Doppler & Polarimetric Statistical Segmentation for Radar Clutter map based on Pairwise Markov Chains

Transcription

1 Doler & Polarmetrc Statstcal Segmentaton for Radar Clutter ma based on Parwse Markov Chans N. Brunel 1 F. Barbaresco 1: DGA Ph.D. Student laboratore CITI Insttut Natonal des Télécommuncatons 9 rue Charles Fourer Evry Cedex France n collaboraton wth Thales Ar Defence & Unversty Pars 6. : Thales Ar Defence 7-9 rue des Mathurns 91 Bagneux Cedex France Abstract: Ths aer deals wth the segmentaton of the radar envronment based on Bayesan statstcal methods n order to rocess adatvely the receved sgnal accordng to the local characterstcs of the clutter. We used frst statstcal models for Doler and olarmetrc sgnal so that the segmentaton wll run on the estmators of the arameters. We used then a generalzed Hdden Markov Chan for the segmentaton of clutter envronment by alyng MPM rule for the estmaton of the hdden states. The usual assumton of condtonal ndeendence of the observatons s relaxed by takng nto account the deendence wth coulas. We roose a generc statstcal estmaton and restoraton rocedure. The methodology s llustrated wth data comng from olarmetrc studes. senstve to abrut changes n the sgnal as CFAR) and can gve oor erformances near the fronters. Hence a good localzaton of these fronters and estmaton of the mean behavor n each area can mrove the erformance n detecton for nstance). Keywords: Unsuervsed segmentaton statstcal restoraton hdden and arwse Markov chans estmaton coula drectonal statstcs auto-regressve models. 1. Introducton The objectve of the statstcal segmentaton s to rovde a ma of the envronment that gves us the statstcal homogeneous areas of the clutter. Data that wll be satally classfed are based on Doler nformaton through resectvely the reflecton coeffcents and reflectvty or Polarmetrc nformaton through the state of olarzaton. Two alcatons of such a ma are : Adataton of waveform accordng to the reflectvty Doler statstcs of clutter data to otmze the robablty of detecton but just enough to reserve the radar tme budget. Adatve rocessng of the receved sgnal : an adatve Constant False Alarm Rate CFAR) could used ths nformaton and avod the use of ACP ma PAC n French). Ths s crucal n case of target detecton near clutter edges because these areas are the most threatenng areas for nstance coast lne n Lttoral Warfare crest lnes ). More recsely the usual radar sgnal rocessng take nto account the satal heterogenety of the envronment by a local adataton of the algorthms. These methods can be Fgure 1 : Statstcal Segmentaton s useful for CFAR adataton near clutter edges most threatenng areas) We use a statstcal modellng of the Radar envronment based on Markov chan rather than on Markov feld. Ths aroxmaton of the D rocess of the envronment by a 1D rocess s made for reasons : on one hand the sgnal s generated and receved azmuth by azmuth and we need an on-lne rocessng. On the other hand we can have analytcal formulas for the osteror robabltes whch dramatcally mroves the seed of the estmaton and restoraton rocedures. We resent frst n ths aer the arametrzaton of the sectral and Polarmetrc nformaton n a way that can be effcently rocessed by the usual arametrc statstcal restoraton algorthms. We gve also some statstcal arametrc models comng from the doman of drectonal statstcs. The classcal Hdden Markov Chan s brefly recalled and we resent then General Hdden Markov Chan. We stress on the ablty of ths model to descrbe new knds of deendences between xels thanks to the use of coulas whch are qute new n mage and sgnal rocessng. Some models used for the segmentaton of real data are resented.

