1 An etended target tracng odel wth ultle rando atrces and unfed neatcs Karl Granströ arxv:146.13v1 [cs.sy] 9 Jun 14 Abstract Ths aer resents a odel for tracng of etended targets, where each target s reresented b a gven nuber of elltc subobjects. A gaa Gaussan nverse Wshart leentaton s derved, and necessar aroatons are suggested to allevate the data assocaton colet. A sulaton stud shows the erts of the odel coared to revous wor on the toc. Inde Ters Target tracng, etended target, grou target, easureent rate, rando atr, gaa dstrbuton, Gaussan dstrbuton, nverse Wshart dstrbuton. I. INTRODUCTION Target tracng can be defned as the rocessng of a sequence of easureents obtaned fro a target n order to antan an estate of the target s current state. In ths contet a ont target s defned as a target whch s assued to gve rse to at ost one easureent er te ste. Wth odern and ore accurate sensors the target a occu ultle resoluton cells of the sensor, thus otentall gvng rse to ore than one easureent er te ste. An etended target s defned as a target that otentall gves rse to ore than one easureent er te ste. Eales of etended target tracng nclude vehcle tracng usng autootve radar and edestran tracng usng laser range sensors. Closel related to etended target s grou target, defned as a cluster of ont targets whch cannot be traced ndvduall, but has to be treated as a sngle object. In ont target tracng the estated state tcall corresonds to the targets oston and ts neatcs veloct, headng, etc. In etended target tracng the ultle easureents ae t ossble to estate also the target s etenson n the easureent doan,.e. to estate the shae, the sze and the orentaton of the target. To estate the target s etenson requres a easureent odel that relates the ultle easureents to the states that govern the etenson. Satal dstrbuton odels n etended target tracng aeared n [1], []. Under ths odel each etended target easureent s a rando sale fro a robablt dstrbuton that s deendent on the etended target state. A nuber of dfferent etended target odels have been resented, where the targets are odeled as, e.g., stcs [] [4], crcles [], ellses [6] [13], rectangles [8], or general shaes [14] [17]. In ths aer we consder state estaton for etended targets whose etensons cannot be aroated b a sle geoetrc shae such as an ellse or a rectangle. The etended Karl Granströ s wth the Deartent of Electrcal Engneerng, Dvson of Autoatc Control, Lnöng Unverst, Lnöng, SE-81 83, Sweden, e-al: arl@s.lu.se. d φ d 3 Fg. 1. Eales n D of etended/grou targets that are reresented b elltc subobjects. Nether one of the eales has a shae that can be descrbed b a sngle ellse. Left: The overall oston of the etended target s, and concdes wth one of the subobject s oston. The ostons of reanng subobjects are gven b offsets d fro the overall oston. Rght: The overall oston does not concde wth ether subobject s oston. target s odeled as a collecton of elltcal subobjects, see Fg. 1, and the ostons and etensons of the subobjects are Gaussan nverse Wshart dstrbuted. The scoe of the aer s lted b the assutons that a there s eactl one target resent; b there are no clutter easureents; and c the nuber of subobjects s constant and nown. To handle ultle targets and clutter, the resented wor can be ntegrated nto a ultle target fraewor, e.g. an etended target PHD/CPHD flter [18] [4]. Estatng the nuber of subobjects s left for future wor. II. PREVIOUS WORK AND PAPER CONTRIBUTIONS A. Overvew of rando atr fraewor Notaton s gven n Table I. In the rando atr etended target odel, orgnall roosed b Koch n [6], the etended target state s the cobnaton of a neatc state vector and an etenson atr X. The vector reresents the target s oston and neatcs, and the atr X reresents the target s sze and shae,.e. ts satal etenson. The atr X s odeled as beng setrc and ostve defnte, whch les that the target shae s aroated b an ellse. The ellse shae a see ltng, however the odel s alcable to an real scenaros, e.g. edestran tracng [1]. In [6] the target state, the target generated easureents, and the transton denst, are odeled as, X Z = X, Z X Z 1a =N ;, P X IW d X ; v, V, 1b z, X =N z ; H, X. 1c 1, X 1, X = 1 X 1, X 1 X, 1d
TABLE I NOTATION R n s the set of real colun vectors of length n, S n s the set of setrc ostve defnte n n atrces, S n s the set of setrc ostve se-defnte n n atrces, and N s the set of non-negatve ntegers. I d s a d d ee atr, and d e s a d e all-zero atr. s absolute value, s Eucldean nor, and F s Frobenus nor. PS n; γ denotes a Posson robablt ass functon f defned of the nteger n N wth rate araeter γ >, PS n; γ =γ n e γ n! 1. G γ ; α, β denotes a gaa robablt denst functon df defned over the scalar γ > wth scalar shae araeter α > and scalar nverse scale araeter β >, G γ ; α, β = β α Γα 1 γ α 1 e βγ, where Γ s the gaa functon. N ;, P denotes a ult-varate Gaussan df defned over the vector R n wth ean vector R n, and covarance atr P S n, N ;, P = e 1 T P 1 π n. detp 1 where det s the atr deternant functon. IW d X ; v, V denotes an nverse Wshart df defned over the atr X S d wth scalar degrees of freedo v > d and araeter atr V S d, [, Defnton 3.4.1] IW d X ; v, V = v d 1 detv v d 1 detx v Γ d v d 1 etr 1 X 1 V, where etr = e Tr s eonental of the atr trace, and Γ d s the ultvarate gaa functon. The ultvarate gaa functon can be eressed as a roduct of ordnar gaa functons, see [, Theore 1.4.1]. W d X ; w, W denotes a Wshart df defned over the atr X S d wth scalar degrees of freedo w d and araeter atr W S d, [, Defnton 3..1] W d X ; w, W = wd detx w d 1 Γ w n etr d detw 1 W 1 X. In ths odel the neatc state conssts of a satal state coonent r target oston, and dervatves of r tcall veloct and acceleraton [6]. Non-lnear dnacs, such as turn-rate, are not ncluded n the neatc vector. The easureent udate s lnear [6], a dervaton of the redcted lelhood can be found n [1, Aend A]. A lnear Gaussan transton denst s used for the neatc state, and for the etenson a sle heurstc s used n whch the eected value s et constant and the varance s ncreased [6]. The etenson transton denst X 1 X n 1d assues ndeendence of the neatc state, whch does not account for, e.g., rotatons durng a turnng aneuver [6]. The rando atr odel 1 was odfed n [7], [8], where the target state, the target generated easureents, and the transton denst, are odeled as, X Z Z X Z a =N ;, P IW d X ; v, V, b z, X =N z ; H, zx R, 1, X 1, X = 1 X 1 X, c d where z s a scalng factor and R s easureent nose. Note the assued ndeendence between the neatc state and X n a, an assuton that cannot be full theoretcall justfed 1. Further the easureent udate s no longer lnear and ust be aroated, see [8] for detals. However, there are consderable ractcal advantages to the odel [8]. Ths odel allows for a ore general class of neatc state vectors, e.g. ncludng non-lnear