A Kernel Partcle Flter Algorthm for Jont Tracng and Clafcaton Yunfe Guo Donglang Peng Inttute of Informaton and Control Automaton School Hangzhou Danz Unverty Hangzhou Chna gyf@hdueducn Huaje Chen Ane Xue Inttute of Informaton and Control Automaton School Hangzhou Danz Unverty Hangzhou Chna axue@hdueducn Abtract For radar urvellance ytem target tracng and clafcaton are two major functon A ernel partcle flter approach wth mproved nformaton mutual feedbac preented for jont tracng and clafcaton Delay Doppler and Radar cro ecton meaurement are ued to etmate target tate and cla repectvely It nvoe the ernel partcle flter and pont model for nonlnear etmaton wth le amount of calculaton Mutual feedbac tructure ued to mprove the clafcaton probablty and etmaton accuracy Smulaton reult how the effcency of the propoed method Keyword-Jont tracng and clafcaton; mutual feedbac; ernel partcle flter; pont model; radar urvellance I ITRODUCTIO Target tracng and clafcaton are two crtcal problem n radar urvellance ytem Tracng etmate the target dynamc tate and clafcaton deal wth determnng target dentfcaton Compared wth treatng them eparately approache of jont tracng and clafcaton (JTC) attract more and more attenton recently [] M I Mller et al frt condered target tracng and target clafcaton n a ngle framewor [] A D Lanterman olved JTC problem n btatc pave radar ytem frtly n 999 [3] S Challa and G W Pulford etablhed the unfed framewor of JTC by ntroducng mutual feedbac [4] The flght envelope etmated by tracer wa ued to auxlary recognton and the output of clafer helpful to target modelng Y Boer et al too JTC a hybrd tate etmaton problem and propoed a partcle flter oluton [5] Further W Me gve a more ntegrated multaneou tracng and clafcaton method [6] Thee algorthm etmate target tate and dentfcaton by nematcal feature only wthout any electro-magnetc or hape nformaton The clafcaton performance of thee method wa retrcted by maneuverng ablty and actual maneuverng behavor of dfferent target [7-8] A more general oluton of JTC cheme baed on Bayean jont decon and etmaton ha been dcued by X R L et al [9] In addton Rgd-body model baed JTC technque [] can mprove clafcaton ablty potentally by etmatng target poe angular yet need pecfc enor and computatonal complexty [6] In th paper a ernel partcle flter baed algorthm wth nformaton mutual feedbac propoed for JTC problem It ue Kernel functon but not Drac functon to contnuouly approxmate the probablty denty functon [-4] Compared wth the tradtonal PF-JTC method fewer partcle are need A mlar urvellance cenaro condered a n [] where a maneuverng target detected by three pave tranmtter and one recever on the ground Delay and Doppler and Radar cro ecton meaurement are ued to etmate target tate and cla repectvely To reduce computatonal complexty a pont model but not rgd model ued and a ernel partcle flter nvoed for nonlnear flterng whch need fewer partcle to approxmate the poter probablty denty The cla probablty e the clafer output ued not only to adjut fuon weght but alo to reagn the partcle number of dfferent cla whch lead to a more reaonable tracng trategy and a better JTC reult The paper organzed a follow Secton II gve a decrpton the JTC problem and a general Bayean oluton The ernel partcle flter algorthm wth mutual feedbac preented n Secton III Smulaton reult and analy are gven n Secton IV and concluon gven n Secton V II PROBLEM DESCRIPTIO A JTC Modelng Jont tracng and clafcaton am to etmate target dynamc tate and cla multaneouly Due to target tate belong to the contnuou doman whle target cla and target moton mode belong to the dcrete doman t eental a hybrd tate etmaton problem [7] X p p ' denote the target tate n 3-dmenonal Let [ ] pace at tme p [ x y ξ ] poton and p x y ξ repreent velocty The target dynamc model gven a follow X f X c m + j w c mj () ( ) ( ) Th reearch wa funded by the project of FSC (6956 and 6744) 386
