Isar

EURASIP Journal on Applied Signal Processing

Inverse Synthetic Aperture Radar

Guest Editors: Marco Martorella, John Homer, James Palmer,Victor Chen, Fabrizio Berizzi, Brad Littleton, and Dennis Longstaff


Guest Editors: Marco Martorella, John Homer,James Palmer, Victor Chen, Fabrizio Berizzi,Brad Littleton, and Dennis Longstaff


Copyright 2006 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in volume 2006 of EURASIP Journal on Applied Signal Processing. All articles are open accessarticles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

Editor-in-ChiefAli H. Sayed, University of California, USA

Associate EditorsKenneth Barner, USA Sren Holdt Jensen, Denmark Vitor H. Nascimento, BrazilMauro Barni, Italy Mark Kahrs, USA Sven Nordholm , AustraliaRichard Barton, USA Thomas Kaiser, Germany Antonio Ortega, USAAti Baskurt, France Moon Gi Kang, South Korea Douglas OShaughnessy, CanadaKostas Berberidis, Greece Matti Karjalainen, Finland Montse Pardas, SpainJose C. Bermudez, Brazil Walter Kellermann, Germany Wilfried Philips, BelgiumEnis Cetin, Turkey Joerg Kliewer, USA Vincent Poor, USAJonathon Chambers, UK Lisimachos P. Kondi, USA Ioannis Psaromiligkos, CanadaBenoit Champagne, Canada Alex Kot, Singapore Phillip Regalia, FranceJoe Chen, USA Vikram Krishnamurthy, Canada Markus Rupp, AustriaLiang-Gee Chen, Taiwan Tan Lee, Hong Kong Bill Sandham, UKHuaiyu Dai, USA Geert Leus, The Netherlands Bulent Sankur, TurkeySatya Dharanipragada, USA Bernard C. Levy, USA Erchin Serpedin, USAFrank Ehlers, Italy Ta-Hsin Li, USA Dirk Slock, FranceSharon Gannot, Israel Mark Liao, Taiwan Yap-Peng Tan, SingaporeFulvio Gini, Italy Yuan-Pei Lin, Taiwan Dimitrios Tzovaras, GreeceIrene Gu, Sweden Shoji Makino, Japan Hugo Van hamme, BelgiumPeter Handel, Sweden Stephen Marshall, UK Bernhard Wess, AustriaR. Heusdens, The Netherlands C. Mecklenbruker, Austria Douglas Williams, USAUlrich Heute, Germany Gloria Menegaz, Italy Roger Woods, UKArden Huang, USA Ricardo Merched, Brazil Jar-Ferr Yang, TaiwanJiri Jan, Czech Republic Rafael Molina, Spain Abdelhak M. Zoubir, GermanySudharman K. Jayaweera, USA Marc Moonen, Belgium

Contents

Inverse Synthetic Aperture Radar, Marco Martorella, John Homer, James Palmer, Victor Chen,Fabrizio Berizzi, Brad Littleton, and Dennis LongstaffVolume 2006 (2006), Article ID 63465, 4 pages

Use of Genetic Algorithms for Contrast and Entropy Optimization in ISAR Autofocusing,Marco Martorella, Fabrizio Berizzi, and Silvia BruscoliVolume 2006 (2006), Article ID 87298, 11 pages

Eigenspace-Based Motion Compensation for ISAR Target Imaging, D. Yau, P. E. Berry,and B. HaywoodVolume 2006 (2006), Article ID 90716, 9 pages

Adaptive Local Polynomial Fourier Transform in ISAR, Igor Djurovi, Thayananthan Thayaparan,and Ljubia StankoviVolume 2006 (2006), Article ID 36093, 15 pages

An Analysis of ISAR Image Distortion Based on the Phase Modulation Effect, S. K. Wong,E. Riseborough, and G. DuffVolume 2006 (2006), Article ID 83727, 16 pages

Target Identification Using Harmonic Wavelet Based ISAR Imaging, B. K. Shreyamsha Kumar,B. Prabhakar, K. Suryanarayana, V. Thilagavathi, and R. RajagopalVolume 2006 (2006), Article ID 86053, 13 pages

Supervised Self-Organizing Classification of Superresolution ISAR Images: An Anechoic ChamberExperiment, Emanuel Radoi, Andr Quinquis, and Felix TotirVolume 2006 (2006), Article ID 35043, 14 pages

Hindawi Publishing CorporationEURASIP Journal on Applied Signal ProcessingVolume 2006, Article ID 63465, Pages 14DOI 10.1155/ASP/2006/63465

EditorialInverse Synthetic Aperture Radar

Marco Martorella,1, 2 John Homer,3 James Palmer,4 Victor Chen,5 Fabrizio Berizzi,1, 2

Brad Littleton,6 and Dennis Longstaff1

1The school of ITEF, The University of Queensland, Brisbane 4072, Australia2Department of Information Engineering, University of Pisa, Via G. Caruso 16, 56122 Pisa, Italy3 School of Information Technology & Electrical Engineering, University of Queensland, Brisbane 4072, Australia4Radar Modelling & Analysis Group, Electronic Warfare & Radar Division, Defence Science & Technology Organisation,P.O. Box 1500, Edinburgh 5111, UK

5Naval Research Laboratory, 4555 Overlook Ave., SW Washington, DC 20375, USA6Centre for Quantum Computer Technology, School of Physical Sciences, University of Queensland, Brisbane 4072, Australia

Received 2 March 2006; Accepted 2 March 2006

Copyright 2006 Marco Martorella et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Introduction to ISAR

Inverse synthetic aperture Radar (ISAR) is a powerful sig-nal processing technique that can provide a two-dimensionalelectromagnetic image of an area or target of interest. Be-ing radar based, this imaging technique can be employed inall weather and day/night conditions. ISAR images are ob-tained by coherently processing the received radar echoes oftransmitted pulses. Commonly, the ISAR image is charac-terised by high resolution along both the range and cross-range directions. High resolution in the range direction isachieved by means of large bandwidth transmitted pulses,whereas high cross-range resolution is obtained by exploitinga synthetic antenna aperture. In ISAR, the synthetic apertureis generated by motion of the target as well as possibly bymotion of the radar platform. In contrast, the related imag-ing technique of Synthetic aperture radar (SAR) has its syn-thetic aperture generated by means of radar platformmotiononly.

Initially, the name ISAR was derived from SAR by simplyconsidering a radar-target dynamic where the radar platformwas fixed on the ground and the target was moving around.Today, however, it is understood that the basis of the dier-ence between SAR and ISAR lies in the noncooperation ofthe ISAR target. Such a subtle dierence has led in the lastdecades to a significant separation of the two areas. The non-cooperation of the target introduces the main problem ofnot knowing the geometry and dynamic of the radar-targetsystem during the coherent integration time. Such a limita-tion leads to the use of blind radial motion compensation

(image autofocusing) and image formation processing thatmust deal with highly nonstationary signals.

The SAR community is very large and the areas of inter-est within SAR grow steadily each year. The ISAR communityis much smaller, in comparison, and it is often dicult tobring together world leaders in this sector. This special issueaims to gather the latest novelties in ISAR in order to pro-vide an updated reference for current and future research inthis area. This has involved a comprehensive peer review pro-cess to guarantee technical novelty and correctness. As dis-cussed below, the presented papers, six in total, are equallydivided amongst the three primary areas of ISAR research,namely: motion compensation (or image autofocusing), im-age formation, and target classification/recognition. Whereasthe first two areas are devoted to the reconstruction of theISAR image, the latter concerns the use of the ISAR imagefor target recognitionone of the principle motivations forISAR development.

Motion compensation

Motion compensation is the first step in the ISAR image re-construction chain. Image focus and clarity strongly dependon the accuracy of motion compensation. Often referred toas image focusing or image autofocusing (blind data drivenmotion compensation), the motion compensation problemhas been largely addressed since the beginning of ISAR. Sev-eral algorithms have been provided that accomplish motioncompensation. Nonparametric algorithms such as promi-nent point processing (PPP) and phase gradient algorithm

2 EURASIP Journal on Applied Signal Processing

(PGA) often, in the past, have been applied in ISAR imaging,largely because they do not need a signal model assumption.More recently, several other nonparametric methods, suchas the maximum likelihood- (ML-) based technique and thejoint time-frequency analysis (JTFA) technique, have beenproposed and are proving to be relatively eective. On theother hand, parametric approaches, such as image-entropyor image contrast-based algorithms, are attracting increasedattention due to the potential enhancements they can pro-vide over nonparametric approaches.

In this special issue, two papers are presented whichaddress the problem of motion compensation. The first,written by Martorella et al., concerns a general exten-sion of two parametric algorithms, namely, the image con-trast based-algorithm (ICBA) and image-entropy-based al-gorithm (IEBA). A second-order polynomial phase model isoften used as the parametric model for motion compensa-tion in algorithms such as the ICBA and the IEBA. Oftensuch a model does not prove to be accurate enough, dueto irregular target motions, such as in the cases of fast ma-noeuvring targets or sea-driven target angular motions inrough sea surface conditions. Motivated by this, researchers,such as those of the Martorella et al. paper, are employ-ing high-order polynomial phase models to achieve accu-rate image focussing. However, estimation of the requiredpolynomial coecients (via solving of an optimisation prob-lem) is typically sensitive to the cost function (image contrastor entropy) and the iterative-search technique employed. Inparticular, solutions provided by classic iterative techniques,such as Newton, quasi-Newton, steepest descent, or gradient,are generally unsuitable due to the multimodal characteris-tics of the cost function (which become more severe as thenumber of polynomial coecients increases). To avoid suchconvergence problems Martorella et al. consider a genetic-based iterative technique, which they apply to the estima-tion/optimisation of a third-order polynomial phase model.