dnacs such as headng and turn-rate, and the Gaussan covarance s no longer ntertwned wth the etenson atr. Ths easureent odel s better when the sze of the etenson and the sze of the sensor nose are wthn the sae order of agntude [8]. The assued ndeendence between and X s allevated n ractce b the easureent udate whch rovdes for the necessar nterdeendence between neatcs and etenson estaton [8]. An alternatve easureent udate for the easureent odel c, based on varatonal Baes aroaton, s gven n [9]. The neatcs transton denst 1 n d s assued ndeendent of the etenson. Ths neglects factors such as wnd resstance, whch can be odeled as a functon of the etenson X, however the assuton s necessar to retan the functonal for a n a Baesan recurson. A lnear Gaussan transton denst s used for the neatc state, and a heurstc transton slar to the one n [6] s used for the etenson. An alternatve to the heurstc etenson redctons fro [6], [8] s to use a Wshart transton denst [6], see also [1], [6], [3]. In [1] transforatons of the etenson are allowed va araeter atrces A, X 1 X =W d X 1 ; δ, A X A T. 3 The araeter atrces corresond to, e.g., rotaton atrces. Ths s generalzed n [6] to allow for transforaton atrces M that are functons of the neatc state, X 1, X =W d X 1 ; n, M X M T n, whch eans that the rotaton angle can be couled to, e.g., the turn-rate and estated onlne. The transton denst 4 relaes the assuton ade n 1d, d, and 3 that the etenson s te evoluton s ndeendent of the neatc state. A coarson of the odels 1d, d, 3 and 4 s gven n [6], where 4 s shown to gve lower errors at lower coutatonal colet. In addton to the transton denst 3, a easureent odel s also suggested n [1], 4 z, X =N z ; H, B X B T where B s a araeter atr. Under the assuton X ˆX 1 = E[X Z 1 ] the odel ncororates c aroatel when B = z ˆX 1/ 1 R ˆX 1/ 1. The rando atr odel has been ntegrated nto the Probablstc Mult-Hothess Tracng PMHT fraewor 1 Condtoned on a set of easureents Z the neatc state and etenson state X are necessarl deendent.
3 [31], see [3] [34]. The odel has also been used n PHD- and CPHD-flters for ultle etended target tracng n clutter, see [1], []. In [16] a sngle etended target odel s gven where the etended target s a cobnaton of ultle subobjects wth neatc state vectors and etenson atrces X. Each subobject s odeled usng 1b, N ;, P X IW d X ; v, V. 6 Usng ultle nstances of a sler shae allevates the ltatons osed b the led elltc target shae, and also retans, on a subobject level, the slct of the rando atr odel [6], [8]. B. Paer contrbutons The ajor contrbutons n ths aer are: A new state sace odel where all subobjects s ostons are odeled as full deendent usng a sngle state vector, there are full unfed neatcs, and where the easureent rate and etenson of each subobject are ndvduall odeled. A dervaton of the redcton udate, the easureent udate, and the redcted lelhood. A coutatonall effcent gaa Gaussan nverse Wshart leentaton, ncludng an ntalzaton ethod that does not rel on an a ror nforaton about the target. A sle and effectve ethod that nzes the nuber of assocaton events that have to be consdered wthout relng on a redcted target estate. The ethod s caable of handlng artal occluson of the etended target,.e. one or a few of the subobjects are hdden fro sensor vew. The roosed etended target odel s valdated on sulated data fro realstc scenaros, and the results are coared to revous wor on the toc. III. PROPOSED MULTIPLE ELLIPSE MODEL A. Etended target state The etended target s ade u of a cobnaton of N s, d- densonal subobjects, where N s, s nown. Each subobject s descrbed b a oston R d, a easureent rate γ > and an etenson state X S d, where subnde refers to dscrete te ste t. The easureent rate governs how an easureents the subobject generates er te ste, and the etenson descrbes the sze and the shae of the subobject. Because etended targets n ost cases can be assued to be rgd bodes the subobjects have unfed dnacs, b whch we ean that all subobjects ove forward wth the sae veloct and the sae headng, turn wth the sae turn-rate, etc. The unfed dnacs are denoted c R nc, where c ncludes araeters for, e.g., veloct, acceleraton, headng and turn-rate. Note that c, n addton to araeters for unfed dnacs, also a nclude araeters for ndvdual subobject dnacs. Ths s useful for grou tracng, where the ndvdual targets n the grou a shft ther ostons wthn the grou. As the nuber of ellses grows, ther cobnaton can for nearl an gven shae. The ostons of the subobjects are 1 = and = d, =,..., N s,,.e. the ostons of subobjects =,..., N s, are offset b vectors d R d fro the frst subobject s oston R d. The frst subobject s also referred to as the an subobject, and the oston s referred to as the overall oston. The unfed neatcs are defned w.r.t. the overall oston. For lnear dnacs t does not atter whch subobject s denoted the frst subobject. However, ths s ortant for non-lnear dnacs, e.g. a turnng aneuver that causes the etended target to rotate. The ostons and dnacs of all subobjects are jontl descrbed b a neatc state R n, [ T T ] T = T c T.... 7 d d N s, Note that the oston of one of the subobjects ust concde wth the overall oston, because f = d for all then s not observable. For brevt the easureent rates, neatc state and etenson states are abbrevated as follows ξ = γ 1,..., γn s,,, X 1,..., XN s, 8a = Γ,, X 8b where ξ s referred to as the etended target state. Let Z be a set of target generated easureents Z = {z j }n z,, z j R d, j, and let Z be a sequence of easureent sets fro te t to te t. The dstrbuton of the etended target state ξ, condtoned on the hstor of easureent sets, s ξ Z = Γ, X, Z X, Z X Z, 9 The followng assutons are ade about the state ξ. Assuton 1: The easureent rates are ndeendent of the neatc vector and the rando atrces. Rear 1: In realt the easureent rate s often deendent on the dstance to the sensor.e. on the oston and on the sze of the target.e. the etenson. However, the dstrbuton of the easureent rate, condtoned on the neatc and etenson states, s unnown n an alcatons. The varance of the estated easureent rate s suffcent to odel the varatons over te [3], and the assuton ensures ractcal coutatonal tractablt. Assuton : The neatc vector s aroated as ndeendent of the rando atrces. Rear : Ths assuton s analogous to a, and slarl t neglects deendence between the subobjects ostons and etenson states an assuton that cannot be full theoretcall justfed. However, the assuton s necessar to enable the subobject ostons and unfed neatcs to be odeled as a sngle state vector. Just as n [6], [8], the te udate and easureent udate that are derved n ths aer wll rovde for the necessar ractcal nterdeendenc between the neatc state estate and the etenson state estate. Assuton 3: The easureent rates are ndeendent of each other. The sae holds for the rando atrces.