Where f a nonlnear functon and c one of nown target cla I m j repreent one of probable moton mode wthn the th cla j J ( ) m j Marov wth a pror nown tranton matrx { π j π j p( m )} mj Π w proce noe wth nown dtrbuton p w A general target meaurement model gven a z h X + v () Where h ( ) repreent nonlnear map from tate to meaurement v meaurement noe wth nown dtrbuton p v and ndependent wth proce noe w From () and () we can ee that X + related to not only lat tate X but alo the target cla c and the moton mode wthn that cla m j Defne Z { z z z } z only depend on X a meaurement et up to and ncludng tme the ultmate goal of JTC to etmate multaneouly the condtonal dente ( ) p ( c Z) or ther mean E( X Z) and E ( c Z) p X Z and They anwer two crtcal queton for urvellance e where and what the target B Bayean General Soluton Accordng to Bayean theorem ( ) ( ) can be expreed a follow p( X Z) p( X c Z) p( X c mj Z) j p( c Z) p( c mj Z) p c Z j j X ( ) p X c m Z dx j p X Z and Obvouly the problem converted to olve a jont probablty denty functon Suppoe that the jont poteror dente at tme p( X c m Z ) are avalable for all I and J ( ) p ( X c mj Z) can be obtaned recurvely by two tage: predcton and update ) Predcton Stage: p X c m Z ( j ) J() X ( ) (3) (4) p( X c mj X c m ) (5) p X c m Z dx J() X ( j ) p X c m X c m ( ) ( ) p c m X c m p X c m Z dx j J() X ( ) ( ) ( ) J() X p X X c m pm m px cm Z dx j ( ) p X X c m j p( X c m Z ) dx Where p( X X c m ) determned by () ) Update Stage: π tate trantonal denty p( X c mj Z) p( z X c mj Z ) p( X c mj Z ) p( z Z ) p( z X) p( X c mj Z ) p( z Z ) Where ( ) (6) p z Z a normalzed contant In mot practcal cae the optmal Bayean oluton wll not be poble to mplement [9] due to ntegraton n (5)-(6) In Secton III a ernel partcle flter baed uboptmal oluton wth mutual feedbac gven III KEREL PARTICLE FILTER JTC ALGORITHM Gven the pror probablty denty functon theoretcally we can get the optmal Bayean etmaton va equaton (5)- (6) In practce for the nonlnear non-gauan cae the optmal Bayean etmator cannot be approached n mot cae (manly due to the ntegral factor n (5) and (6)) A varety of uboptmal method have been propoed uch a the Extended Kalman flter the Uncented Kalman flter and a varety of partcle flter (PF) etc [5][7] PF a technque for mplementng a recurve Bayean flter by Monte Carlo mulaton It repreent the poteror probablty denty functon by a et of weghted random ample/partcle and can approach the true probablty denty functon theoretcally a the partcle number grow nfntely PF ha a uperor performance to other at the cot of expenve computatonal complexty There are lot of dervatve for t uch a the equental mportance amplng algorthm the amplng mportance reamplng algorthm the Auxlary Partcle flter the Regularzed Partcle flter the Multple Model Partcle Flter [][] etc 387
All the above PF mut face a problem of cot computatonal afford To approxmate the true probablty denty functon wth hgh precon the PF need more partcle However there are many partcle located n the nondomnant mode regon and they contrbute lttle to the probablty denty functon and the eventual etmaton Whle the reamplng tep mae thee partcle to be coped le tme than that n the domnant mode regon the partcle number tll enormou If the domnant mode of the probablty denty functon are nown a pror or can be etmated we can chooe more partcle n thee regon and fewer partcle n other regon Aumng the requred partcle number n the domnant mode regon (the major dtrbuton to the etmaton) determned to get an expected etmaton precon the partcle number n the whole dtrbuton regon wll decreae n the cae that the mode can be etmated The ernel partcle flter (KPF) uch a flter It change the Drac-Delta approxmaton of the poteror probablty denty functon to the ernel approxmaton uch that the reamplng tep changed nto mulaton from a contnuou probablty denty functon but not a dcrete probablty denty functon In addton t nvoe the mean-hft procedure to ee domnant mode and move partcle toward thee mode whle tll mantanng a far ample from the