A Multi Target Track Before Detect Application

Transcription

1 Multi Target Trac Before Detect pplication Y. Boers and J.N. Driessen F. Verschure, W.P.M.H. Heemels and. Julosi JRS-PE-Innovation Electrical Engineering THLES Nederland Eindhoven University of Technology 7554 RR Hengelo 56 MB Eindhoven The Netherlands The Netherlands bstract This paper deals with a radar trac before detect application in a multi target setting. Trac before detect is a method to trac wea objects (targets) on the basis raw radar measurements, e.g. the reflected target power. In classical target tracing, the tracing process is performed on the basis of pre-processed measurements, that are constructed from the original measurement data every time step. In this way no integration over time taes place and information is lost. In this paper we will give a modelling setup and a particle filter based algorithm to deal with a multiple target trac before detect situation. In simulations we show that, using this method, it is possible to trac multiple, closely spaced, (wea) targets.. Introduction Classical tracing methods tae as an input so called plots that typically consist of range measurements, bearing measurements, elevation measurements and range rate (doppler) measurements, see [] and []. In this classical tracing setting tracing consists of estimating inematic state properties, e.g. position, velocity and acceleration on the basis of these measurements. In the classical setup the measurements are the output of the extraction, see figure. In this setup there is a processing chain before the tracing, this processing chain can consist (e.g. in case of radar) of a detection stage, a clustering stage and an extraction stage, see figure. In the method, that we propose here, we will use as measurements the raw measurement data, e.g. reflected power, see figure. The method is called Trac Before Detect (TBD), see also [6], [9] and []. If we loo at figure we see that in classical tracing (i.e. the separate blocs) a detection decision is made directly on the basis of the raw measurement from one single scan. This video detect cluster extract trac tracs TBD Figure : Classical data and signal processing (seperate boxes) and TBD (large box) means that already at the beginning of the processing chain a hard decision is made w.r.t. the presence of a possible target. Note that this decision is made instantaneously, i.e without using information from the (near) past. In TBD the decision is made at the end of the processing chain, i.e. when all information has been used and integrated over time. Note that information in a trac has been obtained by integrating over time). This method is especially suitable for tracing wea targets, i.e. targets that in the classical setting often will not lead to a detection. particle filter will be used to perform the TBD. In [8, 9, ] it has been shown that for a single target the detection can be based on the output of this filter. Somewhat related is the wor in [7], a multiple target approach for TBD on the basis of camera observations for well separated targets has been presented. In this paper we will present an algorithm that deals with a multiple, closely spaced target, setup in which targets can popup and disappear (birth/death). System setup Consider the general system s + = f(t,s,d,w ), N () Prob{d + = i d = j} =[Π(t )] ij () z = h(t,s,d,v ), N (3) Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

2 Where s S R n s(d ) is the base state of the system. d D N is the modal state of the system. z R p is the measurement. t R is time. w is the process noise and p w(,d )(w) is the probability distribution of the process noise. v is the measurement noise and p v(,d )(v) is the probability distribution of the measurement noise. f is the system dynamics function. h is the measurement function. Π(t ) is the Marov transition matrix. define Z = {z,...,z } (4) The optimal hybrid filtering problem can be formulated as follows. Problem. (Optimal filtering problem) Consider the system represented by the equations (), () and (3). ssume that the initial pdf p(s(),d()) is available. The hybrid filtering problem is the problem of constructing the a posteriori pdf p(s,d Z ) (5) Note that given the solution to problem., the mean of the state (s,d ) is obtained as E p(s,d Z ))(s,d ) (6) ctually for any function of the state, φ(s,d ), the mean E p(s,d Z ))φ(s,d ) (7) can be calculated on the basis of the filtering solution. 3 Particle filter solution solution to the above hybrid filtering problem is to extend the state in a straightforward manner with a modal state and then apply a particle filter to the extended state, see e.g. [4, ] The following algorithm will result in an approximation of the a posteriori filtering distribution p(s,d Z ) lgorithm 3. Consider the system represented by the equations (), () and (3). ssume that an initial pdf p(s,d ) is given. Choose an integer N, the sample size.. Draw N samples according p(s,d ), to obtain {( s i, d i )} i=,...,n. Generate {d i } i=,...,n on the basis of { d i } i=,...,n and Π(t ). 3. Draw {w i } i=,...,n according to p w (w) and obtain {(s i,di )} i=,...,n using 4. Given z, define 5. Normalize s i = f(t, s i, d i,w i ) q i = p(z s i,d i,t ), i =,...,N q i := 6. Resample N times from ˆp(s, d) := q i N, i =,...,N i= qi N qδ((s, i d) (s i,d i )) j= and obtain {( s i, d i )} i=,...,n to construct goto. ˆp(s Z ):= N j= N δ((s, d) ( si, d i )) Remar 3. The above algorithm is a standard particle filter implementation. Different and better algorithms for multiple model particle filters exist, see e.g. [4]. Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

