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N ational Taiwan University. 1990 THESIS S ubm itted in partial fulfillment of the requirem ents for the degree of Doctor of Philosophy in Electrical Engineering in the G raduate College of the University of Illinois at U rbana-C ham paign. 1998 U rbana. Illinois Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 9912210 UMI Microform 9912210 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. © Copyright by Fu-C hiarng C hen, 1998 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN THE GRADUATE COLLEGE S E P T E M B E R 1998 (date) W E H E R E B Y R E C O M M E N D T H A T T H E T H E S I S BY F U -C H IA R N G C H E N ENTITLED. U L T R A -W ID E B A N D M IC R O W A V E IM A G IN G R A D A R SY ST E M BE A C C E P T E D IN P A R T I A L F U L F I L L M E N T O F T H E RE(JClREMENTS FOR D O C T O R OF P H IL O S O P H Y THE DEGREE OF. v ___ ector of T h esis R esearch H ead o f D epartm ent Committee on Final Examinationf C <.0 C h a irp e rs o n VC. C t R equired for doctor's d eg ree but not for m aster's. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U ltra-W ideband Microwave Imaging R adar System Fu-C hiarng Chen. Ph.D D epartm ent of Electrical and C om puter Engineering University of Illinois a t U rbana-C ham paign. 1998 W eng Cho Chew. Advisor This thesis presents a new low-power tim e-dom ain (TD) ultra-w ideband (UWB) microwave imaging rad ar system . The TD-UW B microwave imaging radar system successfully combines the UW B impulse radar system and nonlinear inverse scat tering imaging algorithm s for nondestructive evaluation (NDE) and rem ote sensing purposes in the near-field region. Both system hardw are and inverse scattering algorithm s of the system will be described in this study. The tim e-dom ain ultraw ideband radar system has several advantages over conventional narrow -band CW rad ar system s. First, the UWB signal is able to provide much more inform ation for targ et sensing purposes. Second, it is cheaper and takes less m easurem ent tim e than the coherent step-frequency rad ar system. Moreover, the TD -UW B imaging radar system can detect dielectric or m etallic targets located in air or shallow subsurface w ith a high resolution image reconstruction capability. R econstructed images of different types of targets are shown in good agreem ent w ith the target geometry and composition. The experim ental results dem onstrate the potential of the system as a cost-effective tool for NDE and rem ote sensing applications. For the first tim e, the super-resolution phenomenon in nonlinear inverse scat tering is verified experim entally in this thesis. The imaging experim ent is based on our TD-UW B imaging rad ar system . The experim ental d a ta were collected from a lim ited viewing angles using a sw itched Vivaldi antenna array which consists of five tran sm itters and six receivers. A novel way to reduce the C P U run tim e and m em ory requirem ent of the nonlinear inverse scattering algorithm s is also proposed in this study. We propose a code-division m ultiplexing scheme for the nonlinear inverse scattering algorithms to reduce the com putational com plexity by a factor proportional to the num ber of tran sm itters. This effectively enables the nonlinear inverse scattering algorithm s to solve much larger practical problem s. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOW LEDGEM ENTS First and foremost. I would like to thank Professor Weng Cho Chew, my thesis advisor, for his support, encouragement and guidance throughout the course of this research. His profound knowledge in physics and m athem atics inspires me to see w ith a broader vision. I would like to th an k the members of my doctoral com m ittee. Professors J. M. Jin. E. Michielssen. J. Schutt-Aine. and K. C. Yeh. for numerous helpful discussions and suggestions. During the years of my graduate study, my colleagues, past and present, in the Electrom agnetics Laboratory' and Center for C om putational Electrom agnetics have been very helpful in m any aspects. James Bowen. Si\ruan Chen. T ie-Jun Cui. A rif Ergin. Eric Forgy. Olivier Franza. Vikram Jandhyala. Jin Seob Kang, Guy Klemens. Jiun-Hwa Lin. Caicheng Lu. W ei-Choon Ng. K aladhar R adhakrishnan. Jim ing Song. Fernando Teixeira. and R obert Wagner provided valuable technical discussions. I am particularly indebted to my predecessors. Yiming Wang. M ahta M oghaddam . Gregory O tto. Jiun-H w a Lin and Bill Weedou for providing the foundation on which much of the work in this thesis is based. I also would like to thank many friends and colleagues outside the L aboratory who provided so much support and encouragement over the years. There are too m any to name them all: however. I would like to acknowledge Professor Tah-Hsiung Chu at N ational Taiw an University for his encouragem ent. Haw-Jvh Liaw for his personal help during my first three years of study, and Jam es Andrews from the Picosecond Pulse L aboratory for several useful suggestions. I am truly grateful to my wife. Hsin-Yi. for her love, patience and support. I deeply thank her for leaving her teaching job in Taiwan and coming to the US to take care of me. I am deeply indebted to my parents, brother and sisters for their love, encour agem ent and support. It would have been impossible to finish my studies w ithout iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. their unconditional support, their faith in me. and their encouragement. T his work was supported by the ARM Y Research Office under grant CPA R CRD A 0720. th e AFOSR under grant F49620-96-1-0025. the NSF under g ran t ECS 93-02145. an d the ONR under g rant N00014-95-1-0872. The com puter tim e for some of this work was provided by a grant from the N ational Center for Supercom puting A pplications (NCSA) at the U niversity of Illinois. U rbana-C ham paign. and th e San Diego S upercom puter C enter (SDSC) at U niversity of California. San Diego. v Reproduced with permission of the copyright owner. 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TABLE OF C O N T E N T S CHAPTER PA G E 1 I N T R O D U C T I O N ........................................................................................ 1 2 1.1 U ltra-W ideband R a d a r ............................................................................. 1.2 Microwave Im a g in g ...................................................................................... 3 1.2.1 Nonlinear inverse s c a t t e r i n g ..............................................................4 1.2.2 S u p e r -r e s o lu tio n ..................................................................................10 1.3 M ultiplexing Schemes for Nonlinear Inverse S c a t t e r i n g ..................... 11 1.4 Thesis O r g a n i z a t i o n ..................................................................................... 11 1.5 References............................................................................................................12 2 U L T R A -W ID E B A N D M IC R O W A V E IM A G IN G R A D A R S Y S T E M .......................................................................................... 16 2.1 System Block D i a g r a m .................................................................................20 2.2 Signal S o u r c e .................................................................................................. 22 2.3 Impulse Forming N e tw o rk ............................................................................ 22 2.4 Digitizing Sampling O scillo sco p e............................................................... 25 2.5 A n t e n n a ...........................................................................................................26 2.6 A m p lifie r ...........................................................................................................29 2.7 D ata Acquisition and C a l i b r a t i o n ...........................................................36 2.8 S N R ................................................................................................................... 38 2.9 C o n c lu s io n s...................................................................................................... 40 2.10 R eferences.......................................................................................................... 42 3 E X P E R IM E N T A L V E R IF IC A T IO N OF S U P E R R E S O L U T IO N IN N O N L IN E A R IN V E R S E S C A T T E R IN G . 45 3.1 D istorted-B orn Iterative M e th o d ...............................................................48 3.1.1 Conjugate gradient m inim ization s c h e m e .................................. 52 3.1.2 Image reconstruction results using D B I M .................................. 53 3.2 Local Shape Function M e th o d ................................................................... 59 3.2.1 Image reconstruction results using L S F ...................................... 65 3.3 C o n c lu s io n s ......................................................................................................65 3.4 References.......................................................................................................... 68 4 M U L T IP L E X IN G SC H E M E S F O R N O N L IN E A R IN V E R S E S C A T T E R I N G ......................................................................... 71 4.1 Form ulation of CDM and FDM Approaches to D B IM .........................73 4.2 Imaging R econstruction R e s u l t s ...............................................................79 4.3 C o n c lu s io n s ......................................................................................................90 4.4 References.......................................................................................................... 97 C O N C L U S IO N S A N D F U T U R E D I R E C T IO N S ..............................98 V I T A ............................................................................................................... 101 5 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 IN T R O D U C T IO N Microwave im aging is intim ately related to the inverse scattering problem, which is to probe th e profile of an unknown object from the m easurem ent d a ta collected away from th e object. It is one of the most challenging problems in electrom agnetics research. T he im portance of inverse scattering is outlined in applications such as medical im aging, seismology, subsurface and ground-penetrating radar (G PR ). nondestructive evaluation (NDE) and testing, and target identifications [1-10]. In the p ast, academ ic effort concentrated m ainly in the development of algo rithm s for inverse scattering. Much progress has been made in developing nonlinear inverse scatterin g im aging algorithms [11-20], which account for b o th the m ultiple scattering effect and strong scattering. The need has arisen for practical im plem en tation and experim ental verification of these m ethods. It is im perative to have a practical an d cost-effective inverse scattering imaging hardware system. Not only does a cost-effective m easurem ent system provide the experim ental verification and evaluation of the algorithm s, but it can also be applied to solve practical problems such as q u a n tita tiv e NDE and remote sensing. This thesis is about the design and development of a new low-power cost-effective tim e-domain (TD) ultra-w ideband (UWB) microwave im aging radar system. We hope th a t this system will serve as a new inverse scattering imaging tool for quantitative nondestructive evaluation (NDE) and rem ote sensing application for targets located in air or shallow subsur face in the near-field region. The specific problems studied in this thesis include the following: (1) design and im plem entation of a new low-power TD -UW B microwave imaging rad ar sys tem. (2) experim ental verification of super-resolution in nonlinear inverse scattering using our new TD -U W B imaging radax system , and (3) a novel scheme employing the code-division m ultiplexing (CDM) for nonlinear inverse scattering a lg o r ith m s . The new CD M scheme reduces the memory requirem ent and C PU run tim e by 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a factor p ro p o rtio n al to th e num ber of tran sm itters in the experim ental setup of the microwave im aging rad ar system. It allows the nonlinear inverse scattering algorithm s to solve m uch larger problems in m any applications. 1.1 U ltra-W id eb an d Radar UWB rad ar has a fractional bandw idth greater th an 0.25. regardless of the cen ter frequency or the signal tim e-bandw idth product [21]. The fractional bandw idth is defined as 2( fh —f i ) / ( f h + //)• where f h is the upper limit of the frequency band w idth and fi is th e lower lim it of the frequency bandw idth. The relative bandw idth is also called percentage bandw idth when converted to percentage. Recently. UW B rad ar system s have been proposed for many applications such as ground p en etra tion. terrain profiling, foliage penetration, and synthetic aperture rad ar imaging [22-24]. There are several ways to implement the UWB radar, such as frequency m odu lated continuous wave (FM CW ). stepped frequency and phase coded radars in the frequency dom ain m eth o d and impulse radars in the tim e-dom ain m ethod. T he UW B rad ar operates a t a very high percentage bandw idth greater th an 25% and usually up to 100%. We adopt the tim e-dom ain impulse radar approach to imple m ent a TD -U W B ra d a r system. The tim e-dom ain m easurem ent technology con tinues to evolve an d becomes more attractiv e as b etter pulse sources, detectors, and instrum ents becom e available [25. 26]. The tim e-dom ain ultra-w ideband rad ar system has several advantages over conventional narrow -band CW radar system s. F irst, the UWB signal is able to provide b etter sp atial resolution, which can give more inform ation for targ et sensing purposes. Second, it usually costs less and re quires shorter m easurem ent tim e com pared to the coherent stepped frequency rad ar system. The m ain com ponents of the tim e-dom ain system are an avalanche transis to r and a high-speed sam pler, while the frequency-dom ain system uses a frequency synthesizer and a q u ad ra tu re receiver. Therefore, the tim e-dom ain system usually costs less th a n th e frequency-dom ain system. O ur new TD-UW B rad ar system costs much less th a n th e o th er coherent stepped frequency rad ar system developed a t the University of Illinois [27], which is based on a HP8510B network analyzer. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he new TD -U W B imaging rad ar system has been tested and used to collect m easurem ent d a ta from different types of targets including dielectric targets, m etal lic targ ets, and m etallic targets buried in concrete. These m easurem ent d a ta will be processed w ith nonlinear inverse scattering imaging algorithm s, the distorted-B orn iterativ e m ethod (DBIM) and th e local shape function (LSF) m ethod, to reconstruct th e image. 1.2 M icrow ave Im aging Since microwave easily couples into a variety of dielectric m aterials and is able to p e n etrate clouds and fogs, it has been used for locating minerals, biological di agnosis. terra in profiling, NDE of civil engineering structures, and radar rem ote sensing [1-10]. A lthough microwave imaging and ultrasonic (elastic) wave im aging b o th utilize waves to penetrate th e m aterials in order to probe their characteristics, ultrasonic wave im aging usually requires either the use of coupling m aterial or direct contact w ith th e test object. Unlike ultrasonic wave imaging, microwave im aging does not require direct contact to the test object. T he higher wave propagation velocity of th e microwave allows more rapid inspection than does the ultrasonic wave. T here are several m ajor differences between microwave imaging and u ltra sonic im aging m ethods. Microwaves cannot penetrate m etal because of the skin d e p th lim itatio n while the ultrasonic waves can. The good conductors have a skin d e p th of a few m icrom eters in the microwave frequency range. However, microwaves easily p en e tra te plastics, ceram ics, concrete, and many other m aterials. Therefore, am plitude, phase, and polarization m easurem ents of microwaves will provide infor m atio n ab o u t the internal profile and characteristics of the test object. T he microwave imaging (or inverse scattering) problem is often quite difficult to solve, and its solution is not unique because the high spatial frequency portions of th e object sc atte r evanescent waves which cannot be m easured away from the scatte rer. In addition, the m ultiple scattering effect causes the scattered field to becom e a nonlinear functional of th e objects. Hence, the inverse scattering problem is effectively nonlinear. Besides n o n lin e a r ity and non-uniqueness, inverse scatte r ing problem s also suffer from constraints related to the problem geom etry and the 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m easurem ent setup [11]. In the past, many Inverse scattering m ethods have been proposed to solve different types of problems under certain assum ption of the scat tering process. For example, com puter tom ography (CT) [28, 29] assumes th a t X-rays propagate in straight lines and thus ignores both the diffraction and m ulti ple scattering effects. Diffraction tom ography (D T) [30-32] assumes th a t the object contrast is sm all and ignores multiple scattering effects. Recently, nonlinear inverse scattering algorithm s which take into account b o th the diffraction and m ultiple scat tering effects have been developed [12.15.19.33-37]. Two iterative nonlinear inverse scattering algorithm s, the distorted-Born iterative m ethod (DBIM) and the local shape function (LSF) method, will be applied to our TD-UW B microwave im aging rad ar system . 