2 . Doler Data : Reflecton coeffcents on the shere The am of Doler segmentaton s to dscrmnate clutters havng dfferent Doler sectrum. In a way we want to cluster smlar functons n a sense to defne) by usng the nformaton of satal roxmty. The sgnal receved by the Radar are bursts whch are consttuted of several ulses. A way to have a cture of the Doler content of a artcular steerng angle s to comute the FFT of the ulses n each cell range gate). It corresonds then to a non-arametrc estmaton of the ower sectrum whose accuracy s ncreasng wth the number of ulses under the assumton of temoral statonarty of the sgnal). One drawback of ths method s that the sectral densty functon s comuted ontwse so the Doler rofle s reresented by a vector of hgh dmenson and we have no nformaton about the artcular onts such as the localzaton and the number of maxma. Moreover n ractcal stuatons the small sze of the samles gves sectra wth oor resoluton and estmators wth great varance. To avod these roblems we use Hgh Resoluton Doler Analyss whch s based on an auto-regressve model of the Radar sgnal : t ermts to relace the sectrum densty to the knowledge of a lttle set of arameters. The estmated arameters wll be then the nuts of the segmentaton algorthms. We recall frst some bass of autoregressve model. In the next art we descrbe the data used for segmentaton and some relevant arametrc models..1. Auto-Regressve rocess and characterzaton of clutter We make the assumton that the Radar sgnal can be nstantaneously aroached by an a comlex gaussan autoregressve rocess of order noted AR) see [1]). The sgnal Y t) s wrtten : Y t = a k = 1 k Y t k + ε where ε t s a whte nose wth mean zero and such that V ε ) = σ arameters a ) are the autoregressve t k coeffcents. An autoregressve rocess can be arameterzed by at least) two other sets of coeffcents : the frst + 1 autocovarance coeffcents * [ ] R k) = E X n X n k the reflecton coeffcents or PARCOR coeffcents) defned through the Levnson algorthm. Fnally we use the reflecton coeffcents ) σ to descrbe µ µ and the ower of the nose 1 a rocess. The relatonshs between these 3 sets of arameters are recalled n []. t Dfferent aroaches are ossble for the estmaton of an autoregressve rocess. It s ossble to estmate drectly the autocovarance matrx by the emrcal covarances and to solve the Yule Walker equatons to get the autoregressve coeffcents [1]). Nevertheless n Hgh Resoluton the small sze of the samles can not gve a good estmaton of R so we have to choose an alternatve soluton. The Burg algorthm s an teratve rocedure that estmates drectly the reflecton coeffcents by avodng the comutaton of the autocovarance. It was also shown [3]) that we can regularze ths algorthm by addng a smoothng constrant n order to mrove the estmaton of the ower sectrum n the case of small samles. In the regularzed verson of Burg algorthm the modules of the µ s are constraned to decrease so we can have a artal answer to the roblem of the model order choce because the vanshng of one PARCOR mly the vanshng of hgher order coeffcents. In all generalty the estmaton of ths hyerarameter s qute dffcult because the usual exressons of the classcal crterons as AIC or BIC are based on asymtotc aroxmatons whose valdty are questonable for small samles. Nevertheless n our context we make the assumton of an order equal to 5. Indeed Haykn and al. n [4] made a survey on the ways to descrbe the dynamcs of sea clutter. Among varous aroaches t aears that the use of small order comlex autoregressve rocess wth order equal to 4 or 5) are suffcent to recover locally the comlexty of the backscattered sgnal. Such a concluson s a useful gudelne for us because one of the am of statstcal segmentaton s to have a better estmaton of sea/ground transton. We strengthen ths oston by remarkng that we want to cluster smlar sectral rofle or shae : we do not really want to have a recse descrton of ower sectrum. The concet of smlarty deends then on the way we reresent the Doler rofle and on the selected arametrc models for the laws of reflecton coeffcents..3. Characterzaton of ower sectrum Fnally the roblem s reduced to the segmentaton of observatons comosed of two heterogeneous arts : a ostve scalar σ and a vector of comlex comonents µ = µ 1 µ ) belongng to the roduct unt dsk. At ths stage t s dffcult to use µ ; σ ) manly because of the absence of natural model for µ no asymtotc gaussan aroxmatons are avalable). Then we rely us on the followng decomoston of the densty of the jont law wth resect to an adequate measure) : ν µ σ ) µ σ ) µ σ ) =