The second paper, written by Yau et al., also addresses themultimodal-related convergence diculties associated withmany parametric-based motion compensation approaches.This paper proposes to overcome the diculties by decou-pling the estimation of the first- and higher-order polyno-mial coecients. This is accomplished via an iterative two-stage approach; first a range-profile cross-correlation stepis applied to estimate the first-order coecient, and then asubspace-based technique, involving eigenvalue decomposi-tion (EVD) or singular value decomposition (SVD), is ap-plied to estimate the higher-order coecients. The potentialbenefits of this two-stage approach arise because the optimi-sation process is implemented over two lower-dimensionalspaces, thereby enhancing the likelihood of convergence to aglobally optimal solution.

Image formation

After motion compensation, the received signal is processedto form the ISAR image. The classic way of forming an ISARimage involves a two-step process. The first step concernsthe range compression (or range focussing). Here, either the

received time-domain signals are compressed by means ofmatched filters or the received multifrequency signals arecompressed via the inverse Fourier transformto producecomplex range profiles. It is worth pointing out that in somecases the range compression is achieved before the motioncompensation. The second step consists of cross-range com-pression (azimuth compression). The fastest and simplestway of obtaining cross-range compression is by means ofa Fourier transform. In ISAR scenarios, where the target ismoving smoothly with respect to the radar and when theintegration time is short enough, the Fourier transform rep-resents the most eective solution. Nevertheless, in ISAR sce-narios with fast manoeuvring targets or sea-driven motionedships or with the requirement of high resolution, the ef-fectiveness of the Fourier approach is strongly limited. Forthis reason, several other techniques have been proposed inthe last decades, such as the JTFA, the range-instantaneous-Doppler (RID), the enhanced image processing (EIP) tech-niques, tomography-based techniques and super-resolutiontechniques, such as the CLEAN technique, and the Capontechnique among others.

In this special issue, the paper by Djurovic et al. pro-poses a novel image formation (cross-range compression)technique based on the use of the polynomial Fourier trans-form (PFT) for enhancing the ISAR image quality in complexreflector geometries at a relatively low computational cost. Amodel is introduced that describes the received signal as thesuperposition of contributions from dierent geometrical ar-eas with given characteristics in terms of signal phases. Thelocal polynomial Fourier transform (LPFT) is then used tomatch the signal contributions that come from dierent im-age areas.

The second paper on image formation, by Wong et al.,proposes a method of analysis for quantifying the image dis-tortion introduced by the conventional Fourier transformapproach. This analysis method involves a numerical modelof the time-varying target rotation rate. The analysis impliesthat severe distortion is often attributed to phase modula-tion eects, whereas a time-varying Doppler frequency pro-duces image smearing. Following insights gained from theanalysis, the authors also propose a time-frequency process-ing/analysis basedmethod for deblurring/refocusing conven-tionally generated ISAR images.

Target classification and identification

Radar signatures are often used for target classificationand/or identification. The need for classifying a target hasled to the development of high-resolution radar. ISAR im-ages can be interpreted as two-dimensional (2D) radar sig-natures. Therefore, a 2D distribution of the energy backscat-tered from the target provides a multidimensional way of in-terpreting the information carried by the radar echo. Sev-eral techniques have been proposed for interpreting thisISAR-based information for the purpose of target classifica-tion/identification. These fall into two main philosophies: (i)featurematching and (ii) template or pointmatching, the lat-ter being more oriented towards target identification.

Marco Martorella et al. 3

In this special issue, two papers deal with the problem oftarget classification by means of ISAR images. In the paper ofShreyamsha Kumar et al., a full system for target identifica-tion is proposed. The authors introduce a wavelet-based ap-proach for ISAR image formation followed by feature extrac-tion and target identification by means of neural networks.The use of the wavelet technique is compared with time-frequency techniques in terms of eectiveness and compu-tational cost. In ISAR imaging it is sometimes dicult topredict the target orientation and often even more dicultto rescale the image along the cross-range coordinate. Thisproblem is avoided in the proposed technique as the featuresused for target identification are invariant to translation, ro-tation, and scalingleading to a robust ISAR image-basedidentification system.

The second paper by Radoi et al. proposes a super-vised self-organising feature-based classification techniqueof super-resolution ISAR images. The super-resolution ISARimages are obtained through a MUSIC-2D method, cou-pled with phase unwrapping and symmetry enhancement.The proposed feature vector contains Fourier descriptorsand moment invariants, which are extracted from the targetshape and scattering center distribution of the ISAR image.These features, importantly, are invariant to target positionand orientation. The feature-based classification is then car-ried out via a supervised adaptive resonance theory (SART)approach, which shows improved eciency over the conven-tional MLP and fuzzy KNN classifiers.

Marco MartorellaJohn Homer

James PalmerVictor Chen

Fabrizio BerizziBrad Littleton

Dennis Longsta

Marco Martorella was born in Portofer-raio (Italy) in June 1973. He receivedthe Telecommunication Engineering Lau-rea and Ph.D. degrees from the Universityof Pisa (Italy) in 1999 and 2003, respec-tively. He became a postdoc. Researcher in2003 and a permanent Researcher/Lecturerin 2005 at the Department of InformationEngineering of the University of Pisa. Hejoined the Department of Electrical andElectronic Engineering (EEE) of the University of Melbourne dur-ing his Ph.D., the Department of Electrical and Electronic Engi-neering (EEE) of the University of Adelaide under a postdoc. con-tract, and the Department of Information Technology and Electri-cal Engineering (ITEE) of the University of Queensland as a Vis-iting Researcher between 2001 and 2006. His research interests arein the field of synthetic aperture radar (SAR) and inverse syntheticaperture radar (ISAR). He is an IEEE Member since 1999.

John Homer received the B.S. degree inphysics from the University of Newcastle,Australia in 1985 and the Ph.D. degree insystems engineering from the AustralianNational University, Australia, in 1995. Be-tween his B.S. and Ph.D. studies, he helda position of Research Engineer at Coma-lco Research Centre in Melbourne, Aus-tralia. Following his Ph.D. studies, he hasheld research positions with the Universityof Queensland, Veritas DGC Pty Ltd., and Katholieke Universiteit,Leuven, Belgium. He is currently a Senior Lecturer at the Univer-sity of Queensland within the School of Information Technologyand Electrical Engineering. His research interests include signal andimage processing, particularly in the application areas of telecom-munications, audio and radar. He is currently an Associate Editorof the Journal of Applied Signal Processing.

James Palmer was born in 1979 in Towns-ville, Australia. James received the Bachelorof electrical engineering (Hons I) and Bach-elor of Arts (Japanese) degrees from theUniversity of Queensland and is currentlyfinishing his Ph.D. studies through the sameinstitution. Palmers major research inter-ests are in the field of bistatic radar, SAR andISAR (including the monostatic, emulatedbistatic, and bistatic varieties), and sea sur-face forward scatter RF signal modelling and analysis.

Victor Chen received the Ph.D. degree inelectrical engineering from Case WesternReserve University, Cleveland, Ohio, in1989. Since 1990, he has been with RadarDivision, the US Naval Research Labora-tory in Washington DC and working on ra-dar imaging, time-frequency applications toradar, groundmoving target indication, andmicro-Doppler analysis. He is a PrincipalInvestigator working on various researchprojects on radar signal and imaging, time-frequency applicationsto radar, and radar micro-Doppler eect. He served as TechnicalProgram Committee Member and Session Chair for IEEE and SPIEconferences and served as a Guest Editor for IEE Proceedings onRadar, Sonar, and Navigation in 2003, and Associate Editor for theIEEE Trans. on Aerospace & Electronic Systems since 2004. Hiscurrent research interests include computational synthetic apertureradar imaging algorithms, micro-Doppler radar, and independentcomponent analysis of features for noncooperative target identifi-cation. He received NRL Review Award in 1998, NRL Alan BermanResearch Publication award in 2000 and 2004, and NRL Techni-cal Transfer Award in 2002. He has more than 100 publications inbooks, journals, and proceedings including a book: Time-FrequencyTransforms for Radar Imaging and Signal Analysis (V. C. Chen andHao Ling), Artech House, Boston, Mass, January 2002.

Fabrizio Berizzi was born in Piombino(Italy) on November 1965. He received theElectronic Engineering and Ph.D. degreesfrom the University of Pisa (Italy) in 1990and 1994, respectively. Currently, he is anAssociate Professor of the University of Pisa(Italy)Department of Information Engi-neering. His main research interests are inthe fields of synthetic aperture radar (SARand ISAR), HF-OTH skywave and surface


wave radar, target classification by wideband polarimetric radardata, hybrid waveform design for HRRP radar. He is the author andcoauthor of more than 100 papers published in prestigious interna-tional journals, book chapters, and IEEE conference proceedings.He is the principal investigator of several research projects fundedby Italian radar industries and by the Italian Minister of Defense.He cooperates to several research activities with the University ofAdelaide (AUS), DSTO (AUS), JPL (USA), NRL (USA), ONERA(France), SOC (UK). He is a Member of the IEEE.

Brad Littleton received his Ph.D. in physicsfrom the University of Queensland, in 2004.His research interests are elastic and in-elastic electromagnetic wave/matter inter-actions, and applications to electromagneticimaging, measurement and superresolutiontechniques. He is currently working on sin-gle quantum dot spectroscopy for the UQnode of the Centre for Quantum ComputerTechnology.

