Target Classification by Autoregressive Modeling using Range Extent Profiles

Transcription

1 Target Classfcaton by Autoregressve Modelng usng Range Extent Profles Mahendra K Mallck and Stefano Coralupp ALPHATECH Inc Mall Road Burlngton MA 3 USA mallck@alphatechcom coralupp@alphatechcom Abstract We present a novel algorthm based on autoregressve (AR modelng for analyzng the separablty and classfcaton of ground targets usng range extent data Gven a range extent profle of a target we estmate the approprate model order for the data usng the Akake nformaton crteron ( Ths prevents under-fttng or over-fttng of the data Prevous researchers have shown that even f the actual process s not an AR process the AR model serves as a reasonable model for a wde class of practcal problems Error modelng for the range extent data s extremely dffcult due to the complex nature of the scatterng process uncertantes n the channel sensor state target dynamcs and estmaton of the range extent from a range profle Therefore our data-drven approach provdes a useful algorthm for analyzng target model separablty and classfcaton We apply the algorthm to smulated range extent data and obtan good classfcaton results We plan to test the algorthm further wth real range extent data Keywords: Autoregressve (AR Model Akake Informaton Crteron ( Determnaton Target Classfcaton Range Extent Hgh Range Resoluton (HRR Feature-aded Trackng Introducton One-dmensonal hgh range resoluton (HRR profles of ground targets have been successfully used n automatc target recognton (ATR systems and target trackng n recent years []-[] HRR profles provde sgnfcant operatonal advantage over the twodmensonal synthetc aperture radar (SAR mages of ground targets due to faster formaton tme of HRR profles over the slower formaton tmes of SAR mages For example dwell tmes for an HRR profle s of the order of mcroseconds compared wth 3 seconds for a SAR mage formaton Secondly HRR profles can be formed for a movng target whereas a SAR mage can only be formed for statonary targets Gven a SAR mage of a target an HRR profle for the target can also be generated usng sgnal processng technques [3] The range extent of a target s a functon of the dmensons of the target the relatve geometry of the target and the sensor scatterng propertes of the target and varous error sources nvolved n the scatterng process The range extent can be estmated ether from an HRR profle derved from a SAR mage of a statonary target or from the observed HRR profle of a movng target If the dmensons of two targets are sgnfcantly dfferent then the range extents can be used n a tracker as measurements n order to resolve knematc ambguty of two closely spaced targets wth smlar knematc attrbutes Ths results n mproved data assocaton and trackng accuracy Measurement error modelng for the range extent data s extremely dffcult due to the complex scatterng process uncertantes n the channel platform state target moton and range extent estmaton from the HRR profle [][] Analytcal models descrbng varous error sources n the range extent data and correspondng probablty dstrbutons are not well understood Often smple models wth addtve Gaussan nose for the range extent data based on geometrcal projectons [7] are used n the tracker These models cannot account for the complex scatterng process and the nherent asymmetry n a range extent profle as a functon of the aspect angle In the model-based target classfcaton approach the range extent profles of a set of known targets are stored n a database An estmated range extent at some observed aspect angle derved from a range profle s compared wth the stored range extent profles and the lkelhoods of the target types are computed for a number of targets Gven a set of range extent profles we address the separablty and classfcaton of target models If the varance of the range extent data for a range extent profle at each aspect angle s known then the generalzed dstance between two range extent profles can be estmated The generalzed dstance represents the separablty of the target models The lmtaton of ths approach s that t requres a statstcally sgnfcant number of range extent values for each target at each aspect angle In practce these

