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## Material Information- Title:
- Focus of attention for millimeter and ultra wideband synthetic aperture radar imagery
- Creator:
- Yen, Li-Kang, 1967-
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- vii, 142 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Datasets ( jstor )
Discriminants ( jstor ) False alarms ( jstor ) Mathematical vectors ( jstor ) Matrices ( jstor ) Radar ( jstor ) Signal detection ( jstor ) Signals ( jstor ) Statistics ( jstor ) Stencils ( jstor ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- 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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- University of Florida
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- 40987530 ( OCLC )
ocm40987530
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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 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 |

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(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 Â£, = 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 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 (AÂ¡,fÂ¡) can be found now by two ID searches of DFT in the (AÂ¡,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 / = fÂ¡ 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(XÂ¡,f) N-1 e;: -2 Xn (79) (3)estimate a and update the subspace 1 = [4,4 4,aÂ¡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 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 gÂ¡ 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 Â£]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) = (XÂ¡, 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. Â¡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/Â£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 Â£ 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- [djÂ¡, 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 fÂ¡(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 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_Â¡ 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 Â£*( 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 kÂ¡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 Srinivasan, R., Designing distributed detection systems, IEE Proceedings, Part F: Radar and Signal Processing, vol. 140, no. 3, pp. 191-197, Jun 1993. Smoumekh, M., Echo imaging using physical and synthsized arrays, Optical Engi neering, vol. 29, no. 5, pp.545-554, May 1990 Stolt, R. H., and Weglein A. B., Migration and inversion of seismic data, Geophys ics, vol. 50, no. 12. pp 2458-2472, Dec 1985 Szego, G., Orthogonal Polynomials, American Mathematical Society: Cololquium papers, 1959. Traven, H. G. C., A neural network approach to statistical pattern classification by semiparametric estimation of probability density functions, IEEE Trans. Neural Net works, vol. 2, no.3, pp.366-377, 1991 Umesh, S., and Tufts, D.W., Estimation of parameters of exponentially damped sinu soids using fast maximum likelihood estimation with application to NMR spectroscopy data, IEEE Transactions on Signal Processing, v 44, n 9, p2245-2259, Sep 1996. Yen, L. K., Principe, J. C., and Xu, D., Adaptive target detection using Laguerre net works, Proceedings of ICNN, Vol 4, pp. 2072-2075, June, 1997 Houston Yen, L. K., Principe, J. C., and Fisher, J., Focus of Attention in UWB SAR images, Proceedings of SPIE, Vol 2760, pp. 523-532, June, 1996 Orlando Wang, Y, Chellappa, R., and Qinfen, Z., Detection of point targets in high resolution synthetic aperture radar images, ICASSP v 5 1994. p V-9-12 141 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 = Â£*[] 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)Â¡ 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 (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 Li, J., and Stoica, R, Efficient mixed-spectrum estimation with applications to target feature extraction, IEEE Transactions on Signal Processing vol. 44, no. 2, p281-295, Feb 1996. Li, J., and Zelino, E., Target detection with synthetic Aperture Radar, IEEE Transac tions on Aerospace and Electronic Systems, vol. 32, no. 2, pp. 613-627, Apr. 1996 Li, Y., Liu, J.R., and Razavilar, J., Parameter estimation scheme for damped sinusoi dal signals based on low-rank Hankel approximations, IEEE Transactions on Signal Pro cessing vol. 45, no. 2, pp481-486, Feb 1997. Lippaman, A. Steven, Elements of Probability and Statistics, Holt, Rinehart and Winstion 1971 Mahammad, M. A., and Ahmed, N., Optimal Laguerre networks for a class of dis crete-time systems, IEEE Tran, on Signal Processing, vol. 39, no. 9, pp.2104-2108, Sep. 1991 McCorkle, M., and Nguyen, L., Ultra wideband bandwidth synthetic aperture radar focusing of dispersive targets, Technical Report ARL-TR-305, Army Research Labora tory, Adelphi, M.D., Apr. 1994 McLachlan, G. J., and Basford, K. E., Mixture Models: Inference and Applications to Clustering, New York: Marcel Dekker, 1988 Miller, J. H., and Thomas, J. B., Detectors for discrete time signals in non-Gaussian noise, IEEE Trans. Inform. Theory, vol. IT-18, pp.241-250, Mar. 1972 Morgan, M. A., and Larison, P. D., Natural resonance extraction form ultra wideband scattering signatures, Ultra Wide Band Radar: Proceedings of the First Los Alomos Sym posium, p.p. 203-215, 1991 Nikias, C. L., Signal processing with alpha-stable distributions and applications, Wiley, cl995. Novak, L. M., Owirka, G. J., and Netishen, M., Performance of a high-resolution Polarimetric SAR automatic target recognition system, Lincoln Lab. Journal, vol. 6, no. 1, pp.11-23, 1962 Park, H., Huffel, S. V., and Elden, L., Fast algorithms for exponential data modeling, 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 'Â£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 