computational simplicity and better per
formance.
Finally we tested the GLRT against quadratic discriminant functions to investigate their
relative performance. The GLRT can be considered a matched filter in the signal subspace
so it is able to produce a linear discriminant function. As it is well known, matched filters
are optimal discriminants only for the case of signal and clutter with equal covariance
matrices. Only by chance this is the case in real target and clutter, but the issue is how
much do we gain in performance over the GLRT when a quadratic discriminator is uti
lized?


46
curlarly symmetry is imposed so that the change of reflectivity due to pose changes can be
avoided, and the analysis is simplified.
3.3.1 Mathematical Background
Suppose we have a 2D continuous image f(x, y) with background clutter mean removed.
The statistics of correlating f(x,y) with 2D circularly symmetric kernel h(x,y) can be
written as
t = jj h(x,y) f(x,y) dxdy
.. (42)
= jj h(r,Q) f{r, 0) rdrdQ
where /z(r, 0) and f(r, 0) are the corresponding image and kernel function in the polar
coordinate domain, respectively. Since h(r, 0) is circularly symmetric, it is independent
of the variable 0, then (42) can be written as
t = j rh(r) fR(r) dr = j h(r) fR(r) dr
where
(43)
h(r) = rh(r) fR(r) = 0) dQ (44)
Note that h(r) is still a symmetric kernel. Suppose the kernel fR{r) can be modeled by
some deterministic signal p(r) and noisy mismatch signal e(r) as fR(r) = p(r) + s(r).
Then (43) can be written as
t = j h(r) (p(r) + £(/*)) dr
(45)


CHAPTER 6
QUADRATIC LAGUERRE DISCRIMINATOR
In the previous two chapters, different detection schemes have been proposed for target
detection in UWB SAR images. However, they always utilized the GLRT as the detection
statistics, which is a special case of the quadratic discriminant function in the feature
space. Actually, target detection is a two-class classification problem for the discrimina
tion between target and non-target data samples. All the previously proposed GLRTs are in
the form of a quadratic discriminant function under the assumption that both the target
class and the non-target data samples have the same Gaussian distribution with the same
unknown variance. Obviously, a more general quadratic discriminant functions would be
more powerful without the limitation of equal variance [Duda and Hart, 1973]. In addi
tion, we have the evidence [Principe et al, 1998b] that a quadratic discriminant function
improves the performance of target detection in the clutter. Therefore, this chapter
explores the extension of the proposed GLRT detector in previous chapters. The GLRT is
extended into a general quadratic discriminant function for the classification of the target
and clutter class without the limitation of equal variance.
Let y is a N x 1 column vector in the down-range profile of the UWB SAR image. The
measurement y of metallic objects in UWB SAR images can be divided in two temporal
regions as y = \^>Td r, where yd is the Nd x 1 driven response, and yr is Nr x 1 reso
nance response. Obviously, N = Nd + Nr. Nd and Nf. are usually target dependent. The
106


42
A 2 ,
where c = 1/(1 hTchc) is the coefficient needed for unbiased estimation of a Substi
tuting the above estimators of m and a2 into (29) and absorbing the constant Jc in the
denominator, the detector becomes
(32)
JhTc[ii]
The detector is essentially an intensity detector, which functions as a normalized correlator
between the kernel and the testing image with the background clutter mean taken out.
Therefore, the success of the intensity detector is highly dependent on the shape of the ker
nel ht. For optimality, ht should match the target intensity signature and it must be appro
priately designed so that the maximal detectability is obtained.
Both the y -CFAR detector and CFAR detector can be framed into a kernel matching
detector. To see this, the variance estimator can be expanded as
/icr[0] = Yj hci-(xi-)2
i = 0...N-1
(33)
ci ~xi+ 2 X hci
i = 0
After direct simplification, (33) can be written in vector form as
ct2 = c[hTc{x x) (hTcx)2~\
(34)
Plugging (34) into (32) yields
t =
(35)


132
, since E{x¡Xj} = E{x¡}E{Xj} = m2. Plugging (83) into (82), then (82) can be simpli
fied into
£{(A£*)2} =
f
X
N
2.
"ci
a2 +
(
X
\
G = o...at-i
G = 0..W-1
y
m-
(84)
= m2 + (h£hc)o2
Suppose x = [x0, with its elements, x0, ...,xN_l being indpendent Gauss
ian random variables with mean m and variance a2. It can be shown that, for a weighting
vector hc [hcQ, hc^N_{)]T with the constraint hTci = 1,
E{(hTcx)2} = m2 + (hTchc)<52.
Note that
V l
X
hJxrm)L
N- 1
X
r = r
/zc((x-/w) + (/w-m))2
N-1 A N-1
X hci(Xi~)2+ X hci(-m)2
i = 0
i = 0
N-\
X hCi(xi-)2 + (m-m)2
i = o
(85)
, since
N1 a N-1
X hci(xi-)(-m) = XM¡-)('-)
i = 0 i = 0
JV- 1
= (m m) X hcixi m2 + mm
i = 0
= 0


102
The ID down range profiles of the single target for simulation are the same as shown in
Figure 43. The Laguerre network with 15 taps and p = 0.7 is employed, since the previ
ous investigation shows that the subspace expanded by the fifteen Laguerre kernels is
appropriate for signal representation. The detection statistics of the usual GLRT based on
the ID resonance model implemented by (67) is shown in Figure 53-(a) (first 3,600 down
range cells) and Figure 53-(b) (remaining down range cells). The algorithm is able to
detect the targets around sample 3000, while producing a false alarm around 4,000 with
detection statistics as high as those of the target.
(a) The detection statistics around the (b) The detection statistics of clutter
target
Figure 53. The detection statistics based on ID resonance model (a) the detection
statistics around the target along down range 1801 ~ 3600 (b) the
detection statistics of clutter along down range
The detection statistics of the GLRT based on the sequential fusion scheme are shown in
Figure 54-(a) and Figure 54-(b), respectively. Comparing the detection statistics of target
in Figure 53-(a) with that of the foliage in Figure 53-(b), we can see that the foliage detec-


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49
(3) find the intensity kernel with maximal projection for the data set by means of princi
pal component analysis (PCA).
The essence of the intensity detector is correlation between signal fR(r) and intensity
kernel p(r), and, equivalently, the projection of signal fR(r) into p(r). To implement the
maximal projection, PCA can be applied to find the eigenvector corresponding to the larg
est eigenvector of the data correlation matrix of the collected data chips. The largest eigen
vector can be used as the intensity kernel p(r). Then the intensity detector can be
regarded as performing maximal eigenfiltering [Haykin, 1991].
Figure 21. The 3-step intensity kernel modeling procedure
Let the elements in <&N be sorted in the ascending order and form the radius vector
r [r0,..., r(Ar+1)Af/2_i]. That means r = [0 1.414 2.236 2.828, ...]r. Notice that the
first element of r is 0. Without affecting generality, we can just use a smaller number like
0.7 instead. Suppose we have the 2D aligned data chips X0,..., XN_,, and the corre
sponding ID radial projection vectors are denoted as Jt0, ...,xN_l, where the i-th com-