3 4 TBD system setup In this section we will describe the models that will be used in the TBD application. We will describe the system dynamics models and the measurement models. 4. System dynamics widely used model is the constant velocity model, see e.g. [, ]. This model is used to describe the position and velocity using Cartesian coordinates. Furthermore the model has an additive process noise term. The discrete-time system dynamics of this model is of the form: s + = f(t,s,d )+g(t,s,d )w (8) f(t,s,d )= T T s (9) with the state vector s =[x,y, ẋ, ẏ ] T x and y are the positions and ẋ and ẏ are the velocities. The process noise w is assumed to be standard white Gaussian noise. T is the revisit time, which has been assumed to be constant here, this assumption can be relaxed, however. The process noise input model is given by g(t,s,d )= ( 3 a x,max)t ( 3 a y,max)t 3 a x,maxt 3 a y,maxt with maximum accelerations a x,max and a y,max. 4. Measurement model () The measurements are measurements of reflected power. One measurement z consists of N r N d N b power measurements, N r, N d and N b are the number of range, doppler and bearing cells. In figure we have plotted power measurements for a fixed bearing angle, but as a function of different range and doppler. The power measurements in this figure correspond to a target that has a power of and a noise level such that the signal to noise ratio is 3dB. The power measurements per range-doppler-bearing cell are defined by =, N () Figure : Power of target in noise (SNR 3dB) as a function of range and doppler cells 5 8, is the complex amplitude 6 data of the target 5 4 which is z, = h (s,t )+n(t ), N () = Ãe ıφ, φ (, π) (3) is the complex amplitude of the target and h (s,t ) is the reflection form that is defined for every range-dopplerbearing cell by (s,t )=e (r i r ) h ijl R L r (d j d ) D L d (b l b ) B L b, (4) i =,...,N r,j=,...,n d,l=,...,n b and N with r = d = ṙ = b = arctan x + y (5) x + y (x ẋ + y ẏ ) (6) ( y x ) (7) which are the range, doppler and bearing respectively of the target. R, D and B are related to the size of a range, a doppler and a bearing cell. L r, L d and L b represent constants of losses. The noise is defined by n(t )=n I (t )+ın Q (t ) (8) Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