4 Rear 3: For setrc targets, e.g. arlanes, the subobjects that corresond to the wngs are not full ndeendent snce the wngs are setrc. Modelng how rando atrces, or easureent rates, deend on each other s dffcult n a general case, and the assuton slfes further analss greatl and ensures ractcal coutatonal tractablt. An ortant toc for future wor s to consder how setr can be used to rela ths assuton. For the rando atrces, the sae s assued n [16], see 6. Ths gves the followng etended target state dstrbuton, ξ Z = Z N s, =1 γ Z X Z. 1 Because of the an uncertantes nvolved the etended target state dstrbuton can also be reresented b a dstrbuton ture. In ths case the dstrbuton s ξ Z J = w l l ξ Z, 11 where J s the nuber of coonents n the ture, the weghts w l su to unt, and each l are of the for 1. The weghts can be nterreted as robabltes that the l:th ture coonent l s the true dstrbuton. B. Predcton For brevt and ncreased readablt, n ths secton we dro sub-nde and wrte sub-nde 1 as sub-nde,.e. we wrte ξ and ξ nstead of ξ and ξ 1. The state transton denst ξ ξ descrbes the te evoluton of the etended target state fro te t to te t. The transton denst decooses as follows ξ ξ = Γ, X, Γ X, Γ, X Γ,, X, 1 where we have used Baes rule and Marov-roert assutons. We now ae the followng assutons: Assuton 4: The easureent rates can be redcted ndeendentl of the neatc vector and the rando atrces. Rear 4: Ths assuton s also ade n [3]. In realt the easureent rates tcall deend on both the sze of the target, and ts dstance fro the sensor.e. deends on the oston, and ths assuton neglects such deendences. However, constructng a general odel for these deendences sees dffcult, and the assuton slfes further analss sgnfcantl. Furtherore, the varances of the estated easureent rates are suffcent to cature the varatons over te [3]. Assuton : The neatc vector and the rando atrces can be redcted ndeendentl of the ror easureent rates Γ. Rear : Ths assuton can be justfed n ost f not all ractcal cases, snce nether the neatcs nor the sze and shae of a target evolve dfferentl deendng on how an sensor detectons the target generates. Assuton 6: The neatc vector can be redcted ndeendentl of the rando atrces. Rear 6: Ths assuton s analogous to d, and slarl t neglects asects such as wnd resstance, whch can be odeled as deendent on the target s sze.e. deendent on the rando atrces. In ths wor the assuton s necessar for the redcted state dstrbuton to be of the sae functonal for as the osteror state dstrbuton, whch s a tcal requreent n Baesan estaton. Both the easureent udate and the redcton udate that are used n ths wor rovde for nterdeendenc between the redcted neatc vector and the rando atrces. For sngle ellse targets t s shown n [6] that estaton erforance s not negatvel affected b the assuton. Assuton 7: Each easureent rate can be redcted ndeendentl of the other easureent rates. The sae holds for the rando atrces. Rear 7: In ractce the easureent rates and etensons evolve over te deendent on the oston and neatcs. Because the rates and etensons are arts of the sae object.e. subobjects, the evoluton over te should be deendent. Under ths assuton ths deendence s not odeled, however the assuton s necessar for the redcted state dstrbuton to be of the sae functonal for as the osteror. For the easureent rates the sae oton odel s used for all estated rates, and the estated varances are suffcent to odel the te varatons. For the etenson states, the neatc state s used n the redcton, whch ntroduces suffcent deendence. Ths gves the followng transton denst for ξ, N s, ξ ξ = =1 γ γ X, X. 13 Wth a osteror 1 and a transton denst 13 the Baes redcted dstrbuton s ξ Z = ξ ξ ξ Z dξ N s, = γ γ Z dγ =1 N s, =1 γ X Z d N s, = Z =1 γ N s, =1 X 14a 14b, X X Z dx, Z Z d 14c We want the redcted dstrbuton to be of the sae functonal for as the osteror 1, however n general the followng
equalt does not hold, s, N =1 X = Z N s,, Z Z d =1 X Z. 1 Therefore, to obtan a redcted dstrbuton of the sae functonal for as the osteror 1, followng the dscusson n [6] we solve ndeendent ntegrals nstead, Z = Z d, 16a X Z = X, Z Z d. 16b Wth a ture dstrbuton, the redcted ture s the ture of the redcted coonents, ξ Z J = ξ ξ w l l ξ Z dξ 17a C. Correcton = J w l l ξ Z. 17b Let θ denote a ossble easureent-to-subobject assocaton event, and let Θ denote the set of all ossble assocaton events. For easureent generaton, we assue the followng: Assuton 8: The subobjects generate easureents ndeendentl of each other. For each subobject, the generated easureents are ndeendent. Each easureent s generated b eactl one subobject. Measureent orgn s unnown. Rear 8: These assutons are analogous to ultle target tracng, where t s tcall assued that each target generates easureents ndeendentl of the other targets, that the target generated easureents are ndeendent, that each easureent s generated b eactl one target, and that easureent orgn s unnown, see e.g. [36]. Under an assocaton event θ the easureent set Z can be arttoned nto N s, ossbl et subsets, N s, Z = =1 { Z θ,, Z θ, = z θ,,j } n θ, z,, 18 where the th subset Z θ, was generated b the th subobject. Condtoned on θ the easureent lelhood s N s, Z ξ, θ = =1 Z θ, γ,, X. 19a If the th subset s et.e. n θ, z, = the subset lelhood s sl the lelhood of an et set of easureents, γ,, X = P n θ, γ z, =. Z θ, If n θ, z, > the subobject lelhood s γ,, X Z θ, =n θ, z,!p n θ, z, γ nθ, z, z θ,,j Let the redcted ture dstrbuton be ξ Z 1 =, X. 1a J w l l ξ Z 1. B the total robablt theore the denst ξ Z s ξ Z = θ Θ ξ Z, θ P θ Z, 3 where ξ Z, θ s the Baes udated dstrbuton for the assocaton event θ, and P θ Z s the robablt of the assocaton event θ. As noted n [16, Assuton 3], wthout an ror nforaton the assocaton events can be assued to be equall lel,.e. P θ Z 1 = Θ 1. In ths case we have P θ Z Z θ, Z 1 P θ Z 1 = θ Θ Z θ, Z 1 P θ Z 1 4a J = wl l Z θ, Z 1 J θ Θ l =1 wl l Z θ, Z 1, 4b where we have agan used the total robablt theore n the second equalt. For the assocaton event θ the Baes udated dstrbuton s ξ Z, θ = Z ξ, θ ξ Z 1 Z ξ, θ ξ Z 1 dξ = a J wl l Z θ, Z 1 l ξ Z, θ J wl l Z θ, Z 1. b Cobnng 3, 4 and gves the osteror dstrbuton ξ Z = θ Θ w l θ = J w l θ l ξ Z, θ, 6 θ Θ w l l Z θ, Z 1 J l =1 wl l Z θ, Z 1, 7 where, followng the assuton that the subobjects generate easureents ndeendentl, for the redcted lelhood we have l Z θ, Z 1 = Z Z 1 = 1 Θ N s, =1 θ Θ l Z θ, Z 1. J w l l Z θ, Z 1, 8a 8b The redcted lelhood Z Z 1 s useful n a ultle target tracng scenaro, e.g. f the resented etended target odel s used n an leentaton of an etended target PHD or CPHD flter [18], [].
