AN IMPROVED PARTICLE FILTER ALGORITHM BASED ON NEURAL NETWORK FOR TARGET TRACKING

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1 AN IMPROVED PARTICLE FILTER ALGORITHM BASED ON NEURAL NETWORK FOR TARGET TRACKING Qn Wen, Peng Qcong 40 Lab, Insttuton of Communcaton and Informaton Engneerng,Unversty of Electronc Scence and Technology of Chna, Chengdu, Chna Abstract: Key words: To the shortcomng of general partcle flter, an mproved algorthm based on neural networ s proposed and s shown to be more effcent than the general algorthm n the same sample sze. The mproved algorthm has manly optmzed the choce of mportance densty. After recevng the samples drawn from pror densty, and then adjust the samples wth general regresson neural networ (GRNN), mae them approxmate the mportance densty. Apply the new method to target tracng problem, has made the result more precse than the general partcle flter. partcle flter, target tracng, general regresson neural networ. INTRODUCTION In practce of target tracng, the dynamc system usually s non-lnear and non Gaussan. The general method (e.g. Kalman flter) s unable to reach the optmal estmate of target. However, partcle flter algorthm based on Bayesan rule unformly made very good result n state estmate of nonlnear and non-gaussan system. So, t wll be a good choce to apply partcle flter to the target tracng system. But, the shortcomng of partcle flter can not be gnored ether. Because the mportance densty functon s dffcult to realze n project practce, so usually adopt pror densty and substtute t. Ths nd of method wll reduce

2 Qn Wen, Peng Qcong the precson of state estmate. So, people are always studyng a better method to approxmate the mportance densty. Ths paper explores the possblty of usng neural networ to get optmal result. The below wll descrbe ths nd of method and analyze the performance mprovement brought by t.. PARTICLE FILTER. Bayesan Rule The problem of tracng s a process of the state sequence estmated. The dynamcs of sngle state vector at tme s descrbed by a stochastc dfference equaton x = f( x, w ) () where w - s an..d. process nose vector wth a nown dstrbuton and f s a possbly nonlnear functon of the state x -. At each dscrete tme pont an observaton z s obtaned, related to the state vector by z = h( x, v ) () where v s an..d. measure nose vector and h s called the measurement functon. Then the state predcton equaton px ( z ) = px ( x ) px ( z ) dx (3) : : The state update equaton px ( z ) : pz ( x) px ( z ) pz ( z ) : = (4) : where the normalzng constant pz ( z ) = pz ( x) px ( z ) dx (5) : :

3 An Improved Partcle Flter Algorthm Based on Neural Networ for Target Tracng 3 The above descrbes the Bayesan estmate rule. Generally, the analytc soluton n (3) does exst n some dynamc system. But, t cannot be determned analytcally n nonlnear and non-gaussan system. Therefore, partcle flter s a good choce to approxmate optmal Bayesan soluton.. Partcle Flterng Method Partcle flter s a technque for mplementng a recursve Bayesan flter by Monte Carlo smulatons. The ey dea s to represent the requred posteror densty functon by a set of random samples wth assocated weghts and to compute estmates based on these samples and weghts. As the number of samples becomes very large, ths MC characterzaton becomes an equvalent representaton to the usual functonal descrpton of posteror probablty densty functon, and the SIS (Sequental Importance Samplng) flter approaches the optmal Bayesan estmate []. N s x, w denote a random measure that characterzes the posteror = Let { } px ( z ), where { x, 0,, Ns} densty functon : = L s a set of support ponts wth assocated weght{ w, =, L, Ns}. Then, as Ns, the posteror densty at can be approxmated as N s : δ = px ( z ) w ( x x) (6) In the SIS algorthm, the samples x are drawn from an mportance densty qx ( x, z: ), then the weghts n (6) update equaton can be shown as w pz ( x) px ( x ) (7) w qx ( x, z) In the paper [], the optmal mportance densty functon that mnmzes the varance of the true weghts w condtoned on x and z has been shown qx ( x, z) = px ( x, z) (8)