3 We wrte then = µ µ when µ s strctly non υ / negatve and µ = 0 corresonds to whte nose. Practcally small values of estmated nterreted as roxmty to whteness. µ wll be.5. Statstcal model In ths aragrah we gve some arametrc models for the modelng of the Doler arameters σ ) The varable σ ) µ and ν. µ belongs to the ostve quadrant and each comonent s the sum of square of varables that can be gaussan n an deal case gaussan assumton s true and asymtotc case). As a consequence a gamma famly mght be adated to the descrton of the law of each varable because t contans the Ch Square law whch s the law for the deal case). Nevertheless a ragmatc model should be the gaussan or log-normal) model because t s the smlest oston scale model: the estmated arameters wll corresond to mean levels µ σ and scatterng ndex. of ) A lot of arametrc famles exst for data on the real 1 1 shere S of R or the comlex shere CS of C see [5]). The most famous one s the Von Mses Fsher law whch s robably the older one and the best studed model. We resent t only for the real shere but t easly extends to the comlex shere thanks to the 1 1 corresondence between CS and S ). The densty on the real shere dstrbuton s the followng : 1 S wth resect to the unform 1 κ / 1 1 ν S f ν x κ) = ) ex κ. ν' x) Γ / ) I / 1 κ) x and κ are arameters of the law whch wll be noted M x κ). x belongs to the shere and s the mean drecton κ s a scalar called the concentraton arameter. Ths descrton of the law s based on the 1 embeddng aroach : the shere S s regarded as a subset of R that means x ν are exressed through ther Cartesan Eucldean) coordnates. The mean drecton s defned by: E[ ν ] = κ x where E [.] denotes mathematcal exectaton. Ths law s qute smle to nterret because t s a locaton model. and a lot of aealng roertes see [5][6]). Fnally we suose that µ ndeendent that s: ν ) µ ) σ ) µ σ ) = σ and ν are because they corresond resectvely to 3 dfferent content of a sgnal : the sectral rchness the resdual nose ower and the shae of the sectrum. In order to temer ths assumton we recall that for the HMC model used for segmentaton ths wll corresond to an assumton of condtonal ndeendence: f we ntegrate out the state the varables are no more ndeendent. It s also ossble to segment data by usng only art of the Doler nformaton : on σ whch s related to the classcal segmentaton on the reflectvty of Radar Sgnal) µ σ segmentaton on the entroc on the vector ) content of the sgnal) and the full nformaton ; σ ) 3. Polarmetrc Data µ Ellse angles on the Poncaré s unt shere Classcally n olarmetry the ellse descrbed by the electromagnetc vector s arameterzed by two angles [7]): Φ : angle between the longer axe of the ellse wth the horzontal axe 1 b τ = tan : angle deduced from the rato between a the two man axes szes of the ellse elongaton). The corresondence between an ellse and a ont on the shere s shown n fgure 4. In a statstcal framework the receved olarmetrc sgnal s suosed to be a centered comlex gaussan rocess so that all nformaton les n the covarance matrx. Fgure 4 : Polarzaton Ellse & the Poncaré shere 3.. Statstcal Model As for the Doler case we need to roose statstcal models adated to an unsuervsed context. A lot of results have been obtaned for the law of the covarance matrx or for the laws of the comonents of the Stokes vector deduced from the covarance matrx). For the latter to our knowledge the laws are deduced comonentwse so that we do not nether know the jont robablty densty functon nor the law of the angles of olarzaton. Moreover these knd of dervaton are not suted to our urose because the samles are suosed to be statonary contrary to us because we need statstcal laws for descrbng the satal) non-statonarty of the olarmetrc characterstc of the Radar envronment. Fnally the am s to segment drectonal data ν = τ ϕ) that belongs to the real shere S. So we roose to use