Dennis Longsta is currently TechnologyConsultant to Filtronic PLC and Emeri-tus Professor with the School of Informa-tion Technology and Electrical Engineer-ing at the University of Queensland. Dur-ing that time at the University of Queens-land, Dennis cofounded the CooperativeResearch Centre for Sensor Signal and In-formation Processing (CSSIP). He was alsothe Founder and Director of GroundProbe,now a thriving global company marketing products invented byhim and developed by his research group. He also served as Headof Department of Electrical and Computer Engineering for threeyears. From 1988 to 1991, he was at the Defence Science and Tech-nology Organisation (DSTO) in Australia, where he was ResearchLeader to the Microwave Radar Division in Adelaide. Previous tothis he spent 18 years as Senior Scientific Ocer, then Principal Sci-entific Ocer at the Royal Signals and Radar Establishment (nowQintiQ), Malvern, England, where he worked on airborne radarsystems. His work has attracted a number of awards and prizes andhis spino company, GroundProbe, received an Engineering Excel-lence Award from the IE(Aust) Qld 2003. He was granted a Queens-land Government Smart State Award in 2004, and an AustralianEmerging Exporter Award in 2005 (see www.groundprobe.com).


Use of Genetic Algorithms for Contrast and EntropyOptimization in ISAR Autofocusing

Marco Martorella, Fabrizio Berizzi, and Silvia Bruscoli

Department of Information Engineering, University of Pisa, Via Caruso, 56126 Pisa, Italy

Received 4 May 2005; Revised 25 October 2005; Accepted 21 December 2005

Image contrast maximization and entropy minimization are two commonly used techniques for ISAR image autofocusing. Whenthe signal phase history due to the target radial motion has to be approximated with high order polynomial models, classic op-timization techniques fail when attempting to either maximize the image contrast or minimize the image entropy. In this papera solution of this problem is proposed by using genetic algorithms. The performances of the new algorithms that make use ofgenetic algorithms overcome the problem with previous implementations based on deterministic approaches. Tests on real data ofairplanes and ships confirm the insight.

Copyright 2006 Hindawi Publishing Corporation. All rights reserved.

1. INTRODUCTION

ISAR image reconstruction has been a widely addressed topicin the last few decades [14]. The exploitation of large band-width signals and the coherent integration of the echoes pro-vide the basis for the ISAR image formation. Before the ac-tual image formation, the signal phase must be compensatedin order to remove the target radial movement. We indicatesuch an operation with image focusing, and, when no an-cillary data are available, with image autofocusing, becauseonly the received signal is used to perform such an operation.

Among the autofocusing techniques proposed in the lit-erature [512], some are based on the use of image focusindicators, such as the image contrast and the image en-tropy [57]. In particular, when the target radial velocitycan be approximated with polynomial models, the optimiza-tion problems that have to be solved are reduced to a searchon a domain of few parameters. In these cases the com-putational cost is strongly reduced and real-time applica-tions are achievable. Optimization problems have often beensolved by using deterministic algorithms such as Steepest De-scent, Gradient, Newton and quasi-Newton, Nelder-Mead,and others. Nevertheless, cost functions that have been usedas image focus indicators, such as the image contrast andentropy, become highly multimodal when the number ofparameters increases. Moreover, deterministic methods canonly be applied when the cost function is continuous anddierentiable. Recently, optimization algorithms based on arandom approach have been introduced in order to over-

come the problem of multimodality and dierentiability. Asubclass of such algorithms is the genetic algorithm (GA).

In this paper we modify two existing autofocusing tech-niques based on image focus enhancement optimization,namely, the image contrast technique (ICT) and the imageentropy technique (IET) by using GAs. Image contrast max-imization and image entropy minimization represent twosimilar optimization problems that encounter the same dif-ficulties when applied to ISAR image autofocusing. Specif-ically, the high number of local maxima in the cost func-tion causes the convergence of deterministic algorithms toa nonoptimal solution. In [13] a solution based on the useof genetic algorithms for ISAR image autofocusing was pro-posed in order to improve the joint time-frequency analy-sis (JTFA) based autofocusing algorithm, which was initiallyproposed in [11].

In this paper the authors confirm and extend the resultsobtained in [13] by applying GAs to two well-known auto-focusing techniques in order to improve their performances.Real data applications will be shown that demonstrate theeectiveness of GAs when applied to image contrast and en-tropy based autofocusing techniques.

Section 2 introduces the signal model and the image aut-ofocusing techniques, namely, the ICT and the IET. Section 3provides a review of classic optimization techniques and in-troduces the genetic algorithms. Section 4 provides a com-parative analysis between classic and genetic optimizationtechniques when used both in the ICT and IET.


x1

x2

x3

R(z, t)

R0(t)

z

z1z2

z3

hr

10(t)

20(t)

30(t)

Figure 1: Reference system.

2. SIGNAL MODEL AND AUTOFOCUSINGTECHNIQUES

2.1. Signal model

After signal preprocessing [6], the received signal, in freespace conditions, can be written in a time-frequency formatas follows:

SR( f , t) =W( f , t)e j(4 f /c)R0(t)V(z)e

j(4 f /c)[zT i(z)R0

(t)]dz,

(1)

where W( f , t) = rect(t/Tobs) rect( f f0/B) and where f0 isthe carrier frequency, B is the transmitted signal bandwidth,Tobs is the observation time, c is the speed of light in freespace. Referring to Figure 1, R0(t) is the modulus of vectorR0(t) which locates the position of a focusing point on thetarget, i(z)R0 (t) is the unit vector of R0(t), z is the vector thatlocates a generic point on the target, and V is the spatial re-gion where the reflectivity function (z) is defined. Functionrect(x) yields 1 when |x| < 1/2, 0 otherwise.

When the target does not undergo significant high-speedmaneuvers, the distance between the radar and the focus-ing point can be approximated by its Taylor series expansionaround the central time instant t = 0:

R0(t) =Ni=0

iti, (2)

where

i = 1i!d(i)

dtiR0(t) |t=0 . (3)

2.2. Autofocusing algorithms

2.2.1. ICT

The ICT attempts to estimate the coecients of (3) by max-imizing the image contrast (IC) with respect to i for i =1, 2, 3, . . . ,N . The zero-order term (0) can be ignored be-cause it only provokes a range shift in the reconstructedimage without producing any defocusing. In the case of an

Nth order polynomial phase, the IC can be expressed as fol-lows:

IC() =

A{[I2(x1, x2;

) A{I2(x1, x2;)}]2}

A{I2(x1, x2;

)} , (4)

where the vector of unknowns can be expressed as =[1, . . . ,N ], the operator A() represents the mean valueoperator over the image coordinates (x1, x2) and whereI(x1, x2;) is the intensity of the image obtained by compen-

sating the signal with the phase term e j(4 f /c)N

i=1 iti and byapplying a two-dimensional Fourier transform (2D-FT). An-alytically, this can be expressed as

I(x1, x2;

) = 2 D-FT [SR( f , t) e j(4 f /c)N

i=1 iti]. (5)

Mathematically, the optimization problem can be formu-lated as follows:

() = arg(max

[IC()

]). (6)

2.2.2. IET

Equivalently to the ICT, the IETminimizes the image entropy(IE) in order to estimate the coecients i.

By following [7]

IE =

I2(x1, x2

)S

lnS

I2(x1, x2

)dx1 dx2, (7)

where S = I2(x1, x2)dx1 dx2. Therefore, the optimizationproblem can be written in an mathematical form:

() = arg(min

[IE()

]). (8)

3. OPTIMIZATION ALGORITHMS

3.1. Deterministic algorithms

Deterministic optimization algorithms, such as Newton,Steepest Descent, Gradient, quasi-Newton, Nelder-Mead[14, 15], are generally ecient methods when the cost func-tion is monomodal and dierentiable in the search domain.Often, when the number of variables increases, monomodal-ity is lost and therefore many local minima appear. In suchcases, the initial guess that has to be provided as startingpoint to the search algorithm is essential for the conver-gence to the global minimum. In this paper, the Nelder-Mead(NM) algorithm [15] has been chosen as a representativeof classical methods to compare to genetic algorithms whenused to solve problems of IC maximization and IE mini-mization. The Nelder-Mead algorithm is chosen because itis a more stable and eective algorithm than other classic ap-proaches, such as Newton and Steepest Descent.


SR( f , t)

1D-FTf

SR(, t)(in)1

(in)2

1

2

Initial guessestimation

IC maximizationIE minimization

Figure 2: Autofocusing algorithm.

3.2. Genetic algorithms

Genetic algorithms, introduced by Holland in [16], belongto the class of approximation (or heuristic) algorithms, andare largely used to solve optimization problems. The geneticalgorithm is a stochastic global search method that mimicsthe metaphor of natural biological evolution. Whereas tradi-tional search techniques use characteristics of the cost func-tion to determine the next sampling point (e.g., gradients,Hessians, etc.), stochastic search techniques do not need it. Infact, the next solution is determined on the basis of stochas-tic decision rules, rather than a set of deterministic ones. Thispeculiarity makes the GAs independent of assumptions likethe dierentiability of the cost function with respect to thevariables that constitute the search domain.

GAs manipulate a family (population) of solutions andimplement a survival of the fittest strategy to produce bet-ter and better approximations of a solution. In general, thefittest individuals of any population tend to reproduce andsurvive. In this sense the successive generations can improve.Such algorithms are able to solve linear and nonlinear prob-lems by exploring all regions of the search domain and byexponentially exploiting promising areas through mutation,crossover, and selection operations applied to individuals inthe population [17].