4 4 Qn Wen, Peng Qcong The SIS algorthm conssts of recursve propagaton of the weghts and support ponts as each measurement s receved sequentally. A pseudo-code descrpton of ths algorthm s gve by Algorthm. Algorthm. SIS Partcle Flter for = : Ns draw x : q( x x, z) assgn the partcle a weght, w accordng to (7) end for N normalze the weght w / sum[{ } s = w w = ] resample the sample w 3. CHOICE OF IMPORTANCE DENSITY BASED ON NEURAL NETWORK 3. Shortcomng of SIS In practce, t s often convenent to choose the mportance densty to be the pror qx ( x, z) = px ( x ) (9) Then, substtuton of (9) nto (7) w w p( z x) (0) In a stuaton that the observe precson s low, ths method can mae better result, but the precson of estmate s not hgh. Because the current measure value z s not consdered n mportance densty functon, the samples drawn from mportance densty and from the real posteror densty have greater devatons. Especally when lelhood functon s at the end of system state transfer probablty densty functon or measure model has very hgh precson, the nds of devaton are more obvous. Ths can see from Fgure.

5 An Improved Partcle Flter Algorthm Based on Neural Networ for Target Tracng 5 px ( x ) pz ( x ) px ( x ) pz ( x ) (a) Fgure. The pror densty and lelhood functon. (a) The lelhood functon s pea. (b) The lelhood functon s at the end of pror densty functon. To the shortcomng of SIS algorthm, A.Doucet proposes to construct suboptmal mportance densty to the optmal mportance densty by usng local lnearzaton technques []. R van der Merwe, A.Doucet, etc. propose to estmate a Gaussan approxmaton to mportance densty usng the unscented transform [3]. Yuan Zejan and Zheng Nannng, etc. propose usng Gauss-Hermte flter to sample and reconstruct the mportance densty n [4]. Peter Torma, etc. propose a local search method to adjust samples, mae t more approxmate mportance densty functon [5]. Ths paper apples artfcal neural networ to the choce of mportance densty. Usng general regresson neural networ (GRNN) to adjust samples after predcton step can get good result, mae samples accord wth the posteror densty further. (b) 3. GRNN GRNN s a novel neural networ proposed by Donald F.Specht n 99[6]. The basc theory s nonlnear regresson analyss. GRNN s dfferent wth the BP networ. The tradtonal BP neural networ s one nd of typcal unversal approxmaton networ. One or more weghts are nfluence to each output n networ. It causes study speed to be slower; moreover, the weght determned s stochastc, whch causes the relatonshp between nput and output after each step of tranng unstable and the forecast result devaton. The GRNN only needs a smple smooth parameter, does not need to carry on the tranng process crcularly, and does not adjust weght between the neurons n the tranng process. The networ s steady and the computaton speed s quc. Therefore, n the real-tme target tracng applcaton, GRNN has the superorty compared to BP.

6 6 Qn Wen, Peng Qcong What show n Fgure s a feed forward networ that can be used to estmate a vector Y from a measurement vector X. x y x y x y Unts Input Pattern Summaton Unts Unts Fgure. GRNN bloc dagram Unts Output Let X be a partcular measured value of random varable x, and X s sample value of x. Defnng the scalar functon T D = ( X X ) ( X X ) () and performng the ndcated ntegratons yelds the followng D exp( ) n Y ˆ( ) = σ = n D exp( ) = σ Y X () where Y s sample value of Y. σ s the wdth of sample probablty for each sample X and Y. GRNN can be used to adjust samples n partcle flter algorthm accordng to measurement value z. Because when n a concrete target tracng scene, the profle of lelhood functon s fxed at any tme of, but ts mathematc expresson s not clear, only can be obtaned by some separate observed values through mage processng methods. So we may tran the networ accordng to the observed value and mae t approach the lelhood functon, and then utlzes ths networ to any samples to carry on the adjustment. Frstly, the nput vector and object vector should be structured to tran the networ. A group of samples are obtaned by samplng equal-space lelhood functon. n neghborhood samples and ther lelhood functon values consttute the nput vector and object vector respectvely. The