CHAPTER 5
NEURAL NETWORK APPROACHES TO TARGET DETECTION
The new ultra wide band (UWB) synthetic aperture radar (SAR) exhibits a different
reflection phenomenology on metallic surfaces which is characterized by a resonant
response. In order to capture the information contained in the resonant response the
Laguerre function space is used as the projection signal subspace, and the subspace CFAR
detectors for resonance response can be derived. However, the large energy contained in
the impulsive driven response (from both metallic and non-metallic objects) preceding the
resonance response degraded the performance of CFAR detectors based on spatial infor
mation. Note that the information that the driven response ahead of the resonance response
is not utilized. To alleviate the problem with degradation due to driven response, it seems
plausible that the subspace detectors can be further enhanced by incorporating that tempo
ral information.
5.1 Target Detection Using Temporal Information
Subspace CFAR detectors based on generalized likelihood ratio test (GLRT) have been
investigated in (56) and (62) for detecting targets resonance response in UWB SAR
images, but they utilize only either the ID or 2D resonance response information. How
ever, the large energy contained in the driven response of the non-metallic objects like
foliage degrades the performance of these CFAR detectors.
96


35
spatial information in the neighborhood of tested pixels, where G. is the k-border
Gamma kernel, and each of its element gk[i,j] is given by
gkiW.I ^ry.(*Ji2+j2)k exp+j2) (ij) e Qv
2 7i k\
where p is the parameter to control the region of support of the Gamma kernels. Suppose
we have the data image X with the pixel under test in the center and two different order
Gamma kernels G; and Gc. The y -CFAR detector can be written as
rj *)2
where x = vect{X}, gt vect{Gt) and gc = vect{ Gc}. Comparing the form of the
CFAR detector with that of the y -CFAR detector, we can see that the two parameter CFAR
detector is extended by replacing the term gfx for the single intensity of the tested pixel
x. Besides, the local mean estimation gTcx and variance estimation JgTc(x x) (gTc x)2
is substituted for p and a, respectively.
Although the extension to y-CFAR detector was heuristic, the y-CFAR detector has
been shown to significantly outperform the CFAR detector [Principe et al, 1998a], with
the choice of gt = gx = vect{G,} and gc = #15 = vect{G]5}. The 3D windowing
function and the corresponding 2D region of support of the y -CFAR detector are shown
Figure 16-(a) and Figure 17-(a), respectively. The extension is intuitive in two aspects: g,
is chosen for gt, since G, has a peaky shape like $[nx, ny], which is used as G; in the
CFAR detector as illustrated in Figure 16. Besides, g15 is chosen for gc in that G15 has


92
Weighted Laguerre
Subspace Detector
2D Laguerre
Subspace Detector
1D Laguerre
Subspace Detector
Figure 49. The ROC curves of subspace detectors
The weighted Laguerre subspace detector has the best performance (5010 false alarms
for Pd = 1), while the ID Laguerre subspace detector performs worst (25368 false
alarms for Pd 1). Although the 2D detection scheme utilizes more information in the
cross range than the ID scheme, the performance enhancement is marginal. The resonance
response theoretically exists along several down range profiles but apparently the driven
response of trees also enhances comparably the clutter detection statistics. The results sug
gest that the more down range profiles in the 2D scheme only provide the merit of statis
tics averaging. However, the detection performance is largely improved by using the
weighting mask in the subspace energy domain, especially in the operating range where
Pd is close to 1 as shown by the ROC curve of the weighted Laguerre subspace detector.
This result suggests us that the better discriminant power between targets and clutter, espe
cially the clutter with severe interference, comes from the weighting in the Laguerre
power spectrum.


61
al, 1998b]. This procedure to determine the guardband will make the CFAR test more
robust and should improve performance even more.


53
1 st order Gamma kernel
1 st Principal component
BMP2
T72
Figure 25. The actual ID slice of the modeled intensity pattern in the radial
direction for the three types of targets in the MSTAR data set
3.4 Individual Target Size Estimation
First, we will present an example to see why the y-CFAR or the CFAR detectors perfor
mance is closely related to the guard-band size. The previous two targets near noisy clutter
with their radial intensity projection are shown in Figure 26. The guard-band size used by
MIT Lincoln Labs ATR system is fixed at 42 x 42. Notice that in both cases the clutter
within sample 42 has higher local mean or variance. It would lead to a contaminated test
ing statistics resulting in the miss detection of those two targets. Suppose the guard-band
size can be set around 30, where most of the target energy are included. Then the esti
mated mean and variance of the local clutter would be smaller, and the testing statistics
would be higher so that the two targets would more probably be detected. Our purpose in
the following is to dynamically determine the guard-band size for the stencil for each tar
get, instead of finding a single guard-band size through off-line training. This goal means
that all quantities must be estimated for each image chip.


87
where S isa NxN diagonal matrix of rank m indicating the locations of known com
ponents. Then the measurement model can be written as
yr = LaNZ + n
- LSa + n
If we apply a linear transform L T to y, then we get
z = LTy = a + v v = LTn
Comparing with the derivation of (56), the testing statistics for this problem can be simpli
fied as
zTSz
" 2
a
5-
<
Hn
(63)
where T is the threshold for tr, and a can be estimated from the neighboring Nu x 1 sam
ple vector u by
cf = (uTu)/Nu.
Basically, (63) utilizes the information in S, and we dont assume any apriori informa
tion about how targets are represented in the signal subspace. To enhance the performance
of the detector, we propose to apply a deterministic MxM diagonal weighting matrix
W = diag([wQ, j]) with ||FF|| = 1 and w0, ...,ww_, >0 to the transform rep
resentation q, so that now the new representation can be written as
Wh = W2LTy = W'/2a+W'/2v


44
Combining (36) and (37), then we get
E{2} = E{c[hy-{h¡xf]}
= c[(/w2 + a2) (m2 + (h£hc)o2)]
= c[l hTchc\a2
= a2
3.2.2 The Detector Preserves the CFAR Property
To say that a detector has the property of a CFAR test requires showing that, under H0,
the associated probability function of the testing statistics is independent of the parameters
for the assumed noise probability density function (PDF). That is the same approach used
by Robey [Robey et al, 1993] to show that the AMF detector preserves the CFAR statis
tics. In our case, the testing statistics t in (32) can be shown as a CFAR test, if, under H0,
its probability function f(t) is independent of the parameters, mean m and variance a2,
of the assumed Gaussian noise.
Notice that (32) can be written as
[(hfx-hTx)/a]
t = (38)
Jh£[x x]/a2
It can be observed that the numerator of t, (hfx hTcx)/a, can be written as
(hjx hTcx)/a
-
= h¡v-hTcv
(39)
where v = (x mi)/a, denoted as v = [v0,..., J7", and its elements v0, ...,
are normalized independent variables with zero mean and variance 1. The numerator can


ACKNOWLEDGEMENTS
There are many people I would like to acknowledge for their help in the genesis of this
manuscript. I would begin with my parents, Chi-Meng Yen and Ai-Jui Sun, for their end
less encouragement and support over the last six years. My little brother Li-Chiang Yen
has also constantly been providing constructive advice. Without their standing strong
behind me, it would be impossible for me to finish this thesis.
I would like to acknowledge my advisor, Dr. Jos Principe, for providing me with an
invaluable environment for the study of target detection and excellent guidance through
out the development of this thesis. His influence will leave a lasting impression on me.
I would also like to thank the students, past and present, of the Computational Neu-
roEngineering Laboratory. The list includes, but is not limited to, Chuan Wang, Doxing
Xu and Quin Zhao for useful discussions on signal processing theory, and Albert Hsiao
and C. Pu for providing much needed recreational opportunities. There are certainly others
and I am grateful to all.
Finally, I would like to thank my girl friend, Bernice, for sharing my joys and tears in the
last two years of my Ph. D. journey. This memory is the thing I will endear forever.
n