6 TABLE II MULTIPLE ELLIPSE PARAMETER INITIALIZATION Fg.. Intalzaton eale. True underlng etended target orange area, easureents red squares, and ntalzed estates blue ellses, ntalzed an subobject shown b thcer lne. IV. A GAMMA GAUSSIAN INVERSE WISHART IMPLEMENTATION In ths secton we gve a gaa Gaussan nverse Wshart leentaton of the ultle rando atr etended target odel outlned above. To handle dfferent tes of oton M dfferent oton odels are used. A. Etended target state dstrbuton In Glhol et al. s etended target odel [1], [] the nuber of easureents that each target generates s Posson dstrbuted wth a araeter ς ξ that s a functon of the etended target state. In ractce ths eans that the eected value of the nuber of easureents generated b a target wth state ξ s ς ξ. Here the Posson odel s adoted for each subobject and the araeters are gven b the easureent rates,.e. ς ξ = γ,. The gaa dstrbuton s the conjugate ror for the Posson dstrbuton s araeter, and the subobjects easureent rates are odeled as gaa dstrbuted. Followng the rando atr odel [6], [8] the subobjects rando atrces are odeled as nverse Wshart dstrbuted. The neatc vector s odeled as Gaussan dstrbuted. The etended target state dstrbuton s ξ Z =N N s, ;, P IW d X =GGIW ξ ; ζ, =1 G γ ; α, β ; v, V 9a 9b where GGIW ; s ntroduced for brevt and ζ denotes all of the nvolved araeters. As noted above, the etended target state dstrbuton s reresented b a dstrbuton ture ξ Z J = w l GGIW ξ ; ζ l, 3 where l wl = 1. When a new target aears the araeters ζ l of the estate ust be ntalzed. Table II gves a sle algorth where ths s erfored usng the frst set of easureents. The algorth ntalzes N hotheses n each oton ode. A sle ntalzaton eale s gven n Fgure a. In ths eale there s a sngle oton ode, and N = 4 hotheses are generated usng onl 8 easureents. 1: Inut: Set of easureents Z = {z } n =1. Desred nuber of ntal hotheses N. Intal neatcs c and ntal covarance P. Intal ean e and varance v for easureent rates. : Defne z c = 1 n n =1 z, r z = 1 a z z c. Set l =. 3: for = 1,..., N do 4: for = 1,..., M do : Set l = l 1 6: γ: α l, = e v, βl, = e v. 7: : P l = P, l = z c, c l = c, d l, = r z cos π N π 1 s 1 N sn π sn N π 1. s 1 N sn = r z4 Id v l, d. 8: X: v l, = d, V l, 9: end for 1: end for 11: Outut: ξ = J wl GGIW ξ ; ζ l B. Predcton where w l = 1 J. Wth a osteror dstrbuton of the for 3 the redcted dstrbuton s ξ 1 Z = M J =1 π, lw l GGIW ξ ; ζ,l 1, 31 where π, l s the robablt of a transton to the current ode fro the revous ode l that coonent l was n. 1 Measureent rates: For the easureent rates the eonental forgettng redcton fro [3] s used. For the th oton odel the araeters are redcted as α,l, 1 = αl,, β,l, η 1 = βl,, 3 η whch corresonds to eeng the eected value of γ constant, whle ncreasng the varance wth a factor η [3]. Ths redcton has an effectve wndow length of w e = η η, where 1 < 1 s the forgettng factor. 1 η Kneatc state: For the th oton odel the neatc state transton denst s odeled as 1 =N 1 ; f, Q 1, 33 where f : R n R n s a state transton functon, and Q 1 s the rocess nose covarance for the neatc state. The transton functon can be arttoned nto N s, arts, [ f = f,c, c T... f d d, c T T...], 34 where f,c descrbes the te evoluton of the overall oston and the neatcs, and f d descrbes the te evoluton of the subobject offsets. The functons f,c and f d are generall nonlnear, see [37] for a thorough overvew of state transton functons.
7 In case f s a lnear functon, the soluton to 16a s gven b the Kalan flter redcton [38]. If f s non-lnear t s straghtforward to solve 16a aroatel. Usng the etended Kalan flter redcton forulas, see e.g. [39], the redcted ean,l 1,l 1 = f l P,l 1 = F,l, P l and covarance P,l 1 are F,l 3a T Q 1 3b where F,l = f l = s the gradent of f evaluated at the ean l. 3 Rando atrces: For the th oton odel we use the transton denst suggested n [6], X 1, X 36 1 = W d X 1 ; n 1, M M T, n 1 X where n 1 > d 1 s a scalar desgn araeter and the atr transforaton M M : R n R d d s a non-sngular atr valued functon of the neatc state. The etenson state s te evoluton s odeled as beng deendent on the neatc state anl because t allows for the odelng of rotaton of etended targets, however n general the onl requreent s that the outut s a non-sngular d d atr [6]. Detals on how the araeters v,l, 1 and V,l, 1 of the solutons to 16b are couted are gven n [6]. C. Generaton of assocaton events The correcton ste, see Secton III-C, nvolves a suaton over the set Θ of all ossble easureent-to-subobject assocaton events. For n z, easureents and N s, subobjects there are N s, n z, ossble easureent-to-subobject assocaton events. For eale, f N s, = 3 and n z, = there are 43 ossble assocaton events, and f n z, = 1 there are 949 ossble assocaton events. Due to the qucl ncreasng sze of the full set of assocaton events aroatons are necessar to acheve tractable coutatonal colet. Usng dfferent ethods to slf the data assocaton roble s coon n target tracng. Poular ethods for ultle ont target tracng nclude robablstc data assocaton PDA, and ultle hothess tracng MHT, see e.g. [36]. Data clusterng ethods are used n etended target PHD/CPHD flters to reduce the nuber of easureent arttons that are consdered, see e.g. [19] []. In [16] t s roosed to reduce the nuber of easureent-to-subobject assocaton events b usng a cobnaton of -eans clusterng, see e.g. [4], [41], and gatng wth a sutable seudolelhood. In ths aer a subset Θ Θ of assocaton events s couted usng a ethod that s based on the Eectaton Mazaton algorth [4] for Gaussan Mtures EM- GM, see e.g. [4, Chater 9]. EM-GM s favored over - eans clusterng because t has been shown that the -eans clusterng algorth often can gve unfavorable results for elltcall shaed etended targets, see [1]. Frst EM-GM s used to artton the current set of easureents nto N c clusters. To accoodate the ossblt that one or ore subobjects does not generate an easureents EM-GM s used for N c [1,,..., N s, ]. Because the soluton to EM- GM has an local otus, for each N c the algorth s gven several dfferent ntalzatons. Note that care s taen to ensure that the set of arttons returned b EM-GM onl contans unque arttons. The net ste s to use the clusters to obtan easureentto-subobject assocatons. Gven a artton of the easureent set wth N c clusters, and an estate wth N s, subobjects, there are N s,!