4 the same model as for the Doler segmentaton wth a Von Mses law. 4. Unsuervsed Segmentaton of Parwse Markov Chans The roblem of the constructon of the ma of statstcally homogeneous areas s formally stated n the followng manner : Let X = X X ) and Y = Y Y ) be two 1 N 1 N stochastc rocesses where X s hdden and dscrete and Y s observed each Y takes ts values n R or C ). The roblem s to estmate the rocess X from an observed samle y of Y. Once the dstrbuton of the jont rocess X Y ) s known we can use Bayesan nference to restore the hdden rocess and the shae of ths dstrbuton nfluence the way we comute the Bayesan estmator. We wll note P x y) for the law of the random varables X Y ). In ths settng the Hdden Markov Chan HMC) model s among the most wdely used [68]). The model s based on three assumtons : The hdden rocess X s a statonary) Markov chan The observatons are ndeendent condtonally on the state The condtonal law of an observaton deend only on the current state : P Y X ) = P Y X ) Then the restoraton s made by comutng the Bayesan estmator Maxmum Posteror Mode MPM) of the rocess X. The comutaton of the osteror robablty s the cornerstone of the statstcal segmentaton. The advantage of HMC s that we have the exstence of analytcal formulas for the osteror robabltes often called «forward-backward» recursons that enables a rad restoraton. However n ractcal cases we never know recsely the law of each observatons P Y X ) and we need frst to estmate them for each classes and the transton robabltes of the rocess X ) and then to erform the segmentaton. Ths s made usually by the use of EM or SEM algorthms and varants of them) whch are teratve aroxmatons of the maxmum lkelhood estmator MLE see [9] [10] [11]). Consequently n most of the cases we can have a correct estmaton of the arameters Removng ndeendence assumton wth coulas We resent here a more general model that enables to relax the classcal ndeendence assumton [11]). Parwse Markov chans were ntroduced to correct the assumton of condtonal ndeendence [1]) that can be questonable because of the comlexty n general of mages or sgnals. If we note Z = X Y ) the defnton of a arwse Markov chan s qute short : The arwse rocess Z ) s a Markov chan We stress then on the fact that classcal HMC are a artcular subcase of Parwse Markov Chan. It was shown t suffces to have ths roerty to get forward backward recursons so we can use the same slghtly modfed) algorthms to comute the estmator of the hdden rocess by MPM). In the same manner the arwse assumton has been aled to Markov trees [13]) and Kalman Flterng [14]). It s nterestng to note that the hdden rocess X s no more a Markov Chan and that the law of the entre rocess s descrbed by the condtonal law Y Y + 1 X X + 1) and the jont robablty X X + 1) or equvalently by the transton kernel Z+ 1Z). Nevertheless we consder a less comlex model that we call a general HMC model wth the followng assumtons : The arwse rocess Z ) s a statonary Markov chan The hdden rocess X ) s a Markov chan The fundamental dfference for a general Hdden Markov Chan s that the condtonal rocesses Y n X ) n and X n Y ) n are both Markovan whereas a classcal HMC assume that the Y n X ) n are ndeendent and deduce that X n Y s Markovan). Ths s the reason why t wll ) n be referable to call a classcal HMC a HMC IN for Indeendent Nose) and use only HMC for the general one. In the case of a HMC the law of the rocess s descrbed by the transton robabltes of the hdden rocess X the margns P Y X ) as for the HMC IN) and also by the jont law P Y Y + 1 X X + 1). So we must choose bvarate laws that resect the margns chosen to reresent each class. A soluton to ths roblem s gven n [15]) thanks to coulas and the