2 measurements are not avalable so that computng the generalzed dstance between two range extent profles s not feasble Under such crcumstances we propose to model the range extent profle of a target as a stochastc process characterzed by an autoregressve (AR process A requrement of our approach s that a large number (compared wth the order of the AR process of range extent values are avalable n the profle and the range extent data are equally spaced n the aspect angle If the data s not equally spaced n the aspect angle then under a Nyqust-type assumpton the data can be nterpolated before processng In Secton we descrbe the AR modelng estmaton of AR model parameters model order determnaton usng the Akake nformaton crteron ( [] and computaton of the generalzed dstance between two AR models In Sectons 3 and we present range extent data and numercal results respectvely We present conclusons and future work n Secton Autoregressve (AR Modelng AR Model Akake [9] showed that AR models provde a reasonable approxmaton to data even though the true model generatng the data s not an AR model We treat the set of range extent estmates x = N as a tme seres A scalar AR model of order k s gven by the dfference equaton [9][][] ( x n = axn + axn + + ak xn k + en where a = k are the constant AR model parameters and e n s a zero-mean Gauusan whte nose wth varance σ : ( ( 3 en e n ~ N( σ E { e m e n } = δ mn σ s assumed to be ndependent of the past values th k xn = n The order AR model denoted by AR(k s completely specfed by (a the order of the model k and (b the parameters a = k andσ Maxmum Lkelhood Estmaton of AR Model Parameters Gven the order k of the AR model we frst address the estmaton of the AR model parameters by the maxmum lkelhood (ML estmator [][] The estmaton of the order of the AR model s addressed n Secton 3 Usng the matrx notaton equatons ( can be wrtten as xk = ( x N xk xk or ( z = Ha + e where ( ( 7 ( xk : H = x N xk xk a : a a = a k : z = ek+ ( 9 : ek+ e = en Snce { e n } s a whte nose sequence ( e ~ N( Iσ x a ek+ x a + ek+ k ak en x x k The ML estmator of a s same as the least squares (LS estmator of a [][3] Snce the varance of en s ndependent of n the LS estmator of a s obtaned by mnmzng the cost functon ( J ( a : = ( z Ha ( z Ha wth respect to a Thsleadsto ( a ( = H z In practce the LS estmator â s determned by usng the sngular value decomposton (SVD [] of H The ML estmator of a s unbased e ( 3 E{} a = a The error covarance assocated wth â s gven by

3 ( P = E{( a a( a a } = σ ( H mnmum and then ncreases wth k In practce the decrease and ncrease are not strctly monotonc P represents the accuracy of the estmator â For easer nterpretaton the smultaneous (-α % confdence ntervals for the true parameters a } are gven by { [ kp F ( ] / = ( a ± k N k α k th where Fk N k ( α s the upper (α percentle of an F dstrbuton wth k and (N-k degrees of freedom Instead of the smultaneous confdence ntervals n (- n practce the followng (-α % confdence ntervals are used by replacng Fk N k ( α wth the one-at-a-tme t value t N k ( α / : ( a ± t N k ( α / p = k Havng obtaned the estmator â the unbased estmator of σ s Substtuton of the maxmum log-lkelhood from (- nto (- gves ( - ( k = ( N k ln(πσ + ( N + k Amaxmumvalue k max << N for k s selected (ks evaluated for k= k max Thevalueofk for whch (k s the mnmum s selected to represent the correct order Power Spectral Densty The power spectral densty (PSD for the AR(ksgvenby [][] ( - 3 σ Γ( f = f N f < f N πkf ak e k ( 7 Thevaranceof σ σ = J ( a /( N k s gven by where s the samplng nterval and frequency defne by f N s the Nyqust ( /( var( σ = σ N k σ /( N k The exact log-lkelhood functon correspondng to the measurement equaton (- s ( 9 N k L( a = ln(πσ σ J ( a Therefore the maxmum of the log-lkelhood functon evaluated at the ML estmate of a s ( N k ( ln( N k L a = πσ 3 Estmaton Akake [][9] ntroduced the Akake nformaton crteron ( based on nformaton-theoretc concepts as a bass for model order estmaton The s applcable to a general class of problems for statstcal model dentfcaton When a model wth k ndependent parameters s ftted to data the s defned by ( - ( k = ln(maxmum lkelhood + k The value of k for whch (k s the mnmum represents the correct order for the tme seres { x n }(kfrst decreases as k gradually ncreases due to domnance of the frst term over the second term on the rght hand sde of (- wth mprovement n fttng (k reaches a ( - f N : = The PSD for a gven AR process can be estmated usng the estmated order and the parameters of the process Generalzed Dstance among Models Consder two AR models wth estmated parameters ( k a σ and( k a σ One approach to dstngush between models s the followng The covarance matrces of â and â are P and P respectvely In general k k In order to compute the generalzed dstance between the two models t s necessary to have covarances wth the same dmenson Suppose k > k Snce we know from the that the AR coeffcents for the second model beyond k elements are zero we can append k k zeros at the end of â to create a new vector β whle β = a Smlarly we create a covarance matrx Σ by addng zeros to P such that Σ s a k k matrx whle Σ = P Defne: β ( - 3 θ = = σ It can be shown (see the Appendx that ( E{ σ a } = σ a