/n s, N c! ossble cluster-tosubobject assocatons. A cluster-to-subobject assocaton defnes a easureent-to-subobject assocaton event θ because each easureent s assocated to a cluster, whch n turn s assocated to a subobject. Let CN c denote the nuber of unque arttons wth N c clusters obtaned usng EM-GM. Then the nuber of easureent-to-subobject assocaton events that has to be consdered s D. Correcton Θ = N s, N c=1 N s,! CN c N s, N c!. 37 Each object generates a Posson dstrbuted nuber of easureents and the easureent odels are lnear Gaussan, P n z, γ =PS, X =N z,j n z, ; γ z,j ; H For the neatc state 7 the odels H 38a, X. 38b are H 1 = [ I d d nc d Ns, 1d], 39a H = [ I d d nc d d I d d Ns, d], 39b for =,..., N s,. For an assocaton event θ Θ the centrod easureent and scatter atr are defned as follows, z θ, = 1 n θ, z, Z θ, = n θ, z, n θ, z, z θ,,j z θ,,j, 4a z θ, z θ,,j z θ, T. 4b The sae easureent odel s used for all oton odels. Wth a redcted dstrbuton ξ Z 1 = J 1 the corrected dstrbuton s ξ Z = θ Θ J 1 w l 1 GGIW ξ ; ζ l 1 w θ,l GGIW ξ ; ζ θ,l 41 4
8 In the sectons that follow the araeters ζ θ,l of the corrected dstrbuton are gven. The dervaton of the easureent udate s gven n the Aend. 1 Measureent rates: The corrected araeters are α θ,l, = α l, 1 nθ, z,, βθ,l, = β l, 1 1. 43 Kneatc state: The corrected araeters are θ,l = l 1 Kθ,l z θ H l 1, 44a P θ,l z θ = = P l [ [ H = K θ,l S θ,l ˆX θ,l H P l 1, 44b T z θ,n T ] T s,, 44c T H N T ] T s,, 44d 1, 44e 1 Kθ,l z θ,1 H 1 = P l 1 HT = H P l S θ,l 1 HT 1 = bldag ˆX l, 1 = V l, 1 l,1 ˆX 1 n θ,1 z, v l, 1 d. 44h ˆX θ,l 1,,..., ˆX l,n s, 1 n θ,n s, z,, 3 Rando atrces: The corrected araeters are v θ,l, V θ,l, =v l, 1 nθ, z,, =V l, 1 Zθ, 1 N θ,l, 1 = ˆXl, 1 ε θ,l, 1 = zθ, S θ,l, ε θ,l, 1 N θ,l, S θ,l, T H l =H P l 1 S θ,l, 1, H 1, 1 ε θ,l, 1 T ˆXl, 1 T ˆXl, 1 n θ, z, T 44f 44g 4a 4b, 4c 4d, 4e where the atr square-roots are couted usng, e.g., Choles factorzaton. 4 Weghts: The weghts are couted as w θ,l = L θ,l, = θ Θ Γ Γ α θ,l, α l, 1 w l Ns, 1 =1 Lθ,l, J 1 l =1 w l 1 β l, 1 β θ,l, n θ, πnθ, z, z, ˆXl, 1 Γ d v θ,l, Γ d v l, 1 d 1 1 d 1 Ns, =1 Lθ,l, α l, 1 α θ,l, d n θ, z, d 1 S θ,l, ˆXl, 1 V l, 1 V θ,l, 1 T v l, 1 d 1 v θ,l, d 1, 46a. 46b TABLE III CHANGE MAIN SUBOBJECT 1: Inut: Coonent GGIW ξ ; ζ. : Center: ˆ c = N [ s, =1 ˆ, where ˆ = E Z ]. 3: Dstances to center: δ ˆ = c ˆ. 4: Closest to center: j = arg n δ : f j 1 then 6: Change an subobject fro 1 to j. 7: = A 1,j, P = A 1,j P A T 1,j, where A 1,j s a erutaton atr that changes lace between subobjects 1 and j. 8: α 1 1, β, ṽ1, Ṽ 1 = α j, βj, vj, V j, α j j, β, ṽj, Ṽ j = α 1, β1, v1, V 1, and α, β, ṽ, Ṽ = α, β, v, V for 1, j. } 9: ζ = { α Ns,, β,, P { }, ṽ =1, Ṽ Ns,. =1 1: else 11: No change of an subobject, ζ = ζ 1: end f 13: Outut: Coonent GGIW ξ ; ζ. Predcted lelhoods: The redcted lelhoods are l Z θ, Z 1 = N s, =1 Z Z 1 = 1 Θ θ Θ E. Mture reducton L θ,l, J 1 N s, w l 1 =1 47a L θ,l,. 47b Wth J coonents, M oton odels and Θ assocaton events there are J 1 1 = Θ M J coonents after one teraton of redcton and correcton. Mture reducton s used n each teraton after the correcton ste to ee the nuber of coonents at a tractable level. Hotheses wth weghts lower than a threshold τ are runed and the weghts are re-noralzed. Mergng s then erfored on the ture, where we have used a cobnaton of the gaa ture ergng fro [3] and the Gaussan nverse Wshart ergng fro [43]. Note that when ultle oton odels are used, ergng s onl erfored wthn the sae oton odes, and not across the oton odes. F. Change of an subobject When the oton odel f s lnear t does not atter whch subobject s defned as the an one. However wth non-lnear oton, e.g. a coordnated turn, t s tcall ost ntutve to defne the an subobject as the subobject closest to the center of the overall etended target. Here center s defned as the ean of the subobjects ostons. It can haen that an estate hothess l ntalzed b the ethod n Table II, after a coule of redctoncorrecton-teratons, converges to a confguraton where the an subobject s not the one closest to the center. In ths case another subobject can be defned as the an one, an oeraton that corresonds to a sle re-orderng of the easureent rates and rando atrces, and a sle lnear transforaton
9 of the neatc vector. Note that an non-lnear dnacs could be re-defned too, however ercall we have found that t s suffcent to ee the neatcs c constant durng the change of an subobject. The algorth that s used s gven n Table III. An ortant secal case s when there are two subobjects, snce n ths case both subobjects are equall close to the center. In ths case the an subobject s arbtrarl assgned uon ntalzaton, and the change of an subobject s never utlzed n the flter recurson. V. COMPARISON TO ALTERNATIVE MODELS In ths secton we dscuss and coare the roosed odel to other etended target odels that are avalable n the ltterature. Models for secfc geoetrc shaes, e.g. stcs, crcles, ellse and rectangles can be found n [] [13]. Because these odels consder secfc shaes we do not coare to the further. Three odels caable of handlng general and rregular shaes can be found n [14] [16]. The odel n [14] consders easureents that are sread along the outlne of the target s shae e.g. laser range easureents, and n ths aer we consder easureent that are sread across the target s surface. The odel n [14] s thus not alcable to