probablty denty functon [3] A Kernel Partcle Flter Let Y [ X c mj ] denote the hybrd tate and t poter probablty denty functon py ( Z ) characterzed a the um of the ernel functon of the partcle () () ( ) α ( ) py Z K Y Y (7) () () K n ( Y Y ) Y α Where () () Y are the th partcle and correpondng weght the partcle number and K α () a ernel functon wth ernel bandwdth α α α αopt the optmal bandwdth opt under the aumpton that the poteror Gauan wth a unt covarance matrx and the ernel Gauan ernel [3-4] 4 n Y + αopt (8) ( ny + ) m Where n the hybrd tate dmenon and Y m the meaurement dmenon To atfy the unt covarance condton there often a whtenng tep on each partcle The emprcal covarance matrx of the partcle gven by ' () () () () ( ) X changed to A X where A( A)' C (9) C X X X X () The partcle to acheve a unt covarance ow we gve the ernel partcle flter tep for JTC algorthm ) Predcton Step (a) reamplng: () Generate I { } wth probablty p( I ) and draw τ from the ernel K The new partcle Y () gven by () ( I) Y Y + αaτ () (b)predcton: Generate the predcton partcle et accordng to the equaton () () () () Y f( Y ) + + w () Where w () drawn from the probablty denty functon p w ) Update Step (a) weght calculaton: Calculate the weght ung z + () ( () ) ( ( () + p z + Y + pv z + h + Y + ) ) () ormalze the weght () () + + (3) () Thu we can get the new ntal weghted ernel partcle et{ () () } X + + at tep + Dfferent wth the general PF whch output the etmaton at th tage the KPF frt etmate the gradent of the probablty denty functon and then move the partcle along the gradent drecton toward the domnant mode lat gve the ytem output (b) Mean hft procedure: The ample mean of the partcle () gven by () ( + ) my α j j + Y + ( ) H Y Y Y ( ) ( j) ( j) ( j) + + + + ( ) H Y Y α () ( j) ( j) + + + (4) Where H another ernel functon If the followng equaton hold h( r) c' ( r) r [ ) c > (5) () () The mean hft vector my ( + ) Y+ n the gradent drecton of the poteror pdf at tep + h are the correpondng profle of the ernel H and K K x x (6) ( ) ( ) ( ) ( ) H x h x (7) It proved that f a Gauan ernel K ued n the denty etmaton the mean hft vector can ue the ame ernel whch atfe (4) [3] The mean hft procedure move each partcle to t ample mean For the partcle poton changed after a mean hft procedure the new partcle do not follow the poteror dtrbuton any more [] The weght need recomputed 388
Denote the partcle et after the l th mean hft procedure at tep () + a { l} Y the new weght of partcle () + l can be recalculated a follow ( Z) + l( Y+ l) ( + ) ( + ) p z Y p Y Y () ( j) ( + l + l) Y + () () ( j) ( j) () l l py () + l j + l () q j K Y Y α (8) It renormalzed to () + l mlar to equaton (3) After the I th teratve ( I the degned maxmum teratve number) the lat weghted partcle et at tep + gven a () () Y Y (9) + + I () () () + + I Thu we can get the lat weghted partcle et { () () Y+ + } The mean hft procedure mae the new partcle concentrate more on the domnant mode In mplementaton a maxmum teratve number I of the mean hft need to be degned In our mulaton I enough to move the partcle to the domnant mode regon For the partcle have the ablty to automatcally move to the domnant mode regon of the poteror probablty denty functon the requred partcle number fewer than that of the general PF to get the ame precon (c) output From new weghted partcle et { () () } Y+ + poteror probablty denty functon py ( Z ) p( X c mj Z ) we can get the + + ealy e + + + Then we can get the poteror tate etmaton clafcaton probablty though ntegraton The jont poteror probablty of the j th moton model wthn the th cla p( c mj + Z+ ) p X c m Z dx () ( ) j j + q + j + + + and the poteror probablty of the th cla p c Z ( + ) J() p( c mj + Z+ ) j J() j j q j + The tate etmaton at tme + I J() j + + j + j + j q ( ) () E X Z X (3) B Mutual Feedbac In prevou wor the nformaton mutual feedbac between tracer and clafer mple and fxed For example the cla probablty a feedbac nformaton of clafer only play