The crossover operator is used to exchange genetic infor-mation between pairs, or larger groups, of individuals. Mu-tation causes the individual genetic representation to changeaccording to some probabilistic rule (such an operator en-sures that there is a nonzero probability of searching a givensubspace). This has the eect of inhibiting the possibility toconverge to local maxima, rather than to the global maxi-mum.

3.3. Implementation of Nelder-Mead algorithm forIC and IE optimizations

The ICT that makes use of NM technique has been pro-posed in [5, 6]. In Figure 2, a flow chart of such an algorithmis depicted. The ICT makes use of IC maximization to fo-cus ISAR images. The IET has been derived from the ICTsimply by replacing IC maximization with IE minimization.Both algorithms use an initial guess that is estimated by us-ing an initialization technique based on the radon transform

(details can be found in [6]). The use of the radon transformhas proved to bemore ecient than other techniques for esti-mating the initial guess. The Nelder-Mead algorithm is basedon the simplex method for the search of the minimum of agiven cost function. Such a method fully described in [15]was implemented in MATLAB by defining two parameters:the maximum number of iterations (MNI) and the tolerancevalue (TV). The explanation of the former is straightforwardand it concerns the stop condition for the iterative algorithm,whereas the second represents the minimum dierence al-lowed between the last two values of the cost function. Alsothis parameter is used for defining the algorithm stop condi-tion, that is, the algorithm stops iterating when the dierencebetween the last two values of the cost function is smallerthan the TV.

3.4. Implementation of genetic algorithms forIC and IE optimizations

The GA replaces both the estimation of the initial guess andthe final focusing parameters. In fact, GAs do not need aninitial guess. This may represent an additional advantage be-cause the performance of the algorithm is not aected by theestimation of the initial guess. The implementation of theGA used in our analysis is the genetic algorithm optimiza-tion toolbox (GAOT) [18], a free toolbox developed at theDepartment of Industrial and Systems Engineering, NorthCarolina State University.

The algorithm, implemented in MATLAB, iterates untila stop condition applies. The stop condition can be definedas the MNI or by means of the TV. The MNI is needed inorder to control the computational load (CL). Because realtime ISAR image reconstruction is often needed, the CL isa parameter to be kept as small as possible. At each itera-tion the population size (PS) is kept constant by equallingthe number of discarded elements to the number of new el-ements. The elements are discarded by comparing the valuesof the IC, which represents the fitness function. The newelements are generated by cloning, combining, and mu-tating the surviving elements (remaining after the discardprocess). The operation of cloning is performed by choosingthe most fit elements (with the largest IC or smallest IE) andcopying them into the next generation set. The operation ofcombining is obtained by choosing two elements within thesurvivors and by genetically combining them. The geneticcombination is a numerical operation that can be performedin many ways [16, 17]. When complex numbers are used, thenumber representation adopted is the floating point. In thiscase, an operation called simple crossover is performed [17].A simple crossover consists of:

(1) dividing the binary representation of N elements intotwo strings of digits of length r and N-r;

(2) concatenating the r digits of the first element with theN-r digits of the second element to create a new ele-ment;

(3) concatenating the r digits of the second element withthe N-r digits of the first element to create another newelement.


Therefore two elements are created from two old ele-ments. The operation of mutating is performed by choosingone or more digits of the binary representation of one ele-ment and replacing them with the relative complement val-ues (e.g., X0X10X becomes X1X01X). The fittest element ofthe last generation represents the solution of the optimiza-tion problem. Several parameters can be defined [18] in or-der to implement ad hoc genetic algorithms. It is worthmentioning the most significant:

(i) population size,(ii) number of iterations,(iii) gene encoding and length,(iv) selection operation,(v) crossover and mutation operations.

For what concerns the experiments carried out insection 4, some parameters were kept fixed whereas oth-ers were changed in order to find an optimal trade-o be-tween maximum search accuracy and computational costin a heuristic sense. Specifically, the gene encoding chosenwas a floating point binary representation on 64 bits. Theselection operation used was the tournament selection. Thecrossover and mutation operations adopted were the heuris-tic cross-over and the multi-nonuniform mutation, respec-tively (see [18] for more details). The population size (PS)is kept constant throughout the generations. Therefore, theinitial population size and PS coincide. The PS plays an im-portant role in the eectiveness of the genetic algorithm anda fine tuning is needed in order to improve the optimiza-tion performance. The same can be said about the num-ber of iterations, which is defined as the number of itera-tions that are needed to obtain the solution of the optimiza-tion problem. In order to limit the number of iterations theMNI has to be defined. The larger the value of the MNI, themore accurate the solution is, although at the expenses of thecomputational load, which is linearly proportional to it. Afew experiments were run in order to provide suitable val-ues for both the PS and the MNI for the eective applicationof genetic algorithms to ISAR image autofocusing. The re-sults showed optimal solutions (in a heuristic sense) whenPS = 50 and MNI = 50 for a second-order signal phasemodel and PS = 100 and MNI = 100 for a third-order signalphase model. Such values have been used in the experimentsshown in Section 4.

4. PERFORMANCE ANALYSIS

4.1. Data set

The two data sets that are considered for the performanceanalysis are relative to an aircraft (737, see Figure 3) anda ship (Bulk Carrier, see Figure 4). Details about the radarparameters for the two data sets can be found in Tables1 and 2, respectively. All data sets were collected by usinga low-power instrumented radar system developed by theAustralian defence science and technology organisation(DSTO). In particular, the first data has been gathered by us-ing a ground-based radar, located near the Adelaide civilian

Figure 3: Boeing 737.

Figure 4: Bulk Carrier photo.

airport, whereas the second data set has been acquired by anairborne radar. In this second configuration, both the air-plane and ship movements contribute to the total aspect an-gle variation.

In this section the eectiveness of the use of genetic al-gorithms for ISAR image autofocusing is tested by meansof real data. Both the ICT and the IET will be consideredto validate the proposed solution for a generic parametrictechnique that makes use of iterative solutions. Moreover, inorder to investigate dierent ISAR scenarios we have cho-sen two data sets concerning two dierent radar-target ge-ometries and dynamics. The algorithm performances will betested by means of three parameters and an image visual in-spection. The three parameters are the IC, IE, and CL (as de-fined in Section 3).

4.2. Test description

The two data sets are analyzed considering both short andlong observation times. The longer is the observation time,the higher is the model order that is able to fit the focusingpoint phase history. We will show that when the integration


Table 1: Radar parameters (aircraft).

N of sweeps 512

N of transmitted frequencies 128

Lowest frequency 9.26GHz

Frequency step 1.5MHz

Range resolution 0.78m

Radar height (hr) Ground level

Target type Boeing 737

PRF/sweep rate 20 kHz/156.25Hz

Table 2: Radar parameters (ship).

N of sweeps 256

N of transmitted frequencies 256

Lowest frequency 9.16GHz


Range resolution 0.98m

Radar height (hr) 305m

Target type Bulk Loader

PRF/sweep rate 20 kHz/78.13Hz

time is short, the second-order model is able to represent thephase history. The IC generally shows a quite regular behav-ior when it is a function of two parameters (IC(1, 2)), as il-lustrated in Figure 5. In such a case, the NM algorithm is ableto solve the optimization problem and find the global max-imum. When a long observation time is used to reconstructthe ISAR image, at least a third-order model is required. Theintroduction of the third parameter causes irregularity in theIC which becomes highly multimodal. In Figure 6, a sectionof the IC(1, 2, 3) along the third-order parameter (3) isillustrated. The presence of many local maxima is clearly vis-ible. In such a case, the NM fails, as the following results willshow, whereas the GA provides a successful image autofocus-ing.

4.3. Test results

4.3.1. Visual inspection

The visual inspection simply consists of a comparison ofISAR images obtained from the same data by means of thedeterministic and genetic algorithms. The ISAR images rel-ative to the Boeing 737 data, obtained by means of the GAand the NM are shown, respectively, in Figures 7 and 8.The two images, reconstructed by coherently processing 128sweeps (0.8 s), show the same features and are equally well fo-cused. The signal phase model used in this case was a second-order polynomial because of the short integration time. Asexpected, the results obtained with NM and GA are quitecomparable. This is due to the fact that the NM algorithmrepresents a good optimization algorithm for the 2D search

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Figure 6: Image contrast section (third-order term).

space represented by the signal phase parameters. The ISARimages shown in Figures 9 and 10 are obtained by coherentlyprocessing 512 sweeps (3.2 s) by means of the GA and theNM, respectively. In this case, it is clearly noticeable that theISAR image, obtained by means of the NM approach, is de-focused, whereas the ISAR image relative to the GA shows agood focus. Because of the long integration time, a third or-der polynomial model was assumed. The results show thatthe NM algorithm is not able to provide a good image fo-cus whereas the GA is able to find an accurate solution. It isworth noting that in all the cases the NM iteration termina-tion was due to the TV and not to the MNI. This confirmsthat the NM algorithm converges to local maxima instead ofthe global maximum.

In order to verify that a second-order model is not accu-rate enough to represent the signal phase history, we show theISAR images relative to the long integration time (512128).Such images were processed by using a second-order model


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Figure 7: ICT-GA128 128 focused with a second-order mod-elBoeing 737.

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Figure 8: ICT-NM128 128 focused with a second-order mod-elBoeing 737.

for both the GA and the NM and are shown in Figures 11 and12, respectively. The image defocus due to the inaccuracy ofthe second-order model is clearly visible in both images.