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) JhÂ£[v <8> v] (hÂ£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 PpLyÂ¡ 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) + Â£(/*)) 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{xÂ¡Xj} = E{xÂ¡}E{Xj} = m2. Plugging (83) into (82), then (82) can be simpli fied into Â£{(AÂ£*)2} = f X N 2. "ci a2 + ( X \ G = o...at-i G = 0..W-1 y m- (84) = m2 + (hÂ£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) = XMÂ¡-)('-) 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- xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E6IZDMDQU_BAK1QH INGEST_TIME 2014-12-08T21:53:13Z PACKAGE AA00024492_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 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-{hÂ¡xf]} = c[(/w2 + a2) (m2 + (hÂ£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) JhÂ£[x x]/a2 It can be observed that the numerator of t, (hfx hTcx)/a, can be written as (hjx hTcx)/a - = hÂ¡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. 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FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ -RVHSK 1 :LOVRQ $VVLVWDQW 3URIHVVRU RI &RPSXWHU DQG ,QIRUPDWLRQ 6FLHQFHV DQG (QJLQHHULQJ 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW :LQIUHG 0 3KLOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ 0 2KDQLDQ 'HDQ *UDGXDWH 6FKRRO 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 Â§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, Parzen, E., On estimation of probability density function and mode, Annual Mathe matical Statistics, vol. 33, pp. 1065-1076, 1962 Picinbono, B., and Duvaut, R, Optimal linear-quadratic systems for detection and estimation, IEEE Trans, on Signal Processing, vol. 34, no. 2, p.p. 304-311, Mar. 1988 Principe, J.C., Radisavljevic, A., and Kim, M., Target prescreening based on 2D gamma kernels, Proceedings of SPIE, vol. 2487, pp.251-258, April, 1995 Principe, J.C., Kim, M., and Fisher, J., Target detection in synthetic aperture radar (SAR) using artificial neural networks., IEEE Trans, on Image Processing, vol. 7, no. 8, p.p. 1136-1149, Aug. 1998. Principe, J.C., Radisavljevic, A., Fisher, J., Hiett, M., and Novak, L., Target pre screening based on a quadratic Gamma discriminator, To appear in IEEE Trans, on Aero space and Electronic Systems Reed, I. S., Mallet, S. D., and Brennan, L.E., Rapid convergence rate in adaptive arrays, IEEE Trans, on Aerospace and Electronic Systems, AES-10, no. 6, p.p. 853-863, Nov. 1974 Reed, I. S., and Yu, X., Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution, IEEE Trans, on Signal Processing, vol. 38, no. 10, p.p. 1760-1770, Oct. 1990 Robey, F. C. Fuhrmann, D. R., Nitzberg, R., and Kelly, E., A CFAR adaptive matched filter detector, IEEE Trans, on Aerospace and Electronic Systems, AES-28, no. 1, p.p. 853-863, Jan. 1993 Sabio, V., Spectral correlation of wideband target resonances, Proceedings of SPIE, vol. 2484, p 567-573. 1995 Scharf, L. L., Statistical Signal Processing, Addison-Wesley. Scott, L. B., Moving-target detection techniques for optical-image sequences, Ph. D. dissertation, University of California, San Diego, CA, 1988 Shnidman, D., Radar detection probabilities and their calculation, IEEE Transac tions on Aerospace and Electronic Systems, AES-31, no. 3, pp 928-950, Jul 1995. Silva, T. O., On the equivalence between gamma and laguerre filters, International Conference on Acoustics, Speech and Signal Processing, vol. 4, pp. 385-388, 1994. 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 MMW SAR tests [Principe et al, 1998b]. In principle, self-organizing neural networks could be made to recognize different clutter environments and improve even further single 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. Ferguson, T.S., Mathematical Statistics: A Decision Theoretic Approach., New York: Acdemic, 1967. Friedlander, B., and Porat, B., Detection of transient signals by the Gabor representa tion, IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, no. 2, pp. 169-180, Feb. 1989 Friedlander, B., and Porat, B., Performance analysis of transient detectors based on a class of linear data transforms, IEEE Trans. Inform. Theory, vol. 38, no. 2, pp. 665-673, Mar. 1992 Frisch, M., and Messer, H., The use of wavelet transform in the detection of an unknown transient signal, IEEE Trans. Inform. Theory, vol. 38, no. 2, pp. 712-645, Mar. 1992 Gabor, D., Theory of communication, J. Electrical Engineering, vol. 93, pp.429-441, 1946. Goldstein, G.M., False-alarm regulation in log-normal and weibull clutter, IEEE Transactions on Aerospace and Electronic Systems, vol. 9, no. 1, pp. 84-92, Jan. 1973 Gottlieb, Concerning some polynomials orthogonal on a finite or enumerable set of points, America Journal of Mathematics, vol. 60, pp. 453-458, 1938 Haykin, S., Adaptive filter theory, Prentice Hall 1991, New York Haykin, S., Neural networks, McMillan 1994, New York Kapoor, R., and Nandhakumar, N., Features for detecting obscured objects in ultra- wideband (UWB) SAR imagery using a phenomenological approach, Pattern Recogni tion vol. 29 no. 11, pp. 1761 -1774, Nov 1996 Kapoor, R., Tsihrintzis, G.A., and Nandhakumar, N., Detection of obscured targets in heavy-tailed radar clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models, Proceedings of the Asilomar Conference, vol. 2, p 863-867, 1997 Kelly, E., Performance of an adaptive detection algorithm: Rejection of unwanted signals, IEEE Trans, on Aerospace and Electronic Systems, AES-25, no. 2, p.p. 122-133, Mar. 1989 Kim, M., and Principe, J.C., A new CFAR stencil for target detection in SAR imag ery, Proceedings of SPIE, vol. 2757, pp. 432-442, April, 1996 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( 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 |