the scenaros consdered here. In the net secton we resent sulaton results that coare the roosed odel to the two odels resented n [1], [16], and n the reander of ths secton we elaborate on the theoretcal slartes and dfferences between the roosed odel and the ones fro [1], [16]. A. Star-Conve odel [1] The Star-Conve Rando Hersurface Model [1], denoted M1, araetrzes the boundar of the target shae as a radal functon. The radal functon s araeterzed usng Fourer seres, and the Fourer coeffcents are estated. Wth ore coeffcents, the radal functon has ore degrees of freedo and ncreasngl cole shaes can be descrbed. In a sense, ths s analogous to how a larger nuber of subobjects can descrbe a ore cole shae. Model M1 does not decoose the etended target nto subobjects, and thus does not need to solve the easureent-to-subobject assocaton roble. B. Mult-Ellse odel [16] The odel n [16], denoted M, odels the target usng ultle elltc subobjects, see 6. The roosed odel s ver slar to odel M as the both etend the rando atr fraewor [6], [8] to odel the subobjects. The two odels also have the followng dfferences: 1 Measureent rates: The roosed odel ncludes a odel of the nuber of detectons er subobject er teste and estates the easureent rates for each subobject, whch M does not. Man subobject: The roosed odel defnes one of the subobjects as the an subobject, around whch reanng subobjects are located. In coarson, M does not defne one of the subobjects as the an one. 3 Poston covarance and unfed neatcs: M s based on the rando atr odel 1, see 6, whle the roosed odel s based on the rando atr odel. Ths dfference s fundaental, because t s what allows the ostons and neatcs of all subobjects to the odeled as a sngle rando vector. Due to the for of the Gaussan covarances n 1 P X, under ths odel the state vectors of the subobjects cannot easl be treated as a sngle rando vector. Modelng wth a sngle rando vector roves the overall odelng n the followng was: 1 The roosed odel estates unfed neatcs c.e. a sngle veloct, a sngle turn-rate etc for the etended target as a whole for all subobjects, and f necessar ndvdual subobject neatcs can be ncluded n c. In coarson, M estates ndvdual veloctes and acceleratons for the subobjects. A ultle oton odel fraewor s used n M where there are soe coon neatcs va the rocess nose araeters, however unfed neatcs are not estated. The roosed odel antans a full covarance atr for the subobjects ostons and the neatcs,.e. the deendences between the subobjects ostons and the neatcs are odeled and estated. In coarson, M does not odel the deendences between the subobjects neatc states, cf 6. 4 Kneatcs-etenson-deendence: The roosed odel does not odel the deendence between the oston and the etenson of a subobject, whch M does. Further, under an assued lnear Gaussan easureent odel the roosed odel s easureent udate s aroate, whle M s s eact. Both these dfferences follow fro the roosed odel beng based on the rando atr odel and M beng based on the rando atr odel 1. Wth regards to the deendence not beng odeled, the easureent udate and the redcton udate rovdes for the necessar nterdeendence between the neatc state estate and the etenson state estate. Ths s analogous to how the easureent udate [8] and the redcton udate [6] rovde for nterdeendence n case the target s odeled usng a sngle ellse. In sulaton coarsons for sngle elltc targets, odels based on [8] outerfor odels based on [6], see [6]. We conclude the coarson b notng that dfferent odels should not be judged and coared onl on ther theoretcal roertes, but also on ther ractcal roertes. In the net secton we resent a sulaton stud that coares the ractcal erforance of the roosed odel and odels M1 and M. VI. SIMULATION RESULTS A. Target etracton and erforance evaluaton To etract a target estate fro a ture 3, ergng s frst erfored, ths te across the oton odes. Eected values of the easureent rates, ostons and etenson atrces are then couted w.r.t. the coonent wth the hghest weght w l. Both the redcted estate ˆξ 1 and the fltered estate ˆξ s coared to the true target state
1 8 6 4 [/s] 4 3 1 [deg/s] 1 1 1 1 1 3 4 6 7 8 9 1 Te 1 3 4 6 7 8 9 1 Te Fg. 3. True target trajector, ntal oston s orgn. Left:, -oston. Mddle: veloct. Rght: turn-rate. ξ. The followng error etrcs are used for the easureent rates, subobject ostons, and rando atrces, N s, d γ = N s, d = N s, d X = γ =1 =1 X =1 ˆγπ ˆπ π ˆX, F [, ˆγ = E γ [, ˆ = E ˆX = E [ Z ] Z ] X 48a 48b Z ] 48c A subobject-to-subobject assocaton π s obtaned b nzng d. Because γ, and X all have dfferent unts we refran fro coutng an overall etrc for the etended target state ξ. B. True tracs and setu We sulate both statonar and ovng targets. In a coarson of shae estaton, statonar targets are used because we wsh to ehasze the shae estaton, not the oton estaton. For the ovng target, the target trajector that was sulated s shown n Fg. 3; the true oston s shown n Fg. 3a and the corresondng seed and turn-rate s shown n Fg. 3b and Fg. 3c. Three dfferent d = densonal etended target shaes were sulated. The frst s T-shaed, wth easureents generated b unforl salng across the shae and addng Gaussan nose wth covarance R = I. The easureent rate was 4γ. The other two shaes consst of three and two subobjects, resectvel. The shae of the targets are consstent wth the eales gven n Fg. 1,.e. the shae resebles that of an arlane and of the letter V, resectvel. For these two shaes, the easureent odel 1c was used. For the lanele target, for the subobject that corresonds to the fuselage the easureent rate was γ, and the etenson atr was X = dag [1, ]. For the subobjects that corresond to the wngs the easureent rates were γ, and the etenson atrces were X = dag [, 1 ]. For the V-shaed target, the subobjects both had easureent rates γ and etenson atrces X = dag [, 1 ]. The scenaros were sulated for