7 An Improved Partcle Flter Algorthm Based on Neural Networ for Target Tracng 7 dmenson of nput vector n and the number of study sample m determne the networ confguraton: n m (n+) n. After the networ s traned, the samples n partcle flter algorthm should be adjusted n form of nput vector, and then t can be apply to the networ. Frstly, construct a n dmenson vector X [, = x x ± jδ ], jδ< L, (j=,,n/ ). The parameter L defnes the adjustment range. Then, transform X h( X) z as the nput vector of GRNN traned. Fnally, through the ndcaton of output vector of networ, the sample x s substtuted by optmal pont x ± jδ. A seres of samples adjusted are more optmally approxmate the mportance densty. 4. SIMULATION Let X represent the state varable at tme, correspondng to the poston of the target n the state space [ ] T X = x y The two-dmensonal target dynamcs s gven by [5] Xˆ = X + S Δ + W + + S = (B ) S + + U = χ( Xˆ K) + + X = U Xˆ + ( U ) X Δ = U ( X X ) + ( U )( X Xˆ ) where W ~ N(0, σ ) are..d. Gaussan random varables, and B s a Bernoull varable. The measure model s Z = X + V, where V ~ N(0, δ )..d. The dynamcs s hghly nonlnear. We test the general SIS algorthm and the mproved algorthm based on GRNN to ths dynamcs model respectvely. Fgure 3 shows a typcal sequence of tracng a ball. The number of partcles was chosen to be as 00 n two nds of algorthms. The dmenson of nput vector n GRNN s 7. Obvously, the two algorthms all fnd the poston of ball precsely along

8 8 Qn Wen, Peng Qcong wth the movement of target. However, the mproved algorthm has hgher accuracy than general one. Fgure 4 shows the tracng precson as a functon of the tme. The mproved algorthm has the obvous superorty compared to the general partcle flter algorthm at the average devaton pxels. (a) (b) Fgure 3. The result of smulaton. (a) The result of tracng wth the general SIS algorthm. (b) The result of tracng wth the mproved algorthm based on GRNN adjustment Fgure 4. Tracng precson as a functon of tme 5. CONCLUSION In ths artcle, an mproved partcle flter algorthm based on GRNN s proposed to solve the problem of mportance densty functon choosng. The

9 An Improved Partcle Flter Algorthm Based on Neural Networ for Target Tracng 9 new method s shown to mprove the performance of samples and to ncrease robustness as compared wth the prevous method proposed, whlst the novel algorthm mnmzes the expected dstorton n the confguraton space. One weaness of the approach s that the GRNN wll spend more tme than general partcle flter algorthm. However, the prce of tme s worth. Moreover, t s n the range that the system can bear. In concluson, we beleve that t s a sgnfcatve exploraton to adopt neural networ to solve the optmal mportance densty. It wll be a valuable step towards the mplementaton of hghly effcent tracng. REFERENCES. M.Sanjeev Arulampalam, Smon Masell, Nel Gordon. A Tutoral on Partcle Flters for Onlne Nonlnear/Non-Gaussan Bayesan Tracng. IEEE Transactons on Sgnal Processng, 00,50():74~88. A.Doucet. On sequental Monte Carlo methods for Bayesan flterng. Dept. Eng., Unv. Cambrdge, UK, Tech. Rep., R.van der Merwe, A.Doucet, N.de Fretas, E.Wan. The Unscented Partcle Flter. Adv. Neural Inform. Process. Syst., Dec Yuan Zejan, Zheng Nannng, Ja Xnchun. The Gauss-Hermte Partcle Flter. Dan Z Xue Bao, 003,3(7):970~ Peter Torma, Csaba Szepesvar. LS-N-IPS:an mprovement of partcle flters by means of local search. In Proc. Non-Lnear Control Systems (NOLCOS 0),00 6. Specht D F. A General Regresson Neural Networ. IEEE Transactons on Neural Networs, 99, (6):568~576