Sklar s theorem [16]) that asserts that every multvarate law can be wrtten as a functon of ts margns. Formally let H y 1 y) be a bvarate cumulatve dstrbuton functon c.d.f.) n R 1 wth F y ) and G y ) the corresondng margnal c.d.f. then there exsts a functon C called a coula such that H y1 y) = C F y1 ) G y)) Moreover f H s contnuous the coula s unque and we can seak of a coula reresentaton of the robablty H. We refer to [16][17] for roertes of coulas and ther use for modelng deendence. We wll only remark that a coula s a c.d.f. on the unt square [ 0 1] and that the denstes are wrtten : h y1 y) = f y1 ) g y) c F y1 ) G y))

5 where the lowercase functons are the robablty densty functons.d.f.) of the corresondng uercase c.d.f. The functon c s called the densty of the coula and s a.d.f. on [ 0 1]. Then a useful examle of famly of coula s the gaussan one. If we wrte ρ as the correlaton matrx of sze R and ζ ' = Φ 1 u ) 1 1 u) ) Φ I the dentty matrx of wth Φ the c.d.f. of the normalzed gaussan densty. The densty of a gaussan coula s: c ) 1/ 1 u u = ρ ex 0.5ς ' ρ I ) ) 1 ς If each margn s gaussan and we use a gaussan coula then we go back to the usual multvarate gaussan law. ulses) gves the Doler-Range ma fgure 6. We can see on the frst art of the burst a clutter wth a negatve seed corresondng to ran also resent at the end) Doler Segmentaton We show n ths examle the segmentaton obtaned by usng only nformaton σ µ ) so we remove the nformaton of shae gven by ν. A 3 class model s selected by an nformaton crteron BIC) whch corresonds to absence of ran clutter and low reflectvty or hgh reflectvty classes) and a thrd class wth resence of ran clutter fgure 6 a)). If we wrte a j j) 4.. Estmaton As for HMC IN models the HMC models can be estmated by maxmum lkelhood through the use of a SEM algorthm. We need to adat the maxmzaton at each ste to the dffculty due the resence of the coula term n the lkelhood. The roblem comes from the resence of the c.d.f. of the margns n the exresson of the densty. The consequence s that the arameters of the margns are ntrcate as the coula ones. So the maxmzaton of the comlete lkelhood M-ste) becomes very dffcult and tme greedy. We use then a two ste trck to do ths maxmzaton whch s well used for the comutaton of maxmum lkelhood estmator [17]) : the dea s to fnd the maxmum by determnng the otmum for a frst set of arameter and then the other. L= L )θ ) η for the comlete loglkelhood wth hdden rocess a j the transton robablty of the θ the arameter of the class and j η j the arameter of the coula of transton ). We gve the synoss of the generalzed SEM: 1. By K-means comute an ntal segmentaton and ntal arameters aˆ 0) ˆ θ 0) ˆ η0. j. Comlete data thanks to the osteror law and comute the MLE of a j and θ as f we had a HMC IN. 3. Comute ηˆ j by maxmzng L= L a j )θ ) ηj ) Then terate ste and 3 untl stablzaton of the sequence a ˆm) ˆ θ m) ˆ ηm) m and take the one that j maxmzes L. It s mortant to note that the extensve use of the osteror robabltes durng SEM or EM) makes crucal the exstence of analytcal formulas and fast rocedures. 5. Exerments on real data We show an examle of Doler and Polarmetrc segmentaton of data collected from an X band radar and comosed of a sngle burst wth 16 ulses) of 56 range cells long. A Doler analyss a FFT made wth 18 a) b) Fgure 6. Doler Range ma of the burst studed. The undmensonal ma are shown below accordng to the nformaton used. We may also note that another segmentaton usng only the sectrum shae ν stll gve 3 classes but corresondng to AR0) AR1) and AR) : the ran clutter has been slt n an area wth ran only one roer frequency) and ran + sol sectrum wth roer frequences) see fgure 6 b). 5.. Polarmetrc segmentaton The same data have also a olarmetrc content and we can make an analyss of the olarmetrc envronment. We segment the data by usng the shae of the olarzaton ellse fgure 7). We show n fgure 7 the scatter lot of the ellse reresented on the shere for better vsualzaton the shere s rojected on a lane thanks to Atoff-Hammer rojecton).