The same data set has been used to conduct an equivalentexperiment by using the IET. Figures 13, 14 show the ISARimages relative to a short integration time and processed byusing a second-order model by means of genetic and deter-ministic algorithms, respectively. Also in this case both ap-proaches achieve the same result. In Figures 15 and 16, theISAR images relative to the long integration time are shown.In this case, the use of a third-order model aects negativelythe results when a deterministic approach is used, whereasthe use of GAs provides a well-focused image.

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Figure 9: ICT-GA512 128 focused with a third-order mod-elBoeing 737.

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Figure 10: ICT-NM512 128 focused with a third-order mod-elBoeing 737.

The second experiment has been conducted for the sec-ond data set relative to a Bulk Carrier. In this case only a longobservation time (3.2 s) has been considered in order to testthe use of a third-order model. Figures 17 and 18 show thetwo ISAR images obtained by using the GA and the NM,respectively. It is clear that the image focused by means ofGAs (Figure 17) is well focused whereas the image obtainedby means of NM (Figure 18) is not focused at all.

4.3.2. Image contrast

The IC is an indicator of the image focusing: the higher theIC, the better the image focusing. In Table 3 we report the IC


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Figure 11: ICT-GA512 128 focused with a second-order mod-elBoeing 737.

for the ISAR images obtained by processing the two data sets.The results confirm the visual analysis. In particular, we notethat a third-order model is needed for longer integrationtimes as confirmed by the image contrast increase. Moreover,the use of GAs is necessary in order to ensure the convergenceof the solution to the global maximum, as shown by compar-ing the IC values in the case of NM and GA, regardless of theparticular ISAR autofocusing technique used (either ICT orIET). It is worth noting that small dierences in the IC canprovoke big dierences in the image focus (compare with vi-sual inspection).

4.3.3. Image entropy

The IE is an indicator of the image focus as well as the IC. Inthis case the smaller the entropy, the better the image focus[6]. In Table 4, the results relative to the IE confirm the resultsfound in both the visual inspection and the IC analysis.

4.3.4. Image peak

The image peak (IP) is another indicator of the image focus-ing. Its definition is as follows:

IP max{I2(x1, x2

)}. (9)

When an image of a rigid body is well focused, the energy rel-ative to any single scatterer is more concentrated around itspeak. Such an indicator of performance could be misleadingwhen used alone but it is a good indicator when it is usedjointly with other indicators such as IC and IE, which con-sider the whole image focus quality. In Table 5, the resultsrelative to the image peak (in dB) strengthen the previousanalyses in most of the cases. It is worth noting that the val-ues relative to the Bulk Carrier data set, when the IET-GA isused, show a dierent trend with respect to the other exper-

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Figure 12: ICT-NM512128 focused with a second-order mod-elBoeing 737.

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Figure 13: IET-GA128 128 focused with a second-order mod-elBoeing 737.

iments. In particular the value relative to the second-orderand 64 256 data set is significantly larger than any othervalues. This behavior can be explained by the fact that a sin-gle scatterer can be highly focused even though the rest ofthe image is not highly focused. This phenomenon occursespecially when low-order polynomial models are sued forrepresenting the signal phase.

4.3.5. Computational load

The CL has been calculated by running the algorithm on aPentium III833MHz processor with 192MB of RAM, andit is reported in seconds. It is worth noting that the algorithmis coded in MATLAB and it is not optimized, hence only acomparative analysis must be considered. In order to speed


Table 3: Image contrast as indicator of image quality (higher values indicate better image focus).

Algorithm Model orderAirplane Bulk Carrier

128 128 512 128 64 256 256 256ICT-NM

(2nd order) 1.27 1.09 2.84 2.61

(3rd order) 1.27 1.09 2.87 2.60

ICT-GA(2nd order) 1.27 1.09 3.03 2.65

(3rd order) 1.27 1.18 3.05 2.92

IET-NM(2nd order) 1.26 1.08 2.97 2.65

(3rd order) 1.25 1.09 2.82 1.48

IET-GA(2nd order) 1.26 1.07 3.02 2.65

(3rd order) 1.27 1.15 3.02 2.92

Table 4: Image entropy as indicator of image quality (lower values indicate better image focus).


128 128 512 128 64 256 256 256ICT-NM

(2nd order) 7.10 9.28 6.33 10.63

(3rd order) 7.10 9.28 6.38 10.62

ICT-GA(2nd order) 7.10 9.29 6.33 7.57

(3rd order) 7.09 8.87 6.17 7.56

IET-NM(2nd order) 6.99 9.28 6.33 10.63

(3rd order) 6.99 9.27 6.37 10.62

IET-GA(2nd order) 6.99 9.27 6.33 7.56

(3rd order) 6.97 8.79 6.17 7.55

Table 5: Image peak as indicator of image quality expressed in dB scale (higher values indicate better image focus).


128 128 512 128 64 256 256 256ICT-NM

(2nd order) 42.1 41.7 55.8 58.7

(3rd order) 42.1 41.7 54.5 58.8

ICT-GA(2nd order) 42.0 41.6 56.1 58.1

(3rd order) 41.9 46.3 55.7 57.2

IET-NM(2nd order) 43.2 41.4 56.4 58.2

(3rd order) 42.6 41.4 55.8 54.9

IET-GA(2nd order) 43.2 42.6 62.4 58.2

(3rd order) 43.4 46.3 56.4 59.9

Table 6: CL-time required to find the solution of the optimization problem (in seconds).


128 128 512 128 64 256 256 256ICT-NM

(2nd order) 4.1 14.1 4.4 17.8

(3rd order) 6.9 30.0 12.8 76.3

ICT-GA(2nd order) 10.9 63.7 6.7 26.9

(3rd order) 13.0 77.5 24.4 117.2

IET-NM(2nd order) 12.5 12.3 4.3 37

(3rd order) 10.8 182.9 10.4 165.7

IET-GA(2nd order) 22.6 238.6 41.4 247.1

(3rd order) 50.4 534.9 52.3 274.8


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Figure 14: IET-NM128128 focused with a second-order mod-elBoeing 737.

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Figure 15: IET-GA512128 focused with a third-order modelBoeing 737.

up the processing for real-time applications both code op-timization and faster processors must be implemented. Theresults relative to the two data sets are shown in Table 6. Thecomputation burden required by the NM algorithm is gen-erally less than the GA. It is worth noting that such a bur-den becomes significant when a third-order model is used.Nevertheless, the results obtainable by using GA justify theincrease of CL.

5. CONCLUSIONS

In this paper an extension of both the ICT and IET is pro-posed by introducing genetic algorithms. The ability of such

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algorithms to solve optimization problems in the case ofhighly multimodal cost functions has been shown by meansof real data for two well-known parametric ISAR autofocus-ing techniques, namely, the ICT and the IET. The improve-ment is noticed when long integration times are used to formthe ISAR image. In fact, in such cases model orders higherthan the second must be used and the cost function becomeshighly multimodal. Even by using accurate initial guesses,classical techniques are not always able to converge to theglobal maximum. In our analysis the NM algorithm has beenused to represent deterministic approaches. The results haveshown an equal performance at short integration times thatleads to the use of deterministic techniques because of their


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Figure 18: IET-NM256 256 focused with a third-order mod-elBulk Carrier.

less expensive computational load. In a generic case, whenarbitrary integration times are used, the GA approach showsbetter performances and robustness, and hence it is preferredto deterministic approaches.

ACKNOWLEDGMENTS

The authors acknowledge the Defense Science and Technol-ogy Organisation (DSTO) for the use of real data and theUniversity of North Carolina for sharing the GAOT toolbox.Special thanks to Petrina Kapper for English language sup-port.

REFERENCES

[1] J. L. Walker, Range-doppler imaging of rotating objects,IEEE Transactions on Aerospace and Electronic Systems, vol. 16,pp. 2352, 1980.

[2] D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, andE. C. Poggio, Developments in radar imaging, IEEE Trans-actions on Aerospace and Electronic Systems, vol. 20, no. 4, pp.363400, 1984.

[3] W. C. Carrara, R. S. Goodman, and R. M. Majewsky, SpotlightSynthetic Aperture Radar: Signal Processing Algorithms, ArtechHouse, Boston, Mass, USA, 1995.

[4] D. R. Wehner, High Resolution Radar, Artech House, Nor-wood, Mass, USA, 1995.

[5] F. Berizzi and G. Corsini, Autofocusing of inverse syntheticaperture radar images using contrast optimisation, IEEETransaction on Aerospace and Electronic System, vol. 32, no. 3,pp. 11851191, 1996.

[6] M. Martorella, B. Haywood, F. Berizzi, and E. Dalle Mese,Performance analysis of an ISAR contrast based autofocusingalgorithm using real data, in Proceedings of IEE Radar Confer-ence, pp. 200205, Adelaide, Australia, September 2003.

[7] L. Xi, L. Giosui, and J. Ni, Autofocusing of ISAR images basedon entropyminimisation, IEEE Transactions on Aerospace andElectronic Systems, vol. 35, no. 4, pp. 12401252, 1999.

[8] B. Haywood and R. J. Evans, Motion compensation for ISARimaging, in Proceedings of the IEEE Australian Symposium onSignal Processing and Applications (ASSPA 89), pp. 113117,Adelaide, Australia, April 1989.

[9] J. Li, R. Wu, and V. C. Chen, Robust autofocus algorithmfor ISAR imaging of moving targets, IEEE Transactions onAerospace and Electronic Systems, vol. 37, no. 3, pp. 10561069,2001.

[10] W. Haiqing, D. Grenier, G. Y. Delisle, and F. Da-Gang, Trans-lational motion compensation in ISAR image processing,IEEE Transactions on Image Processing, vol. 4, no. 11, pp. 15611571, 1995.