4 which is complex Gaussian, n I (t ) and n Q (t ) are independent, zero-mean white Gaussian with variance σ n. In this way the power measurements per range-dopplerbearing cell are defined by =, = (9) = h ijl (s,t )+n I (t )+ın Q (t ), N These measurements, conditioned on s, are now exponentially distributed p( s )= e () = E[ ] = E[ Ãe iφ h ijl (s,t )+n I (t )+ın Q (t ) ] = E[(Ãh ijl (s,t ) cos(φ )+n I (t )) with +(Ãhijl sin(φ )+n Q (t )) ] = Ã (h ijl (s,t )) +σn = Ph ijl P (s,t )+σn () h ijl P (s,t )=(h ijl (s,t )) = () = e (r i r ) R L r (d j d ) D L d (b l b ) B L b which describes the power contribution of a target in every range-doppler-bearing cell. 5 Multiple target setting In this section we consider the (possible ) presence of two targets, one target can originate (spawn) from the other. Thin, e.g. of a missile being fired from a fighter airplane. We can consider the general system introduced in section, i.e. s + = f(t,s,d,w ), N (3) Prob{d + = i d = j} =[Π(t )] ij (4) z = h(t,s,d,v ), N (5) Furthermore, for this problem it is convenient to define a base state vector that consists of the base states of both targets. Thus, ( ) S = s () s () (6) The discrete mode d represents one of three hypotheses d =: There is. d =: There is one target present. d =: There are two targets present. 5. Measurements models For the multi target setup the measurement models need to be extended. The complex amplitude data that is received from two targets can be modelled by z, = () h() (s,t )+ () h() (s,t )+n(t ) (7) (), h() and (), h() are the amplitude and reflection form of the first and second target respectively. Thus, the power measurements are =, = () h ()ijl (s,t )+ () h ()ijl (s,t ) +n I (t )+ın Q (t ) (8) These will, again be exponentially distributed p( s )= e (9) = E[ ] = E[ Ã() eıφ h ()ijl (s,t ) +Ã() eıφ h ()ijl (s,t )+n I (t )+ın Q (t ) ] = E[(Ã() +Ã() +(Ã() +Ã() cos(φ() )h()ijl (s,t ) cos(φ() )h()ijl (s,t )+n I (t )) sin(φ() )h()ijl (s,t ) sin(φ() )h()ijl (s,t )+n Q (t )) ] = (Ã() h()ijl (s,t )) +(Ã() h()ijl (s,t )) +σn = P () h ()ijl P (s,t ) +P () h ()ijl P (s,t )+σn (3) using this we obtain for the mode dependent lielihood ijl p v( ) for d = p(z S,d )= ijl p(zijl S ) for d =,s () = s () ijl p(zijl S ) for d = (3) 4 Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

5 6 Multiple target tracing simulations In this section we give a demonstration of a particle filter that is capable of tracing multiple targets. In the scenario, initially there is, the first target appears after 5 seconds at a position of 88.6m from the sensor and flies at a constant velocity of ms directly to the sensor. t t =5a second target spawns from the first, accelerating to a velocity of 3ms over three scans, see figure 3 for an illustration of the scenario..5 First target Second target a max,y = 5ms. For the second target we choose a max,x =35ms and a max,y =5ms. We assume that the power of both targets is nown, P () =and P () =. The noise level is to be assumed such that a specified SNR for the first (strongest) target is realized according to ( ) P () SNR =log [db] σ n Simulations are performed for two different SNR values, SNR =3dB and SNR =7dB. This implies that the SNR of the second target becomes 3dB and 3dB respectively =5 y position [m] y position [m] x position [m].3.4 Figure 3: Target trajectories We consider range cells in the interval [8, 9]m, doppler cells in the interval[.34,.]ms. We only consider one bearing cell in this example. We therefore have N r N d N b cells, N r =5,N d =6and N b =. Initially, 96 particles are uniformly distributed in the state space, in an area between [85, 9]m and [.,.]ms in the x-direction and [.,.]m [.,.]ms and in the y-direction. Furthermore, initially we uniformly distribute the particles over all modes. The transition probability matrix is assumed to be Π(t )= The update time, T, is set to one second. The dynamics of both targets are captured by a constant velocity model, but with different values for their maximal acceleration. For the first target we set a max,x = x position [m] Figure 4: Particles at =5, SNR=3dB In the figures 4, 5 and 6, we see, for the case of SNR=3dB, the particle clouds at different time steps. In figure 7, we see the corresponding mode probabilities. Until time step =5, the cloud remains more or less uniformly distributed. When the first target appears the cloud quicly concentrates on the target and remains there, see figure 5. The filter however, is somewhat indecisive whether one or two targets are present, see figure 7. fter the birth of the second target the filter is very well able to distinguish the two targets. In the figures 8, 9 and, we see for the case of SNR=7dB, the particle clouds at different time steps. In figure, we see the corresponding mode probabilities. gain until time step =5, the cloud remains more or less uniformly distributed. When the first target appears the filter pics it up quite good, but is again quite indecisive on the 5 Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