APPENDIX B
THE STATISTICAL PROPERTY OF THE SUBSPACE DETECTOR
Let denote the known component subspace by Hs = SH, and the unknown component
subspace by Hc = (I-S)H. Then [HS,HC] is the matrix whose columns form a basis for
the full signal space. Lets also partition L the compatibly with Hs and Hc, and denote its
respective blocks by Ls and Lc. Its obvious that the projection matrix PL can be given
by
pL = pl,*pls)t[lspls)tT'lsp,
Notice that L is chosen as the left inverse of H. That means
PLHS = 0 and PHS = H*
Then PLH, the projections of the known component subspace H onto the column space
of Ls can be given by
PpLH = pUL)TWPi,W}'vPl,
It can be shown that the numerator of the proposed testing statistics t could be written as
ztQ 'z zcTQc~lzc = yTPplHy
134


Page
3.4.2Experiment Results 56
3.5 Conclusion 60
4 SUBSPACE DETECTION OF TARGETS IN UWB SAR IMAGES 62
4.1 Target Detection Using Laguerre Networks 63
4.1.1 The 1D Resonance Response Model 64
4.1.2 Formulation of the Proposed ID CFAR Detector 66
4.1.3 Detection Scheme and Simulation Results 68
4.2 Target Detection Utilizing A Spatial Template 79
4.2.1 The Spatial Template Model and the Detector Derivation 80
4.2.2 The Detection Scheme and Simulation Results 82
4.3 Subspace Detectors Extended with Data-Driven Templates 86
4.3.1 Formulation of the Weighted Subspace Detector 86
4.3.2 Subspace Parameter Tuning and Template Finding 88
4.3.3 Simulation Results 91
4.4 Conclusion 94
5 NEURAL NETWORK APPROACHES TO TARGET DETECTION .... 96
5.1 Target Detection Using Temporal Information 96
5.2 The Targets Temporal Template Model 97
5.3 Sequential Detection Fusion Using A Neural Network 99
5.4 Training for the Fusion Neural Networks 100
5.5 Simulation results 101
5.6 Conclusion 105
6 QUADRATIC LAGUERRE DISCRIMINATOR 106
6.1 Two Stage Detection Scheme 108
6.1.1 Prescreener: 1D Gamma-CFAR Detector 108
6.1.2 Quadratic Laguerre Discriminator (QLD) 112
6.1.3 Temporal Training for the Discriminator 115
6.2 Discrimination Using Damped Sinusoidal Subspace 117
6.3 Simulation Results 119
6.4 Conclusion 121
7 CONCLUSION 124
IV


50
ponent of xn is the radial projection along radial distance r, the corresponding element in
r. The criteria used for principal component analysis (PCA) is given by
J = argmax
u
N 1
S h-""7*!2
i = 0
(49)
where u is the eigenvector with the largest eigenvalue of the data correlation matrix
^xj(xj)T. For the three types of the targets BMP2, BTR70, and T72 in MSTAR data set,
the largest eigenvectors are shown in the most right hand side of Figure 22, Figure 23, and
Figure 24, respectively. To implement the intensity detector in (42) by means of DFT fil
tering, h(r) rather than h(r) = rh(r) in (43) is needed, and the corresponding signatures
in the discrete spatial domain are shown in the Figure 25.
The previous results show that the intensity detector with the Gamma kernels works
well, so we rather prefer to use the Gamma kernels to model the intensity pattern since it is
a computationally simpler operation. The projection modeling can be written as
J =
argmax
k, p
N-
\\Xi-ai tffcJ
i = 0
Q:
i = Xi
(50)
where gk, n = rgkyL. The components in gk, M can be specifically written as
_ M*+1 / .
gk,r) = r gK^r) r (r)i_1 exp( pr).
2nk\
To implement the intensity detector in (42) by means of DFT filtering, its necessary to
get h(r) rather than h(r) = rh(r) in (43). Thats the reason why gk,^ is used. The best
gk, n is found through parameter searching with respect to both k and p. In our experi-



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Page
7.1 Summary 124
7.2 Future Work 129
APPENDICES
A THE STATISTICAL PROPERTY ASSOCIATED WITH THE LOCAL
INTENISTY KERNEL TEST 131
B THE STATISTICAL PROPERTY OF THE SUBSPACE DETECTOR ... 134
REFERENCES 137
BIOGRAPHY SKETCH 142
v


56
3.4.2 Experiment Results
The enhanced image chips, their radial projections and the estimated target size for the
two targets BMP2 and T72 in MSTAR data set are shown in Figure 28, respectively, while
those for the two embedded targets in TABLIS data set are shown in Figure 29. From the
radial representations, it can be immediately observed that the SNR is improved due to the
spatial filtering. The SNR for the different data are listed in Table 1 where the average
SNR is improved approximately by 4.6 db. Form Table 3 the estimated size for each data
set is obtained with NMSE around 15%. That provides us with enough information for the
setting of the guard-band size, since the guard-band size can be set 15% larger. More
importantly, most of the information in our scheme exists already to compute the y -CFAR
detector. There is not too much overhead involved.


119
, and
(81)
6.3 Simulation Results
The ROC curves of three detection schemes are shown first in Figure 65: the ID y-
CFAR detector in (71), the weighted Laguerre subspace detector in (64), and the subspace
detector in (56) with Gabor subspace. The data for simulation is the same data set as
described in 6.1.1. The parameters used in ID y-CFAR detector are p = 0.9 for gt, and
p = 0.35 for gc. The weighted Laguerre subspace detector uses the Laguerre subspace
with kernel up to 15 and the feedback parameter equal to 0.7. The first 15 damped sinuso
idal bases are used for the Gabor subspace detector, while the damping factor X is fixed at
0.9 to save the computation complexity.
Pd
o.o
O. 6
O 8
1 O
1 O
1 o
1 o
FAJkm2
1D y CFAR detector
Weighted Laguerre
Subspace Detector
Gabor subspace detector
Figure 65. The ROC curves of the subspace detectors


8
Figure 3. Typical response of a metallic object (a dihedral) in UWB.
Magnitude
Figure 4. Typical response of a non-metallic object (a tree) in UWB.
The response of a resonant scatter to an incident wideband pulse will generally be com
posed of two temporally distinct parts, referred to as the early-time or driven response and
the late-time or resonance response. The driven response is the echo of the incident pulse,
caused by local currents driven on the surface of the object; alone, it does not convey a
great deal of information about the scatter. The resonance response is a ring down of the
natural frequencies of the target excited by the incident pulse. These natural frequencies
are a function of the electrical dimensions of the metallic object.