4 Therefore the cross-covarance between therefore between β and â and σ s exactly zero The θ s gven by generalzed dstance between θ and ( [ / ( θ θ ( R + R ( θ ] d = θ σ and 7 3 Tgt Type 3 7 Aspect Angle (radan Tgt Type Aspect Angle (radan where ( R = var( σ A second frequency-doman approach to dstngush between models s to consder the power spectral densty of the AR models and consder the Eucldean dstance between them 3 Smulated Range Extent Data The relatve geometry of the target and sensor s shown n Fgure The range profle and hence the range extent vares as a functon of the aspect angle Local North Projecton of Sensor Poston on Local East-North Plane Aspect Angle α Target Local East Fgure Relatve Geometry of the target and Sensor Ffteen target types were used to smulate the HRR profles Smulated nose-free SAR data were generated frst and then SAR chps were sub-apertured and sub-banded nto range Doppler chps representng multple coherent processng ntervals (CPIs Complex Gaussan whte nose was added to each CPI The fnal step was a non-coherent sum of ndvdual CPIs HRR profles were generated from range- Doppler chps by takng a range slce centered on the brghtest pxel An estmate of the range extent was obtaned from each HRR profle usng a threshold-crossng algorthm For each target type range profles were generated for aspect angles 3 degrees Four dfferent sets of range extent data sets were generated for our analyss The frst data set had a low nose level The next three data sets had a hgher and constant nose level than the frst The seeds were dfferent for the last three data sets Sx sample range extent profles for target types 3 and from the frst data set are shown n Fgure Aspect Angle (radan Fgure Sample Range Extent Profles from Data Set Computatonal Approach and Numercal Results For the gven set of reference range extent profles we found that the k max = was satsfactory for estmatng the AR model parameters For each range extent profle the model order k was vared from one to and for each model order the AR model parameters a σ and(kwere estmated The model order for whch the (kwasa mnmum was selected as the model order for the data We present the varaton of the wth the model order for four dfferent sample cases n Fgure 3 Model orders determned for varous target types usng the are shown n Fgure Tgt Type 3 7 Aspect Angle (radan Tgt Type 7 7 Tgt Type Aspect Angle (radan Tgt Type Aspect Angle (radan 9 9 Tgt Type Fgure 3 Varaton of wth for Data Set 3 Tgt Type

5 Tgt Type 9 Tgt Type Tgt Type Tgt Type - - Tgt Type Tgt Type 3 3 Fgure 3 (contnued Varaton of wth Fgure llustrates the estmated AR coeffcents and 99% confdence bounds for a selecton of targets Note that the confdence bounds ncrease wth ncreasng order of the AR model Indeed a fxed number of measurements wll lead to more accurate estmates of a smaller number of parameters than a larger number of parameters Target Type Index Fgure Estmated for Varous Targets Tgt Type 3 3 Tgt Type Tgt Type Tgt Type 3 3 Fgure Estmated AR Model Coeffcents wth 99% Confdence Bounds for Data Set Fgure (contnued Estmated AR Model Coeffcents wth 99% Confdence Bounds for Data Set We frst tred to analyze the effect of dfferent nose levels on target classfcaton In order to do ths we computed the generalzed dstance matrces between data set pars ( ( 3 and ( usng the approach descrbed n Secton A target type from the second set member of the par (j was assumed to be correctly assocated and hence classfed f the mnmum dstance lead to correct assocaton Then the average percentage of correct classfcaton for all target types was determned for the two data sets We found that the dfference n nose levels lead to a very low percentage (~% of correct classfcaton Treatng the PSD at varous frequences for a target type as vector we also computed the dstance between two PSDs two target types for the par (j We obtaned smlar low percentage of correct classfcaton Ths ndcates that f the nose levels n the reference data and test data sets are sgnfcantly dfferent then target classfcaton usng range extent data s dffcult Next we calculated the average percentage of correct classfcaton usng data set pars ( 3 ( and (3 All the data sets n ths case have the sane nose level The average percentage of correct classfcaton for data set pars ( 3 ( and (3 usng the generalzed dstance metrc of Secton were 33 7 and 733 respectvely Correspondng results usng the dstance between the PSDs were and respectvely The dstance matrces based on PSDs are shown n Fgures 7 and These results ndcate that dstance between PSDs s a better metrc for classfcaton whle usng range extent profles than the generalzed dstance between two AR models Conclusons Ths paper has ntroduced a data-drven approach to target classfcaton based on range-extent profles Our approach s based on treatng the profles as tme seres and fttng each tme seres to an AR model of approprate order Ths order s determned by an nformaton-theoretc metrc the The models can then be compared n a robust manner based on ther power spectral denstes Our data-drven approach s superor to a drect comparson of the range extent profles snce the latter requres the knowledge of