dfferent values of γ :, and. For the resented flter, a constant turn-rate CT oton odel f,c wth olar veloct, see [37, Eq. 7], was used for the overall oston and neatcs. In ths case the unfed neatcs are gven b c = [ ] T v φ ω, where v s the seed, φ s the headng and ω s the turn-rate. For ths te of oton odel the te evoluton of the subobject offsets s d 1 = f d d, c = R T ω d 49 where T s the salng te and R s a rotaton atr. The atr transforaton functon s also a rotaton atr, M = R T ω. Two CT odels were leented, one wth sall rocess nose corresondng to non-aneuver, and one wth larger rocess nose corresondng to aneuver. The transton robabltes were set to 9% robablt to sta n the sae ode, and % robablt for ode swtch. The flter araeters that were used n the leentaton are lsted n Table IV. TABLE IV PARAMETERS FOR PROPOSED METHOD Paraeter Value Sale te T 1 Nuber of ntal hotheses N N s, 1 Intal neatcs c 3 1 Intal covarance P 1 I n Measureent rate ntal ean e 1 Measureent rate ntal varance v 1 Measureent rate redcton factor η 1., Predcton degrees of freedo n 1 1, Prunng threshold τ.1 For M1 an leentaton avalable onlne was used 3. Model M was leented as nstructed n Secton VI Sulaton studes n [16], and augented to nclude estaton of the easureent rates, see [3] for detals. The ethod was araetrzed wth three oton odels: the frst corresonds to constant veloct oton; the second corresonds to a turn wth turn-rate ψ; the thrd corresonds to a turn wth turn-rate ψ. In the sulatons the araeter ψ was set to degrees er second. For the lower easureent rates γ = and γ = soetes durng the aneuvers one of the subobjects would searate fro the other two subobjects. To correct ths, an hothess l wth a subobject ore than eters fro the other subobjects was deleted. In case all hotheses were deleted, the target estate was rentalzed. C. Results: statonar target The T-shaed and the lane-shaed targets were sulated for easureent rates γ = and γ =. A coarson of the roosed odel and the odels M1 and M s shown n Fg. 4. As eected all three ethods converge uch faster when there are ore easureents.e. hgher γ. All three 3 Thans to M. Bau and R. Sanduehler for rovdng code. htt://www.cloudrunner.eu/algorth/1/rando-hersurface-odel/verson//
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 = 1 1 1 1 1 = 1 1 1 1 = 1 1 1 1 1 = 1 1 1 1 = 1 = 3 = = 1 = 1 1 1 1 1 = 1 1 1 1 = 3 1 1 1 1 = 4 1 1 1 1 = 1 = = 3 = 1 Fg. 4. Eale results for tracng of a lane-shaed and a T-shaed statonar target. To row γ =, botto row γ =. True target gra area coared to roosed odel sold blue lne, an subobject ndcated b thcer lne, odel M dashed orange lne, odel M1 dash-dotted red lne. Te ste wrtten n lower left corner. As eected convergence s uch faster when there are ore easureents.e. hgher γ. 1 d γ 1 d 3 d X d γ 8 d d X 1 d γ 4 d d X 9 8 1 1 1 7 6 18 16 9 8 3 3 7 6 7 6 8 8 14 1 7 6 6 1 6 4 1 4 4 3 4 1 4 3 8 6 4 3 1 1 3 1 1 4 1 1 a γ = b γ = c γ = Fg.. Estaton errors for lane-shaed target. Proosed odel n blue, odel M n orange. -labels F and P denote flter errors d and redcton errors d 1. On each bo, central ar s edan, edges of bo are th and 7th ercentles, whsers etend to ost etree dataonts the algorth consders to be not outlers. ethods gve reasonable results, however the roosed odel and the M are closer to the ground truth than M1. Because of ths, for the ovng target we onl coare the roosed odel and odel M. D. Results: ovng target The lane-shaed and the V-shaed targets were sulated for γ =, γ =, and γ =. For each value of the easureent rate γ the scenaros were sulated 1 3 tes. For the lane-shaed target three subobjects the flter errors d and redcton errors d 1 are shown n Fg., for the V-shaed target two subobjects the results are shown n Fg. 6. Eale flter and redcton oututs for the laneshaed target for γ = are shown n Fg. 7. Fro the results the followng observatons can be ade: Both the redcton errors and the flter errors are saller for the resented ethod for all γ. The bggest dfference s for the subobject oston errors, esecall durng aneuvers. Note that, even f the estated rando atrces have the correct sze and orentaton, the subobject ostons are ore ortant for the overall etended target etenson estate. The larger the subobject oston errors are, the ore dstorted the overall shae becoes, whch can be seen n Fg. 7. The lower errors for the resented ethod, esecall the lower oston errors, are a drect effect of a usng a sngle state vector for the subobject ostons and the neatcs ncludng a full covarance atr; and b havng unfed neatcs for the subobject ostons. E. Coutatonal colet The code used n ths wor was leented n MATLAB and run on a.83ghz Intel Core Quad CPU wth 3.48GB of RAM runnng Wndows. Note that the code has not been otzed for seed. In each te ste aroatel 1 to dfferent arttons of the set of easureents were couted. The average nuber of easureent-to-subobject assocaton events are gven n Table V. A coarson to the nuber of assocaton events f there are E[n z, ] easureents shows that the set of assocaton events s reduced b several orders of agntude. It s noteworth that for γ = the reducton n nuber of assocaton events s b far greatest, et the estaton errors are saller for γ = than for γ = and γ =. The nuber of ture coonents ncrease n each te ste. However, n the ture reducton ste an coonents