2
Automated target-recognition (ATR) systems generally require a large number of resolu
tion cells on the target to achieve a specific performance level. Thus, the desire for
improving ATR capabilities has motivated increases in system bandwidths, operating
wavelength, and aperture sizes. To improve §range, the system bandwidth A/ can be
increased by the use of a chirp signal with high carrier frequency in MMW radar, or
directly by the use of an impulsive signal with large bandwidth in UWB radar.
Shorter wavelength X suffers from increased scattering and atmospheric attenuation.
Therefore wavelength must be operated in some desirable range. For a fixed operating
wavelength X of the radar, one can improve azimuth resolution by decreasing the range to
the target or increasing the aperture size in the azimuth dimension. Since target range is
usually dictated by operational constraints, and thus not subject to alteration, aperture
modification remains the only option for improving azimuth resolution 5azimutf,
In the real-aperture case, it is not practical to achieve high resolution by simply increas
ing the physical aperture; for example, at X = 1 ft and R=1000 ft. (short range in most
applications), a 1000-ft. antenna would be required to achieve an azimuth resolution of 1
ft. Aperture sizes such as these are often impractical at fixed sites, and entirely unreason
able in mobile applications. Thus, the ability to synthesize a large aperture by modeling it
as a coherent, linear array of smaller antennas is critical to achieving reasonably high azi
muth resolutions.
1.2 SAR Imaging Model
The two-dimensional SAR imaging model, as shown in Figure 1, is considered. The
vectors (x,y) and (kx, ky) are used to identify the spatial coordinate and the spatial fre
quency, where x and y -coordinate corresponds to the slant range and cross range, respec-


54
Figure 26. The two targets of Tablis24 data embedded in the clutter of Mission 90
data set. The vertical line shows the MIT stencil to estimate the local
statistics.
3.4.1 Individual Target Size Estimation Procedure
Essentially, the intensity detector match the intensity pattern with the radial projection of
data image above the estimated noise mean. In that sense, the target extent meaningful
to the intensity detector is where most of the intensity of targetss radial projection lies.
We select this radius as the mean intensity of the local clutter level. Inspired by the results
in the last section, the following 4-step scheme is proposed to estimate the target size by
determining where targets radial projection falls off the estimated noise mean.


86
4.3 Subspace Detectors Extended with Data-Driven Templates
In the previous two sections, the ID and 2D subspace detectors are presented. We
showed that there is a chance to improve discriminability of targets if the right signal sub
space is chosen appropriately. The question is how to find the most appropriate signal sub
space for a collection of targets. In the proposed subspace detectors, there exist unknown
template matrices such as S in (56) to describe the energy distribution in the subspace for
each target in the data set. Its implied by the previous transient signal detection algo
rithms that the we can use the 1 component in the matrix S to indicate if the correspond
ing axis extracts the targets projection energy or not. It is a rough representation since a
hard-limit decision instead of soft decision is used. But most importantly, a procedure to
find the relevant basis is not easy due to the variability of responses among the targets.
This section demonstrates how the subspace detector can be derived by incorporating a
soft decision template matrix, where the elements are rational numbers between 0 and 1,
and that corresponds to the target subspace. A data-driven approach is proposed to build
the parameters and template matrix S. In the end, we will compare the performance of all
the proposed detectors in terms of their receiver operational characteristics (ROC) curves.
4.3.1 Formulation of the Weighted Subspace Detector
Lets assume that the ideal resonance response y belongs to a known M-dimensional
orthogonal signal subspace represented by a N x M matrix, L, i.e. y = La + n, where a
is the representation vector. Suppose y concentrates on only m out of the M bases. Let
the nonzero representation vector be denoted by aNZ. Then it can be given by aNZ = Sa,


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140


66
Then the measurement model is given by
y HaNZ + w
= HSa + w
Consequently, the transformed vector z can be rewritten as
Z = Ly = LHSa + Lw
v
= Sa + v
where v = Lw is the colored Gaussian noise vector with covariance g2Q, where
Q = LLT. (54) is a well known subspace model described by [Scharf, 1988],
4.1.2 Formulation of the Proposed ID CFAR Detector
Based on the previous signal model, we have the following two hypothesis testing prob
lem
H0 : z =v v ~ iV(0, a2Q)
H, : Z = Sa + v v~N(0,o2Q)
The GLRT statistic is defined by
t = max{2\ogfx(z)} 21og/0(z)
where /,(z) and /0(z) are the corresponding probability functions under H0 and Hj,
respectively. Then, we have
21og/i(z) = -log(27t)-log|0|-(z-a5)r Q \z-as)


105
Through these preliminary results, the idea of exploiting the structure of the UWB
response from metallic objects by fusing the driven response with the resonant response
seems to improve the accuracy of the focus of attention. It is worthy making more efforts
to explore the temporal information to enhance the performance.
5.6 Conclusion
The large energy contained in the impulsive driven response of clutter would degrade
the performance of CFAR detectors based on spatial information. To alleviate this prob
lem, the strategy of temporally combining two sequential subspace CFAR detectors is
explored, where the front one is to detect the early driven response and the successive one
is to detect the delayed resonance response. The two detectors are integrated by a neural
network to capture the temporal information. The testing results of this detection scheme
combining two successive subspace detectors is improved, compared to that of the intrin
sic subspace detector, but it is still not satisfactory enough.


130
However, their optimal training for detection is still under intensive research. But there
has been recent evidence that ANNs outperform the quadratic discriminant function in
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discrimination based classifiers.
In terms of testing on SAR imagery, there is a lot of work to be done. I have not tested
the performance of the adjustable y-CFAR and this should be done to analyze the poten
tial improvement advantage of the technique. The testing on UWB SAR of all the detec
tors would also require more diversified and larger databases. The discriminators utilize
information from the training set, so their performance is highly dependent upon data
availability.


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138


10
For example, consider a resonator illuminated by a wideband pulse occupying a band
from flow and fhigh If the resonator has fundamental frequency and radiating harmonics
at (on, then the resonators response can be described by
(9)
n
From the above equation, we can see that only metallic man-made objects in UWB radar
produce a damped sinusoidal response due to the wideband excitation. This phenomenol
ogy of the reflection of the target is quite different from that in the conventional SAR. As
we can see in Figure 3 and Figure 4, the resonance response is a key feature for metallic
objects. It seems that a very promising approach for concealed target detection in UWB
SAR images is to exploit the resonance response of metallic objects.
1.5 UWB SAR Image Focusing
In this section, the principles of focusing the SAR images with some pulse p(t) trans
mitted is discussed. In UWB radar, the transmitted pulse p{t) is a wideband signal with an
impulsive waveform, and the range resolution requirement can be met with the large band
width. Now, the p{ oo ) in (2) is of large bandwidth and no longer can be assumed to be nar
rowband. Thus, to focus the UWB SAR image, the first step is to produce
s(u, co) = s(u, co)/P( the simpler Fresnel approximation is no longer appropriate. Instead, the more complicated


95
feedback parameter in the Laguerre kernels by finding the value that provides the most
peaky response. We presented the first realistic results for targets in clutter using the UWB
data. The data set contains 50 targets and covers 2 {km)1 of ground, which is limited but
it is the only data available to conduct these tests. For this reason we had to utilize the
same targets for training and testing, which can not be considered a true performance test.
However, the clutter was not utilized for our design.
We found out that the ID detector performs poorly, producing almost 30,000 false
alarms for a Pd=l. Adding the cross range information barely improves performance,
which means that the 3 dB gain in target response is matched by an equal increase in clut
ter response, eliminating the potential advantage. However, we showed that the weighting
of the response in the Laguerre subspace by the eigenvector of the target responses
improves performance by an order of magnitude (at Pd=l false alarms are now 5,000). We
can explain this improvement by the matching of energies.
Overall we consider that the performance of the weighted Laguerre detector is still not
appropriate because it creates too many false alarms. All the detectors developed in this
chapter operate with the driven and resonance response information. We know that one of
the discriminant aspects of UWB is the resonance response of metallic objects, so we have
to exploit this information to improve detectability.


