Gaussian Mixtures Proposal Density in Particle Filter for Track-Before-Detect

Transcription

1 12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 29 Gaussian Mixtures Proposal Density in Particle Filter for Trac-Before-Detect Ondřej Straa, Miroslav Šimandl and Jindřich Duní Department of Cybernetics, Research Centre Data-Algorithms-Decision Maing, Faculty of Applied Sciences, University of West Bohemia, Czech Republic Abstract The paper deals with state estimation for the trac-before-detect approach using the particle filter. The focus is aimed at the trac initiation proposal density of the particle filter which considerably affects estimate quality. The goal of the paper is to design a proposal based on a Gaussian mixture using a ban of extended Kalman filters. This leads to root mean square error lower than that achieved by usual simple trac initiation proposals. Due to application of several developed techniques reducing computational requirements of the designed proposal, the Gaussian mixture particle filter also achieves lower computational requirements than ordinary particle filter. Performance of the proposed Gaussian mixture trac initiation proposal in the particle filter is demonstrated in a numerical example. Keywords: Tracing, nonlinear filtering, estimation, tracbefore-detect, Gaussian mixture, proposal density, particle filtering 1 Introduction Classical tracing approaches estimating target state consider target measurements, typically position, range, bearing and so forth, that are extracted by thresholding from the output of the sensor signal processing unit. The purpose of the thresholding is to simplify subsequent target state estimation by eeping only the data exceeding the threshold and thus reducing the data flow. These approaches are not suitable for tracing targets with low signal-to-noise ratio (SNR), typically stealthy military aircraft and cruise missiles, for which thresholding has an undesirable effect of disregarding potentially useful data [1]. To cope with tracing low SNR targets, the tracing approach woring with raw unthresholded data is used. This approach thus has to simultaneously detect and trac target and is nown in literature as the trac-beforedetect (TBD) approach [2]. Due to nonlinearity and/or non-gaussianity of the model considered in the TBD approach, it is necessary to use a global nonlinear estimation technique which computes probability density function (pdf) of the target state. The particle filter () [3] is a global state estimation method providing an approximate of the pdf of the state estimate in the form of a set of particles and corresponding weights. The has dominated in recursive nonlinear state estimation due to its easy implementation in very general settings and cheap and formidable computational power. This is also the reason for popularity within the TBD approach [4], [1], [5]. This paper focuses on the proposal density of the for the TBD approach. The proposal density, which serves for particles sampling, considerably affects quality of estimates and therefore its careful design may increase quality of the estimates. Traditional designs of the proposals use no or little information (bootstrap proposal) coming from the available measurement in the process of sampling the particles. Utilizing full information from the measurement in the proposal density should improve the positioning of the particles and consequently estimate quality. A natural choice for a better proposal is to use the composite approach [6], which is based on utilizing another nonlinear estimation technique to obtain a proposal density. The paper follows the aforementioned idea and designs a Gaussian mixture (GM) proposal density for trac initiation (TI). The proposal is computed by a ban of extended Kalman filters (EKF s). The proposal density approximates the true filtering pdf which implies a very simple and fast computation of the weights corresponding to the particles. The paper also discusses reduction of high computational requirements of the proposed algorithm. The reduction of the requirements follows two directions. The former consists in limiting the number of measurements processed by each EKF and the latter in limiting the number of EKF s utilized and precomputing a large amount of data processed in the EKF s. By utilizing these techniques, in many cases the proposed Gaussian mixture () achieves lower computation demands than the with bootstrap proposal density [7]. The paper is organized as follows: Section 2 introduces the model considered in the TBD approach, Section 3 provides an algorithm of the for the TBD and Section 4 presents the GM proposal for the. In Section 5 the reduction of computational requirements of the GM proposal is discussed, Section 6 provides a numerical example demonstrating the properties of the proposed in compari ISIF 27