1 d γ 18 d 1 1 d X 18 16 d γ 16 14 d 9 8 d X 4 3. d γ 1 9 d 4 3 d X 16 14 1 7 3 8 3 1 14 1 8 1 1 1 6. 7 6 1 6 8 8 4 1 8 6 4 4 6 4 6 4 3 1. 1 4 3 1 1 1. 1 a γ = b γ = c γ = Fg. 6. Estaton errors for V-shaed target. Proosed odel n blue, odel M n orange. -labels F and P denote flter errors d and redcton errors d 1. On each bo, central ar s edan, edges of bo are th and 7th ercentles, whsers etend to ost etree dataonts the algorth consders to be not outlers. TABLE V NUMBER OF ASSOCIATION EVENTS, MEAN ± STANDARD DEVIATION γ Θ N s, E[n z,] 114 ± 9 6.6 1 3 18 ± 4 3. 1 9 13 ± 3 1. 1 38 can be runed, and the reanng coonents can be erged such that tcall onl to 6 coonents rean. The average ccle tes are gven n Table VI. We see that the tes for redcton and reducton are ndeendent of γ. The te to coute clusters for data assocaton ncreases when γ ncreases, because wth ore easureents t taes ore te to cluster the. The correcton te decreases when γ ncreases, because wth ore easureents the scenaro s less abguous and the robablt ture tcall has fewer coonents. Note that these two ncreases/decreases n te offset each other such that the average total ccle te s about 1.7 seconds for all values of γ that were tested. The average ccle te for the ethod fro [16] s.1 ± 1.4 seconds for γ =, 1.7 ±.7 seconds for γ =, and 1.6 ±. seconds for γ =. When coarng the ccle tes, reeber that nether leentaton was otzed for seed. TABLE VI CYCLE TIMES [SECONDS], MEAN ± STANDARD DEVIATION γ Predcton Clusters Correcton Reducton Total.4 ±..7 ±.1.6 ±.4. ±. 1.7 ±.6.3 ±.1.9 ±.1. ±.3.1 ±. 1.8 ±.4. ±.1 1. ±.1.3 ±..1 ±. 1.7 ±.3 VII. CONCLUSIONS AND FUTURE WORK The aer has resented an etended target odel n whch the target etenson s odeled usng a collecton of elltcal subobjects. The sulaton results show that the roosed odel outerfors revous wor. The resented odel can be reduced to the cases where ether the easureent rates γ, the etenson atrces X, or both, are nown. The sulaton stud consdered etended targets, however the odel s alcable also to grou targets. In case the targets n the grou are ovng relatve to each other, n addton to the unfed grou oveent, ndvdual neatcs can be estated along wth the unfed neatcs. An ortant toc for future wor s to nclude estaton of the nuber of subobjects. Interestng lnes for future wor also nclude ntegratng the roosed odel nto a ultle target algorth, such as an etended target PHD- or CPHD flter, see e.g. [18], []. Furtherore alternatve easureent odels can be consdered, see e.g. [1], [8], [9] for recent wor on ths toc wthn the rando atr fraewor. APPENDIX The ror dstrbuton and the easureent lelhood are ξ =N ;, P N s G γ ; α, β IW d X ; v, V, =1 N s n Z ξ = n!ps n ; γ N z j ; H, X. =1 a b The roble at hand s to derve the osteror dstrbuton ξ Z and the corresondng lelhood L, Z ξ ξ = L ξ Z, 1 where the osteror ξ Z s of the sae functonal for as the ror a,.e. ξ Z =N ;, P N s G γ ; α, β =1 IWd X ; v, V, The roduct of Gaussan dstrbutons n b can be
13 8 78 76 74 7 7 8 78 76 74 7 7 3 3 4 4 6 6 7 1 1 3 3 4 4 6 6 7 1 1 Fg. 7. Eale results for γ = for the trajector n Fg. 3. To: fltered estates. Botto: redcted estates. Ground truth gra area, coared to odel M dashed orange lne and roosed ethod sold blue lne, an subobject drawn wth thcer lne. rewrtten as follows n N z j ; H, X 3 = π nd/ X n/ etr 1 n z j H z j H T X 1, where etr = e Tr s eonental trace. Defne the centrod easureents as and the scatter atrces as z 1 n n z j 4 n Z z j z z j z T, and rewrte the suaton as n z j H z j H T =Z n z H z H T. 6 Insertng 6 nto 3 gves n N z j ; H, X = π n d X n etr 1 Z X 1 etr 1 1 z X H z H T n = π n 1d X n 1 n d etr =A N 1 Z X 1 z ; H, X n z ; H, X n 7a 7b N 7c. 7d The roduct of Gaussan dstrbutons n b s thus rewrtten as n N z j ; H, X = A N A = X n 1 etr 1 Z X 1 π n 1d z ; H, X n, 8a n d. 8b The roduct of the lelhood and the ror dstrbuton s Z ξ ξ =N ;, P Ns Ns A N z ; H, X IW d X ; v, V =1 n!ps n ; γ G γ ; α, β. 9 =1 For the easureent rates we have the followng where n!ps n ; γ G γ ; α, β = βα γ αn 1 e β1γ Γ α n 6a Γ α n β α =G γ ; α n, β 1 6b Γ α β 1 αn =L γ G γ ; α, β, 6c α = α n, 61a β = β 1, 61b L γ = Γ α β α Γ α β α, 61c The lelhood L γ s roortonal to a negatve bnoal dstrbuton, see e.g. [44]. For the neatc vector and the rando atrces we have N s N ;, P A N z ; H, X IW d X ; v, V n =1 N s =N ;, P N z ; H, X A IW d X ; v, V, =1 6a
14 where z = [ z T 1 z T z T N s ] T 6b H = [ ] H1 T H T HN T T s 6c X1 X X Ns X = bldag,...,,...,. 6d n 1 n n Ns Usng the Kalan flter [38] easureent udate we get N s N ;, P N z ; H, X A IW d X ; v, V 63a =N where =1 ;, P N z ; H, HP H T X N s A IW d X ; v, V =1 = K z H, P =P KHP, K =P H T S 1, S =HP H T X. 63b 63c 63d 63e 63f At ths ont we ae two aroatons. Aroaton 1: In S n 63f the rando varable X s aroated b ts eected value ˆX1 ˆX ˆX = E [X] = bldag,...,,..., n 1 n ˆX = E [X ] = V v d ˆX Ns n Ns, 64a 64b Rear 9: Ths aroaton s analogous to an aroaton ade b Feldann et al., see [8, Equatons 31 and 33]. Aroaton : In N z ; H, HP H T X n 63b the atr HP H T s aroated b the bloc-dagonal atr bldag H 1 P H T 1,..., H P H T,..., H Ns P H T N s. 6 Rear 1: Ths aroaton s necessar to obtan a osteror dstrbuton that s of the sae functonal for as the ror dstrbuton. Under these aroatons nstead of 63b we have N ;, P 66 N s A N z ; H, H P H T X IW d X ; v, V n =1 where = K z H, P =P KHP, K =P H T S 1, S =HP H T ˆX. 67a 67b 67c 67d For the factors n the roduct n 66, the Gaussan covarances H P H T X n can be eanded e.g. usng Choles Factorzaton as H P H T X 1 X 1 X X T H P H T n X T. n 68 A thrd aroaton s now ade. Aroaton 3: Equaton 68 s aroated b S 1 ˆX 1 X ˆX T S T, S = H P H T ˆX n. 69 Rear 11: Ths aroaton s analogous to an aroaton ade b Feldann et al., see [8, Equatons 38 and 39]. Under ths aroaton the Gaussans n 66 can be rewrtten as N z ; H, S 1 ˆX 1 X ˆX T S T = π d etr = π d etr S 1 1 7a ˆX 1 X ˆX T S T 1 z H T S 1 ˆX 1 X ˆX T S T 1 z H S 1 ˆX 1 1 X 1 1 z H T S T ˆX T S T 1 7b ˆX T X 1 ˆX 1 S 1 z H = π d ˆX T S ˆX 1 1 X 1 etr 1 ˆX 1 S 1 z H z H T S T = π d ˆX T S ˆX 1 where 1 X 1 etr 1 N X 1 N = ˆX 1 S 1 z H z H T S T ˆX T 7c ˆX T X 1 The factors n 66 can now be rewrtten as A N z ; H, S 1 ˆX 1 X ˆX T S T IW d X ; v, V = X n 1 etr 1 Z X 1 π n 1d n d π d ˆX T S ˆX 1 v d 1 V v d 1 Γ d v d 1 1 X v etr 7d 7e 7f 71a X 1 etr 1 N X 1 1 X 1 V 71b
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