26
transfer function L0 ^(z). Let the response of each stage k be denoted by xk(n). The dif
ference equations for computing the response of each stage recursively are as follows
**+i () = AX*+i(II-l)+JC*(-1)-Jjfc()
I (20)
x0() = px0(n 1) + Jl p2x() 0 The complexity of the above recursive formula is only O(K), where K is the highest
order of Laguerre kernels. Compared to the complexity of Gabor bases, 0(N?> ), the com
putation load is largely reduced. Due to the simplicity of recursive implementation and
good modeling for the damped signals, we will focus on using Laguerre recurrent net
works to extract the projection information of the original resonance response and develop
our detection algorithms based on it.
In the following, we will see the general subspace representation of a targets response
using Laguerre bases, while the signal representation by the Gabor functions is also shown
for comparison. Suppose we have the Nx 1 signal vector
x(n) = [x(), ...,x( + iV l)]r, and it belongs to a general NxM subspace S, where
the column vectors of S are the modeled kernels like Laguerre bases or Gabor bases. That
means Jt(n) = Sa, where a is the M x 1 projection vector, and it can be obtained by
a = 0STSY'STx{n).


80
4.2.1 The Spatial Template Model and the Detector Derivation
The resonance response of a target in UWB is spatially contained in J neighboring
down range profiles yx,y2, ---^j, where each down range profile y- is a Njx 1 vector.
So, we can collect these J different down range profiles to constitute a N x 1 target tem
plate vector
y = Ly[,y[, JjY
where N is given by N = + ... + Nj Notice that the components yj of y do not need
to be adjacent to each other, and Nj doesnt need to be equal to N¡ when i*j. Then the
resonance response template vector y can be written as
y = X + W (58)
where x = [jcf,jcJ, ...,*J]r and W = [w\, w>\, ...,ivj]r are the corresponding ideal
signal vector and the noise vector respectively. Both are of dimension N. Usually, is
assumed to be white and Gaussian, with zero mean and unknown variance ct2 and w is
independent of w-, when i *j.
Each ideal signal component x of X is assumed to belong to a known Mj -dimensional
signal subspace represented by a x Nj matrix, Lj If we apply a linear transform Lj
to the yj, then we get
/.>, Sja.+ Vj (59)
where a¡ is the representation vector, and S a N- x TV. matrix of rank K¡ describing the
J J J J J
locations of known components. At most one element in each row and each column would


CHAPTER 7
CONCLUSION
7.1 Summary
In this study, we concentrate on developing the focus of attention stage in a multi-stage
ATD/R system for MMW and UWB SAR images. The focus of attention stage in an ATD/
R system is composed of two substages: a front-end detection stage and a false alarm
reduction stage.
The front-end detection stage employs a prescreener to nominate potential target loca
tions in the image. For MMW and UWB SAR images, targets always contain point scat
ters with large reflectivities. So detectors based on local intensity tests are recommended
for their simplicity and robustness. In this dissertation, the generalized likelihood ratio test
(GLRT) is extensively utilized because of its relation to the Neyman-Pearson test and ver
satility. In fact, the formulation of all the detectors proposed in this work was done based
on the GLRT. Moreover, the analysis of the detectors properties is also done from GLRT
ideas.
Using the GLRT approach, I showed that both the y-CFAR and the two-parameter
CFAR detectors can be cast into the same framework of a matched filter between the local
image and the detection kernel. To find the best intensity matching kernel, we apply prin
cipal component analysis (PCA) to the radial intensity profile of several targets. We exper
imentally show that the first eigenfunction of the radial intensity profile of targets can be
124


94
4.4 Conclusion
This chapter introduces our first formulation of a detector for UWB SAR. We utilized
the well known concepts of transient detection using GLRT and applied them to the design
of targets in UWB. Due to the preliminary evidence established in Chapter 2 that Laguerre
basis represent well the resonance response of targets, our detection scheme is centered
around Laguerre kernels, with the added advantage of fast computation.
We first established that the ID Laguerre detector works for nonstationary noise, since
this is a condition found in realistic UWB clutter. We then analyzed the responses of man
made clutter (a power line), a target and tree reflections in a 1 -D profile of a real UWB
image. We conclude that a value of p=0.7 produces good concentration of energy in the
Laguerre subspace, because it matches the combined (driven and resonance) response of
the objects utilized. We also showed that it is possible to find different subspaces where
the responses of man-made clutter, targets and trees are emphasized. This means that sub
space projection should be utilized to enhance discriminability of targets from man-made
clutter and trees. The issue is how to select a general methodology to find these subspaces.
The next step was to investigate if the use of the cross-range information would improve
detectability of targets. We extended the detector formulation to 2D templates, and found
that for a simple example, in fact the detector outputs where higher and more distin
guished from clutter.
Finally, we addressed a more general way to find the subspace matched to the signal sig
natures in the subspace. We propose to utilize again the ideas of eigendecompositions, and
proposed the largest eigenvector of the target training set as the weighting vector to deter
mine the signal subspace. We also presented a methodology to find the best value of the


5
Based on the system model in (2), SAR image focusing algorithms can be regards as to
restore the original reflectivity function f(x,y) from the received signal s(u, co), which
is the deconvolution with the system function H(x, y) = exp \-j2kJx1 + y2]. Apparently,
the difficulty of analyzing the deconvolution is that the transfer function H(kx, ky) (Bessel
function of the first kind) is too complicated. Thus, in different cases, different focusing
algorithms approximate the spherical wave function with different and simpler wave func
tions so that the analysis of the deconvolution becomes easier.
1.3 MMW SAR Image Focusing
In this section, the principles of focusing the stripmap mode MMW SAR for surveil
lance purpose is discussed, which is applied to form the images like ADTS data set of
DARPA.
In MMW SAR, the waveform p(t) in (1) to be transmitted is a chirp signal
exp(j(£>ct +jKt2), where coc is the carrier frequency and k is the modulation constant.
With a high carrier frequency ooc, the chirp signal can be assumed to be operated with nar
rowband characteristics, while its bandwidth is still large enough to improve the range res-


3
tively, The transmitting/receiving airborne radar moves along the line x Xx, and it
makes a transmission and the corresponding reception at each position specified by
(Xx, 7, + u) for u g [L, L], where the slant range Xx and cross range 7, are constants.
Here, we assume that the radar stops, transmits a signal and receives the corresponding
reflection, and then moves to the next position. The assumption is unrealistic due to the
fact that radar is continuously moving. However, it is still a valid one since the aircrafts
moving speed is much smaller than the speed of light. The induced distance difference
between transmitting and receiving positions can be ignored.
While the radar illuminates the target area with a time dependent signal p(t), the round
trip delay x of the reflected signal received by the radar due to a point scatter at (x, y) is
x(x,y) = 2 R{x,y)/C = 2 J(Xx-x)2 + (Yx + u-y)2/C
where i?(x,y) is the distance between the radar and the point scatter. Thus the total ech
oed reflection received s(u,t) at each moving position can be written as
s(u, t) = \\f(x,y)p(t--J(Xx -x)2 + (7, +u-y)Adxdy
V 7 (1)
where f(x, y) is the target areas reflectivity function, and the integral is taken all over the
target region illuminated by the radar in the (x, y) domain.


72
ure 35-(a) and Figure 36-(a), while the resulting JL(n) for the foliage is shown in Figure
37-(a) for comparison. The Jpeak for the three objects closest to 1 occur at n = 564,
n = 3256, and n = 4096. Theoretically speaking, the power lines act like a metallic
object also producing a strong resonance response. It can be observed that the tree pro
duces a response which has an amplitude comparable but smaller than the other two
objects. However, we would like to investigate if the energy is concentrated in the same
bases as the target.
The corresponding distribution of the projection energy are shown in Figure 35-(b), Fig
ure 36-(b) and Figure 37-(b), respectively. We can see that the projections of the three
objects have a large overlap in the bases, but there are some differences. The power line
has a dominant component in the first Laguerre basis, with a second emphasis on bases
around 7. The target has a clean peak around order 7 and a second concentration around
order 15, while the tree has the largest projection around order 15. So, there are significant
differences in the energy pattern in the subspace that can be further explored to improve
the discrimination of the targets from the clutter.