6 Fgure 7. Scatter lot of the olarzaton state. The undmensonal olarmetrc ma s shown below. We obtan 3 classes by BIC crteron) : the frst one s mean-olarzed n rectlnear horzontal state oston 00) on the lane) and well concentrated t corresonds to the ran clutter). The second one s also centered on 00) but wth a hgh dserson scattered houses and greenery). The thrd class s small and has an erratc olarmetrc resonse corresondng to hgh sloes. 6. Concluson In ths aer we recall the am of Doler and Polarmetrc segmentaton of the Radar envronment. We show how the comlexty of the radar sgnal can be decomosed n order to be roerly rocessed n the classcal frame of statstcal mage segmentaton. By modelng the Radar sgnal by an autoregressve rocess we roose an orgnal method for obtanng statstcal ma of the clutter accordng to the reflectvty and entroc content of the clutter and also accordng to ts sectral shae. In the same manner we show that the clutter can be segmented accordng to the shae of the Polarzaton ellse. We want to stress on the orgnalty of the Markov model used for segmentaton that enables to ntroduce deendences between observatons that are neglected by the classcal HMC model. The deendence between the observatons are tuned on one hand by the transton robabltes between areas and on the second hand by a qute new quantty : the coulas. Coulas are not known n mage and sgnal rocessng and ther use n our work s a novelty n ths doman. We roose then an algorthm of estmaton based on SEM that s thought to be fast and effcent n a way to be used n real-tme analyss of the radar envronment. 7. Aknowledgment 8. References [1] S. Haykn : Adatve flter theory 3 ème édton Prentce Hall Internatonal Edtons 000. [] S. Haykn : Adatve Radar detecton and estmaton Edted S. haykn and A. Renhardt. [3] F. Barbaresco «Calcul des varatons et analyse sectrale : équatons de Fourer et de Burgers our modèles autorégressfs régularsés» Revue Tratement du Sgnal Vol [4] S. Haykn R. Bakker B. Curre : Uncoverng nonlnear dynamcs : the case study of sea clutter. IEEE on Sgnal Processng 00. [5] K. Marda P. Ju : Drectonal Statstcs Wley Seres n Probablty and Statstcs [6] Mac Lachlan and Peel : Fnte mxture models Wley seres n Probablty and Statstcs 000. [7] C.Aubry : Polarmétre Radar Bases théorques revue technque Thomson CSF vol 0.1 n décembre [8] L. R. Rabner: A tutoral for Hdden Markov models and selected alcatons n seech recognton. Proceedngs of the IEEE [9] Mac Lachlan and Krshnan : The EM algorthm and Extensons Wley seres n Probablty and Statstcs [10] G. Celeux J. Debolt : The SEM algorthm : a robablstc teacher algorthm derved from the EM algorthm for the mxture roblem Com. Stats. Quarterly [11] N. Brunel W. Peczynsk : Unsuervsed Sgnal Restoraton usng Hdden Markov Chans wth coulas submtted. [1] W. Peczynsk : "Parwse Markov chans" IEEE Transactons on Pattern Analyss and Machne Intellgence Vol. 5 No [13] W. Peczynsk : Arbres de Markov Coule Comtes Rendus de l'académe des Scences - Mathématque Sére I Vol. 335 Issue [14] F. Desbouvres et W. Peczynsk : Modèles de Markov Trlet et fltrage de Kalman Comtes Rendus de l'académe des Scences - Mathématque Sére I Vol. 336 Issue [15] N. Brunel and W. Peczynsk Unsuervsed sgnal restoraton usng Coulas and Parwse Markov chans IEEE Worksho on Statstcal Sgnal Processng SSP 003) Sant Lous Mssour Setember 8-October [16] R. Nelsen : An ntroducton to coulas Lectures Notres n Statstcs Srnger-Verlag 000. [17] H. Joe : Multvarate models and deendence concets Monograhs on statstcs and aled robablty 73 Chaman and Hall London. The authors acknowledge Prof. W. Peczynsk laboratore CITI Insttut Natonal des Télécommuncatons INT) for hs advce. Ths PhD study s funded by DGA/DSP.