2 son with the with special focus on choice of parameters, root mean square (RMS) error and computational aspects and Section 7 concludes the paper together with several remars on performance of the. 2 Trac-Before-Detect The state of the target at time instant is defined by position (x, y ) and velocity (ẋ, ẏ ) of the target in the x and y directions and by the return intensity I of the target, which is considered to be unnown. Thus the target state is given by x = [x, ẋ, y, ẏ, I ] T. The target state evolves according to the discrete-time linear Gaussian model [1] as x +1 = Fx + e, (1) where e is a zero mean white Gaussian noise with covariance matrix Q, i.e. p(e ) = N {e :, Q}. As the constant velocity process model is considered, the transition matrix F is given by F = 1 T 1 1 T 1 1 and the covariance matrix of the state noise by q s T 3 /3 q s T 2 /2 q s T 2 /2 q s T Q = q s T 3 /3 q s T 2 /2 q s T 2 /2 q s T, q i T (3) where T is the sampling period, q s is the power spectral density of the acceleration noise in the spatial dimensions and q i is the power spectral density of the noise in the rate of change of target return intensity. The prior pdf of the target state upon its appearance at time is denoted as p b (x ). Indication of the target presence in the measured data is modelled through a target existence variable E [8], with E = 1 if the target is present in data or E = if the target is absent. Probability of the target presence is modelled by a Marov chain with two states and the following transition matrix [ ] 1 Pb P = b, (4) Pd 1 P d where P b is probability of target birth and P d is probability of target death. The measurements obtained as a result of sensor signal processing are in the form of a sequence of images [1]- Ch. 11. The measurement z at time instant is assumed to be a two-dimensional image consisting of n x cells in the (2) x direction and n y cells in the y direction. Hence, each measurement z is a set {z } n x,n y i=1, j=1 of values of measured intensity at each cell. Each cell contains a contribution of the target denoted as h (x ) and noise denoted as v. If the target is not present, only contribution of the noise is considered, i.e. z = { h (x ) + v, if target present v, if target not present. The measurement noise v is considered to be a white zero-mean Gaussian noise with variance R for all cells, i.e. p(v ) = N {v :, R}. Note that the noise is independent between cells. The nonlinear function h (x ) representing contribution of the target to each cell is given by h (x ) = x y I 2π 2 exp ( ( xi x ) 2 + ( y j y ) ) (5), (6) where x and y represent the size of a cell in the x and y directions respectively. It can be seen that the spread of the measurement contributed by the target is modelled by a two-dimensional Gaussian distribution. As the state estimation methods assume the measurement to be a vector, let us denote the measurement z and the set of measurement functions {h (x )} n x,n y i=1, j=1 in (6) staced up as columns of n x n y elements as z and h(x ), respectively. The aim of estimation here is to find an estimate of the state x based on all the measurements up to time in the form of the conditional pdf p(x z ), where z = [z T, zt 1,...,zT ]T. This pdf completely describes the filtering estimate of x, conditioned by z. For the purpose of estimation, the state of the target x must be extended by the existence variable E representing the presence of the target, which is also unnown. The extended state x is defined as x = [x T, E ] T. Thus the state estimation problem of x changes to the hybrid estimation problem of x with continuous part x and discrete part E. The general solution to the state estimation problem is given by the Bayesian recursive relations which provide filtering estimate of the state x in the form of the filtering probability density function p( x z ) as p(x, E =1 z ) = p(z x, E =1)p(x, E =1 z 1 ) p(z z 1. ) (7) 271