99
where Td is the threshold. With the Laguerre functions as the bases of the projection
space, the implementation of our subspace detectors is shown in Figure 51. The down
range profile is sent to two Laguerre delay lines that implement the subspace projection.
Each tap output is a projection on a basis. In order to implement (67) and (68), one has to
select what are the taps that contain most of the information about the resonant response
(and the driven response). These taps constitute the signal subspace to implement the
GLRT. The remaining taps represent the null space for the signal.
5.3 Sequential Detection Fusion Using A Neural Network
The sequential detection fusion can be viewed as a two-hypothesis detection problem
with two individual detectors for the driven and resonance responses, respectively. The
optimum decision rule to fuse the detection results td and tr is given by
/(Wj, w2) = sign(a0 + <2jU| + a2u2) (69)
where w, = sign(tdTd) u2 = sign(trTr) The optimum weights are given by
[Chair and Varshney, 1996]
a0 = logCPj/Po)
, l-p
ai lQg P
^Fi
if U¡ = + 1
1 ~Pp.
ai ~ l0§ n '
JC
II
where PM and PF are the miss detection probability and false alarm probability for the
i th detector, respectively. However, the above optimal fusion rule implies fixed local
detectors with preset thresholds. Moreover all the weights are pre-calculated based on the
theoretical signal distributions which is unrealistic for UWB SAR. The neural network


107
realization of detecting resonance response yr is a subspace detector which tests the ratio
of the projection energy of resonance response in the signal subspace to that of the back
ground noise. For foliage or non-metallic objects, the only response is the driven response
yd, and no resonance response yr in y.
The energy contained in the driven response is usually larger than that contained in the
resonance response. Although detecting the resonance response directly seems to be a rea
sonable approach, the normal GLRT detectors suffer from degradation due to the driven
response of foliage. In the previous chapter, a neural networks was proposed to integrate
the information in two temporally successive subspace detectors: one for the driven
response followed by the other for the resonance response.
The proposed temporal fusion approach is similar to the detection scheme in the ATR
system [Novak et al, 1993], where, to cope with the non-Gaussian clutter environment, the
detection system is divided into two stages: prescreening and then discrimination. It has
been shown that in MMW SAR images, the y-CFAR detector followed by Quadratic
Gamma Discriminator is a very effective two stage scheme. Both y-CFAR and QGD uti
lize the spatial information of the reflection of targets point scatters, which only corre
sponds to the driven response information in UWB radar. Since it is the resonance
response information of targets that differs from that of the foliage, the discriminator has
to be redesigned to utilize the temporal information in UWB SAR images.


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.
Joseph N. Wilson
Assistant Professor of Computer
and Information Sciences and
Engineering
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.
August 1998
Winfred M. Philips
Dean, College of Engineering
M. J. Ohanian
Dean, Graduate School


97
A strategy to temporally combine two subspace CFAR detectors is explored, where one
detects the early driven response and the other detects the delayed resonance response.
The two detections are further integrated by a neural network to capture the temporal
information. Although some algorithms are proposed to optimally fuse several statisti
cally independent detections, they are either only optimal for fixed threshold local detec
tors or for unrealistic distributions. The previous fusion rule in [Chair and Varshney, 1996]
is extended using a neural network, so that all the weights and threshold could be adapted
by the training data to improve performance.
5.2 The Targets Temporal Template Model
It is known that the ideal response x of a metallic object in UWB SAR image can be
temporarily decomposed into two components: the driven response and the resonance
response. The driven response always precedes the resonance response, which only con
tains damped sinusoids. The total response x can be temporarily divided into 2 separate
regions like x = x jj where xd is the Nd x 1 driven response, and xr is Nr x 1
resonance response. Let y be an N x 1 column vector in the down-range profile of the
UWB SAR image. The measurement vector y is assumed to be formed by the total
response x and a Gaussian noise vector w. Obviously, N = Nd + Nr, where Nd and Nr
are usually target dependent. Now, y can be written as
(66)


52
Figure 23. The Modeling of the Radial Projection of BTR70
Figure 24. The Modeling of the Radial Projection of T72


58
Image Enhancement
J
Radial Projection
i
Figure 29. The associated processed data involved in target size estimation for the
two embedded targets in TABLIS24 data set.


98
where wr and wd are the corresponding noise vectors. Note that for foliage or non-metal-
lie objects only the driven response yd exists. The energy contained in the driven response
is much larger than that contained in the resonance response, which hinders the design of
detectors based on the resonant response.
Lets assume that the ideal resonance response xr belongs to a known Mr -dimensional
orthogonal signal subspace represented by a Nr x Mr matrix, Lr. If we apply linear trans
form Lj to the yr, then we get
zr = LTryr = Srar + vr
where ar is the representation vector, and Sr a Mr x Mr matrix of rank Kr describing the
locations of known components. At most one element in each row and each column would
be equal to one, and the remaining of the elements are zero. vr = Ljwr is the colored
Gaussian noise vector with covariance Lj.Lr o?Qr. Following the same approach
leading to (56), the GLRT testing statistics for this problem is given by
T-\
t, =
z,Q
r r Zr
T -1
: Qcr zcr
< Tr
(67)
where Tr is the threshold for tr. QLr is the correlation for the null space of the resonant
response (dimension MrKr). In our UWB SAR scenario, we assume is unknown,
A
and can be estimated from the neighboring p x 1 sample vector u by = (uTu)/p.
Similarly, for the driven response, the GLRT testing statistic can be written as

zTdQd*d ~ zcdQcd
d
<
Hn
(68)


6
olution. Discarding the carrier frequency components of the chirp signal, s(u, eo) in the
system model is given by
s(u, co) = s(u, co)/P(go) = s(u, (o) P*(co)
In other words, s(u, co) is formed by matched filtering the raw data with the chirp signal
in the frequency domain. With the approximation J\ + q 1 + q/2 for q 1, the Fresnel
approximation used in the system model for the spherical wave function H(x, y) in the far
field is given by
2kJ[X,-x)l + (u-y)l 2k(X,-x) +
tAl ~~X)
(3)
The minus sign in (2) can be absorbed into k without loss of generality, and then the sys
tem model with Y, = 0 becomes
s(u, co) w {]/(*, T)exP[/2£(^ ~x)] exp
Xu-y)2'
L ~x)J
dxdy
With direct algebraic simplification, one can get the following relationship
s(u, co) w exp[/2&Wj] exp(ju2/Xx) Fx(2k,u) (4)
where Fx(2k, u) = j/(x, y) exp [2jkx] dx and denotes convolution in the u domain.
Notice that (4) suggests s(u, co) is the signal generated by blurring the signal Fx(2k, u)
with the chirp signal exp [ju2/X] ] and its carrier frequency exp [j2kX] ]. Taking a sim
ilar matched filtering approach to form s(u, co), we can match filter the signal s(u, co)
with (exp(ju2/X{))* as
Fx(2k, u) exp[j2kXx] Qxp(ju2/Xx) s(u, co)
(5)


55
(1) p = gTcx. The local mean ¡a is estimated by gTcx, where the guard-band size of the
Gamma kernel gc includes the largest target to be detected.
(2) Xg Gt <8> X, where means 2D circular convolution. Get enhanced image chip
by filtering the original image chip X with the first order Gamma kernel G,, the
approximate principal eigenvector. This processing basically can be viewed as spatial
eigenfiltering.
(3) jc*[r] = x[/,y']/N(r). Get the radial projection representation xs[r] of
i,j 3 i2 + j2 = r2
the filtered image data chip Xg .
(4) Find the smallest r, such that xR[r] < p.
The above scheme can be illustrated by
Figure 27. Target size estimation scheme in MMW SAR images.