[11] Y. Wang, H. Ling, and V. C. Chen, ISAR motion compensa-tion via adaptive joint time-frequency technique, IEEE Trans-actions on Aerospace and Electronic Systems, vol. 34, no. 2, pp.670677, 1998.

[12] I.-S. Choi, B.-L. Cho, and H.-T. Kim, ISAR motion com-pensation using evolutionary adaptive wavelet transform, IEEProceedings on Radar, Sonar andNavigation, vol. 150, no. 4, pp.229233, 2003.

[13] J. Li and H. Ling, Use of genetic algorithms in ISAR imagingof targets with higher order motions, IEEE Transactions onAerospace and Electronic System, vol. 39, pp. 343351, 2002.

[14] E. Polak, Optimization: Algorithms and Consistent Approxima-tions, vol. 124 of Applied Mathematical Sciences, Springer, NewYork, NY, USA, 1997.

[15] J. A. Nelder and R. Mead, A simplex method for functionminimisation, Computer Journal, vol. 7, pp. 308313, 1965.

[16] J. Holland, Adaptation in Natural and Artificial Systems, Uni-versity of Michigan Press, Ann Arbor, Mich, USA, 1975.

[17] Z. Michalewicz, Genetic Algorithms +Data Structures= Evolu-tion Programs, Springer, New York, NY, USA, 1994.

[18] C. R. Houck, J. A. Joines, and M. G. Kay, A genetic algorithmfor function optimization: a MATLAB implementation,North Carolina State University, http://www.ie.ncsu.edu/mirage/GAToolBox/gaot/.

Marco Martorella was born in Portofer-raio (Italy) in June 1973. He receivedthe Telecommunication Engineering Laureaand Ph.D. degrees from the University ofPisa (Italy) in 1999 and 2003, respectively.He became a Postdoctoral Researcher in2003 and a Permanent Researcher/Lecturerin 2005 at the Department of InformationEngineering of the University of Pisa. Hejoined the Department of Electrical andElectronic Engineering (EEE) of the University of Melbourne dur-ing working on his Ph.D., the Department of Electrical and Elec-tronic Engineering (EEE) of the University of Adelaide under apostdoctoral contract, and the Department of Information Tech-nology and Electrical Engineering (ITEE) of the University ofQueensland as a Visiting Researcher between 2001 and 2006. Hisresearch interests are in the field of synthetic aperture radar (SAR)and inverse synthetic aperture radar (ISAR). He is an IEEEMembersince 1999.


Fabrizio Berizzi was born in Piombino,Italy, in 1965. He received the ElectronicEngineering Laurea and Ph.D. degrees atthe University of Pisa (Italy) in 1990 and1994. Since October 2000 he has been anAssociate Professor at the Department ofInformation Engineering of the Universityof Pisa (Italy). He currently lectures nu-merical communications in the computerengineering course, project and simulationof remote sensing systems in the telecommunication engineeringcourse, and signal theory and applications at the Italian Navy. Hehas published more than 60 scientific papers. Since 1998, he hasbeen the principal investigator of two Italian Space Agency (ASI)projects on sea remote sensing. His research interests are in thefields of radar systems, synthetic aperture radar (SAR and ISAR),sea remote sensing by means of active sensors. He is a Member ofIEEE.

Silvia Bruscoli was born in Cecina, Italy, inAugust 1977. She received the Laurea de-gree in telecommunication engineering atthe University of Pisa (Italy), in 2003. Sheis currently a Ph.D. student in methodsand technologies for environmental moni-toring at the Department of InformationEngineering of the University of Pisa. Herresearch interests include inverse syntheticaperture radar and target classification inSMR environments.


Eigenspace-Based Motion Compensation forISAR Target Imaging

D. Yau, P. E. Berry, and B. Haywood

Electronic Warfare and Radar Division, Department of Defence, Defence Science and Technology Organisation (DSTO),Australian Government, Edinburgh, South Australia 5111, Australia

Received 8 June 2005; Revised 17 October 2005; Accepted 24 November 2005

A novel motion compensation technique is presented for the purpose of forming focused ISAR images which exhibits the robust-ness of parametric methods but overcomes their convergence diculties. Like the most commonly used parametric autofocustechniques in ISAR imaging (the image contrast maximization and entropy minimization methods) this is achieved by estimatinga targets radial motion in order to correct for target scatterer range cell migration and phase error. Parametric methods generallysuer a major drawback, namely that their optimization algorithms often fail to converge to the optimal solution. This dicultyis overcome in the proposed method by employing a sequential approach to the optimization, estimating the radial motion of thetarget by means of a range profile cross-correlation, followed by a subspace-based technique involving singular value decomposi-tion (SVD). This two-stage approach greatly simplifies the optimization process by allowing numerical searches to be implementedin solution spaces of reduced dimension.

Copyright 2006 D. Yau et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Imaging of targets using inverse synthetic aperture radar(ISAR) exploits the large eective aperture induced by therelative translational and rotational motion between radarand target and has the ability to create high-resolution im-ages of moving targets from a large distance. The techniqueis independent of range if rotational motion is significant,and it therefore has good potential to support automatic tar-get recognition. A target image is formed by estimating thelocations of target scatterers in both range and cross-rangebut the scatterer motion needs to be compensated for in or-der to avoid image blurring which can occur due to scatterermigration between range cells and scatterer acceleration.

The common autofocusing methods can be categorizedinto parametric and nonparametric approaches. Computa-tionally, nonparametric methods are much more ecientand easy to implement. The compensation for translationalmotion normally comprises two separate steps: range cellrealignment and phase-error correction. Range cell realign-ment is considered to be routine and is based upon, for in-stance, the correlation method (see Chen and Andrews [1])or the minimum-entropy method (see Wang and Bao [2]).Phase autofocus is more stringent in its requirements andmany nonparametric methods have been proposed, most of

which track the phase history of an isolated dominant scat-terer (prominent point processing (PPP), see Steinberg [3])or the centroid of multiple well-isolated scatterers (multiplescatterer algorithm (MSA), see Carrara et al. [4], Haywoodand Evans [5], Wu et al. [6], Attia [7]). The phase-gradientalgorithm (PGA, see Wahl et al. [8]) is another popular non-parametric technique, which iteratively estimates the residualphase by integrating over range an estimate of its derivative(gradient). Because nonparametric methods are based on theassumption of well-isolated dominant scatterers, they do notperform satisfactorily in many practical situations. On theother hand, parametric methods (Berizzi and Corsini [9], Xiet al. [10], Wu et al. [11], and Wang et al. [12]) are muchmore robust but more numerically intensive.

Common parametric techniques that use a polynomialmodel to approximate the targets translational motion anduse an image focus criterion to estimate the model parame-ters are the image-contrast-based technique (ICBT, see [13])and entropy-based technique (EBT, see [10, 14]). The chal-lenge for autofocus is to devise algorithms which not onlyfocus adequately for target recognition but are also both ro-bust and ecient. This generally involves a tradeo, betweeneciency and eectiveness.

The nonlinear optimization techniques are employed tosearch for a solution for the parameters by optimizing the


image focus quality which is formulated as an objective func-tion. Depending on the order of the model, the search is nor-mally carried out over a two- or three-dimensional space.One major drawback of these methods is that the optimiza-tion algorithm (minimization/maximization routine or op-timizer) often converges to a suboptimal solution if the ob-jective function is highly multimodal or has a large numberof local minima/maxima. Moreover, most deterministic op-timization methods, such as Newton, gradient, and so forth,are constrained by the fact that the objective function has tobe continuously dierentiable.

In summary, a successful convergence to the optimal so-lution and the numerical eciency of the method very muchdepend on various factors in relation to the nature of theobjective function, such as dierentiability and continuity,number of local minima/maxima, as well as the robustnessof the optimization algorithm, for example its sensitivity tothe initial guessed value.

The motion compensation technique described in thispaper is a parametric method that does not depend upon theassumption of prominent scatterers but estimates the targetsradial velocity based upon the composite of all of the targetscatterers. It uses this to correct the data for the slant-rangeand cross-range phase errors due to the translational motionof the target, thereby significantly improving image quality.

In the proposed optimization procedure, the first-orderand higher-order parameters of the targets radial motionare estimated sequentially by means of a range profile cross-correlation and a subspace-based technique involving eigen-decomposition. By decoupling the first- and higher-order pa-rameter searches, the technique allows the optimizers (min-imization/maximization routines) to be implemented overspaces of lower dimension, and thus reduces the likelihoodof converging to a suboptimal solution as encountered withother parametric methods. An overview of autofocus meth-ods is given by Xi et al. [10] and Li et al. [14], and a recentsurvey is presented by Berizzi et al. [15].

2. PROBLEM FORMULATION

Consider a target with complex reflectivity function (r) inthe imaging plane of the targets frame of reference, that is,in slant-range x and cross-range y (see Figure 1). The targetmotion with respect to the radar line of sight (RLOS) canbe decomposed into radial motion of an arbitrary referencepoint O on the target and rotational motion about the ref-erence point. Let R0(t) denote the radial distance of O fromthe radar at time t then O may be chosen to be the origin ofthe targets coordinate system (see Figure 1).

If the radial motion of the targets reference pointO (dueto translational motion) is defined by the initial velocity v0rand constant acceleration ar , and if the target is rotating atan angular velocity of about O, then the distance from anarbitrary scatterer (xk, yk) on the target to the radar at time tcan be written as

Rk(t) = R0(t) + Rk(t), (1)where R0(t) = R0(0) + v0r t+ art2/2 and Rk(t) = xk costyk sint.