a role of weght for tracer In fact hgher probablty cla and moton mode deerve to be agned more partcle whch wll lead to a better performance For the j th moton model wthn the th cla the reagned partcle number proportonal to t weght Reample j + tme from { } j Y and obtan Y j + j j q j + Where j + j + ( j + Z+ ) ( ) q j + (4) p c m p c m Z (5) j + j + + Go to ) predcton tep ote that the reagned partcle number for the th cla + J() ( ) p c z (6) + j + : + j and the total number of partcle nvarable I + (7) By the mproved nformaton mutual feedbac more partcle are ued for tracng hgher probablty cla whch helpful to get more relable clafcaton reult IV SIMULATIO In the mulaton we tae a mlar urvellance cenaro a n [] where a maneuverng target detected by three FM pave tranmtter and one recever on the ground The recever located at the orgn and tranmtter are located at A (9m -m m) B (m 5m m) and C (-35m m m) Three tranmtter utlze FM gnal at the ame frequency MHz and hence the wavelength 3m Each tranmtter-recever par can provde the delay Doppler and RCS nformaton of the target Delay and Doppler data are ued to etmate the target dynamc tate whle RCS data provded for clafcaton The n th meaurement zn [ τn dn σn ] ( n 3 ) contan three dcrete component τ the gnal tme delay whch relatve to the poton n of target and correpondng enor and can be calculated a follow p pt n + p pr τ n (8) υc Where p ptn and p r repreent target poton the n th tranmtter poton and recever poton repectvely υ c 389
lght velocty d the Doppler hft whch relatve to the relatve n velocty between the target and tranmtter/recever It can be repreented a follow p p tn p pr p dn + (9) λ n p p tn p r p where p the target velocty and λ n the n th tranmtter wavelength σ n the target RCS whch depend on many varable (radar parameter and target parameter) For a pecal urvellance cenaro t relatve to the target cla target poe and relatve poton to enor In mulaton for mplcty we aume that the target trajectory and target poe n each mulaton run the ame; hence the RCS relatve to target cla only For each mulaton run target cla maybe Target (mlar Cone hape) or Target (mlar Rectangular hape) Ther CAD rgd model and deal RCS value by Fnte dfference tme doman method (FDTD) are degned and calculated wth a oft named FEKO Two target CAD model and correpondng RCS graph are gven n fgure (ncdent elevaton angle and catterng elevaton angle equal to 9 horzontal polarzaton) For mplcty the meaurement RCS aumed to normal dtrbuton around deal value provded by fg (a) Target model (c) Target RCS (b) Target model (d) Target RCS Fg Two cla target CAD model and RCS (ncdent elevaton angle and catterng elevaton angle equal to 9 horzontal polarzaton) Whatever cla the target belong to t tae a maneuverng flght at fxed alttude (m) It ntal poton (-m m m) From to t head due eat wth ntal velocty (3m/ ) then t tae a clocwe turn wth angular rate 57rad/ untl after a contant velocty flght t tae an antclocwe turn wth angular rate 34rad/ untl 4 at lat t head due north wth near contant velocty The total tep are 5 The delay and Doppler meaurement of three tranmtterrecever par are gven n Fgure The meaurement noe aumed to be zero mean Gauan proce and wth correpondng tandard devaton σ delay e-6 σ Doppler 5Hz delay() delay() delay() 3 x 4 TR 3 4 tme() 3 x TR 4 3 4 tme() 3 x TR3 4 5 3 4 tme() doppler(hz) doppler(hz) 5 TR 5 3 4 tme() TR 3 4 tme() TR3 5 doppler(hz) 5 3 4 tme() Fg Delay and Doppler meaurement The parameter of algorthm are agned a follow Target modeled a a fghter whch aumed to have two moton mode One near contant velocty mode (CV) wth proce noe covarance matrx Q dag{ σx σvx σ y σvy } whereσ x σ y m and σvx σvy m/ The other near contant turn mode (CT) wth proce noe covarance matrx Q dag{ σx σv x σ y σv y σ} where σx σ y m σvx σvy m/ and σ rad/ Target modeled a a arlner a non-maneuverng or wea maneuverng target whch aumed to have only one moton mode e CV wth the ame parameter of Target CV mode The ernel bandwdth n the KPF α 3 and the ernel partcle number The mean hft teratve number I To llutrate the performance of propoed algorthm we compare t wth other three method The Monte Carlo mulaton number Fgure 3 compare the root mean quare error (RMSE) of range etmaton wth four method Table how the maxmum range etmaton error elaped tme and cla probablty Table performance comparon of four method PF-MF () PF-MF () KPF-MF () KPF-IMF () Range RMSE at 4 (m) 5 6 7 45 Elaped Tme (ec) 48 6 639 68 Cla probablty 75% 89% 86% 9% 39