110
symmetric Gamma kernel for gc. To get the best performance, different values of
pe {0.05, 0.10, 015..., 0.95} are used for and g15 in the y-CFAR detector. The best
combination found by exhaustive search is pi = 0.9 for g}, and p = 0.35 for g!5.
Figure 60 depicts the corresponding results of the ROC curves for UWB SAR images,
where we also show the ROC curve of the y-CFAR detector with the original stencil.
Notice that all the ROC curves of the proposed detectors are reported after clustering. The
raw detections are clustered within a user specified radius 5 m, and only the detection with
the largest statistics is presented for those clustered detections. It can be concluded that the
test with 2D front guard-band stencil has better performance. This test verified our reason
ing: the performance of y -CFAR with rear guard-band clutter stencil would be degraded
because the resonance response in the tails of the targets would increase the estimated
clutter variance.
1 .o
. > 1111111 i in
Pd e
0.4
- / / -
0.2
0.0
1 C
3 1 O 1 1 O^ 1 O'3 1 O * 1 C
FAJkm2
2D yCFAR with front guard-band kernel
-
- 2D yCFAR with full guard-band kernel
Figure 60. ROC comparison of 2-D y -CFAR processing with front and rear
guard-band stencil.


15
88M UWB SAR Images
Clutter Testing Data Chips
Clutter Training Data Chips
\^l
1
first largest 300 l_
false alarm detections L
2588 false alarm
locally peak detections
1., \
-
h n
-
==^
w
o
U-
Figure 7. The formation of the clutter chip sets for training and testing


120
As shown in Figure 65, the ID y-CFAR detector has the best performance among the
three proposed detection schemes. Notice that the weighted Laguerre subspace detector
has better performance than the damped sinusoidal subspace. It shows that the appropriate
template matrix in the subspace contributes to the better performance of the Laguerre sub
space detector than the Gabor subspace detector. However, the weighted subspace detector
and the damped sinusoidal subspace detector both have a poorer performance than the
intensity based ID y-CFAR detector. It hints that the interference of the driven response in
the clutter is so severe that the subspace energy detectors can not work as expected. To be
more conservative to detect a target by its driven response, the intensity based detector
like ID y-CFAR detector is a better choice.
Pd
0.-3-
0.2
0.0
1
Figure 66. The ROC curves of y-CFAR-QLD detectors
Next, the results of y-CFAR detector combined with quadratic Laguerre discriminator
and Gabor subspace discriminator are shown in Figure 66. Like the weighted Laguerre
subspace detector, the QLD use the first 15 Laguerre kernels with p = 0.7. For a compar-
FA/km2
1D y CFAR detector ~ QLD
- Gabor subspace discriminator


81
be equal to one, and the remaining of the elements are zero, v- = Ljwj is the colored
Gaussian noise vector with covariance a2LjLJ.
Lets denote the temporal representation vector by a = [aj, a\, ...,aJ]T. L is the
MxN template signal subspace matrix, which is a block diagonal matrix given by
L, 0
L = Ll
Lj
Then, from (59), the transformed temporal vector z = [z\, z{, ,zj]r can be given by
z = L/y = Sa + v (60)
where S a yxJV component selection given by
Sx 0
*2
Sj
V is a Gaussian noise vector with covariance matrix a2Q, where Q = LrL Based on
the previous signal model, we thus have the following two hypothesis testing problem
H0 = z =v v ~iV(0, a2Q)
Hj = z = Sa + v v ~ iV(0, ct2Q) (61)


16
88M UWB SAR Images
88M UWB SAR Images
Figure 8. The formation of the target chip sets for training and testing


39
3.1 Formulation Of Optimal Intensity Detectors
Observing targets in MMW SAR images, we can see that, in addition to the bright point
scatters, the whole target exhibits brighter reflection compared to the background because
of its metallic material. Although the intensity is not as large as that of the point scatters,
its reasonable to assume that there exists higher intensity around the point scatters of tar
gets, and target detection in the intensity image can be regarded as an intensity detection
problem. As a result, for the (2N+ 1) x (2N + 1) intensity image X, target detection can
be formulated as the two hypothesis testing problem:
Hi : x = aht+w w~N(mi,a2i)
H0 : x = w w ~ N(mi, a2/)
where ht is the intensity kernel, a is the unknown amplitude, x = vect{X}, i is the
(2N+ l)2 x 1 vector with all its component equal to 1, and w is assumed to be white
Gaussian noise, with mean vector mi and variance a2/. Both m and a2 are unknown sca
lar parameters to be estimated. Note that ht is assumed to be a known intensity kernel that
provides maximum detectability, and we will show later how to model the kernel from the
data set.
Without loss of generality, we can impose the constraint hji = 1. So, the likelihood
ratio test can be written as
= PrfrlHQ =
/V(*|H0)


74
(a) The JL(n) with Laguerre subspace
order k
(b) The normalized projection energy in the Laguerre bases
Figure 36. The Laguerre subspace representation for the vehicle in Figure 33 (a)
the JL{n) around the sample from 3150 to 3350 (b) the distribution of
the normalized projection energy in the Laguerre bases for the sample
n=3256


73
(a) The JL{n) with Laguerre subspace
order k
(b) The normalized projection energy in the Laguerre bases
Figure 35. The Laguerre subspace representation for the power lines in Figure 33
(a) the JL{n) around the sample from 550 to 750 (b) the distribution of
the normalized projection energy in the Laguerre bases for the sample
n=654


9
The resonance response phenomenon is best illustrated with a canonical example. The
spatio-temporal distribution of current along a thin-wire dipole of length L is described by
i(x, t) = Ie a,sin^^^jsin(coi)
where / is the value of the current at a current antinode, and co = 2nc/X. The boundary
condition requires i(x, t) = 0 at x = 0 and x = L, which leads to the condition
2kL
X
= kn
k = 1,2,.... For x = L/2, we get
i(L/2,t) = le asin(7i;;/2)sin(cD) k = 1,2,... (7)
The meaningful solutions to (7) exist for k odd. These are the so called radiating frequen
cies. If we use the variable n = (k+ l)/2 ,
i(L/2,t) = ^1 ne~a,'tnt) n = 1,2,... (8)
n
Unless the forcing function that initiated the current flow is reapplied, the factor in e ^
indicates that the current decays with time at a rate proportional to an, which is also
known as the damping factor. Thus, if a dipole in empty space is illuminated by a incident
pulse, resonance at the odd frequencies within the band of the illuminating pulse will be
excited, and will decay exponentially once the forcing function has been removed.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 INTRODUCTION 1
1.1 Motivation for SAR 1
1.2 SAR Imaging Model 2
1.3 MMW SAR Image Focusing 5
1.4 Motivation of UWB SAR: Resonance Effects 7
1.5 UWB SAR Image Focusing 10
1.6 SAR Image Data Sets 13
2 BACKGROUND 17
2.1 Representation of Signals In Gray-Scale Images 18
2.2 Representation of Bipolar Transient Signals 21
2.2.1 Gabor Bases 22
2.2.2 Laguerre Bases 23
2.3 The Neyman-Pearson Tests 29
2.4 Conclusion 32
3 LOCAL INTENSITY TESTS FOR OPTIMAL DETECTABILITY 33
3.1 Formulation Of Optimal Intensity Detectors 39
3.2 Statistical Properties Associated with the Intensity Detector 43
3.2.1 Unbiased Estimators 43
3.2.2 The Detector Preserves the CFAR property 44
3.3 Intensity Modeling of Targets 45
3.3.1 Mathematical Background 46
3.3.2 The Modeling Procedure and Experiment Results 48
3.4 Individual Target Size Estimation 53
3.4.1Individual Target Size Estimation Procedure 54
in