Scatterer onthe target

Point ofreference

Target

RLOS

Radar

y

vt(t)

(x, y) x

r(t)

O

R0(t)R(t)

Figure 1: System geometry.

Let us define the transmitted RF signal for a coherentprocessing interval (CPI) of period T as the real part of

z(t) = u(t)e2 j f0t, (2)

where f0 is the carrier frequency and u(t) is the complex en-velope of the waveform given by u(t) = A(t) exp{ j(t)}, andwith Fourier transform U( f ), where A(t) and (t) are theamplitude and phase modulation of the signal, respectively.Then the received signal after demodulation and downcon-version to baseband can be written as

sR(t) =K

k=1ku(

t k(t))

exp{ j2 f0k(t)

}

, (3)

where K is the number of scatterers on the target, k is thereflectivity of the kth scatterer which has local coordinates of(xk, yk) with respect to O and is of distance Rk(t) from theradar, k(t) is the delay function given by k(t) = 2Rk(t)/c,and c is the velocity of light.

If is small in comparison to T and the targets radialdisplacement is negligible for the duration of fast time sam-pling, then we can write the Fourier transform of the receivedsignal as

S( f , t) = U( f )K

k=1k exp

{ 2 j f k(t)}

, (4)

where t now refers to slow time, that is, pulse-to-pulse. TheFourier transform of the range profile, following the devel-opment in [9, 16], is therefore

SR( f , t) = S( f , t)U( f )

=K

k=1k exp

{ 2 j f k(t)}

. (5)

D. Yau et al. 3

Now k(t) = 2Rk(t)/c = (2/c)[R0(t) + Rk(t)], hence

SR( f , t)

= exp{

4 j f R0(t)c

} K

k=1k exp

{

4 j f Rk(t)c

}

,

(6)

where Rk(t) xk ykt, from which we see that phasechanges occur in slow time for each scatterer but that thephase changes associated with radial target motion are sep-arate from those associated with target rotation. The phasechanges associated with the radial motion of the referencescatterer may therefore be corrected for by making the phaseadjustment 4 j f R0(t)/c to each pulse in the frequency do-main. Since R0(t) = R0(0)+v0r t+art2/2, we need to estimatev0r and ar for the reference scatterer.

The corrected range profile Fourier transform is then

SR( f , t) = exp{

4 j f R0(t)c

}

SR( f , t)

=K

k=1k exp

{

4 j f Rk(t)c

}

,

(7)

fromwhich the realigned and phase-compensated range pro-file may be recovered by means of an inverse FT. Frequencyestimation may then be performed in each range cell forcross-range velocity estimation. In a radar system, f and ttake the digitized forms of f = fm (m = 1, . . . ,M) andt = pT (p = 1, . . . ,N), where fm is themth sample in the fre-quency domain and T is the pulse repetition interval (PRI);M and N are the number of frequency samples and Dopplerpulses, respectively.

The radial motion of the target has the eect of causingscatterers to migrate between range cells, and hence smear-ing of the image in the range dimension; whereas the 1/2art2

term alone causes nonlinear phase variation in slow time,and hence smearing of the image in the cross-range dimen-sion. If the length of the burst is suciently small comparedwith the rotation rate of the target, then the ykt term isapproximately linear and provides the Doppler informationnecessary for cross-range imaging.

3. VELOCITY AND ACCELERATIONCORRECTION TECHNIQUE

The purpose of the present paper is to estimate the ra-dial velocity and acceleration so as to determine the de-lay and hence correct for the phase in the data. The op-timization procedure comprises maximizing the objectivefunctions Fv(v0r) and Fa(ar) separately for the single vari-able v0r and ar spaces, respectively, whilst keeping the othermotion parameter fixed. The objective functions are formu-lated in a way such that Fv(v0r)/Fa(ar) is relatively invari-ant to the changes of the fixed parameter ar/v0r . This notonly allows the optimization to be implemented solely inone-dimensional space, but also guarantees a fast conver-gence rate. The two-stage estimation technique procedure is

Initializev0r & ar

Computez(m, p)

Compute Fv(v0r)to estimate v0r

Fv(v0r)maximized?

Computez(m, p)

Compute Fa(ar)to estimate ar

Fa(ar)maximized?

Recomputez(m, p)

Successiveestimates

within errorbound?

Completed

Yes

Yes

Yes

No

No

No

Figure 2: Flowchart showing the procedure for estimating velocityand acceleration.

summarized in Figure 2 and as follows: initial estimation ofvelocity using a cross-correlation technique followed by es-timation of acceleration using a subspace-based approach.Further refinement is achieved as required by repeating theprevious steps although in practice, at most only three itera-tions are required.

Denote the matrix of complex radar signals organized ac-cording to range cell and pulse, respectively, by z(m, p). Forassumed values of v0r and ar , the previously described pro-cedure for range realignment and phase compensation is ap-plied to the recorded data of (6) . The range profile z(m, p)produced for range cells m = 1, . . . ,M and Doppler pulsesp = 1, . . . ,N is written as the inverse discrete Fourier trans-form of (6) after being adjusted for phase errors as follows:

z(m, p) = IDFT [(v0r , a0r , pT)

SR(

fm, pT)]

, (8)

where (v0r , a0r , p) = exp{ j4 f R0(v0r , a0r , pT)/c} is theterm associated with phase compensation of the received


data, and R0 is the corresponding radial range displacementof the target at time t = pT given some estimates of velocityand acceleration (v0r , a0r). In the following, it is shown howthe objective functions are formulated separately in terms ofthe parameters v0r and a0r for optimization.

The measure of how well the range realignment andphase compensation have been achieved is to compute thecross-correlation function r1,p(0) between the first rangeprofile (i.e., for the first pulse) and each of the remainingrange profiles, and summing their moduli, thus

Fv(

v0r) =

N

p=2

r1,p(0)

, (9)

where r1,p(0) =M

m=1 z(m, 1)z(m, p). The velocity estimateis that value of v0r (given the correct a0r) which maximizesthe objective function Fv(v0r), either by a blind search proce-dure or by formal optimization.

Following the improvement in the estimate of v0r , therange realignment and phase compensation procedure arerepeated. The acceleration ar is estimated as follows. The ac-celeration estimation technique exploits the fact that therewill be many scatterers within the range cells occupied bythe target to be imaged, which have a very similar radial ve-locity, although there will be a relatively small spread dueto the superimposed varying cross-range rotational veloci-ties. Because we are concerned with estimating a radial ve-locity which changes within the duration of a burst, we takea fixed window within which it is assumed that the radial ve-locity is approximately constant and fit a linear model to allof the range cells. This produces a covariance matrix whichis averaged over range cells. This has the advantage of incor-porating all of the energy from the targets scatterers ratherthan having to find and depend upon one of a small numberof prominent scatterers. As a general criterion for spectralestimation techniques, the size of a fixed window of pulsesshould be chosen to be greater than or equal to the size of thesignal subspace.

A data matrix is constructed from range profiles takenwithin a window superimposed on the pulses in slow time,thus

Zi =

z(

1,ni) z(1,ni +Nd 1

)

.... . .

...z(

M,ni) z(M,ni +Nd 1

)

, (10)

where the window begins with the nith pulse and containsNdpulses. Then the covariance matrix Ri = (1/M)ZHi Zi for theith window is formed by averaging over the range cells. Sub-space theory tells us that the principal eigenvectors span thesame subspace as the signal vectors. In general, we will notknow the dimensions of these subspaces (except that theirsum isNd), but it is sucient for our purposes to identify thedominant signals associated with the signal subspace througheigendecomposition.

Therefore, the covariance matrix is subject to the eigen-decomposition

Ri = ViiVHi , (11)

where the i = diag(1, 2, . . . , Nd ) is the diagonal ma-trix of eigenvalues with 1 2 Nd and Vi =[vi,1 vi,2 vi,Nd ] is the matrix containing all the corre-sponding eigenvectors.

Computationally, singular value decomposition (SVD) isa more practical approach to computing the set of eigenvec-tors directly from the data matrix Zi = UiiVHi , where Uiis an M-by-M unitary matrix, i is a diagonal matrix of theformi = diag(1, 2, . . . , Nd ), and k(1, . . . ,Nd) are the sin-gular values which are related to the eigenvalues by k = 2k .The diagonal matrix i (i) is always full rank because of re-ceiver noise. The noise power is small in comparison to thesignal power, and thus the signal subspace can be determinedby examining the singular values.

The assumption behind the method is that the principaleigenvectors contain the Doppler information for the dom-inant scatterers on the target for the ith window. All of thescatterers will be subject to phase changes between pulses:one component will be a linear phase shift due to the com-mon initial velocity v0r and the other nonlinear phase shiftdue to acceleration ar . This may be seen from the rangeresponse function 0e4 j f0/c(R0+v0r t+1/2ar t

2) for the referencescatterer. The Doppler information implicit in two data ma-trices taken from dierent time intervals, say Zi and Zi+ j , willdier by an amount proportional to the change in the veloc-ity or acceleration ar . In mathematical terms, this dierencecorresponds to the rotation of the signal subspace with re-spect to the origin of the vector space. However, when theacceleration has been correctly adjusted, the signal subspaceor the principal eigenvectors associated with the windowsshould coincide (within an arbitrary phase).