1.2.1 N onlinear inverse scattering In this section, the reason th a t we need the nonlinear inverse scattering algo rithm s for our TD -U W B microwave imaging rad ar system will be given. F irst, we briefly review the diffraction tom ography [30. 31. 38] which m ight be the most well known linear inverse scattering algorithm for weak scatterers which employs the Born approxim ation. Then we explain why it is not suitable for our TD-UW B im aging rad ar system . Consider the case of a current source J radiating in the vicinity of a general inhomogeneity. From Maxwell's equations, the electric field satisfies the following vector wave equation: / i V x / i l V x E ( r ) — k 2[ r ) E ( r ) = iuj yJ(r), (1-1) where k 2(r) = u 2ye. T he perm eability y , and perm ittivity e are functions of posi tion inside the inhomogeneous region. The solution of Equation (1.1) can be expressed as follows: 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E(r) = Einc(r) + j ' drG(r, r') - 0{r')E(r) (1.2) where O(r) = [fc2(r) — fc2] is the object function an d G is the dyadic Green's function. E quation (1.2) states th at the total field in the presence of a scatterer is the sum of the incident field Ei nc(r) and the scattered field (the second term on the right side of the equation). Since E( r) is a functional of O(r). it is obvious th a t the scattered field is a nonlinear functional of 0( r ) . In fact, the nonlinear relationship between the scat tered field and the object O(r) is due to the m ultiple scattering effect [11]. Figure 1.1 illustrates th a t as a result of the multiple scattering between two scatterers. the total scattered field is the sum of the two isolated scattered fields plus an additional term due to the m utual interactions between the two scatterers. It shows th a t the multiple scatterin g effect precludes the use of Unear superposition to find the to tal scattered field from many scatterers. The n onlinear relationship between the scat tered field and the object makes the inverse scattering problem difficult to solve. However, under weakly scattering conditions, the problem can be Unearized. The Born and R ytov approxim ations are two approxim ations used to linearize the nonUnear inverse problems. In the Born approxim ation, th e scattered field am phtude is assum ed to be a Unear functional of the object, whereas in the Rytov approxi m ation. the phase pertu rb atio n is assumed to be a linear functional of the object [11]. In the case of a weak scatterer. [fc2(r) —k%] is very smaU. Hence, the second term in E quation (1.2) is smaller th an the first term, which aUows the approxim ation E( r) % E inc{r). Subsequently, the volume integral Equation (1.2) can be expressed as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (1.3) F ig u r e 1.1 M ultiple scattering between scatterers gives rise to nonlin earity in the inverse scattering problem. This nonlinearity precludes the superpositions of (a) scattered field E \s solely from object 1. (b) scattered field E-2.s solely from object 2, and (c) multiple scattered field E m s when b o th objects are placed next to each other. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E(r) = Einc(r) + ^ dr'G{r, r') • 0{r')Einc{r'). (1.4) This is known as th e first-order Born approximation. Consider a transm itter-receiver m easurem ent in diffraction tom ography as shown in Figure 1.2. T he scattered field in the far zone has been found to be related to the Fourier tran sfo rm of the object as [11] <t>sca{p) « 7 —r— 8t t k 0y/ .------- 0 { k R - k T ) prpR (1.5) where O( k ) represents the Fourier transform of O (r). Since the receivers have the same frequencies as the tran sm itters, the lengths of the vectors k r and k& are equal to k 0 . Therefore, (fcy — k n ) can only span a finite space in the Fourier space. In addition. 0 ( k f t — k r ) is known only for jfe| < k Q, only a low-pass version of 0 ( r ) can be o btained [11. 31]. However, by sweeping both the receiver and transm itter, inform ation on 0 ( k R — fc-r) can be obtained within a circle of radius 2ko. In diffraction tom ography, the basic idea is to collect d a ta from several viewing angles so as to increase the size of the A;-space data. For example, in synthetic aperture ra d ar (SAR) or inverse synthetic aperture radar (ISAR) imaging problems, more A:-space d a ta collected will effectively increase the m ore sp atial resolution of the reconstructed image. The fc-space d a ta can be increased by using different diversity techniques such as frequency, angle and polarization diversities [3. 29]. In the frequency diversity m ethod, a m ultiple frequency broadband signal is used for d a ta collection. T he angle diversity refers to the use of multiple viewing angles to collect th e data. T he polarization diversity m ethod makes use of different polarizations for the tran sm ittin g and receiving antennas. Our experim ental setup in Figure 1.3 depicts our tim e-dom ain (TD) ultrawideband (U W B) im aging rad ar system w ith five tra n sm itters and six receivers. The system only collects 30 sets of d a ta from limited viewing angles, hence the k Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transmitter Receiver F ig u re 1.2 The transm itter-receiver measurem ent in diffraction to mography. space d a ta collected is quite lim ited. An example of the received fc-space d ata is presented in Figure 1.4. D iffraction tom ography will not be able to reconstruct an accurate object profile from the measured d ata which were obtained from the lim ited viewing angles. Therefore, it is desirable to use the nonlinear inverse scattering algorithm s, the DBIM and the LSF. to solve the image reconstruction problem . Two iterative nonlinear inverse scattering algorithm s, the distorted-B orn itera tive m ethod (DBIM) [12. 33] an d the local shape function (LSF) m ethod [15. 19. 34], can be used to process the tim e-dom ain m easurem ent d a ta in order to reconstruct the image of the target object. T he DBIM and LSF account for m ultiple scattering effects of the test targets and show a high resolution image reconstruction capability. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a cm [rT| [tTJ [rxJ (tT] [rT ] [t*J [j^T] (tT] [ rT ] R7] [r7 | F ig u r e 1.3 The experim ental setup of the switched V’ivaldi antenna array. T he eleven Vivaldi antennas are focused a t th e range of 40 cm. F ig u r e 1.4 The k space d ata from kn — fc-r in the switched Vivaldi antenna array experim ental setup. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e D BIM and LSF are sim ilar to N ew ton-type m ethods where th e grad ien t of a nonlinear cost function is sought in order to m inim ize th e cost function. Such a gra dient can be com puted by invoking a forward scatterin g solver. For our T D -U W B im aging ra d a r system case, the finite difference tim e dom ain (F D T D ) a lg o rithm is used as the forward solver. In the DBEM. th e background m edium is n o t con strain ed to be homogeneous and is updated a t each iteration. It has been show n th a t DBIM has second-order convergence and is hence b etter suited for objects w ith large co n trast [11. 12. 33]. On the other hand, th e LSF m ethod has been show n to work well for strong scatterers [19. 34]. More in-depth treatm ent of th e D B IM and LSF m ethods is covered in C hapter 3. 1.2.2 Super-resolution O ne of the aim s of this thesis is to experim entally verify the super-resolution phenom enon in nonlinear inverse scattering based on our newly developed tim edom ain ultra-w ideband microwave imaging rad ar system and the tim e-dom ain non linear inverse scattering algorithms. T he super-resolution phenomenon in nonlinear inverse scattering has been re p o rted previously using numerically sim ulated d a ta [16, 17]. W hat has been shown was th e ability of a nonlinear inverse scattering m ethod to resolve features th a t are m uch less th an half a wavelength, the criterion d ictated by the R ayleigh criterion. T he phenom enon has been attrib u ted to the m ultiple scattering effect w ithin an inhom ogeneous body. It has been shown that tom ographic techniques can provide super-resolution w hen the targ et d a ta is collected from full viewing angles. However, resolution ten d s to degrade to the Rayleigh resolution when th e target d ata is collected from lim ited viewing angles [39]. We use lim ited viewing angles experim ental setup based on our T D -U W B im ag ing ra d a r system to verify the super-resolution phenom enon in nonlinear inverse scatterin g . More details of this subject will be discussed in C hapter 3. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3 M u ltip lex in g Schem es for N on lin ear Inverse S ca tterin g It has been widely known th a t nonlinear inverse scattering imaging a lg o r ithm s need large m em ory storage and tremendous C PU run tim e. To speed up the C PU ru n tim e an d reduce the large memory storage requirem ent of the nonlinear inverse scatterin g algorithm s, we propose to incorporate frequency-division m ultiplexing (FDM ) and code-division multiplexing (CDM) approaches for the nonlinear inverse scatterin g algorithm s. O ur FD M and CDM approaches for the nonlinear inverse scattering algorithm are sim ilar to the m ultiple access schemes in wireless com m unication system s. FDM is sim ilar to the frequency division multiple access (FDM A) in wireless communi cation system s. T he FDM scheme divides the whole available RF bandw idth into several segm ents an d assigns different segments to different tran sm itters. Each tra n sm itte r is allocated a unique RF bandw idth. CDM is sim ilar to the code divi sion m ultiple access (CDMA) in wireless com m unication system s. T he CDM scheme assigns each tra n sm itte r a unique code to encode the tra n sm itte d signal. All the tra n sm itte rs in the CDM sheme share the sam e frequency bandw idth and the same tim e slot. O ur proposed m ethod, which is described in further d etail in C hapter 4. will shorten the C P U rim time and reduce the m em ory requirem ent of the n o n lin ear in verse scatterin g algorithm s by a factor proportional to the num ber of the tran sm it ters. This will allow nonlinear inverse scattering algorithm s to solve large problems for p ractical applications. 1.4 T h esis O rganization In this thesis, we are prim arily concerned w ith designing and developing the TD -U W B im aging radar system which combines the UWB ra d a r system hardware and the nonlinear inverse scattering algorithm s for 2-D objects. In C h ap ter 2. we present the newly developed prototype of the TD -UW B mi crowave im aging rad ar system and discuss th e m easurem ent techniques of the sys11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tem . Each com ponent of the system, such as the antenna, signal pulse, detector, and amplifiers will be described. The signal acquisition and calibration procedure of the au to m ated TD -UW B imaging radar system will also be discussed. In C h ap ter 3. we discuss the super-resolution phenom enon and present the ex perim ental verification of super-resolution in nonlinear inverse scattering. We use two nonlinear iterative inverse scattering solvers, th e tim e-dom ain distorted-B orn it erative m ethod (DBIM) and the tim e-dom ain local shape function (LSF) m ethod, to reconstruct images from measurement d a ta collected by our new TD-UW B im ag ing ra d ar system . The reconstructed images m atch well w ith the geom etry and com position of the test objects. In C h ap ter 4. we present the code division m ultiplexing (CDM) and frequency division m ultiplexing (FDM) schemes for speeding up the C PU run tim e and reduc ing the m em ory requirem ent of the nonlinear inverse scattering imaging algorithms such as the DBIM and LSF. In our proposed m ethod, the complexity of th e non linear inverse scattering algorithm s will be reduced by a factor proportional to the num ber of the tran sm itters in the experim ental setup of the imaging radar system . This will enable the nonlinear inverse scattering algorithm s to solve larger problem s and reconstruct the image faster. In C h ap ter 5. we conclude the the research findings in this dissertation. F uture research directions will be also discussed. 1.5 R eferen ces [1] D. J. Daniels. D. J. Gunton, and H. F. Scott, “Introduction to subsurface ra d a r.’’ IE E Proc. Communication, Radar, and Signal Processing, vol. 135, P t. F. pp. 278-320. Aug. 1988. [2] M. I. Skolnik. Introduction to Radar Systems. New York: McGraw-Hill, 1980. [3] D.-B. Lin and T.-H. Chu, "Bistatic frequency-swept microwave imaging: P rin ciple. m ethodology and experim ental results.” IE E E Trans. Microwave Theory Tech., vol. M TT-41, pp. 855-861. May 1993. [4] C. Pichot, L. Jofre. G. Peronnet, and J. C. Bolomey, "Active microwave im ag ing of inhomogeneous bodies,” IE E E Trans. A ntennas Propagat.. vol. AP-33, pp. 416-425. 1985. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [5] S. Y. Semenov et al.. "Microwave tom ography: Two-dim ensional system for biological im aging,” IE E E Trans. Biomed. Eng., vol. 43. no. 9. pp. 869-877, 1996. [6] L. Peters Jr. and J. D. Young. "Applications of subsurface transient rad ar,'’ in Tim e D om ain M easurements in Electromagnetics. E. K. Miller. Ed. New York: Van N ostrand Reinhold. 1986. pp. 297-351. [7] L. Peters Jr.. J. J. Daniels, and J. D. Young, “G round ppenetrating rad ar as a subsurface environm ental sensing tool.” Proc. IE E E . vol. 82. no. 12, pp. 18021281. 1994. [8] D. A. Noon, D. Lognstaff. and R. J. Yelf. "Advances in the developm ent of step frequency ground penetrating rad ar.” in Proc. o f the 5th Int. Conf. on Ground Penetrating Radar. Kitchner. O ntario. C anada. June 1994. pp 117-132. [9] Y. Michiguchi, K. Hiramoto. M. Nisi. Toshihide. and M. O kada, ‘‘Advanced subsurface rad ar system for imaging buried pipes,’’ IE E E Trans. Geosci. R e mote Sensing, vol. 26. pp. 733-740. 1988. [10] J. E. M ast, ‘‘Microwave pulse-echo radar imaging for the nondestructive eval uation of civil structures.” Ph.D . dissertation. U niversity of Illinois a t U rbanaCham paign. 1993. [11] W. C. Chew. Waves and Fields in Inhomogeneous Media. N ostrand. 1990. New York: Van [12] Y.-M. Wang and W . C. Chew. "An iterative solution of two-dim ensional elec trom agnetic inverse scattering problem.” Int. J. Imaging Syst. Tech.. vol. 1. pp. 100-108. 1989. [13] Wr. C. Chew and Y.-M. Wang. "Reconstruction of two-dimensional p erm ittiv ity using the distorted born iterative m ethod.’’ IE E E Trans. Med. Imaging, vol. MI-9, no. 2, pp. 218-225. 1990. [14] Y.-M. Wang and W . C. Chew. "Limited angle inverse scattering problem s and their applications for geophvsical explorations.” Int. J. Imaging Syst. Tech.. vol. 2. no. 2. pp. 96-111. 1990. [15] W. C. Chew and G. P. O tto. "Microwave imaging of multiple conducting cylin ders using local shape functions.” IE E E Microwave Guided Wave Lett., vol. 2, pp. 284-286. July 1992. [16] M. M oghaddam , W. C. Chew, and M. O ristaglio, "Com parison of the born iterative m ethod and Tarantola’s m ethod for an electrom agnetic tim e-dom ain inverse problem .” Int. J. Imaging Syst. Tech.. vol. 3. pp. 318-333. 1991. [17] M. M oghaddam and W. C. Chew, ‘‘Nonlinear two-dimensional velocity pro file inversion using tim e dom ain d a ta ,” IE E E Trans. Geosci. R em ote Sensing, vol. 30, pp. 147-156. Jan. 1992. [18] M. M oghaddam and W. C. Chew. ‘‘Study of some practical issues in inver sion w ith the born iterative m ethod using tim e-dom ain d a ta ,” IE E E Trans. Antennas Propagat., vol. 41. no. 2. pp. 177-184, 1993. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [19] G. P. O tto and W. C. Chew. "Microwave inverse scattering: Local shape func tio n im aging for im proved resolution of strong scatterers." IE E E Trans. M i crowave Theory Tech., vol. 42. no. 1. pp. 137-141. 1994. [20] W . H. Weedon and W. C. Chew, "Tim e-dom ain inverse scattering using the local shape function (LSF) m ethod." Inverse Probl.. vol. 9. pp. 551-564. 1993. [21] O SD /D A R PA . "U ltra-w ideband ra d a r review panel,’’ in A ssessm ent o f UltraWideband. (U W B ) Technology, A rlington. VA: DARPA. 1990. [22] J . D. Taylor. Introduction to Ultra-wideband R adar System . Boca R aton. FL: C R C Press. 1995. [23] I. J. LaHaie. Ultrawideband Radar. Los Angeles. CA: SPIE P roc.. 1992. [24] R. Vickers. Ed.. Ultrahigh Resolution Radar. SPIE Proc.. vol. 1875. B e llin g h a m , WA: S PIE . 1993. [25] E. K. Miller. Ed.. Tim e D om ain M easurem ents in Electromagnetics. New York: V an N ostrand Reinhold. 1986. [26] M. A. M organ. "U ltra-w ideband impulse scattering m easurem ents," IE E E Trans. A ntennas Propagat.. vol. 42, pp. 840-846. June 1994. [27] W . H. Weedon and W . C. Chew. "B roadband microwave inverse acattering for nondestructive w valuation (N D E ),’’ in The 20th Annual Review o f Progress in Q uantitative NDE. Brunswick. M E. 1993. [28] A. C. Kak. "C om puterized tom ography w ith x-ray. emission and ultrasound sources." Proc. IEEE. vol. 67. no. 9. pp. 1245-1272. 1979. [29] A. C. K ak and M. Slaney. Principles o f Computerized Tomographic Imaqinq. New York: IEEE Press. 1987. [30] A. J. Devaney. "A filtered backpropagation algorithm for diffraction tom ogra phy." Ultrason. Imaging, vol. 4. pp. 336-360. 1982. [31] A. J. Devaney. "A com puter sim ulation stu d y of diffraction tom ography.’’ IE E E Trans. Biomed. Eng., vol. BM E-30. pp. 377-386. 1983. [32] A. J. Devaney. "Geophysical diffraction tom ography." IE E E Trans. Geosci. R em ote Sensing, vol. GE-22. pp. 3-13. Jan . 1984. [33] W . C. Chew and Y.-M. W ang, "R econstruction of two-dim ensional p erm ittiv ity using the distorted b orn iterativ e m ethod.” IE E E Trans. Med. Imaging, vol. MI-9, no. 2. pp. 218-225. 1990. [34] G. P. O tto and W. C. Chew. "Inverse scattering of H z waves using local shape function imaging: A T -m atrix form ulation.” Int. J. Imaging Syst. Tech., vol. 5, pp. 22-27. 1994. [35] R. E. K leinm an and P. M. van den Berg. “An extended modified gradient technique for profile inversion.” Radio Sci., vol. 28. pp. 877-884, 1993. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [36] S. Barkeshli an d R. G. Lautzenheiser, “An iterative m ethod for inverse sc a tte r ing problem s based on an exact gradient s e a r c h , Radio Sci.. vol. 29. pp. 11191130, 1994. [37] N. Joachimowicz. C. Pichot. and J.-P. Hugonin, “Inverse scattering: A n iter ative num erical m ethod for electrom agnetic im aging," IE E E Trans. A n ten n a s Propagat.. vol. AP-39. no. 12. pp. 1742-1752, 1991. [38] A. J. Devaney. “Reconstructive tom ography w ith diffracting wave f i e l d , In verse Probl.. vol. 2. pp. 161-183. May 1986. [39] D. R. W ehner. High-Resolution Radar. Norwood. MA: Artech House. 1995. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 U L T R A -W ID E B A N D M ICROW AVE IM A G IN G R A D A R SY ST E M U ltra-w ideband (UWB) terminology and definitions are quite new [l. 2]. UWB rad ar has a fractional bandw idth greater th a n 0.25, regardless of the center fre quency or the signal tim e-bandw idth product [3]. The fractional bandw idth is defined as BW , = ' ~ ^ lK ( h + fl) (2.1) ' ’ where fh is the upper fimit of the frequency bandw idth and fi is the lower limit of the frequency bandw idth. The relative bandw idth is also called percentage bandw idth when expressed in the form of a percentage. In contrast, there are two other radar types: w ideband radar and narrow-band radar. W ideband rad ar refers to radar w ith percentage bandw idth from 1 to 25. N arrow -band radar refers to radar w ith percentage bandw idth less th an 1. About 95 percent of all radar systems have instantaneous bandw idths of less th an a few percent [1], It is to be noted th a t ultra-w ideband radar may also be referred to as nonsinusoidal radar, impulse radar, baseband rad ar, or ultrahigh-resolution radar. In the past few years, researchers have proposed UWB rad ar system s for many applications such as ground penetration, terrain profiling, foliage penetration, and synthetic ap ertu re radar imaging [1. 2. 4. 5]. The advantages of the UWB radar over the conventional narrow-band radar have been reported in the literature [1. 2. 5]. The m ajor advantages of using UWB signals for imaging radar systems are as follows: (i) b etter spatial resolution, which provides more target inform ation from reflected signals, and (ii) lower probability of intercepted signals com pared to conventional narrow -band signals. The are two ways to implement the UWB rad ar systems: the frequency-domain and the tim e-dom ain m ethods. The frequency-dom ain system uses th e stepped frequency or frequency m odulated continuous waveforms (FM CW ) to measure the 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. steady-state response of the environment. The resu ltan t signal is the coherent sum of these waveforms over the waveform period. O n th e other hand, the tim e-dom ain impulse radar system s use short pulses. A fast analog-to-digital converter (ADC) is needed to convert the analog signal to a digital form at. In m any UWB radar system s, the equivalent tim e sampling m ethod [6] is used to acquire the data since most ADCs are not fast enough to keep up w ith th e required pulse repetition rate. In equivalent tim e sam pling, only one discrete d a ta is acquired during each cycle. In a frequency-dom ain stepped frequency ra d a r system , the tr a n sm ittin g fre quency is swept th ro u g h a certain range of frequencies via a frequency synthesizer. The m agnitude an d phase of the received signal a t each frequency are detected by a qu ad ratu re receiver. The quadrature receiver is used in coherent rad ar systems to decompose th e received signal into two com ponents, in-phase (I) com ponent and qu ad ratu re (Q) com ponent. As shown in Figure 2.1, the reference LO signal is split into in-phase (I) and qu ad rature (Q) com ponents via a 90° phase shifter. The incoming signal is split into two parts through a 0° power divider and then mixed w ith I and Q com ponents of the reference LO signals. Two low-pass filters are used to ex tract the dc portion, and then the signal is fed into an ADC which obtains the am plitude and phase of the scattered signal. The frequency-dom ain rad ar system needs a frequency synthesizer and a quadra ture receiver, while the tim e-dom ain radar system needs an avalanche transistor and a high-speed sam pler. The architecture of the tim e-dom ain system is less compli cated th an th a t o f the frequency-domain system. As a result, the frequency-domain system usually costs more th an the time-domain system . Frequency-dom ain rad ar system s have a higher signal-to-noise ratio (SNR) th an the UW B tim e-dom ain system s due to their reduced bandw idth. Frequency-domain rad ar system s w ith super-heterodyne receiver designs use narrow -band interm ediatefrequency com ponents and result in higher SNR [7, 8]. To improve the SNR in the tim e-dom ain system , averaging (stacking), tim e-gating, and wavelet techniques [9-14] can be applied to suppress the noise level. W avelets are a family of basis functions th at have been used for the representation of functions, sequences, im17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LO LFF LPF P ow er D iv id e r F ig u r e 2.1 Q uadrature detector. ages, and operators. The contrast of the spatial. Fourier, and wavelets bases is shown in Figure 2.2. The spatial basis represents a function using a set of spatially localized basis functions which is global in spectral frequency. The Fourier basis represents a function using a set of spectrally localized functions which is global in sp atial frequency. The wavelets are m ultiresolution basis functions. Wavelets have been useful for signal, image, and m atrix compression and denoising because of th eir efficient m ultiresolution representation of functions. We adopt the tim e-dom ain impulse radar approach to im plem ent a TD -UW B rad ar system . As high-speed and fast rise-time pulse source, detectors, and instru m ents become available [15-20]. tim e-dom ain electrom agnetic m easurem ent tech nology becomes more attractive. This im provem ent results in m any applications which require ultra-w ideband measurement system s. Those applications include targ et identification and classification, rem ote sensing, NDE. electrom agnetic in terference (EM I) and electrom agnetic com patibility (EM C) in high-speed digital system s. We intend to develop a low-cost, low-power. and ultra-w ideband imag ing radax system for quantitative nondestructive evaluation of targets located in 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X (b) F o u rie r <c) W av e le ts F ig u r e 2 .2 (a) Spatial basis functions are local in space and global in sp atial frequency. (b)Fourier basis functions are global in space and local in spatial frequency, (c) Wavelets are m ulti-resolution basis functions. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. air o r in shallow subsurface. The low-power design will reduce the EM I and EM C problem s in nearby environments. T here are several reasons why we use th e tim e-dom ain radar approach to build a UW B im aging rad ar system for q u an titativ e NDE and remote sensing applications. F irst, th e TD-UW B imaging radar system directly uses the tim e-dom ain nonlinear inverse scattering algorithm s to probe the inner profile of the targets. T he inverse problem is usually b etter conditioned and "well-posed” in the tim e-dom ain because the ultra-w ideband transient pulse provides more available information content th a n those available from CW measurement at a few discrete frequencies. Second, the tim e-gating window in the tim e-dom ain UW B approach can be used to elim in ate unw anted early-tim e and late-tim e arrival signals. Third, this approach does not suffer from the "phase wrapping” problem in traditional diffraction tom ography [21]. Fourth, the tim e-dom ain UWB radax system costs much less th a n an o th er stepped frequency rad ar system based on the HP8510 network analyzer [22]. T h is chapter presents our newly developed tim e-dom ain UWB im aging ra d a r system . All the com ponents in the TD -U W B rad ar system will be discussed. T he d a ta acquisition and signal calibration issues will also be addressed. 2.1 S y stem B lock Diagram T h e block diagram of the TD-UW B microwave imaging radar system is shown in F igure 2.3. The TD-UW B system consists of a Hewlett-Packard (HP) 54120B digitizing oscilloscope mainframe, an H P 54121A 20-GHz four-channel test set, a Picosecond Pulse Lab (PSPL) 4050B voltage step generator, a PSPL 4050RPH rem ote pulse head, two PSPL 5210 impulse forming networks, two UWB am plifiers, a d u al stepping m otor positioning system , and an UWB antenna system . T he synchronization of the pulse source and the digitizing oscilloscope is achieved by connecting the PSPL 4050B trigger o u tp u t port to the trigger input p o rt of the HP54121A test set. The pulse repetition rate of the pulser is 100 KHz. T here are two different types of UWB antenna system s employed in the UW B microwave NDE system : a Vivaldi antenna [23-25] array and a ridge horn synthetic ap ertu re 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HP 54120B Digitizing Oscilloscope HP 20GHz Four-Channel Test Set PSPL 4050B Step Generator picosecond step generator Trigger Output f Source Output PSPL 4050RPH Remote Pulse Head IEEE-488 Bus Two PSPL 5210 Impulse Forming Networks Low Noise UWB Amplifier UWB Amplifier Computer F ig u r e 2 .3 system . A n te n n a S y s te m Block diagram of the tim e-dom ain ultra-w ideband rad ar 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. radax (SA R) system . T he whole NDE system is controlled and autom ated by a personal com puter via th e IEEE-488 bus. 2.2 S ign al Source T he P S P L 4050B step generator with the P S P L 4050RPH remote pulse head generates a 10-V pulse w ith a rise tim e of 45 ps. The rise tim e is defined as the t im e interval betw een 0.1 and 0.9 of the pulse am plitude. A 50-ps FW HM (full w idth at half m axim um ) im pulse is obtained by passing the rem ote pulse head output through a PSPL 5210 impulse forming network (IFN). A monocycle pulse is generated by passing th e FW H M im pulse through a second IFN . T he transient measurement d a ta of th e pulse, impulse, monocvcle pulse and their respective frequency spectra are shown in Figure 2.4. From the figure, we can clearly observe th at the monocycle pulse signal does satisfy the UW B definition. 2.3 Im p u lse Form ing N etw ork The P SP L 5210 IFN is a passive network which provides an approxim ate deriva tive of th e in put waveform while m aintaining an im pedance m atch at the input and o u tp u t ports. The coupling coefficient of the IFN is m atched to the risetime of th e P SP L 4050 to maximize the output impulse am plitude while maintaining a distortion-free impulse (d V /d t) output. The PSPL 5210 IFN is a four-terminal pas sive netw ork which gives a good approxim ation to th e derivative function (dV /dt) while providing excellent input and output im pedance m atching. The input step pulse from th e 4050 is attach ed to the "Input" port. T his step pulse passes through the 5210 an d again exits from the 5210 and is term inated in a 50-fi SMA load. W hile passing th ro u g h the 5210. half of the high-frequency content of the step pulse is fil tered off a n d sent out the “O u tput" port of the 5210 as the desired impulse output. H alf of th e high frequencies are still present on the step pulse which exits from the 5210. P SP L has designed the frequency response of the IFN 's filtering to maximize the am p litu d e of the o u tp u t impulse. Figure 2.5 shows the frequency response of the IFN s as m easured by the HP8510B netw ork analyzer. The frequency bandw idth of the HP8510B network analyzer is 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) 10 Voltage -—'6 0 0 1 2 Time (1 ns/div) (b) x 10 0.5 1 1.5 2 Frequency (5 GHz/div)x 1Qio (e) -9 Voltage 40 1 20 Q. 0 2 Time (1 ns/div) (c) 0 1 x 10 -9 1 0.5 2 1.5 Frequency (5 GHz/div) 1Qio (f) Voltage -2 1 2 Time (1 ns/div) 0.5 1 1.5 Frequency (5 GHz/div)x x 1Q-9 F ig u re 2 .4 T he tim e-dom ain measurement d a ta of (a) pulse, (b) im pulse. and (c) monocycle pulses. Their frequency sp ectra are shown in (d), (e) an d (f). respectively. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 (a) One IFN (c) Two IFNs -1 0 Amplitude (dB) -1 0 S -20 -3 0 -3 0 -4 0 -4 0 -5 0 -50 -6 0 -60 Frequency (GHz) Frequency (GHz) (b) One IFN (d) Two IFNs 150 150 100 100 50 ® ® to C O <0 CO ■C ■C Q_ Q. -5 0 -50 -1 0 0 -100 -150 -150 0 5 10 0 15 Frequency (GHz) 5 10 Frequency (GHz) 15 F ig u r e 2.5 M easured S 21 d a ta of the IFNs. (a) Am plitude and (b) phase are m easured d a ta of one IFN. (c) A m plitude and (d) phase are m easured d a ta of two IFNs. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from 500 MHz to 18 GHz. The reason for using the monocycle pulse as the tra n s m itting signal source is to m atch the operational frequency bandw idth. 2 GHz to 12 GHz. of the Vivaldi antennas. This ensures th a t there is no ringing effect after transm ission by th e antenna. 2.4 D ig itizin g Sam pling O scilloscope B andw idth an d sam pling speed are the two m ain criteria used in selecting th e oscilloscope [26]. T he bandw idth of the scope is typically defined as the frequency range over which the wave am plitude falls by less th an 3 dB. If the signal under consideration has a frequency component outside th e bandw idth of the scope, errors are introduced in am plitude a n d /o r tim e-interval m easurem ent. There are two types of bandw idths. namely, the real-time (or single-shot) bandw idth and the repetitive (or equivalent tim e) bandw idth [6]. Real-tim e bandw idth is the highest frequency a scope can cap tu re in a single acquisition. R eal-tim e bandw idth is tied to th e sam pling rate. R epetitive bandw idth applies only to repetitive signals, and th e display is built from samples taken during m ultiple signal acquisition. T he repetitive bandw idth should typically be at least three tim es greater than the bandw idth of the signals to be m easured [6]. T here are basically two types of signals to be viewed w ith a scope: signals th a t occur repetitively an d those th at happen infrequently. For repetitive waveforms, a scope can either take all the samples in th a t occurrence (real-time sam pling), or it can take a few sam ples each time (repetitive sam pling). Figure 2.6 shows the sequential repetitive sam pling mechanism. Each trigger samples one point after a tim e delay t td in each repetitive waveform. A fter each point is acquired, the tim e delay is increased by tse and the acquisition cycle is repeated until th e entire waveform has been captured. Two im p o rtan t param eters in choosing the DSO are the noise and jitte r [17]. Noise is th e voltage uncertainty and jitte r is th e tim e uncertainty. Usually noise is determ ined by the sam pling head, and jitte r is determ ined by the trigger and th e tim e base circuits. Typical values for noise and jitte r in an HP 54121A are 1.2 mV 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig u r e 2 .6 The sequential repetitive sampling captures one point per trigger and increases the tim e delay t^s after each trigger. and 1.6 ps. respectively. T h e HP 54121A sam pler uses a two diode (GaAs) sam pling bridge and is internally term in ated at 50 Q. The rise tim e is 17.5 ps. T he 54121A sam pler includes four sam pling channels, one TD R pulser. the trigger and tim e base circuits. The tim ing jitte r of 1.6 ps was the lowest available on the m arket at th a t tim e. The HP 54120 series offers 0.25-ps tim e resolution and 10-ps tim e accuracy. 2.5 A n te n n a A switched Vivaldi an ten n a array [22. 25] is adopted in the TD -U W B imaging ra d ar system . The Vivaldi antenna is shown in Figure 2.7. The Vivaldi antenna is a flared slotline an ten n a etched on both sides of a microwave lam inated board. Figure 2.8 shows two tap ered metaflic wedges attached to b o th sides of the copper clad board for th e coax-to-slotline (C /S ) transition to achieve a broad bandw idth 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65mm thick edge of wedge 75mm thin edge of wedgp in n e r conductor of coax 75 mm 0.085 coaxial cable F i g u r e 2 .7 Vivaldi antenna. Wedge Transition C/S Transducer W/ Brass Plug Sloiline F i g u r e 2 .8 T he Vivaldi antenna uses two tapered metallic wedges a t tach ed to b oth sides of the copper clad board for the coax-to-slotline (C /S ) transition. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) 90 "I 130 24! 270 270 (c) (d) 901 901 100 130 24! 130 24i 270 270 100 F igure 2.9 M easured normalized antenna p a tte rn of the Vivaldi an tenna a t different frequencies, (a) 2.02 GHz. (b) 4.06 GHz. (c) 6.1 GHz. (d) 11.9 GHz. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. g reater th a n 10:1 [25]. Figure 2.9 shows the m easured antenna p atterns of th e Vi valdi a n ten n a a t several different frequencies. From Figure 2.9. we can observe th a t the an ten n a has a higher directivity at a higher frequency because the beam w idth decreases w hen the frequency increases. T he Vivaldi antenna also has a good po larization isolation which is higher th an 20 dB. The switched Vivaldi an ten n a array consists of five tra n sm ittin g antennas and six receiving antennas. Figure 2.10 shows the sw itched Vivaldi an ten n a array. There are two single-pole, six-throw (SP6T) m echanical switches to control the transm itting and receiving antennas. T he two switches are controlled by a m ultichannel relay ac tu a to r card in the P C control. Those an ten n as are sep arated by 8 cm each, as shown in th e m easurem ent setu p in Figure 2.11. T he eleven Vivaldi antennas are focused a t a range of 40 cm . an d the test targ ets are p u t close to the focused zone. We use th e tim e division m ultiplexing (TDM ) approach to implement the switched Vivaldi a n ten n a array in our TD-UW B rad ar system . Two microwave sw itches are used to control the an ten n a elements in the switched Vivaldi antenna array. One swatch controls five tra n sm itte rs while the other switch controls six receivers. Each sw itch chooses one an ten n a a t a given tiine slot. We have five tran sm itters an d six receivers for a to ta l of 30 tim e slots, as shown in Figure 2.12. At tim e slot 1. only T xi and R xi are functioning. At time slot 2. only T xi and R X2 are on. and so on. Therefore, the TD -U W B system will collect 30 sets of m easurem ent d ata. 2.6 A m p lifier Am plifiers are typically used to increase the system dynam ic range. A UWB am plifier is a tta ch e d to the transm itting port, and a UW B low-noise am plifier is attach ed to th e receiving p o rt. The amplifier is chosen to have a b andw idth large enough to m atch th a t of the monocycle pulse source. T his ensures th a t th e o u t p u t waveform will be free of the ringing effect. Two different UWB am plifiers. HP83006A an d M IT E Q MPN-01001800-23P. have been tested by connecting them to th e tra n sm ittin g p o rt. HP83006A is a 20-dB gain, 13-dBm output power am pli fier. Its o p eratio n al frequency is from 0.0l GHz to 26.5 GHz. Figure 2.13 an d 2.14 shows th e tim e-dom ain m easurem ent d ata of a metallic cylinder and a plastic pipe 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig u r e 2 .1 0 Switched Vivaldi antenna array containing five tra n sm it ting antennas, six receiving antennas and two microwave switches. Rx Tx Rx Tx Rx Tx Rx Tx Rx Tx Rx 8 cm 4 0 cm T — _______ 8 .7 5 cm i 8 .7 5 c m F ig u r e 2.11 Experim ental setup of the switched Vivaldi an ten n a ar ray. The eleven Vivaldi antennas are focused at the range of 40 cm. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Txi Txi Txi Txi Txi Txi TxS Txs Txs Txs R xi R x2 R *3 Rx4 Txs Txs -------------------------------- > . R xi R x2 R x3 Rx4 R x5 Rx6 R xs Rx6 Measurement Time Slots F ig u r e 2.1 2 M easurem ent time slots for our switched Vivaldi antenna array which has five transm itters and six receivers and two microwave switches. At each tim e slot, only one transm itter and one receiver are functioning. It is sim ilar to the TDM A scheme. w ith the HP83006A connected to the transm itting port. Figure 2.13(a) is the mea sured d a ta of the c lu tter w ithout the test target. The clutter includes background noise and coupling between the transm itting antenna and receiving antenna. Figure 2.13(b) shows the m easured d a ta of the metallic cylinder. Figure 2.13(c) shows the m easured d a ta of the cylinder after subtracting the clutter. A single reflected pulse corresponding to the presence of the metallic cylinder is clearly observed. In Figure 2.14, (a) shows the m easured d a ta of the clutter w ithout the test target, (b) shows the m easured d a ta of the plastic pipe. Figure 2.14(c) shows the measured d a ta of the plastic pipe after su btracting the clutter. In this case we can observe a multiple reflected pulse caused by the presence of the plastic pipe. The results suggest th a t the p rototype tim e-dom ain pulse radar system is working correctly. However, the o utput power was still not high enough. The above m easurem ent used the largest average factor 2.048 in the HP54121T digitizing sampling oscilloscope in order to increase the SNR. It took about 14 min to do one tim e trace of 1,024 m easurem ent d a ta points. We can reduce the m easurem ent time by adopting a higher gain and higher o u tp u t power amplifier w ith a lower averaging factor in the m easurem ent. The M PN-01001800-23P is a 30-dB gain. 23-dBm o u tput power amplifier. Its operational frequency is from 1 GHz to 18 GHz. Figures 2.15 and 2.16 show the 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02 0 - 0.02 4 Time(ns) (c) 4.5 5 5.5 6.5 Time(ns) F ig u r e 2 .1 3 Tim e-dom ain measurement d a ta of fa) clutter and (b) m etallic cylinder using the HP83006A am plifier, (c) is the subtraction of the c lu tte r from the metallic cylinder. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02 o oo o > - 0.02 3 3.5 4 4.5 5 5.5 S 6.5 7 5.5 6 6.5 7 Time (ns) (c) x 10 5 o > ■5 3 3.5 4 4.5 5 Time (its) F ig u r e 2 .1 4 Tim e-dom ain m easurem ent d a ta of (a) clu tter and (b) plastic pipe using the HP83006A amplifier, (c) is the subtraction of the c lu tter from the plastic pipe. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) 0.04 p « 0 .0 2 Q Io o-> - 0.02 - -0.04 L 1.5 2 4 Time(ns) (b) 0.04 r0. 0 2 - a® 2 > 00.02 -0.04 1.5 - 2 4 5 Time(ns) (c) 0.02 0 - 0.02 1 4 Time (ns) F ig u r e 2.15 Tim e-dom ain measurement d ata of (a) c lu tter and (b) metallic cylinder using the M ITEQ MPN-01001800-23P amplifier, (c) is the su b trac tio n of the clutter from the metallic cylinder. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) 0.02 a o S > -0.02u 1.5 n L 15 3 3.5 4 4.5 5 3.5 4 4.5 3 3.5 4 4.5 5 Time(ns) (b ) 0.02 a a o > -0.02L 1.5 2 15 3 Time(ns) (c) 0.01 a a o > - 0.01 1.5 9 15 3 Time (ns) F ig u re 2.16 Tim e-dom ain m easurem ent d a ta of (a) clutter and (b) plastic pipe using the M ITEQ MPN-01001800-23P amplifier, (c) is the subtraction of the clu tter from the plastic pipe. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tim e-dom ain m easurem ent d a ta for the sam e metallic cylinder and plastic pipe w ith the M PN-01001800-23P connected to th e tran sm ittin g port. Figure 2.15(a) is the m easured d a ta of the clu tter w ithout te st target. Figure 2.15(b) is the m easured d a ta of th e m etallic cylinder. Figure 2.15(c) shows the m easured d a ta after sub tra ctin g th e clutter. Once again, we can clearly observe the single reflected pulse indicating th e presence of the m etallic cylinder. In Figure 2.16. (a) shows th e m ea sured d a ta of the clutter, (b) shows the m easured d a ta of the plastic pipe and (c) shows the d a ta of the measured plastic pipe after subtracting the clutter. T he m ul tiple reflected pulse shapes due to the plastic pipe can be clearly observed. T he M PN-01001800-23P was adopted in the final design because of its larger gain and larger o u tp u t power. Although its operational frequency bandw idth is narrow er th a n th e HP83006A. its operational frequency is still wide enough to suppress ring ing in the o u tp u t waveform. To increase the system dynam ic range, another lownoise UW B amplifier, a 20-dB gain and 13-dBm output power amplifier M IT E Q AFS3-001010000-32-S-4. is connected to the receiving port. 2.7 D a ta A cqu isition and C alibration T he evaluation of the TD-UW B rad ar system has been done by collecting m ea surem ent d a ta (30 sets) from the five tra n sm ittin g and six receiving antennas in the Vivaldi an ten n a array. Figure 2.17(a) shows the 30 m easurem ent d a ta sets of the clu tte r (C ) w ithout the test target. The c lu tte r includes the background noise and the coupling between the tran sm itting an ten n a and receiving antenna. A strong coupling effect between the tran sm itting a n d receiving antennas of the Vivaldi an ten n a array is observed a t 1 ns. Figure 2.17(b) shows the 30 m easurem ent d a ta sets of th e m etallic cylinder (Tm). We can identify a small target signal em bedded in range of 3 to 4 ns. Figure 2.17(c) shows the m easurem ent d a ta after su b tractin g the c lu tter (Tm —C). The single reflected pulse from the m etallic cylinder is clearly observed. However, the received reflected signals do not estim ate the actual value of the scattered fields from the m etallic cylinder. They represent the scattered fields convolved w ith the system response (S ) of th e impulse rad ar system as shown in th e following equation: 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) metallic cylinder (a) clutter (c) metallic cylinder - clutter 6 —v— —\ r — - \ 5 r— — V s- —v '-V '~ -VA— 4 —V 'v~- v "— Voltage - V ' ----- 3 §,3 o> —V ' — ' -\A — o > -v ~ —v — 2 — v - -V '— — V '- 1 -v ~ — -V'—— - V '--------- — v ~ --- ----- V — 0 0 2 Tim e (ns) 4 0 2 Tim e (ns) 4 2 Tim e (ns) F ig u r e 2 .1 7 M easurem ent d a ta (30 sets from the 5 transm itting anten nas and 6 receiving antennas) of the (a) clutter, (b) m etallic cylinder, an d (c) su b tractio n of (a) from (b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tm - C = Tc * S (2.2) where Tm is the measured d a ta of the target. C is the clutter. S is the system response, and Tc is the calibrated d a ta of the target to be solved. Therefore, to obtain actual scattered fields, th e effects of the system response should be removed by carrying out a deconvolution process [27]. To calibrate the system , we use an approach sim ilar to th at in [28]. We first measure the response of a known reference targ et for which the actual scattered field R t is known. The m easured d a ta of th e reference targ et satisfies the equation Rm — C = R t * S (2-3) where Rm is the measured d a ta of the reference target. C is the clutter. S is th e system response, and R t is th e theoretical value of the reference target. From Equation (2.3). the system response is (Rm — C) deconvolved w ith R t . Once th e system response S is estim ated, we can plug it into Equation (2.2) to obtain the actual scattered field value of th e metallic cylinder, which is (Tm — C) deconvolved with S . The calibrated results are shown in Figure 2.18(a). Excellent agreement is seen between the calibrated results and the theoretical results com puted using the finite-difference tim e-dom ain (FD TD ) [29. 30] shown in Figure 2.18(b). 2.8 SN R The tim e-dom ain radar system has a lower SNR than the frequency-domain radar system . This is because a frequency-domain rad ar system employing super heterodyne receiver design uses narrow -band IF (interm ediate-frequency) amplifiers and com ponents and therefore has much higher SNR [7. 8]. In order to improve the SNR of the tim e-dom ain system , tim e-gating functions, averaging (stacking), and wavelet techniques can be applied to the tim e-dom ain m easurem ent data. The tim e-gating function can be applied to the m easurem ent data in order to remove the unwanted early tim e and late tim e arrival signal. The statistical errors can be reduced by an internal averaging function of the digital sampling oscilloscope. Averaging can be done over up to 2,048 frames in the H P 54120B digitizing oscillo scope m ainfram e. For example, the noise level is reduced by a factor of y/n, th a t is, 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Electric Field (a) Calibrated Data (b) Theoretical Data 0.5 0.5 0.4 0.4 0.3 u. 0.3 0.2 0.2 0.1 0.1 0 2 4 0 6 Tim e (ns) 2 4 Tim e (ns) F ig u r e 2.1 8 (a) C alibrated d a ta of the 30 sets m easurem ent d ata, (b) Theoretical d a ta of th e same target calculated using FD T D . 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 18 dB reduction for n = 64. T he wavelet technique can also be applied to suppress th e noise in the tim e-dom ain m easurem ent d ata [9-14]. Figure 2.19 shows the noise reduction when the wavelet technique is applied to th e m easured tim e-dom ain signal. We use the Daubechies w avelets [9] to do the noise reduction in this case. 2.9 C onclusions A low-power tim e-dom ain (TD ) ultra-w ideband (UW B) microwave im aging ra d a r system has been developed for quantitative nondestructive evaluation and rem ote sensing purposes. All the com ponents have been discussed in the chap ter. T he d ata collection and signal calibration processes have been also addressed. T his system successfully com bines the TD-UW B impulse rad ar system s hardw are an d inverse scattering im aging software. The system shows a high po ten tial for nondestructive evaluation an d quality control of civil engineering stru ctu res. This TD -U W B radar system has several attractive features com pared to o th er com m er cial system s on the m arket, such as low-power and ultra-w ide b andw idth. The low o u tp u t power will reduce the electrom agnetic com patibility (EM C ) and electro m agnetic interference (EM I) problem s in nearby environm ents. In addition, it will reduce the electrom agnetic wave radiation im pact on the hum an o p erato r of the system . The system can achieve high-resolution images w ith te st targ ets located in air or in shallow subsurface. Its ultra-wide bandw idth can provide much more inform ation for target sensing, classification, and evaluation purposes. We use the TD-UW B im aging rad ar system to collect useful d a ta from several different targets. The collected m easurem ent d a ta from our T D -U W B imaging ra d a r system will be processed using the tim e-dom ain nonlinear inverse scattering algorithm s. Two tim e-dom ain nonlinear iterative inverse scatte rin g solvers, the distorted-B orn iterative m eth o d (DBIM) [31-33] and the local shape function (LSF) m eth o d [34-37], will be applied to process the measurem ent d a ta to o b ta in a highresolution reconstructed image. T he details will be described in th e next chapter. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) original signal V o lta g e 0.01 0.01 - 10 Time (ns) (b) denoised signal Voltage 0.01 I I ' " I I------------ 0.01 I j _____________I_____________ i_____________ i_____________i— - 0 I I T ir 0 - I 1 2 3 4 - i_____________ I_____________ 1_ 5 6 Time (ns) (c) noise I 7 , 8 . 9 . 10 , F i g u r e 2.19 Applying the wavelet technique to denoise the tim e-dom ain m easurem ent d ata, (a) Original signal, (b) Denoised signal, (c) Noise. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.10 References [1] J. D. Taylor, Introduction to Ultra-wideband Radar System . Boca R aton, FL: CR C Press. 1995. [2] R. Vickers, Ed.. Ultrahigh Resolution Radar. SPEE Proc.. vol. 1875. Bellingham. WA: SPIE . 1993. [3] O SD /D A R PA . "U ltra-w ideband rad ar review panel,"’ in A ssessm ent o f UltraWideband (U W B ) Technology. Arlington. VA: DARPA. 1990. [4] G. S. Gill. H. F. Chiang, and J. Hall. "Waveform synthesis for u ltra wideband ra d a r.” in IE E E National Radar Conference. A tlanta. GA. M arch 1994. pp. 240-245. [5] I. J. LaHaie. Ultrawideband Radar. Los Angeles. CA: SPIE Proc.. 1992. [6] J. C. F. Coombs. Electronic Instrum ent Handbook. New York. NY': McGrawHill. 1995. [7] S. H aykin, Com munication System s, Second Ed. New York: Wiley. 1983. [8] M. I. Skolnik. Introduction to Radar Systems. New York: McGraw-Hill. 1980. [9] I. Daubechies. Ten lectures on wavelets. Philadelphia: SLAM. 1992. [10] I. Daubechies. "The wavelet transform , time-frequencv localization and signal analysis.” IE E E Trans. Inf. Theory, vol. 36. pp. 961-i005. Sep. 1990. [11] G. S tran g and T. Nguyen. Wavelets and Filter Banks. W ellessley-Cambridge Press. 1996. Wellesley, MA: [12] G. S trang. "Wavelets and dilation equations: A brief introduction." S IA M R e view, vol. 31, pp. 614-627. Dec. 1989. [13] D. Donoho. "Progress in wavelet analysis and WVD: A ten m inute to u r,” in Progress in Wavelet Analysis and Application. Y. Meyer and S. Roques, pp. 109-128. Frontieres Ed. [14] D. Donoho. "De-noising by soft-thresholding,” IE E E Trans, on Inf. Theory. vol. 41. no. 3. pp. 613-627. 1995. [15] E. K. Miller. Ed.. Time D om ain M easurements in Electromagnetics. New York: Van N ostrand Reinhold. 1986. [16] J. R. Andrews. "Pulse m easurem ents in the picosecond dom ain.” Appl. Note AN-3a. Picosecond Pulse Labs 1988. [17] J. R. Andrews. "Com parison of ultra-fast rise sampling oscilloscopes,” Appl. N ote AN-2a, Picosecond Pulse Labs 1989. [18] J. R. Andrews. "TDR, step response "S” param eter m easurem ents in the tim e dom ain,” Appl. Note AN-4, Picosecond Pulse Labs, 1989. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [19] M. A. M organ and B. W. McDaniel. "Transient electrom agnetic scatter ing: D ata acquisition and signal processing,’’ IE E E Trans. Instrum . Meas.. vol. EME-37, pp. 263-267, June 1988. [20] M. A. M organ, “U ltra-w ideband impulse scattering m easurem ents,” IE E E Trans. A nten n a s Propagat.. vol. 42. pp. 840-846, Ju n e 1994. [21] T. J. Cavicchi an d W. D. O ’Brien. “Numerical stu d y of high-order diffra c tio n tom ography via the sine basis moment m ethod.” Ultrason. Imaging, vol. 11. pp. 42-74. 1989. [22] W. H. W eedon an d W. C. Chew. "Step-frequency ra d a r imaging for nde and gpr applications.” in S P IE Advanced Microwave and M illim eter Wave Detectors. San Diego. CA. 1994. pp. 156-167. [23] P. J. Gibson. "The Vivaldi aerial.” in Ninth European Microwave Conference. Brighton. U.K.. 1979. [24] K. S. Yngvesson. D. H. Schaubert. T. L. Korzeniowski. E. L. Kollberg, T. T hungren. and J. F. Johansson. "Endfire tapered slot antennas on dielectric sub stra te s.” IE E E Trans. Antennas Propagat.. vol. A P-33. no. 12. pp. 1392-1400. 1985. [25] K. M. Frantz. "An investigation of the Vivaldi flared rad iato r.” M.S. thesis. University of Illinois a t U rbana-Cham paign. 1992. [26] H.-P. Com pany. Test & Measurement Catalog. USA: Hewlett-Packard Com pany. 1997. [27] S. Riad. "The deconvolution problem: An overview.” Proc. IEEE. vol. 74. no. 1. pp. 82-85.1986. [28] D.-B. Lin and T.-H . Chu. "Bistatic frequency-swept microwave imaging: P rin ciple. m ethodology and experim ental results.” IE E E Trans. Microwave Theory Tech.. vol. M TT-41, pp. 855-861 May 1993. [29] K. S. Yee. "N um erical solution of initial boundary value problems involving M axwell's equations in isotropic media.” IE E E Trans. Antennas Propagat.. vol. AP-14. pp. 302-307. 1966. [30] A. Taflove. "Review of the formulation and applications of the finite-difference tim e-dom ain m ethod for numerical modeling of electrom agnetic wave interac tions w ith a rb itra ry stru ctu res.” Wave M otion, vol. 10, pp. 547-582. 1988. [31] W. C. Chew. Waves and Fields in Inhomogeneous Media. N ostrand, 1990. New York: Van [32] Y.-M. W ang and W. C. Chew. "An iterative solution of two-dimensional elec trom agnetic inverse scattering problem.” Int. J. Imaging Syst. Tech.. vol. 1. pp. 100-108. 1989. [33] W . C. Chew and Y.