RMSE of range etmaton(m) 5 5 5 Performance comparon of four JTC algorthm PFMFnum PFMFnum KPFMFnum KPFIMFnum 3 4 5 tme () Fg3 Range etmaton RMSE of four method It how that the range etmaton RMSE of each algorthm around 4 tae the maxmum value (note that the etmated error before can be reduced by a better tate ntal value) It correpond to a tronger target maneuverng behavor For the tradtonal partcle flter JTC algorthm wth mutual feedbac (PF-MF) t etmaton accuracy ncreae wth more partcle Wth the ame partcle the ernel partcle flter JTC algorthm wth mutual feedbac (KPF-MF) mprove the etmaton precon greatly at the cot of le tme The man reaon that KPF ue ernel functon to contnuouly approxmate the poteror probablty denty functon whle PF ue Drac functon for dcrete approxmaton Meanwhle Table how that wth more nformaton mutual feedbac between tracer and clafer the mproved mutual feedbac (KPF-IMF) can further mprove tracng and cla probablty COCLUSIOS In th paper a ernel partcle flter wth mproved mutual feedbac propoed for the jont tracng and clafcaton problem It ue a ernel functon but not a Drac functon to contnuouly approxmate the probablty denty functon Fewer partcle are need n the KPF-JTC algorthm compared wth the tradtonal PF-JTC method For further reducton of the computatonal complexty the target regarded a a pont model but not a a rgd model A reaonable aumpton on the target RCS alo taen Compared wth the tradtonal mutual feedbac trategy the propoed method ue the feedbac nformaton completely and adaptvely whch contrbute to mprove the etmaton accuracy and cla probablty Smulaton comparon how the effectvene and effcency of the propoed algorthm REFERECES [] Angelova D Mhaylova L Jont target tracng and clafcaton wth partcle flterng and mxture Kalman flterng ung nematc radar nformaton Dgtal Sgnal Proceng Vol 6 6 8-4 [] Mller M I Srvatava A and Grenander U Condtonal-mean etmaton va jump-dffuon proce n multple target tracng/recognton IEEE Tran on Sgnal Proceng Vol 43 o 995 678-69 [3] Lanterman A D Tracng and recognton of arborne target va commercal televon and FM rado gnal Proc of SPIE Acquton Tracng and Pontng Vol 369 999 89-98 [4] Challa S and Pulford G W Jont tracng and clafcaton ung radar and ESM enor IEEE Tran on Aeropace and Electronc Sytem Vol 37 o 3 39-55 [5] Boer Y and Dreen H Hybrd tate etmaton: a target tracng applcaton Automatca Vol 38 53-58 [6] We Me Gan-Ln Shan and X Rong L Smultaneou tracng and clafcaton: a modularzed cheme IEEE Tran on Aeropace and Electronc Sytem Vol 43 o 7 58-599 [7] Rtc B Gordon and Beell A On target clafcaton ung nematc data Informaton Fuon Vol 5 4 5- [8] Blom H A P and Bloem E A Partcle flterng for tochatc hybrd ytem Proc of the 43 rd IEEE Conference on Decon and Control 4 4-7 [9] X Rong L Mng Yang J feng Ru Jont Tracng and Clafcaton Baed on Baye Jont Decon and Etmaton th Informaton Fuon Internatonal conference 7-8 [] Herman S and Mouln P A partcle flterng approach to FM-band pave radar tracng and automatc target recognton Proc of the IEEE Aeropace Conference Vol 4 789-88 [] C Chang R Anar Kernel partcle flter for vual tracng IEEE Tran IEEE Sgnal Proceng Letter vol o3 5 4-45 [] Yunfe GuoAne Xue and Donglang Peng A recurve algorthm for bearng-only tracng wth gnal tme delay Sgnal Proceng Vol88 o6 8 539-55 [3] Y Z Cheng: Mean Shft Mode eeng and Cluterng: IEEE Tran on PAMI 995 79-799 [4] D Comancu V Pameh P Meer: Kernel-baed object tracng IEEE Tran PAMI 3 564-577 39