We suppose that only two windows are chosen. A mea-sure of how well the principal eigenvectors coincide betweenthe first and second windows of the burst is the sum of themoduli of their respective inner products:

Fa(

ar) =

Np

k=1

vH1,kv2,k

, (12)

where Np is the number of principal eigenvectors chosen torepresent the signal subspace. This objective function Fa(ar)is to be maximized over ar . For simplicity, we choose theeigenvector which corresponds to the largest eigenvalue foreach window so that Fa(ar) = |vH11v21|. In fact, the number ofwindows chosen is arbitrary and is always a compromise be-tween accuracy and eciency. For example, if we choose Nwwindows and use the first window as the reference, then theobjective function is reformulated as Fa(ar) =

Nwi=1 |vH11vi1|.

It can be easily shown that the number of computer op-erations required for calculating Fv(v0r) isM(N 1) and forFa(ar) is approximately 4MN2p 4N3p/3 (the number of op-eration required for SVD).

4. RESULTS

A summary of the radar parameters used in the simulationis given in Table 1 and a diagram showing the configurationof point scatterers on the test target is displayed in Figure 3.The reflectivity of the scatterers is indicated by the size of

D. Yau et al. 5

Table 1: Radar parameters.

Pulse compression Stepped frequency

Number of sweeps 64

Number of transmitted frequencies 64

Centre frequency 10GHz


Bandwidth 150MHz

PRF 74.46 kHz

x

y

4

4

4

43

3

3 3 3

3

2

2

Figure 3: Simulated target point reflector configuration.

the circles in the drawing. The relative sizes are 0.5, 1 and2, respectively. The target is travelling towards the radar withan initial velocity of 5m/s and with constant acceleration of2m/s2, with self-induced rotation of 0.16 rad/s.

Our technique is compared to Haywood-Evans MSA [5]and PGA [8] motion compensation techniques with a signal-to-noise ratio (SNR) of 20 dB. The results are shown inFigure 4 and it can be seen that the present technique gen-erates a better focused image than the other techniques. Us-ing our technique, v0r and ar are estimated to be 5.15m/sand 2m/s2, respectively. For a detailed examination of thebehavior of the objective functions, Fv and Fa are plottedagainst v0r and ar in Figures 5(a) and 5(b). The objectivefunctions are also plotted with respect to fixed parameters inFigures 6(a) and 6(b). It can be seen that their variation is rel-atively insignificant as compared to the previous figures. An-other similar set of data but with lower signal-to-noise ratio(SNR = 10 dB) was tested and used for comparison betweenthe dierent techniques. The resulting images are shown inFigure 7. Again our technique outperforms MSA and PGA.

In computing these plots, the subspace technique wasimplemented using two windows of size 8. Only the eigen-vector which corresponds to the largest eigenvalue was usedto maximize Fa(ar). The computation time taken was lessthan 1 second on a Pentium IV 2.5GHz computer and tookroughly 10 times longer than the PGA method. This is com-parable to the ICBT and EBT methods as stated in [15].The algorithm was run on Matlab and the maximization wasimplemented by a toolbox function called fminbnd (used tominimize the negative of the objective functions). It took

only two iterations for the optimization algorithm (Figure 2)to converge to the desired results. On average, it requiredabout 10 iterations for fminbnd to maximize each objectivefunction.

Next, we show an experimental example of a Boeing737 (Figure 8) ISAR image reconstructed using MSA andthe proposed subspace algorithm in Figure 9. The radar isat ground level and the parameters are lowest frequency= 9.26GHz, frequency step = 1.5MHz, range resolution= 0.78m, PRF/sweep rate = 20 kHz/156.25Hz; and the sizeof the data matrix is 64 by 64. Again it is seen that the pro-posed technique produces a much better image.

5. CONCLUSIONS

This paper has proposed a new parametric autofocusmethodfor simultaneously realigning range and compensating forphase by estimating radial velocity and acceleration using acombination of range profile correlation for velocity estima-tion and subspace eigenvector rotation for acceleration esti-mation. The method does not suer from the limitation ofassuming the existence of prominent scatterers as in othernonparametric methods.

As shown in the paper, by formulating the objective func-tions with respect to only a single variable and implementingthe optimization in two separate steps, the problem of con-vergence to a suboptimal solution suered by other paramet-ric methods can be avoided. It has proven to be both robustand has demonstrated that good results can be achieved interms of image quality.


454035302520

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151050

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plerfrequency

(Hz)

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9

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0dB

(c)

Figure 4: Range-Doppler image of simulated target (SNR = 20) using (a) MSA technique [5]; (b) PGA technique [8]; (c) proposed subspacetechnique.

109876543210

v0r

30

35

40

45

50

55

60

Fv

(a)

43.532.521.510.50

ar

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fa

(b)

Figure 5: Plots of objective functions against the estimated parameters (a) Fv versus v0r (ar = 2m/s2) and (b) Fa versus ar (v0r = 5.15m/s2).

D. Yau et al. 7

43.532.521.510.50

ar

30

35

40

45

50

55

60Fv

(a)

109876543210

v0r

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fa

(b)

Figure 6: Plots of objective functions against the fixed parameters (a) Fv versus ar (v0r = 5.15m/s2) and (b) Fa versus v0r (ar = 2m/s2).

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Figure 7: Range-Doppler image of simulated target (SNR = 10) using (a) MSA technique [5]; PGA technique [8]; (c) proposed subspacetechnique.


Figure 8: Schematic of a Boeing 737 (top view).

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Figure 9: ISAR image of a Boeing 737: (a) MSA technique [5] (3 reference cells used); (b) proposed subspace technique.

REFERENCES

[1] C.-C. Chen and H. C. Andrews, Target-motion-inducedradar imaging, IEEE Transactions on Aerospace and ElectronicSystems, vol. 16, no. 1, pp. 214, 1980.

[2] G. Wang and Z. Bao, The minimum entropy criterion ofrange alignment in ISAR motion compensation, in Proceed-ings of the IEEE International Radar Conference, pp. 236239,Edinburgh, UK, October 1997.

[3] B. D. Steinberg, Microwave imaging of aircraft, Proceedingsof the IEEE, vol. 76, no. 12, pp. 15781592, 1988.

[4] W. C. Carrara, R. S. Goodman, and R. M. Majewski, SpotlightSynthetic Aperture Radar: Signal Processing Algorithms, ArtechHouse, Boston, Mass, USA, 1995.

[5] B. Haywood and R. J. Evans, Motion compensation for ISARimaging, in Proceedings of Australian Symposium on SignalProcessing and Applications (ASSPA 89), pp. 113117, Ade-laide, Australia, April 1989.

[6] H. Wu, D. Grenier, G. Y. Delisle, and D.-G. Fang, Transla-tional motion compensation in ISAR image processing, IEEETransactions on Image Processing, vol. 4, no. 11, pp. 15611571,1995.

[7] E. Attia, Self-cohering airborne distributed arrays on landclutter using the robust minimum variance algorithm, in Pro-ceedings of IEEE Antennas and Propagation Society Interna-tional Symposium (APS 86), vol. 24, pp. 603606, June 1986.

[8] D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and C. V. JakowatzJr., Phase gradient autofocusa robust tool for high reso-lution SAR phase correction, IEEE Transactions on Aerospaceand Electronic Systems, vol. 30, no. 3, pp. 827835, 1994.

[9] F. Berizzi and G. Corsini, Autofocusing of inverse syntheticaperture radar images using contrast optimization, IEEETransactions on Aerospace and Electronic Systems, vol. 32, no. 3,pp. 11851191, 1996.

[10] L. Xi, L. Guosui, and J. Ni, Autofocusing of ISAR images basedon entropyminimization, IEEE Transactions on Aerospace andElectronic Systems, vol. 35, no. 4, pp. 12401252, 1999.

[11] H. Wu, D. Grenier, G. Y. Delisle, and D.-G. Fang, Transla-tional motion compensation in ISAR image processing, IEEETransactions on Image Processing, vol. 4, no. 11, pp. 15611571,1995.

[12] Y. Wang, H. Ling, and V. C. Chen, ISAR motion compensa-tion via adaptive joint time-frequency technique, IEEE Trans-actions on Aerospace and Electronic Systems, vol. 34, no. 2, pp.670677, 1998.

D. Yau et al. 9

[13] F. Berizzi, E. Dalle Mese, and M. Martorella, Performanceanalysis of a contrast-based ISAR autofocusing algorithm, inProceedings of the IEEE International Radar Conference, pp.200205, Long Beach, Calif, USA, April 2002.

[14] J. Li, R. Wu, and V. C. Chen, Robust autofocus algorithmfor ISAR imaging of moving targets, IEEE Transactions onAerospace and Electronic Systems, vol. 37, no. 3, pp. 10561069,2001.

[15] F. Berizzi, M. Martorella, B. Haywood, E. Dalle Mese, and S.Bruscoli, A survey on ISAR autofocusing techniques, in Pro-ceedings of IEEE International Conference on Image Processing(ICIP 04), vol. 1, pp. 912, Singapore, October 2004.

[16] D. A. Ausherman, A. Kozmer, J. L. Walker, H. M. Jones, andE. C. Poggio, Developments in radar imaging, IEEE Trans-actions on Aerospace and Electronic Systems, vol. 20, no. 4, pp.363400, 1984.

D. Yau received the B.E. degree in electricalengineering from The University of Sydney,Sydney, Australia, and the M.Eng.Sc. andPh.D. degrees in electrical engineering fromThe University of Queensland, Queensland,Australia. He is currently working as a Re-search Scientist at DSTO, Australia. His re-search interests include radar imaging andsignal processing.

P. E. Berry works in the Electronic War-fare and Radar Division of DSTO and hasinterests in estimation, optimization, andcontrol applied to microwave radar engi-neering. He has previously worked in re-search lab

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