-M. Wang, ‘‘Reconstruction of two-dim ensional perm ittiv ity using the d isto rted born iterative m ethod.” IE E E Trans. Med. Imaging. vol. MI-9, no. 2, pp. 218-225, 1990. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [34] W. C. Chew an d G. P. O tto. "Microwave im aging of m ultiple conducting cylin ders using local shape functions.’’ IE E E M icrowave Guided Wave Lett., vol. 2, pp. 284-286. Ju ly 1992. [35] G. P. O tto and W. C. Chew. "Microwave inverse scattering: Local shape func tion im aging for improved resolution of strong scatterers.” IE E E Trans. M i crowave Theory Tech.. vol. 42. no. 1. pp. 137-141, 1994. [36] G. P. O tto an d W. C. Chew, ‘‘Inverse scatterin g of H z waves using local shape function imaging: A T -m atrix form ulation.’’ Int. J. Imaging Syst. Tech.. vol. 5. pp. 22-27. 1994. [37] W. H. YVeedon and YV\ C. Chew. "Tim e-dom ain inverse scattering using the local shape function (LSF) m ethod." Inverse Probl.. vol. 9. pp. 551-564. 1993. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 E X P E R IM E N T A L V E R IF IC A T IO N OF SU P E R -R E S O L T U IO N IN N O N L IN E A R IN V E R S E SC A T T E R IN G This chapter presents image reconstruction results using th e tim e-dom ain non linear inverse scattering algorithm s: distorted-B orn iterative m ethod (DBIM) [1-3] and local shape function (LSF) m ethod [4-6]. We use the tim e-dom ain DBIM and LSF m ethods to process the m easurem ent d a ta collected from our newly developed tim e-dom ain ultra-w ideband im aging rad ar system which was described in the pre vious chapter. An experim ental verification of the super-resolution phenomenon in nonlinear inverse scattering is also presented. Super-resolution is the term used to refer to resolutions b e tte r th a n the Rayleigh resolution. Tomographic techniques can provide super-resolution if d a ta can be col lected from a full set of viewing angles surrounding the target. However, resolution tends to degrade to the Rayleigh resolution when data can be collected only from a lim ited set of viewing angles [7]. Spectral estim ation is an o th er m ethod by which super-resolution can be achieved [8]. T he spectral estim ation an d tom ographic tech niques produce super-resolution based on the spatial-frequency bandw idth. How ever. they do not take m ultiple scattering effects into account. The super-resolution phenom enon in nonlinear inverse scattering has been re p o rted previously using num erically sim ulated d ata [9. 10]. W h at was shown was the ability of a nonlinear inverse scattering m ethod to resolve features th a t are much less th an half a wavelength, the limit dictated by th e Rayleigh criterion. The phenom enon has been a ttrib u te d to the multiple scattering effect within an inhomogeneous body. The high spatial frequency (high resolution) inform ation of the object is usually contained in the evanescent waves when only single scattering physics is considered. M ultiple scattering converts evanescent waves into propagat ing waves and vice versa. Hence, in an inverse scattering experim ent— even though an object is interrogated w ith a propagating wave, and only scattered waves cor- 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. responding to propagating waves can be m easured—the scattered waves contain high-resolution inform ation about the scatterer because of the mode conversion o f evanescent-propagating waves in a m ultiply scattered field. Therefore, an inverse scattering m ethod th at unravels multiple scattering effects can extract the highresolution inform ation on a scatterer. Figure 3.1 shows the mode conversion phenomenon of the propagating waves and evanescent waves in a hollow plastic pipe. W hen the incident wave, a p ro p a gating wave, h its the outer boundary of the plastic pipes, it generates two reflected waves (a propagating wave and an evanescent wave) and two transm itted waves (a propagating wave and an evanescent wave). The transm itted waves generate o th er reflected and tran sm itted waves. The am plitudes in the figure are not to scale an d only serve to illustrate the mode conversion process. Propagating Wave Evanescent Wave F ig u re 3 .1 Mode conversions of propagating and evanescent waves in a hollow plastic pipe. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he DBIM is an iterative optim ization scheme which applies the distorted-B orn approxim ation a t each iteration. The distorted-B orn approxim ation linearizes the nonlinear integral equation of scattering for inhomogeneous background m edia by replacing th e to ta l field inside the object w ith the incident field. It is sim ila r to the B orn approxim ation. The basic difference is th a t th e Born approxim ation assumes a homogeneous background m edium [1 1 ]. T he distorted-B orn approxim ation is fre quently used in diffraction tom ography on objects w ith weak contrast as com pared to the background medium. Instead of applying the distorted Born approxim ation ju st once, this approxim ation can be applied repeatedly as the background m edium is u p d ated a t each iteration step. This is th e essence of the DBIM . It has been shown th a t th e strength of the DBIM is its ability to model inhomogeneous back ground m edia w ith second-order convergence: hence the m ethod is well suited for objects w ith larger contrast [1-3.10]. DBIM is a modification of the Born iterative m ethod (BIM) [1]. T he differ ence between th e BIM and the DBIM is th a t the Green's function of the former rem ains th e sam e while the G reen's function of the latter is changed in each it eration step. Since the BIM assumes a constant background m edium throughout the entire itera tio n sequence, the G reen's function remains invariant at each iter ation. T he DBIM . however, uses the object profile from the last itera tio n step as the background medium resulting in the u p dating of the Green's function a t each iteratio n step. It has been shown th a t the BIM and DBIM have first- and secondorder convergence rates, respectively [1 . 10]. B oth th e BIM and DBIM are similar to N ew ton-type m ethod where the gradient of a nonlinear cost function is sought in order to m inim ize the cost function. Such a gradient m ethod can be found by invoking a forward scattering solver. T he DBIM has been im plem ented for b o th CW and tra n sien t excitations. For the C W case, fast-forward scattering solvers like the recursive aggregate T m atrix algorithm (RATM A) [1 2 ] and C G -F F T m ethod [13. 14] can be used to reduce the com putational tim e and m em ory requirem ent. For the tran sien t case, a finite difference tim e dom ain (FDTD) [15] algorithm is norm ally used as the forward solver. On th e oth er hand, the local shape function (LSF) m ethod is used to solve 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the nonlinear inverse scattering problems for strong scatterers. T he tim e-dom ain version of th e LSF algorithm has been developed using the T -m atrix form ulation [6 ]- 3.1 D isto r ted -B o rn Iterative M ethod The tim e-dom ain distorted-B orn iterative m ethod (DBIM ) for 2-D inverse scat tering problem s is described in this section. F irst, we briefly review the tim e-dom ain DBIM form ulation [16]. Second, we present the reconstructed images using the DBIM to process the tim e-dom ain m easurem ent d a ta from our newly developed TD -U W B im aging ra d ar system. Figure 3.2 illustrates a 2-D transverse m agnetic (TM ) or E z scatterin g prob lem. A 2 -D cylindrical scatterer excited by a line current source J z,n produced the scattered electric field E z,n {r,t). The subscript n stan d s for the tra n sm itter num ber since there are m ultiple transm itters in our experim ental setup. T he scat terer is characterized by the perm ittivity and conductivity profiles e (r) + Se(r) and cr(r) -t- S a ( r ) . respectively, in an inhomogeneous background m edium e (r),c r(r). The scatte rer can be viewed as consisting of a p ertu rb atio n Se(r). 5cr(r) in the in homogeneous background. It is assumed th a t th e d'e(r) and Scr(r) are nonzero only w ithin the su p p o rt volume V of the scatterer. W ith such notations, the perm ittiv ity and conductivity can be everywhere expressed as e(r) + S e( r) and cr(r) + Scr(r), respectively. T he E z%n( r . t ) can be found from the scalar wave equation V d2 d - ^ o e (r)— - n Qa ( r) — dt2 (3.1) + fi0Se( r)— z E z,n {r, t) + ^ 06 a { r ) — E z^ ( r . t). dt dt2 U nder th e d istorted Born approxim ation, th e solution of the p artial differential equation (P D E ) (3.1) is approxim ated as E z,n(r , l ) ~ £*,n,o(r, t) + S E ze n (r, t) -I- 6 E ° n (r, t). 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.2) L ine c u rre n t source E lectric field S c a tte re r F ig u r e 3.2 Two-dimensional (2-D) TM or E z scattering problem. Twodim ensional cylindrical scatterer excited by a z line current source J z,n . The subscript n stands for tran sm itter number. In the above equation. roo ro c Q E ,n,o(r - t ) = -A*o / dr' dt' g(r, r ' , t — t') — J z,„ (r', t') J oc Joe ut (3.3) is the incident field in the presence of the background inhomogeneous m edium e(r),< r(r). T he p ertu rb atio n terms 8 E ze n ( r , t ) and S E z n ( r , t ) are the scattered field induced by the perm ittivity and conductivity p ertu rb atio n 8e(r) and Scr(r) respectively. They are given as 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ro o S E t,n(r ^ ) = - V o roc dr> Q -i dt' 9 ( r -r ' . t - t ') 5e{r') — E z^ o { r ' , t ' ) (3.4) and 6E The inhomogeneous Green's function g ( r . r ' . t ) contained in E quations (3.3). (3.4). and (3.5) satisfies the inhomogeneous wave equation d d V 2 - /i 0 e ( r) — - n 0cr(r)— g ( r , r ' A ) = - 8 ( r - r ' ) 8 ( t ) . (3.6) Under the distorted Born approxim ation, the integrals in E quation (3.2) are linearized assum ing th at the scattered fields 5E% n and 8 E ° n are weak com pared to the incident field E z-n o. The incident field E z_n,o has been su b stitu te d in place of the to tal field E z n inside integrals in Equations (3.4) and (3.5). In the practical case, the tim e integral is performed from 0 ~ T. where all the E field is zero when t' > T at any point r of interest in the model space. Equation (3.6) can be com puted as a finite-difference time-domain (FD TD ) [15. 17] forward solver. In the DBIM. e*;(r) and crfc(r) represent the perm ittivity and conductivity at the fcth iteration. Therefore. E z^n^ is the incident field at the k th iteratio n in the presence of the background medium Cfc(r) and a^ir) . E z n k is com puted num eri cally using the FD TD forward solver. The object param eters ejt(r) and cr^(r) can be u pdated at each iteration using a conjugate gradient optim ization scheme. Equations (3.4) and (3.5) can also been viewed as operator forms of the Frechet derivatives th a t m ap perturbations Sek(r) and 6crk(r) into the field variations 5 E \ n>fc(r, t) and 8 E ° n k ( r , t ) . It has been shown [6 . 9] th a t Frechet transposed o perators corre sponding to these Frechet derivatives map the field perturbations 8 E \ n fc(r, t) and 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 E ° n k (r, t) back into the perm ittivity and conductivity spaces via th e following relations: N t T Q2 and T he above two equations can be im plem ented by a tim e reversal backpropagation of the induced sources caused from the field perturbations, followed by a correlation w ith the second time derivative or first tim e derivative of the incident field inside the object. The process is repeated for every tra n sm itter and the result is sum m ed. Note th a t in the second fine of Equations (3.7) and (3.8), linearity al lows th e exchange of the sum m ation and intergal operation: th a t is. we can sum the induced sources SEl n fc( r m. t') at all receivers and run th e FD T D ju st once instead of running it N r times. B o th the Frechet derivative and transposed operators are required in the conju gate gradient optim ization m ethod in th e following m a n n er. The Frechet derivative o p erato r is required in com puting the conjugate gradient step size for updating along th e given search direction. On the other hand, the Frechet transposed opera to r is used to com pute the gradient and hence the search direction. T his allows the optim ization search direction and u p d ate step size to be com puted w ith three calls to an FD T D forward solver. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.1 C onjugate gradient m inim ization scheme The inverse scattering problem is a nonlinear problem because of the m ultiple scattering phenom enon within the test object. T h a t is. the measured scattered field is a nonlinear functional of the object. A cost function could be defined to solve this nonlinear problem as S(£) = i (||**“ (<0 - 4^“ .11 + 7 1|£ - £611), where (3.9) e an d et contain the discretized values of e (r') and e&(r'), respectively, ^meas represents the measured scattered field at the receivers and <£sca(e) depicts the com puted scatte red field under the current estim ate for the object param eter e a t the sam e location. The vectors an d <£sca(e) are indexed as a function of tra n sm itte r location, receiver location, and tim e. The second norm term on the right-hand side of Equation (3.9) is the Tikhonov regularization term [18-20] used to overcome ill-conditioning of the inverse scatterin g problem, where 7 is a real num ber regularization param eter. To reduce the com putational cost, we use th e conjugate gradient (CG) optim iza tion algorithm [21-23]. which is a N ew ton-type m inim ization m ethod, to m in im ize the cost function. The conjugate gradient optim ization scheme as applied to the DBIM can be sum m arized as follows: (i) For n = 0 . 1 . 2,... (ii) <$>nsca = / ( e n ); If \\$3nca - < A . th en stop (iii) h n = gn + Cnhn - i , where Cn = {9n30T9n~lf nn-l*yn-l (iv) en + 1 = en - an hn , an = ^ r ^ n\ n • S° to 0 ) where / is the tim e-dom ain forward solver. gn is the gradient of S , hn is th e con ju g ate direction. Cn is the step size for com puting the conjugate direction, an is 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the step size for updating the param eter, and H n is the Hessian m atrix. In the first step, th e initial guess usually assumes a homogeneous background. A for ward solver is then invoked to calculate th e scattered fields a t the receivers. T he calculated scattered fields are subsequently com pared to th e m easured scattered fields. The iterations are stopped when the error A becomes less th an a prescribed tolerance value. 3.1.2 Im age reconstruction results using DBIM We use the TD DBIM to process the m easurem ent d a ta collected from the TDUWB im aging radar system. For each test target. 30 sets of measurem ent d a ta were collected from the five transm itters and six receivers in the switched Vivaldi antenna array. Figure 3.3 shows the experim ental setup. The distance between the transm itting antenna and receiving antenna is 8 cm. All the eleven antennas of the array are focused a t th e range of 40 cm. The test scatterers are put close to the focused zone. The distance between the scatterer and the antenna array is large enough so th a t only propagating waves are collected. In the image reconstruction procedure, the geom etry of the inverse scattering problem can be depicted in Figure 3.4. Figure 3.4 show's the lim ited angles recon struction setup from our new TD-UW B imaging radar system . The five transm itters and six receivers are located at one side in front of the targ et zone (optim ization subgrid region). In Figure 3.4. Ni = 229. N j = 341, Oi — 35. and Oj = 35. The num ber of tim e steps used in the FDTD solver is 1,500. T he grid param eters used are A x = A y = 2.5 mm and A t = 5 ps for FD T D . Figure 3.5 shows the measurement d a ta of a plastic pipe. The outer and inner diam eters of the plastic pipe are 4.8 cm and 3.7 cm, re spectively. Figures 3.6 and 3.7 show the reconstructed p erm ittiv ity image of the hollow plastic pipe using the DBIM at different iteration steps. The reconstruction result of the first iteration step shown in Figure 3.6(a) is the reconstruction using 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rx Tx Rx Tx Rx Tx Rx Tx Rx Tx Rx AAAAAAAAA \ '■- V \ n \ , ; ■ ,i ' ! \\ \ \ . ' / / ■/ // s/ ' // , • /■ ' / - / / \ \ \ 40 cm \ •.\ > 1, i ;/ / ' // / ji ; / / ' .. . Object ’ All elements focus at the range of 40 cm. " The elements locations and mam beam directions are adjustable. F ig u r e 3 .3 Experim ental setup of the switched Vivaldi antenna array. T he eleven Vivaldi antennas are focused at th e range of 40 cm. Nj aaaaaaaaaaa F ig u r e 3 .4 Lim ited angle inverse scattering configuration. FD T D grid contains TV* x N j grid points which contain an optim ization subgrid th a t contains Oi x 0 3 grid points, tran sm itters and receivers located around the optim ization subgrid, and an absorbing border region. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 3 3.5 4 4.5 5 Time (ns) F ig u re 3.5 M easurem ent d a ta of a hollow plastic pipe. (4.8 cm in outer diam eter. 3.7 cm in inner diam eter) using the TD -U W B rad ar system. the first-order Born approxim ation. The reconstructed shape and th e p erm ittiv ity values are inaccurate because th e Born approxim ation does not account for th e m ul tiple scattering effects. It is clear th a t when the num ber of iteratio n steps increases, the reconstructed image of th e hollow plastic pipe can be clearly observed and the reconstructed p erm ittiv ity value of the plastic pipe is close to th e actual value of 2.5 as shown in Figure 3.7(d). T he reconstruction results of Figures 3.6 and 3.7 dem onstrate th a t the D BIM provides a much b e tte r result th a n first-order B orn approxim ation algorithm when m ultiple scattering effects are significant. The frequency sp ectru m of the scattered field from the plastic pipe is shown in Figure 3.8(a). The half w avelength corresponding to the frequency spectrum 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) iteration step = 1 (b) iteration step = 2 1.35 36 Range (cm) S £ 1.3 Range (cm) 1.25 1.2 1.15 42 1.1 1.05 -2 0 44 1 2 - 4 - 2 0 2 Cross Range (cm) Cross Range (cm) (c) iteration step = 5 (d) iteration step = 10 36 2 Range (cm) 1.8 38 £ O §>40 c (0 GC 42 1.6 1.4 1.2 -2 0 2 Cross Range (cm) 4 44 1 -2 0 2 Cross Range (cm) Figure 3.6 DBIM p e rm ittiv ity reconstruction image of a hollow plastic pipe. (4.8 cm in outer d iam eter. 3.7 cm in inner diam eter) at different iteration steps, (a) Ite ra tio n step = 1 . (b) iteratio n step = 2. (c) iteration step = 5. and (d) iteratio n step = 10. •56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) iteration step = 15 Range (cm) (b) iteration step = 20 8,40 -4 -2 0 2 Cross Range (cm) 4 - 4 - 2 0 2 Cross Range (cm) (c) iteration step = 30 Range (cm) (d) iteration step = 50 o 40 -2 0 2 -2 0 2 Cross Range (cm) Cross Range (cm) F ig u r e 3 .7 DBEM perm ittivity reconstruction image of a hollow plastic pipe. (4.8 cm in outer diam eter. 3.7 cm in inner diam eter) a t different itera tio n steps, (a) Iteration step = 15. (b) iteration step = 2 0 . (c) itera tio n step = 30. and (d) iteration step = 50. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) -1 0 -2 0 -3 0 H- —40 -5 0 -6 0 -7 0 -8 0 -9 0 0 2 4 6 8 10 8 10 F re q u e n c y (GHz) (b) 10 8 6 4 2 0 0 2 4 6 F re q u e n c y (G H z) F ig u r e 3 .8 (a) Frequency spectrum of the scattered field from the plastic pipe, (b) H alf wavelength corresponding to the frequency spec trum . 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the sca tte red field is shown in Figure 3.8(b). From Figure 3.8(a), it is seen th a t the sc atte re d field lies predom inantly in th e range of 3 to 8 GHz. Besides, th e m agnitude of th e sp ectru m at the high-end frequency. 10 GHz. is about 40-dB lower th an the peak value. From Figures 3.6. 3.7, and 3.8. we observe th a t the D BIM achieves a 7.5-m m resolution, which is b etter th a n the q u arter wavelength for a high-contrast scatterer. However, the first-order Born approxim ation reconstruction result is incorrect because it does not account for m ultiple scattering effects. In addition, the experim ent dem onstrates th a t the DBIM can recover in form a t inn on the backside o f th e scatterer even though scattering d a ta are collected only from lim ited viewing angles on the front side of the scatterer. T he reconstruction results also d em o n strate th a t we are able to surpass the m axim um contrast recovered by the first-order Born inversion [24] by a factor of ab o u t 25. Note th a t no a priori inform ation a b o u t th e object is needed in the reconstruction. Figure 3.9 shows the m easurem ent d a ta of two plastic rods. The diam eters of the plastic rods are b o th 2 cm. and they are separated by 2.5 m m . This is m uch finer th a n the experim ental setup's Rayleigh resolution of 1.5 cm for 10-GHz R F bandwddth or 1.875 cm for 8 -GHz RF bandw idth. Figures 3.10 and 3.11 show the reconstructed im age of two plastic rods using DBIM p erm ittiv ity reconstruction. The reconstruction result of the first iteration step as shown in Figure 3.10(a) shows the reconstruction using first-order Born approxim ation. T he reconstructed shape and p e rm ittiv ity values are not correct because th e Born approxim ation does not account for m ultiple scattering effects. Figure 3.11 shows the reconstructed results of several different steps in the iterations of DBIM . Figures 3.10 and 3.11 clearly show th a t as th e num ber of iteration steps increases, the reconstruction result of the two plastic rods is more correct. 3.2 Local S h a p e F un ction M ethod The local sh ap e function (LSF) m ethod [4. 5. 6 ] was developed for strong m etal lic scatterers. T h e BUM and DBIM work well for dielectric and conductive m edia w ith contrasts as high as 10:1. But the contrast for m etallic scatterers is infinity. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Time (ns) F ig u r e 3 .9 M easurem ent data of two plastic rods (2-cm diam eter, sep ara ted by 2.5 mm) using the TD-UWB ra d a r system . Therefore, th e linearizing Born approximations applied at each iteration step of the BIM and DBIM m ay not be valid. The LSF m aps a scatterer w ith infinite con ductivity into a problem w ith a binary function which ranges between 0 and 1. In this m anner, th e problem becomes more linearized although it is still a nonlinear problem. T he LSF discretization of the scattering object is shown in Figure 3.12. The object region V. which contains all possible scatterers 5 . is discretized on a regular grid into N subvolumes V* for i = function 7 1 .2 ,..., N . We assign a binary local shape i to each subvolume Vi such that 7i 1, 0 , Ft fj.S = 0 . 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.10) (a) iteration step = 1 (b) iteration step = 5 1.25 36 1.2 38 E 1.15 §>40 c a 1.1 GC 42 1.05 - 4 - 2 0 2 Cross Range (cm) 44 1 - 4 - 2 0 2 Cross Range (cm) (c) iteration step = 10 (d) iteration step = 20 36 2 38 1.8 if 1.6 §>40 c as 1.4 GC 42 1.2 -2 0 2 44 1 -2 Cross Range (cm) 0 2 Cross Range (cm) F ig u r e 3 .1 0 DBIM p erm ittivity reconstructed image of two plastic rods (2-cm diam eter, separated by 2.5 mm) at several iteration steps, (a) Ite ratio n step = 1 . (b) iteration step = 5. (c) iteration step = 1 0 . and (d) iteration step = 2 0 . 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) iteration step = 30 Range (cm) (b) iteration step = 60 -4 -2 0 2 C ro ss R an ge (cm) -2 4 0 2 C ross R an ge (cm ) (c) iteration ste p = 8 0 (d) iteration ste p = 1 0 0 36 Range (cm) ^38 * E o § ,4 0 ♦ a a cc 42 -2 0 2 -2 C ross R an ge (cm) 0 2 C ross R ange (cm ) F ig u r e 3.11 DBIM perm ittivity reconstructed image of two plastic rods (2-cm diam eter, separated by 2.5 mm) at several iteration steps, (a) Iteratio n step = 30. (b) iteratio n step = 60. (c) iteration step = 80. and (d) iteration step = 1 0 0 . 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 ¥ T j\ 1 tu t/ Vs. 1 d y SS/ SSj □ r□ < Vn F i g u r e 3.1 2 Discretization of strong scatterers on a finite difference grid. The shaded regions indicate 7 , = 1. Inversion is perform ed in object region V. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The local shape function can be im plem ented as a volum etric boundary condi tion in an FD T D forward solver [6 ]. In the LSF discretization, we assume th a t the scatterer has a homogeneous perm ittivity and conductivity in each subvolume V*. One way to deal with the metallic scatterers is to m anually enforce the boundary condition such th a t E z (ri) = 0 at the location t \ where 7 *= 1. This boundary condition is called “volumetric boundary condition" since it is applied a t arbitrary locations where 7 i = 1 inside the volume V . Intuitively it can be viewed as placing a filam ental m etallic scatterer at each location V\ where 7 * = 1. The LSF volumetric boundary condition can be expressed m athem atically as E z A ^ i . t ) = (l + l i Ti{l)) E l n {r i ,t) (3.11) where Xqx) is the single-scatterer T -m atrix and E ^ n ( r , t ) is th e incident field on the scatterer a t position rt which includes m ultiple scattering effects from other cells Vj. j ^ i. Tt(1) = 7 , = 1 —1 for the case of filemental metallic scatterers. Therefore. enforces the volumetric boundary condition E z_n ( r i , t) = 0 in Equation (3.11). Field E%n {r.t} is called the “ghostv field because it represents the to tal field th a t would be produced at r* assum ing is not present at 7 7 i = 0 . or th a t a metallic scattering ,. In a practical iterative optim ization scheme, it is necessary to relax the binary variable 7 * into a continuous real variable on the interval [0,1]. The inverse scattering algorithm will produce an image of the variable 7 j as a function of 2-D space [6 ]. The LSF m ethod can be implemented in an iterative algorithm w ith a m ethodol ogy sim ilar to th a t of the DBIM. The m ajor difference of the LSF and DBIM is th a t the Frechet derivative and Frechet transposed operators are different. The Frechet derivative and Frechet transposed operators of the LSF m ethod can be expressed as follows: S E z_n^k {rn ,t) = J dr' dt' h(r. r', t - ^/)<b'fc(0-E'f,nJfc(r , ’ t>) 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3 -12) and Nt rT ^ 7 4 (0 = 1 ; / d t E z9 n k { r ' . T - t ) n = 1 7° where h [ r , r ' . t ) is the inhomogeneous medium G reen's function in the presence of lk { r ) . T he right-hand side of Equation (3.13) is a backpropagation of the tim e-reversed field S E Zjn'k{rm , T — t') from the receivers back into the object space and th en correlated with a time-reversed version of the ghost field inside the object w ith a final sum over all the transm itters. It is sim ilar to the DBIM Frechet derivatives. 3.2.1 Image reconstruction results using LSF We use the LSF m ethod to process the collected tim e-dom ain measurem ent d a ta from our TD-UW B imaging radar system. Figure 3.13 shows the reconstructed image of a metallic cylinder using the LSF m ethod. The diam eter of the cylinder is 3.2 cm. The front curvature of the cylinder facing the Vivaldi antenna array is clearly observed. Figure 3.14 shows the reconstructed image of two m etallic cylindrical rods using the LSF m ethod. The two rods are separated by 3.2 cm. The diam eters of the rods are b o th 4.5 m m . Figure 3.15 shows th e reconstructed images of a sm all metallic cylinder (1 cm in diam eter) em bedded 2 cm beneath a concrete cement block using the LSF algorithm . The upper curvature is the shape of the concrete cement block. T he lower part of the image reveals the image of the sm all m etallic cylinder em bedded 3.3 2 cm beneath the concrete cem ent block surface. Conclusions This chapter presented the image reconstruction results using the tim e-dom ain nonlinear inverse scattering algorithm s, the distorted-B orn iterative m ethod (DBIM ) 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 - 0 03 1 - 0 0 35 1 - 0 04 1 - 0 045 1 - 0 05 Cross Rang* (cm) F ig u r e 3 .1 3 Shape function reconstructed image of a m etallic cylinder (3.2-em diam eter). Cross Rang* (cm) F ig u r e 3 .1 4 Shape function reconstruction image of two m etallic cylin der rods (4.5-m m diam eter). The two rods are separate by 3.2 cm. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x 10~3 0 (a) iteration step = 1 (b) iteration step = 2 -2 Range (cm) 32 -4 E o *34 o> c (0 ^ 36 -6 -8 38 -10 Cross Range (cm) Cross Range (cm) (c) iteration step = 3 (d) iteration step = 5 Range (cm) -0.005 - 0.01 -0.015 0 Cross Range (cm) 5 - 0.02 Cross Range (cm) F ig u r e 3 .1 5 Shape function reconstruction image of a small metallic cylinder ( 1 -cm diam eter) embedded 2 cm beneath a concrete cement block at different iterations steps. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the local shape function (LSF) m ethod. We have successfully applied these two iterative nonlinear inverse scattering solvers to process the m easurem ent d a ta using our newly developed tim e-dom ain ultra-w ideband microwave imaging radar system. T he im age reconstruction results are well m atched to the geometries and com positions of the objects. An experim ental verification of the super-resolution phenomenon in nonlinear inverse scatte rin g imaging was also presented. The inverse scattering experim en tal setup is based on our newly developed tim e-dom ain ultra-w ideband imaging radar system . The experim ental data are processed w ith the distorted Born it erative m eth o d (DBIM ). and show th at it can resolve features sm aller th a n the half-wavelength d ictated by the Rayleigh criterion. Moreover, we have dem on strated th a t th e DBIM can recover inform ation on the backside of the penetrable scatterer even though the scattering data are collected only from limited viewing angles on the front side of the scatterer. T his is also a consequence of extracting inform ation from the m ultiple scattering physics inherent in the d a ta collected, and is not possible in the inverse scattering m ethod where only single scattering physics is assumed. 3.4 R eferen ces [1] W. C. Chew. Waves and Fields in Inhomogeneous Media. N ostrand, 1990. New York: Van [2] Y.-M. W ang and W. C. Chew. “An iterative solution of two-dimensional elec trom agnetic inverse scattering problem." Int. J. Imaging Syst. Tech.. vol. 1 . pp. 100-108. 1989. [3] W. C. Chew and Y.-M. Wang. “R econstruction of two-dimensional p erm ittiv ity using the distorted born iterative m ethod.” I E E E Trans. Med. Imaging, vol. MI-9, no. 2. pp. 218-225. 1990. [4] G. P. O tto and W. C. Chew, ‘‘Microwave inverse scattering: Local shape func tion im aging for improved resolution of strong scatterers.” I E E E Trans. M i crowave Theory Tech.. vol. 42. no. 1. pp. 137-141. 1994. [5] G. P. O tto and W. C. Chew. “Inverse scattering of H z waves using local shape function imaging: A T -m atrix form ulation,” Int. J. Imaging Syst. Tech., vol. 5, pp. 22-27. 1994. [6 ] W. H. W eedon and W. C. Chew. “Tim e-dom ain inverse scattering using the local sh ap e function (LSF) m ethod,” Inverse Probl., vol. 9. pp. 551-564, 1993. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [7] D. R. W ehner. High-Resolution Radar. Norwood, MA: Artech House, 1995. [8 ] S. L. M arple, Digital Spectral Analysis With Applications. Englewood Cliffs. NJ: Prentice-H all, 1987. [9] M. M oghaddam . W. C. Chew, and M. Oristaglio, "Comparison of the born iterative m ethod and tarantola's m ethod for an electrom agnetic tim e-dom ain inverse problem .” Int. J. Imaging Syst. Tech.. vol. 3. pp. 318-333. 1991. [10] M. M oghaddam and W. C. Chew. "Nonlinear two-dimensional velocity pro file inversion using tim e dom ain d a ta .” I E E E Trans. Geosci. Remote Sensing, vol. 30. pp. 147-156. Jan. 1992. [11] M. Born an d E. Wolf. Principles of Optics, Sixth Ed. New York: Pergam on Press. 1980. [12] Y.-M. W ang and W. C. Chew. "Accelerating the iterative inverse scatterin g algorithm s by using the fast recursive aggregate T -m atrix algorithm ." Radio Sci.. vol. 27. no. 2. pp. 109-116. 1992. [13] D. T. B orup and O. P. Gandhi. "Calculation of high-resolution sar distributions in biological bodies using te fft algorithm and conjugate gradient m eth o d .” I E E E Trans. Microwave Theory Tech.. vol. 33. no. 5. pp. 417-419. 1985. [14] T. K. Sarkar. "Application of fft and the conjugate gradient m ethod for th e so lution of electrom agnetic radiation from electrically large and sm all conducting bodies.” I E E E Trans. Antennas Propagat.. vol. 34. no. 5. pp. 635-640. 1986. [15] K. S. Yee. "Numerical solution of initial boundary value problems involving M axwell’s equations in isotropic m edia.” I E E E Trans. Antennas Propagat.. vol. AP-14. pp. 302-307. 1966. [16] W. H. W eedon. "Broadband microwave inverse scattering: Theory and experi m ent.” P h.D . dissertation. University of Illinois at U rbana-C ham paign, 1994. [17] A. Taflove. "Review of the form ulation and applications of the finite-difference tim e-dom ain m ethod for numerical modeling of electrom agnetic wave interac tions w ith a rb itra ry structures.” Wave Motion, vol. 10. pp. 547-582. 1988. [18] A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-Posed Problems. W ashing ton DC: V. H. W inston k. Sons. 1977. [19] A. N. Tikhonov. "On the problems w ith approxim ately specified inform ation,” in Ill-Posed Problems in the Natural Sciences, A. N. Tikhonov and A. V. Goncharsky. E d.. Moscow: MIR Publishers. 1987. [20] A. V. Goncharsky. "Ill-posed problems and their solution m ethods.” in Ill-Posed Problems in the Natural Sciences, A. N. Tikhonov and A. V. Goncharsky. E d.. Moscow: M IR Publishers. 1987. [21] J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Opti mization and Non-linear Equations. Englewood Cliffs. NJ: Prentice-H all, 1983. [22] E. Polak. Computational Methods in Optimization. New York: Academic Press, 1971. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [23] W . H. Press. B. P. Flannery. S. A. Teukolsky. and W . T . Vetterling. Numerical Recipes in C. New York: C am bridge University P ress. 1990. [24] M. Slaney. A. C. Kak, and L. E. Larsen. “L im itations of im aging with pirst-order diffraction tom ography.’’ I E E E Trans. Microwave Theory Tech.. vol. M TT-32. pp. 860-874. 1984. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 M U L T IP L E X IN G SC H E M E S F O R N O N L IN E A R IN V E R S E SC A T T E R IN G Nonlinear inverse scattering imaging algorithm s are notorious for their large m em ory consum ption and C P U rim tim e requirement. To reduce the C PU rim tim e and m em ory requirem ents, we propose frequency-division m ultiplexing (FDM) and code-division m ultiplexing (CDM) approaches for nonlinear inverse scattering algorithm s. The FD M and CDM schemes can be incorporated into nonlinear inverse scattering algorithm s such as the distorted-B orn iterative m ethod (DBIM ) [1 , 2] and the local shape function (LSF) m ethod [3-6]. This reduces the C P U run tim e and m em ory requirem ents by a factor proportional to the num ber of the tran sm itters. It allows us to use th e nonlinear inverse scattering algorithms to solve large problems for practical applications. T he FDM an d CDM approaches for nonlinear inverse scatterin g algorithm s are sim ilar to the m ultiple access schemes in wireless com m unications system s. M ulti ple access schemes are used to allow several mobile users to sim ultaneously share an available RF bandw idth spectrum . There are several types of m ultiple access schemes in wireless com m unication systems. Frequency division m ultiple access (FDM A). tim e division m ultiple access (TDM A). and code division m ultiple access (CDMA) are the m ain m ultiple access schemes [7]. The differences between the TD M A. FDM A. an d CDMA are shown in Figure 4.1. In tim e division m ultiple access (TDM A). each user tran sm its at a different tim e slot while sharing the same R F bandw idth. Frequency division multiple access (FDMA) divides the available RF bandw idth into several subdivisions and assigns different subdivisions to differ ent users. Each user is allocated a unique RF bandw idth and shares the same tim e slots. In code division m ultiple access (CDMA) [8 ], each user is assigned a unique code to encode th e tra n sm itted signal. All the users in CDM A system s shaxe the sam e frequency bandw idth and the same tim e slots. Besides th e tim e, frequency, and code diversity techniques, there is also a space division m ultiple access (SDMA) 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [7] scheme which explores the spatial diversity resulting from different users trans m itting and receiving a t different spatial locations. SDMA system s require the use of spatially separated m ultiple antennas or antenna arrays. TDMA FDMA CDMA F ig u r e 4.1 M ultiple access schemes. TDM A divides the R F spectrum into tim e slots; only one user is allowed to transm it or receive. FDM A divides the R F bandw idth into several divisions for different users. CDMA lets all users use the sam e RF bandw idth w ith th eir own codes and tran sm it simultaneously. 