6 .5 =4 Estimation of the mode y position [m] x position [m] True mode target present targets present target present targets present Figure 5: Particles at =4, SNR=3dB Figure 7: Mode probabilities, SNR=3dB.5 =5.4.5 = y position [m]... y position [m] x position [m] Figure 6: Particles at =5, SNR=3dB number of targets, see figure. fter the birth of the second target, the filter is able to trac both the first and second target, but is still quite indecisive on the number of targets. The explanation for this that the second target is a really wea one, i.e. SNR=-3dB. 7 Conclusions In this paper we presented a modelling setup and an algorithm for a multiple target TBD application. The problem has been solved by using a multiple model particle filter x position [m] Figure 8: Particles at =5, SNR=7dB Simulations show that the method can deal with fairly wea targets, that in a classical target tracing setup never would have been detected. In the future, the method should be extended to the case the SNR of the targets is unnown and fluctuation of the strength of the targets, so called radar cross section fluctuation, is taen into account. Furthermore, it should be studied to what extent other (better?) multiple model particle filter algorithms can improve the results. 6 Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.

7 .5.4 =4.8 Estimation of the mode target present targets present y position [m] x position [m] True mode target present targets present Figure 9: Particles at =4, SNR=7dB Figure : Mode probabilities, SNR=7dB y position [m] =5 [4]. Doucet, J.F.G. De Freitas and N.J. Gordon, Eds., Sequential Monte Carlo Methods in Practice Springer-Verlag, New Yor NY,. [5] N.J. Gordon, D.J. Salmond and.f.m. Smith, Novel approach to nonlinear/non-gaussian Bayesian state estimation, Proceedings IEE. F, vol. 4, no., pp. 7-3, 993. [6] D.J. Salmond and H. Birch, Particle Filter for Trac- Before-Detect, In Proceedings of the merican Control Conference, rlington, V, June 5-7, x position [m] References Figure : Particles at =5, SNR=7dB [] S. Blacman and R. Popoli, Design and nalysis of Modern Tracing Systems, rtech House, Norwood, M, 999. [] Y. Bar-Shalom and X.R. Li. Estimation and Tracing Principles, Techniques and Software, rtech House, Norwood, M, 993. [3] D. Crisan and. Doucet. Survey of Convergence Results on Particle Filtering Methods, IEEE Transactions on Signal Processing, vol. 5, no. 3, pp ,. [7] D.J. Ballantyne, H.Y. Chan and M.. Kouritzin. Branching Particle-Based Nonlinear Filter for Multi-Target tracing. In Proceedings of the FUSION Conference, Montreal, Canada, ugust 7-,. [8] Y. Boers. On the number of samples to be drawn in particle filtering. In proceedings of the IEE colloquium on Target Tracing, London, UK, 999. [9] Y. Boers and J.N. Driessen, Particle Filter Based Detection For Tracing, In Proceedings of the merican Control Conference, rlington, V, June 5-7,. [] Y. Boers, J.N. Driessen and K. Grimmerin. Particle filter based detection schemes. In Proceedings of the Small Targets Conference at SPIE EROSENSE, Orlando, Fl., pril -5,. [] Y. Boers and J.N. Driessen. Particle Filter Based Detection Scheme. ccepted for publication in IEEE Signal Processing Letters, 3. [] Y. Boers and J.N. Driessen. Hybrid state estimation: a target tracing application. UTOMTIC, vol. 38, no., pp ,. 7 Proceedings of the 3 Conference on Computer Vision and Pattern Recognition Worshop (CVPRW 3) /3 $ 7. 3 IEEE uthorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 9, 9 at 7:3 from IEEE Xplore. Restrictions apply.