6 Fgure PSD Based Dstance Matrx between Data Sets and 3 Percentage of Correct Classfcaton s 733 Fgure 7 PSD Based Dstance Matrx between Data Sets and Percentage of Correct Classfcaton s 7 Fgure PSD Based Dstance Matrx between Data Sets 3 and Percentage of Correct Classfcaton s complex error models as well as a statstcally sgnfcant number of range extent profles We have compared two dfferent metrcs the generalzed dstance between two AR models and the dstance between two PSDs for target classfcaton Based on the current smulated data sets we observe that the dstance between two PSDs serves as a robust metrc for target classfcaton usng range extent profles We beleve our data-drven approach s preferable to modelbased classfcaton approaches snce ( these approaches are lmted by the avalablty of models for targets ( such models are often exceedngly complex and (3 classfcaton based on range-extent profles s not robust to nosy data We plan to test our approach usng more extensve smulated range-extent data and analyze the effect of nose on target classfcaton Also we plan to extend our approach to more general autoregressve movng average (ARMA and autoregressve ntegrated movng average (ARIMA models [] to more effectvely model nonstatonary tme seres Appendx A In ths Appendx we prove that σ and â are uncorrelated We note that ( ( (A- z Ha z Ha σ = N k (A a = a + H( H For convenence defne the matrx ( H H (A 3 B : = Substtutng (A- n (A- we get ( ( v I HB I HB v (A σ = N k Usng(A-and(A-wehave ( ( ( { v I HB I HB v a + v B (A E σ a } = E{ } N k Notng that v s a zero-mean Gaussan random varable and usng the matrx trace dentty tr ( XY = tr(yx whch holds for arbtrary matrces X and Y of approprate dmensons (A- can be smplfed In partcular we have: (A E { σ a } { ( v ( I HB = E tr ( I HB v } a N k = E{ tr( vv ( I HB ( I HB } a N k σ a = tr( I HB ( I HB N k We note the followng propertes of HB: v

7 (A 7a (A 7b (A 7c HB = H( H ( HB = HB ( HB = ( HB Usng (A-7 n (A- we get (A E Therefore (A 9 HB s symmetrc HB s dempotent { σ a σ a } = ( N k tr( H( H N k σ a = ( N k tr( H( H E Snce E{ } σ N k σ a = N k { σ a } = σ a ( N k σ = and E{ } = a a t follows that σ and â are uncorrelated Ths s an mportant and somewhat surprsng result Not only are the estmates σ and â based on the same data z but σ s an explct functon of â It can be shown that ths property does not hold for the (based maxmum lkelhood estmator for σ [7] T Kuren J Horn L Palumbo and D Looze Multple Hypothess Tracker: Implementaton and Evaluaton TR-9 ALPHATECH Inc December 999 [] H Akake A new Lookat the Statstcal Model Identfcaton IEEE Transactons on Automatc Control AC-9: [9] H Akake Fttng autoregressve models for predcton Annals of the Insttute of Statstcal Mathematcs : [] G E P Box and G M Jenkns Goossens Tme Seres Analyss: forecastng and control Holden-Day 97 [] M B Prestley Spectral Analyss and Tme Seres Volumes and Academc Press 9 [] J M Mendel Lessons n Estmaton Theory for Sgnal Processng Communcatons and Control Prentce Hall 99 [3] R A Johnson and D W Wchern Appled Multvarate Statstcal Analyss Communcatons Prentce Hall 99 []G M Jenkns and D G Watts Spectral Analyss and Its Applcatons Holden-Day 9 Acknowledgement The authors thank Dr Steven Crooks of ALPHATECH Inc for smulatng the range extent data used n ths paper References [] R Wllams J Westerkamp D Gross and A Palomno Automatc Target Recognton of Tme Crtcal Movng Targets Usng D Hgh Range Resoluton (HRR Radar IEEE AES Systems Magazne 37-3 Aprl []KBEomNon-cooperatve target classfcaton usng herarchcal modelng of hgh range resoluton radar sgnatures Proc SPIE [3] M Mrkn and B Hodges ATR for MTE Program MIT Lncoln Laboratory QPR September 997 [] R Popp N Sandell R Washburn H Maney B Hodges A Baley M Mallck and B Johnson MTE Ground Staton Testbed 999 [] T G Allen D A Castanon I A Farber W C Karl and M Predny Multresoluton Fuson of MTI HRR and SAR for Enhanced Target Trackng and Classfcaton of Ground Targets TR-9 ALPHATECH Inc January 3 [] S M Verbout Analyss of Features for ATV ALPHATECH Inc February