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Formulation of CDM and FDM Approaches to DBIM This section describes th e form ulation for the tim e-dom ain DBIM and LSF m ethods. As shown in C h ap ter 2. the DBIM Frechet transposed o p erato r equation is Sek (r) = iV-j' Ho ^ 2 n = 1 Jo Nr (4.1) rT x E j dt1gk ( r \ r m , t - t ^ S E l ^ i r ^ T - t'). m=i ’ Equation (4.1) represents a back-propagation in time followed by a correlation. The induced sources at each receiver Jq dt'8 E\ n k (rm, T —t') are back-propagated into the object space. (FD TD ) solver [9. 10] This requires us to rim the finite-difference tim e-dom ain for each tra n sm itte r separately. However, we can invoke lin earity to backpropagate all th e induced sources simultaneously to the object space. This is accomplished by sw apping the order of integration over t and .sum m ation over N r as shown in the equation below: Sek (r) = - f i 0 ^ 71=1 / d t — E z,n,k ( r ' . T - t ) Jo at-* T NR 1"“ “ / d f £ gk ( r ' . r m A - t ' ) S E l ^ k ( r ^ T - t'). m= 1 The next step in com puting the Frechet transposed operator is th e correlation of th e backpropagated field w ith the second tim e derivative of the incident field, o ^ E Ztn,k {r',t). This process is repeated for each transm itter location. If we can swap the sum m ation over N t w ith the integration over t. we can further lim it the num ber of times we need to ru n the FD T D solver. Unfortunately, this will alter the correct solution as shown in th e following steps. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If we let 5e'k (r) denote the result obtained after swapping the order of integration and s u m m a t io n , we get T nt Se'k (r) = - n o Jq dt £ q2 — E z.ntk{ r ' . T - t) r T ” = 1 nv r (4.3) dt' Y , x / 0 9 k{r'.rm ,t - t') 6El n k ( r m, T - t') m =l T he above equation can be further expressed as the following: rT <*4 (r ) = / jo rT + / Jo Q2 rT N r d t — E Z' i , k { r \ T - t) dt' ^ gk { r ' , r m , t - t')SEl l k ( r m , T - t ' ) at Jo m=1 q 2 rT N r d t — E z a ,k ( r ' . T - t) / dt' ^ 2 dt Jo ^ 9 k ( r ' . r m , t - t’)S Eez 2 k( r m , T - t') 9 k { r ' , r m , t - t')8EZ n k ( r m , T - t?) + ... rT + / Jo T ^2 ^R /•T’ d t — E z,n_k ( r ' . T - t ) / dt' dt Jo ^ ^2 t + lo d t d i ^ E z 'l 'k^r ' ' T ~ ^ I o N r dt> 1 2 Sk(r , , r m , t - t ' ) x m= 1 { S E l 2 k ( r m , T - t ' ) + . . . + 6 E l n .fc( r m, T - t')) B rT Q2 rT + Jo d t f o 2 E z n k (r ' ' T ~ t i' J ® N r dt>m= E 9 k { r \ r m , t -- t t') ' x 1 (<S££ltB( r m, T - ? ) + . . . + SEt n_ l k (rjn, T - t')) (4.4) In th e above equation, p art A is equal to ek (r) from Equation (4.3) and p a rt B represents th e ex tra cross-product terms. If we can somehow elim inate these cross pro d u ct term s, then e'k (r) reduces to e*;(r). T his allows us to swap th e order of su m m atio n over N t and the integration. T h e FD M and the CDM approaches are 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. used to elim inate the undesirable cross product terms. By doing so, we can reduce th e num ber of tim es we need to run the FD TD solver from N t to 1. This will save th e large m em ory requirem ent and CPU run time by a factor proportional to N t tim es. We use the TD M approach to implement the switched Vivaldi antenna array in our TD -U W B rad ar system as described in Chapter 2. Two microwave switches are used to control the antennas elements in the switched Vivaldi antenna array. O ne sw itch controls five tran sm itters while the other switch controls six receivers. Each sw itch chooses one antenna a t a given time slot. Using the TDM approach to im plem ent the Freechet transposed operator in the 2-D TD DBIM. we need to declare two 4-D arrays of size N x x N y x N t x N t • where N x is the num ber of unknow ns along the x axis (cross range), N y is number of unknowns along the y axis (range), N t is the num ber of tim e steps in the FDTD m ethod, and N t is the num ber o f tran sm itters. For FD M and CDM approaches, we only need to declare two 3-D arrays of size N x x N y x N t . The memory requirement for applying the TD M , FDM, an d CDM to a 2-D tim e-dom ain DBIM with respect to the num ber of tr a n sm itte r s is shown in Figure 4.2. In this case, the param eters are N x = 100. N y = 100, and N t = 1500. From the figure, we can clearly see that the m emory requirem ent for FD M an d CDM schemes is independent of the number of tra n sm itte r s, while the m em ory requirem ent of the TDM scheme is linearly proportional to the num ber of tran sm itters. In the FD M approach, different parts of the frequency spectrum are assigned to different tran sm itters. For our switched Vivaldi antenna array, the whole frequency b an d w id th is divided into five subbands because we have five tran sm itters in our sw itched Vivaldi an ten n a array. Each tran sm itter is allocated a different frequency b an d w id th as shown in Figure 4.3. Because each tran sm itter transm its a different R F b an d signal, the values of cross product term s appearing in p art B of E quation (4.4) are zero. In the CDM approach, we can assign a set of orthogonal codes to the sam e frequency spectrum for the transm itters as shown in Figure 4.4 and Figure 4.5. Since the FDM codes are also orthogonal over the given frequency spectrum , th ey can be viewed as a special case of the more general CDM approach. A lthough 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iah TDM CDMi FDM 16F 1 , 21- i<y £• 8 1 2 4f- - 2F O'---------------—— —----------------------- ---------------- ----------------------------------------------------------------0 5 10 15 20 25 30 35 40 Number of Transmitters F ig u r e 4 .2 M emory requirement of applying TDM. CDM. FDM to a 2-D tim e-dom ain DBIM Frechet transposed operator. The num ber of unknown variables for a 2-D object is 10.000. There are 1.500 tim e steps in FD TD . th e CDM A has been used in the communication systems for a long tim e, to the best of our knowledge it has not been used in the FD TD solver and nonlinear inverse scatterin g for EM problems. A pplying the FDM and CDM schemes to the DBIM. the CPU rim tim e com p ared to the original TD M approach is reduced by a factor less th an N t (num ber of tra n sm itte rs). In FDM . the whole frequency spectrum is divided by N t resulting in a pulse w idth N t tim es wider than the original transm itted signal. This increases th e to ta l tim e steps needed in the FDTD forward solver as shown in Figure 4.6. In the figure, we assum e the original pulse w idth for the TDM to be 50 tim e steps. The tra n sm ittin g pulse sta rts a t tim e steps equal to zero, and the receiving pulse s ta rts a t the tim e steps equal to 1.500. In the FDM case, the pulse w idth becomes 250 tim e steps after the frequency spectrum is divided by the num ber of tr ansm itte rs (five in this case). As a result, the FDTD solver has to be rim for an additional 200 steps. So the reduction of the CPU run tim e for FDM in this case is N t x tim es. In CDM , the tran sm itted pulses are widened after being coded w ith the orthogonal codes. The widened pulses will also cause additional steps for FD T D 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F ig u r e 4 .3 Basis functions for FDM. F ig u r e 4 .4 H aar basis functions for CDM. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 3 F ig u r e 4 .5 6 S 7 a 9 10 9 10 y X 2 4 y \ 3 s' 4 5 6 Frequency (GHz) 7 s 8 Sinusoidal basis functions for CDM. (a) TDM 1500 1550 Time Step* (b> FDM O 250 Time Steps 1500 1750 F ig u r e 4 .6 Com parison of tim e steps in the FDTD solver needed for th e TD M and FDM schemes. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solver, sim ilar to th e FDM . The sam e FD M and CDM approaches can also be applied to the LSF because the LSF m ethod has the same form of Frechet transposed operator as the DBIM. 4.2 Im aging R econ stru ctio n R esult This section presents some of the imaging reconstruction results using the FDM and CDM approaches. First, we show the image reconstruction results using the limited angle con figuration as shown in the Figure 3.4 of C hapter 3. Figure 4.7 shows the image reconstruction result obtained by using an FDM -based LSF on a m etallic cylinder of diam eter 3.2 cm. The figure shows the image reconstruction results at different iterations steps: (a) step = 1. (b) step = 2. (c) step = 5. (d) step = 10. (e) step = 15. and (f) step = 20. From the result, we notice th a t using FDM to LSF inverse scat tering causes different tran sm itters to see different resolutions of the image. This is because the different frequency bands are allocated to different transm itters. Figure 4.8 shows the perm ittivity profile of a test object to be reconstructed. In Figure 4.8. (a) shows the 2-D image of the p erm ittiv ity profile while (b) shows th e 3-D profile of the perm ittivity. Figures 4.9-4.13 show the image reconstruction result obtained by using a CDMbased DBIM. T h e figures show the image reconstruction results at different iteration steps. From the results, we see th a t the reconstructed image approaches the shape of the test object as the num ber of iterations increases. The relative residual error a t the Arth iteration is defined as , 1/2 RRE = Z n h Z l u f i dt[Ez ,„,k ( r .t ) Y (4.5) where E z,n(Tm, t ) is the measured electric field a t th e m th receiver due to th e n th 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) iteration step = 1 Range (cm) (b) iteration step = 2 cr 42 -15 -4 -2 0 2 4 -4 (c) iteration step = 5 -2 0 2 4 Range (cm) (d) iteration step = 10 x 10" 0 -5 ® 40 cr 42 -4 -2 0 2 4 -1 0 -4 (e) iteration step = 15 -2 0 2 4 Range (cm) (0 iteration step = 20 cr 42 -4 -2 0 2 4 -4 Cross Range (cm) -2 0 2 4 Cross Range (cm) F ig u r e 4 .7 Image reconstruction result of a metallic cylinder (3.2 cm diam eter) by applying FDM to LSF m ethod. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) Range (cm) F ig u re 4 .8 35 -5 Cross Range (cm) P erm ittiv ity profile of a test object. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) iteration step = 2 (a) iteration step = 1 1.15 o 40 1.05 -2 0 2 Cross Range (cm) -2 0 2 Cross Range (cm) (d) iteration step = 2 (b) iteration step = 1 Range (cm) 35 -5 Range (cm) Cross Range (cm) 35 -5 Cross Range (cm) F ig u r e 4 .9 Im age reconstruction result of the test object in Figure 4.8 by applying CDM to th e DBEM m ethod, (a) and (b) show the profile of the first itera tio n step, (c) and (d) show the profile of the second itera tio n step. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) iteration step = 5 (c) iteration step = 10 -2 0 2 Cross Range (cm) -2 0 2 Cross Range (cm) (d) iteration step = 10 (b) iteration step = 5 Range (cm) 35 -5 Range (cm) Cross Range (cm) 35 -5 Cross Range (cm) F ig u r e 4 .1 0 The image reconstruction result of the test object in Fig ure 4.8 by applying CDM to the DBEM m ethod, (a) and (b) show the profile of the fifth iteration step, (c) and (d) show the profile of the ten th iteration step. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) iteration step = 30 (a) iteration step = 20 1.8 36 1.7 1 .6 1.5 38 1.6 § j4 0 e co tr 42 1.4 E f X 1 .4 1 .3 1 .2 1.2 1.1 44 1 -2 0 2 Cross Range (cm) -2 0 2 Cross Range (cm) ' (d) iteration step = 30 (b) iteration step = 20 Range (cm) ' Range (cm) Cross Range (cm) Cross Range (cm) F ig u r e 4 .1 1 Image reconstruction result of th e test object in Figure 4.8 by applying CDM to D BIM m ethod, (a) and (b) show th e profile of the 20th iteration step, (c) and (d) show th e profile of th e 30th iteration step. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) iteration step = 100 (a) iteration step = 50 36 1.8 38 1.6 E o §>40 § cc 42 1.4 1.2 44 1 -2 0 2 Cross Range (cm) -2 0 2 Cross Range (cm) (d) iteration step = 100 (b) iteration step = 50 Range (cm) Range (cm) Cross Range (cm) Cross Range (cm) F ig u r e 4 .1 2 Image reconstruction result of the test object in Figure 4.8 by applying CD M to the DBIM m ethod, (a) and (b) show the profile of the 50th iteratio n step, (c) an d (d) show the profile of the 100th iteration step. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (c) iteration step = 200 (a) iteration step = 150 2 36 1.8 1.8 Range (cm) 38 "e o. £40 c CO cr 42 1.6 1 .4 1.6 1.4 w 1.2 1.2 44 1 1 -2 0 2 Cross Range (cm) -2 0 2 Cross Range (cm) (d) iteration step = 200 (b) iteration step = 150 Range (cm) 4 Range (cm) Cross Range (cm) Cross Range (cm) F i g u r e 4 .1 3 Image reconstruction result of the test object in Figure 4.8 by applying CDM to the DBIM m ethod, (a) and (b) show the profile of th e 150th iteration step, (c) an d (d) show the profile of the 2 00 th iteration step. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transm itter, and £ z.n.ib(rm , t ) is the com puted solution at the k th iteration. The RRE is shown in Figure 4.14 for each iteration step. The C PU run tim e for each iteration is 1.75 m in on a DEC personal w orkstation 500au. 0.9 0.8 0.7 uj 0.6 S 0.5 I 0.4 0.3 0.2 0.1 40 60 100 120 140 160 180 200 Iteration Step F ig u re 4 .1 4 Relative residual error for DBIM reconstruction shown in Figures 4.9-4.13. Note th a t tru n catio n of the CDM codes in the tim e dom ain will prevent the spurious cross-product term from vanishing completely. This is illustrated in Figures 4.15 and 4.16. which show the tim e domain profiles for the CDM codes shown in Figures 4.4 and 4.5. To save the num ber of tim e steps in the FD T D m ethod, the transm itted signal needs to be truncated. From Figures 4.15 and 4.16. we can see th a t the sinusoidal basis functions are a b e tte r choice than the H aar basis functions. This is because th e time dom ain profiles of the sinusoidal basis functions are more localized in tim e such th a t we can use the sh o rter pulses as the sources for the F D T D solver to save th e time steps. The tru n catio n in time dom ain prevents the signals from being completely orthogonal to each other. This introduces an undesirable cross product 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.1 Tffl 3D a. E < - 0.1 0 0.1 0.5 1 1.5 2.5 2 3 3.5 4 4.5 CL E < 0.1 0.5 0.05 2.5 3.5 4.5 T T -0.05 0 0.05 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -0.05 0 0.05 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Amplitude Amplitude Amplitude - -0.05 0 Time (ns) F ig u re 4 .1 5 CDM. Tim e-dom ain profiles of the H aar basis functions for 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x 1Q-9 Amplitude x 10“3 -5 3.5 4.5 2.5 3.5 4.5 2.5 3.5 4.5 2.5 3.5 4.5 Amplitude 2.5 -5 Amplitude 1.5 Amplitude -5 Amplitude -5 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (ns) F ig u r e 4 .1 6 CDM. Tim e-dom ain profiles of the sinusoidal basis functions for 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 term s which ten d to degrade the reconstructed image. Despite this drawback, it is still worthwhile to try the CDM and FDM approaches because of their huge savings in m em ory requirem ent and C PU run time. They can also provide a quick estim ate which can be used as the initial guess for the TD M approach. Figure 4.17 shows a perm ittivity profile of University of Illinois logo. The di electric constant is 2. Figures 4.18. 4.19. and 4.20 show the image reconstruction result obtained by using a CDM -based DBIM on the UI logo. T he figures show the image reconstruction results at different iteration steps. From the results, we see th a t the reconstructed image approaches the shape of the dielectric UI logo as the num ber of iterations increases. T he relative residual error is shown in Figure 4.21 for each iteration step. This result is based on a full angle configuration as shown in Figure 4.22. In Figure 4.22, iV, = 341, N j = 341, Oi = 35. and Oj = 35. There are four transm itters and eight receivers in full angle arrangem ent. The num ber of tim e steps used in the FD T D solver is 1.500. T he grid param eters used are A x = A y = 2.5 mm and A t = 5 ps for FDTD. The C P U ru n tim e for each iteration costs 2.65 m in on a DEC personal w orkstation 500au. 4.3 C on clusion This chapter has presented a new scheme for nonlinear inverse scattering. We have applied the FDM and CDM schemes to reduce the CPU rim tim e and memory requirem ent of the nonlinear inverse scattering solvers DBIM and LSF by a factor p ro portional to the num ber of transm itters. The memory requirem ent and CPU run tim e have been com pared for the TDM. FDM and CDM -based schemes. We have also shown some prelim inary numerical results based on the FDM and CDM schemes. T he m ain draw back in the FDM -based scheme is th a t different transm itters see different resolutions of the image. This is because different frequency bands are used for different transm itters. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 4 - 2 0 2 Cross Range (cm) 4 (b) Range (cm) F ig u re 4 .1 7 Cross Range (cm) P erm ittiv ity profile of the UI logo. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) iteration step = 2 (a) iteration step = 1 36 1.4 38 "oE §>40 § cr 42 Range (cm) 1.3 1.2 1.1 44 1 -2 -2 0 2 Cross Range (cm) 2 (d) iteration step = 10 (c) iteration step = 5 2 36 1.8 Range (cm) 0 Cross Range (cm) 2 38 1.6 eo. 1.4 §>40 § cc 42 • * 1.8 1.6 1'4 1.2 1.2 44 1 - 4 - 2 0 2 Cross Range (cm) 1 - 4 - 2 0 2 Cross Range (cm) F ig u re 4 .1 8 Image reconstruction result of U I logo by applying CDM to the DBIM m ethod, (a) iteration step = 1. (b) iteration step = 2. (c) iteration step = -5. and (d) iteration step = 10. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) iteration step = 30 (a) iteration step = 20 2.2 36 2 Range (cm) 38 1.8 E tj. §>40 c « £E 42 1.6 1.4 • « 1.2 44 1 -2 -2 0 2 Cross Range (cm) 2 (d) iteration step = 50 (c) iteration step = 40 36 2.2 2 Range (cm) 0 Cross Range (cm) 38 E 1.8 o S)40 c a tr 42 1.6 1.4 1.2 44 1 -2 -2 0 2 Cross Range (cm) 0 2 Cross Range (cm) F ig u r e 4 .1 9 Im age reconstruction result of UI logo by applying CDM to the DBIM m ethod, (a) iteration step = 20. (b) iteratio n step = 30. (c) iteration step = 40. and (d) iteration step = 50. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) iteration step = 100 (a) iteration step = 60 36 2 .2 Range (cm) 2 38 1.8 E o. 1.6 o>40 c (0 oc 1.4 42 1 .2 44 1 -2 0 -2 2 0 2 C ro ss R a n g e (cm ) C ro s s R a n g e (cm) (d) iteration s te p = 2 0 0 (c) iteration s te p = 150 2 .4 36 2 .2 38 Range (cm) 2 1.8 E o. 1.6 8 ,4 0 c o c c 1.4 42 1.2 44 1 -2 0 -2 2 0 2 C ro ss R a n g e (cm ) C ro s s R a n g e (cm) F ig u r e 4 .2 0 Image reconstruction result of UI logo by applying CDM to the DBINI m ethod, (a) iteration step = 60. (b) iteration step = 100. (c) iteration step = 150. and (d) iteration step = 200. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o.z1- 60 80 100 120 140 160 180 200 ((•ration Slap F ig u r e 4.2 1 Figures 4 Relative residual error for DBIM reconstruction shown in Ahttvtxnp hnfcr [ ] : Roxtvcr Uxaoun ^ . rmwn«u/mu»u h F ig u r e 4 .2 2 Full angle inverse scattering configuration. FD TD grid contains Ni x N j grid points which contain an optim ization subgrid th a t contains Oi x Oj grid points, transm itters and receivers located around the optim ization subgrid, and an absorbing border region. 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he CDM approach for inverse scattering is a breakthrough because of the large reduction of the memory requirem ent and the com putational tim e of the nonlinear inverse scattering. However, there will be some errors introduced in the process due to the presence of the unw anted cross-product term s. A careful study of the various codes used in the CDM -based scheme might yield a set of optim al orthogonal codes which will minimize the error. A lthough the image reconstruction results obtained by using the CDM- and FD M -based approaches to DBIM and LSF m ethods are not as good as the TD M approach, there are still m any advantages to use the former. A part from the sig nificant reduction in the m em ory requirem ent and C PU tim e, the CDM approach can also be used to obtained a quick estim ate of the target. This can be used as an initial guess in the more accurate TDM scheme. The accuracy of the initial guess will help us limit the to ta l num ber of iterations in the m ore expensive TD M approach. A nother possible area for improvement is the use of the CDM A scheme for d a ta acquisition. The switched Vivaldi antenna array in our T D UWB radar system adopts the TDM A and SDMA approaches. The TDM A approach is used to reduce the cost of our imaging rad ar system since we need to use ju st two UWB amplifiers a t th e two microwave switches. One switch controls five tra n sm itters and the other sw itch controls six receivers. Only one tran sm itter and one receiver operate a t a given tim e slot. All the antennas in the switched Vivaldi antenna array are focused at a distance of 40 cms from the center of the array. This SDMA approach ensures th a t the m ain beam will focused on the targ et zone. To improve the system perform ance, we can use a CDM A-based scheme to carry out the d ata acquisition. This will allow us to o perate all the tran sm itters and receivers at the same tim e. This will reduce the m easurem ent tim e by a factor of N r x N t - 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4 References [1] W. C. Chew. Waves and Fields in Inhomogeneous Media. N ostrand. 1990. New York: Van [2] Y.-M. W ang and W. C. Chew. "An iterative solution of tw o-dim ensional elec trom agnetic inverse scattering problem." Int. J. Imaqinq Syst. Tech.. vol. 1. pp. 100-108. 1989. [3] W. C. Chew and G. P. O tto. "Microwave imaging of m ultiple conducting cylin ders using local shape functions." IE E E Microwave Guided Wave Lett., vol. 2. pp. 284-286. July 1992. [4] G. P. O tto and W . C. Chew. "Microwave inverse scattering: Local shape func tion im aging for im proved resolution of strong scatterers." IE E E Trans. M i crowave Theory Tech.. vol. 42. no. 1. pp. 137-141. 1994. [5] G. P. O tto and W. C'. Chew. "Inverse scattering of H z waves using local shape function imaging: A T -m atrix formulation." Int. J. Imaging Syst. Tech.. vol. 5. pp. 22-27. 1994. [6] W. H. W eedon and W. C. Chew. “Tim e-dom ain inverse scatterin g using the local shape function (LSF) m ethod." Inverse Probl.. vol. 9. pp. 551-564. 1993. [7] T. S. R ap p ap o rt. Wireless Communications Principles & Pracitce. U pper Sad dle River. NJ: Printice-H all. 1996. [8] A. J. V iterbi. CDMA: Principles o f Spread Spectrum C om m unication. Reading. MA: Addison-W es ley. 1995. [9] K. S. Yee. "Num erical solution of initial boundary value problem s involving Maxwell's equations in isotropic media." IE E E Trans. A ntennas Propagat., vol. AP-14. pp. 302-307. 1966. [10] A. Taflove. "Review of the form ulation and applications of the finite-difference tim e-dom ain m ethod for numerical modeling of electrom agnetic wave interac tions w ith arb itrary structures." Wave Motion, vol. 10. pp. 547-582. 1988. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 C O N C L U S IO N S A N D F U T U R E D IR E C T IO N S In this dissertation, we have covered a wide variety of topics ranging from exper im ental work in the low-power tim e-dom ain (TD) ultra-w ideband (UW B) imaging rad ar system to num erical work in nonlinear inverse scattering algorithm s. Three m ajor contributions of this thesis are summarized below. The first contribution is the development of the low-cost low-power tim e-dom ain (TD) ultra-w ideband (UW B) microwave imaging radar system . The system is au tom atically controlled by a personal computer via the IEEE-488 bus. The d a ta acquisition and system calibration procedures have been presented and verified. T he system is designed for a q u antitative nondestructive evaluation and im aging of dielectric or m etallic targets located in air or shallow subsurface. The low-power TD -UW B im aging rad a r system has dem onstrated its utility by collecting several useful m easurem ent d a ta from several different kinds of targets including dielectric targets, m etallic targets, and m etallic targets buried in concrete. These m easure m ent d a ta have been processed by the nonlinear inverse scattering algorithm s, the DBIM and the LSF. to obtain the reconstructed images which m atch well w ith the geom etry and com position of the targets. The UWB tim e-dom ain m easure m ent d a ta are usually preferred for inverse scattering because the UWB pulse has more inform ation content th an th a t of frequency-domain d a ta collected at several discrete frequencies. The “phase wrapping" problem present in diffraction tom og raphy is not present when transient data are used along w ith nonlinear iterative inverse scattering algorithm s such as the DBIM and the LSF m ethods. In addition, a tim e-gating function can be used to eliminate undesired early-tim e and late-tim e arrival signals. The second contribution is the experimental verification of the super-resolution phenom enon in nonlinear inverse scattering. We have used our newdy developed tim e-dom ain ultra-w ideband microwave imaging radar system to obtain the super- 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. resolution im aging reconstruction result from an experim ental setup w ith lim ited viewing angles. This is the first tim e th a t the super-resolution phenom enon has been verified experim entally in nonlinear inverse scattering using d a ta collected from lim ited viewing angles. The experim ental d a ta is processed with the DBIM and shown to be able to resolve features smaller th a n the half-wavelength dictated by the Rayleigh criterion. In addition, we have dem onstrated th a t the DBEM can recover inform ation on the backside of the penetrable scatterer even though the scatterin g d a ta is collected only from limited viewing angles on the front side of the scatterer. This is a consequence of extracting inform ation from the m ultiple scatterin g physics inherent in the measurem ent d ata. T he th ird contribution is the application of the code-division multiplexing (CDM ) scheme to nonlinear inverse scattering algorithm s. T he CDM -based nonlinear in verse scattering algorithm s has reduced the C PU run tim e and memory requirem ent by a factor proportional to the num ber of tran sm itters in the imaging experim ental setup. T he CDM -based scheme for nonlinear inverse scattering is sim ilar to the code-division m ultiple access (CDMA) technique for wireless com m unication sys tems. Each tran sm itter is assigned a unique code for transm itting. All the coded transm ission signals are sim ultaneously used as the excitation sources in the finitedifference tim e-dom ain (FD TD ) forward solver. Consequently, we need to run the FD TD forward solver ju st once instead of running it N t times. Since the forward scatterin g problem m ust be solved a t each iteration step of the nonlinear iterative inverse scattering algorithm s, our inverse scattering algorithm s are lim ited by the com putational com plexity of the forward scattering solver. Therefore, applying the CDM scheme to forward solvers is one of the keys to im proving the nonlinear inverse scatterin g algorithm s. The CDM scheme for nonlinear inverse scattering will allow nonlinear inverse scattering algorithm s to solve much larger problems for practical applications. A lthough we have successfully designed and developed a cost-effective timedom ain ultra-w ideband imaging radar system for the purpose of quantitative non destructive evaluation, there is still much work to be done in the future. direction for possible future work is briefly discussed as follows. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T he Several issues need to be investigated in th e proposed CD M -based n o n lin ear inverse scatterin g algorithm s, such as the choice of the optim al codes for each tra n sm itte r and the error introduced a t the Frechet transposed o p erato r from the approxim ated orthogonal codes due to the tim e-dom ain truncation. Since th e forward solver is the bottleneck in our nonlinear inverse scatterin g solver, fast forward solvers such as tim e-dom ain fast m ultipole m ethods (FM M ) could b e incorporated into the tim e-dom ain nonlinear inverse scattering a lg o r ith m . In ad d itio n , the possibility of applying the CDM scheme to the FM M m eth o d for n on lin e a r inverse scattering needs to be explored. O ur TD -U W B imaging rad ar system currently uses the tim e-dom ain m ultiplex ing (TD M ) scheme for d a ta acquisition due to its sim plicity in im plem entation. Using th e TD M scheme, two microwave switches are used to control the five tra n s m ittin g antennas and six receiving antennas in the Vivaldi antenna array. Each switch chooses one antenna at a given tim e slot. To speed up the m easurem ent process, th e CDM scheme m ust be em ployed in the d ata acquisition procedure. In CDM schem e, all the tran sm ittin g a n d receiving antennas transm it and receive in the sam e tim e slot. This will speed up the d a ta acquisition procedure by a factor of Nt x N r tim es as com pared to th e TD M scheme. The system should also be extended and im plem ented in 3-D for more practical applications. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V IT A Fu-Chiarng Chen was born on May 7. 1966. in Taiwan. R.O .C. From Septem ber 1984 to June 1988. he atten d ed N ational Taiwan University (N TU ) and received the Bachelor of Science degree in electrical engineering. From Ju ly 1988. Mr. Chen began his graduate studies at NTU. During the studies, he participated in designing and implem enting a polarim etric rad ar sys tem under the guidance of Professor Tah-Hsiung Chu. In June 1990, he received the M aster of Science in electrical engineering with a thesis entitled “M onostatic W ideband A utom atic Polarim etric Scattering M easurement System .” From July 1990 to M ay 1992. Mr. Chen served the two years compulsory m ilitary service as a second lieutenant com m unication engineering officer in Taiwan, R.O.C. From June 1992 to Ju ly 1993. Mr. Chen joined the In stitu te of Inform ation Science. Academia Sinica. Taiwan. He participated in the research of a large-scale and high-speed asynchronous transfer mode (ATM) switching network. Mr. Chen began his P h.D . study a t University of Illinois at U rbana-C ham paign (UIUC) in August 1993. Since then, he has been working for Professor Weng Cho Chew as a research assistant in the D epartm ent of Electrical and Com puter Engineering. W hile at UIUC. Mr. Chen had the unique opportunity to work on both experim ental system s and theoretical aspects of electrom agnetics. To gain some teaching experience, he volunteered as a teaching assistant in th e undergraduate an ten n a course. In June 1998. Mr. Chen won the Best Student Paper Award a t the 1998 IEEE A ntennas and P ropagation International Symposium. A list of Mr. Chen's technical publications and presentations is given below. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. F.-C. Chen and W . C. Chew. “M ultiplexing schemes for nonlinear inverse scat tering,” subm itted to 1999 Progress In Electromagnetics Research Symposium. 2. T. C. Cui. W. C. Chew, and F.-C. Chen. “R adar antenna." to be published in Encyclopidia o f Electrical and Electronics Engineering. New York: John Wiley Sons. 3. F.-C. Chen and W. C. Chew. "Experim ental verification of super-resolution in nonlinear inverse scattering." Appl. Phys. Lett., vol. 72. No. 23, pp. 3080-3082, June 1998. 4. F.-C. Chen and W. C. Chew. "Time-domain ultra-w ideband microwave imaging rad ar system." to be su b m itted for publication. 5. F.-C. Chen and W. C. Chew. “CDM and FDM for nonlinear inverse scattering,” to be subm itted for publication. 6. F.-C. Chen and W. C. Chew. "Development and testing of the tim e-domain mi crowave nondestructive evaluation system." Review o f Progress in Quantitative Nondestructive Evaluation, vol. 17. pp. 713-718. 1998. 7. F.-C. Chen and W. C. C hew ."Experim ental verification of super-resolution in nonlinear inverse scattering." in Progress In Electromagnetics Research Sympo sium. Nantes. France. July 1998. 8. F.-C. Chen and W. C. Chew. "Ultra-wideband rad ar im aging experiment for verifying super-resolution in nonlinear inverse S cattering.” in IE E E A P -S In ternational Sym posium and U R SI National Radio Science Meeting, A tlanta, GA. June 1998. pp. 1284-1287. 9. F.-C. Chen and W. C'. Chew ."Tim e-dom ain ultra-w ideband microwave imag ing radar." in IE E E Instrum entation and M easurement Technology Conference Proceedings . St. Paul. MN. May 1998. pp. 648-650. 10. F.-C. Chen and W. C. Chew. "A portable nondestructive evaluation system." Technical Report for ARM Y CPAR-CRDA 0270 project. Sep. 1997. 11. F.-C. Chen and W. C. Chew. "Microwave imaging rad ar system for detecting buried objects." in IE E E International Geoscience and Rem ote Sensing Sym posium. Singapore. Aug. 1997. pp. 1474-1476. 12. F.-C. Chen and W. C. Chew. "Ultra-wideband im aging rad ar system .” in U RSI Radio Science Meeting. M ontreal. Canada. July 1997. p. 13. 13. F.-C. Chen and W. C. Chew. "An impulse rad ar nondestructive evaluation system ." Review o f Progress in Quantitative N ondestructive Evaluation, vol. 16. pp. 709-715. 1997. 14. F.-C. Chen. W. C. Chew, and W. H. Weedon, "Inverse scattering imaging using tim e-dom ain ultra-w ideband radar." in U RSI Radio Science Meeting, Baltimore, MD. July 1996. 15. F.-C. Chen and W. C. Chew. "Time-domain ultra-w ideband radar system for nondestructive evaluation." in U RSI Radio Science Meeting. Baltimore, MD, July 1996. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16. F.-C. Chen and W. C. Chew. "Tim e-dom ain pulse system for inverse scattering and imaging,” A n ten n a Applications Symposium . M onticello. IL, Sep. 1996. 17. W . H. Weedon, F.-C. C hen and W. C. Chew. "D etection and imaging of defects in concrete stru ctu res using microwave nondestructive evaluation,” Prog. In E M Research Sym . Proc.. the Universitv of W ashington. Seattle. July 1995, p. 821. 18. T .-J. Chen. T.-H. C hu an d F.-C. Chen. "A new calibration algorithm of wide band polarim etric m easurem ent system .” IE E E Trans. Antennas Propagat., vol.39. no. 8. pp. 1188-1192. Aug. 1991. 19. T .-J. Chen. T.-H. C hu an d F.-C. Chen. "A new calibration algorithm of wide band polarim etric m easurem ent system." in IE E E A P -S /U R S I International Symposium, O ntario. C anada, June 1991. pp. 764-767. 20. T.-H. Chu and F.-C. Chen. "Broadband polarim etric m onostatic m easurem ent system .’’ in Telecommunication Symposium, Taipei. Taiwan. Dec. 1989. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IMAGE EVALUATION TEST TARGET ( Q A - 3 ) 1.0 “ IS l £ 12.0 l.l 1.8 1.25 1.4 15 0 mm IM/4 G E . Inc 1653 East Main Street Rochester. NY 14609 USA Phone: 716/482-0300 Fax: 716/288-5989 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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