76
Figure 38. The effect of changing the number of bases to the subspace energy for
the three objects (a) the power lines (b) the vehicle (c) the tree
From these results, we can assume that the resonance responses of the power lines and
the vehicle are within the subspace expanded by first 15 order Laguerre kernels. Thus, we
can use the Laguerre recurrent networks of order 15 and parameter p equal to 0.7, since it
was observed that the Laguerre subspace constructed by the first 15 kernels captures 90%


40
Taking the log of both sides of the above equation, we get
t = -j^(\\x-aht-mi\\2-\\x-mi\\2)
Expansion of the right hand side of the above equation leads to
t Y^2(2hTtx ah*ht2hTtmi)
(26)
where r| = \\ht\\2. Here, we would take the generalized likelihood ratio test (GLRT)
approach. That means we would maximize (26) by using the maximum likelihood (ML)
estimate of a:
t = argmaxl
a [cr
h^x mi)'^a
Straightforwardly, the ML estimate of a is given by
a = hUx mi)
r\
Then, the estimated a can be plugged into (26) to yield
t
(27)
(28)


32
2,4 Conclusion
In this chapter, we introduced the Gamma bases for the signal representation in intensity
images like MMW SAR images. Moreover, Laguerre bases and Gabor bases are intro
duced for representation of transient signals such as the resonance response in UWB. In
the following chapters, depending on the different reflectivity characteristics of targets,
different bases are utilized to represent the target reflections, and then we will take the
GLRT approach to formulate the detectors based on the representation. All the proposed
detection schemes are simulated on the extensive data set: the MMW Moving and Station
ary Target Acquisition and Recognition (MSTAR) data set and UWB SAR image data set.


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
FOCUS OF ATTENTION FOR MILLIMETER AND ULTRA
WIDEBAND SYNTHETIC APERTURE RADAR IMAGERY
By
Li-Kang Yen
Aug 1998
Chairman: Dr. Jos C. Principe
Major Department: Electrical and Computer Engineering
The major goal of this research is to develop efficient detectors for Synthetic Aperture
Radar (SAR) images, exploiting the reflectivity characteristics of targets in different radar
types. Target detection is a signal processing problem whereby one attempts to detect a
stationary target embedded in background clutter while minimizing the false alarm proba
bility. In radar signal processing, the better resolution provided by the Millimeter Wave
(MMW) SAR enhances the detectability of small targets. As radar technology evolves, the
newly developed Ultra Wideband (UWB) SAR provides better penetration capabilities to
locate concealed targets in foliage.
In this thesis we demonstrate that local intensity kernel tests can be formulated based on
the generalized likelihood ratio test (GLRT), while preserving constant false alarm rate
(CFAR) characteristics. Both the widely used two-parameter CFAR and the y -CFAR can
be viewed as special cases of the local intensity tests with different intensity kernels. It is
vi


14
For the neural network based detectors, it is necessary to prepare the training and testing
data set, where target and clutter examplars have to be included. To prepare the clutter
training and testing data set, one single data run is prescreened by ID y-CFAR detector to
generate about 8000 false alarm detection points. These raw detections are clustered
within a user specified 5m radius, and only the detection with the largest statistics is pre
sented for those clustered detections. Therefore, there are 2588 clustered false alarm
detections being reported. Among these false alarm points, the 300 points corresponding
to the largest 300 detection statistics are chosen to make the training clutter chips. As
described earlier, there are 50 vehicles in the UWB SAR data set. The locations corre
sponding to the least 15 detection statistics are chosen for making training target chip sets.
Each image chip is of size 250 x 30. All the false alarm points and target locations are
used for generating the testing data set.


48
points whose indices are (0,1), (0,1), (0,-1) and (-1,0), respectively, with radial distance
r = 1, Thus we have A^( 1) = 4. In the same manner, we can get A^(1.414) = 4,
Nr(2.236) = 8, and etc.

(-2,2)
(-1,2)
(0,2)
(1,2)
(2,2)
(-2,-1)
(-1,1)
(0,1)
(1,1)
(2,1)
(-2,0)
(-1,0)
(0,0)
(1,0)
(2,0)
(-2,1)
(-1,-1)
(0,-1)
(1,-1)
(2,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)



2.828
2.236
2
2.236
2.828
2.236
1.414
1
1.414
2.236
2
1
0
1
2
2.236
1.414
1
1.414
2.236
2.828
2.236
2
2.236
2.828


Figure 20. The usual image indices with (0,0) in the image center (left), and the
corresponding radial indices (right).
3.3.2 The Modeling Procedure and Experiment Results
The problem left out is how to build a good intensity kernel model for p[r] using the
information fR[r\ computed from the target data set we had. As we mentioned earlier, the
features of targets which we are interested in for detection are around bright point scat
ters. Therefore, we propose the following 3-step procedure to build the intensity kernel
model p(r)
(1) align each target chips so that the brightest point is in the center
(2) get the ID radial projection of each target chips


REFERENCES
Ansari, N., Hou, E. S. H., Zhu, B., and Chen, J. G., Adaptive Fusion by Reinforce
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137


43
The main difference between the two-parameter CFAR and the y -CFAR detector is the
intensity kernel of the point scatters to be matched: In the two-parameter CFAR detector
case, the intensity kernel ht is simply ht[i,j] = 8[z,y], while ht = gt in the y-CFAR
detectors case.
3.2 Statistical Properties Associated with the Intensity Detector
3.2.1 Unbiased Estimators
The estimator 0 would be called unbiased for 0 if £{0} = 0. Let
x = [x0, i ]T. We assume that the elements of x, x0,..] are independent
Gaussian random variables with mean m and variance a2. That means E{x} = mi and
£{ac2} = a2i. In the following, we show that the two estimators m and a2 given in (31)
are unbiased.
First, we show that m is unbiased:
E{m} = E{hTcx} = hTcE{x)
= mhTci m
Secondly, we show that ct2 is unbiased. The first term in the right hand side of (34) can be
written as
E{hpc2} = hTcE{x2}
= (m2 + a2)hTci (36)
= m1 + a2
Then it can be shown (see Appendix A) that the second term in the right hand side of (34)
can be written as
E{(hTcx)2} = m2 + (hTchc)G2
(37)


24
CD 3
CD -<4-
CD ^2
Magnitude o.o
CD Z2.
CD .
CD e>
CD "I CD ^ CD 30 -n- CD S. CD
n (sample index)
Figure 12. the Laguerre kernels of different order k=0, 1, 2, 3 in the discrete time
domain
Magnitude
Normalized Frequency
Figure 13. the Laguerre kernels of different order k=0, 1, 2, 3 in the frequency
domain
The Laguerre sequence lk ^[] satisfies the relation
£ hJnVj^n] = Sy
n = 0