3 The predictive pdf p(x, E =1 z 1 ) in (7) is given as p(x,e =1 z 1 ) = p(x, E =1 x 1, E 1 =1) p(x 1, E 1 =1 z 1 )dx 1 + p(x, E =1, E 1 = z 1 ) = p(x x 1, E =1, E 1 =1)[1 P d ] p(x 1, E 1 =1 z 1 )dx 1 + p b (x )P b. (8) The transition pdf p(x x 1, E = 1, E 1 = 1) in (8) is given by the state equation (1) as p(x x 1, E = 1, E 1 = 1) = N {x : Fx 1, Q} (9) and the measurement pdf p(z x, E =1) in (7) is given by the measurement equation (5) in the following form n x n y p(z x, E = 1) = N {z : h (x ), R} (1) i=1 j=1 p(z x, E = ) = p(z E = ) = n x n y = N {z :, R}. (11) i=1 j=1 Note that for the purpose of state x estimation by means of the it is suitable approximate the measurement pdf p(z x, E = 1) to reduce the computational requirements. As the target affects only the cells within the vicinity of its location (x, y ), it is possible to approximate the measurement pdf as p(z x, E = 1) i C(x ) j C(y ) i / C(x ) j / C(y ) N {z : h (x ), R} N {z :, R}, (12) where C(x ) and C(y ) represent sets of cell indices within the vicinity of the target position in the x direction and y direction, respectively. 3 Particle filter for trac-beforedetect The idea of the in nonlinear state estimation is to approximate the target filtering pdf p( x z ) by the empirical filtering pdf r N ( x z ), which is given by N random samples of the state { x (i) }N i=1 and associated weights {w ( x (i) )}N i=1, w ( ), N i=1 w ( x (i) ) = 1, as r N ( x z ) = N w ( x (i) )δ( x x (i) ), (13) i=1 where δ( ) is the Dirac function defined as δ(x) = for x = and δ(x)dx = 1. The general algorithm of the can be found in [9]. In the case of the TBD approach the particles are divided at each time instant into two groups, the alive particles, with E = 1, and dead particles, with E =. The state part x of the dead particles is not defined. For alive particles the state part x is drawn from the proposal densities π(x x (i) 1, E(i) 1 = 1, z ) for living particles (i.e. E (i) = 1 and E (i) 1 = 1) and π b(x E (i) 1 =, z ) for newborn particles (i.e. E (i) = 1 and E (i) 1 = ). Consider an empirical pdf r N ( x 1 z 1 ) with uniform weights {w 1 ( x (i) 1 )}N i=1 and N a out of N total samples being alive. For the purpose of the TBD approach, the fundamental steps of the general algorithm are modified as (see [1]-Ch.11): Sampling: Each of N a alive particles will die with probability P d and survive with probability 1 P d, i.e. The target existence variable of N a P d randomly chosen alive particles that will die will become zero, E (i) =. The remaining N a (1 P d ) alive particle will survive with E (i) = 1 and their state part x (i) is drawn from the proposal density π, i.e. x (i) π(x x (i) 1, E(i) 1 =1, z ) Each of (N N a ) dead particles will remain dead with probability 1 P b and be born with probability P b, i.e. The target existence variable of (N N a )(1 P b ) randomly chosen dead particles that will remain dead will be zero, E (i) =. The remaining (N N a )P b dead particle will be born with E (i) = 1 and the state x (i) will be drawn from the TI proposal density, i.e. x (i) π b (x E (i) 1 =, z ) Note that the existence variable part E (i) the corresponding state part x (i) x (i). together with form the new particle Weighting: The weights of the particles that survived or were born, i.e. with E (i) = 1, are computed as Survived particles w ( x (i) ) = p(z x (i), E(i) = 1)p(x (i) x(i) 1, E =1, E 1 =1) π(x (i) x(i) 1, E(i) 1 =1, z ) (14) 272

4 Newborn particles w ( x (i) ) = p(z x (i), E(i) = 1)p b (x (i) ) π b (x (i) E(i) 1 =, z ) (15) The weights of the particles that remained dead or have died are given by w ( x (i) ) = p(z E (i) = ). (16) All the weights are then normalized, i.e. w ( x (i) ) = w ( x (i) )/ N j=1 w ( x ( j) ), Resampling: Generate a new set { x (i) }N i=1 by resampling with replacement N times from { x (i) }N i=1 with probability P( x (i) = x(i) ) = w ( x (i) ) and set w ( x (i) ) = N 1. Replace the sets { x (i) }N i=1 and {w ( x (i) )}N i=1 by the resampled sets { x (i) }N i=1 and {w ( x (i) )}N i=1 respectively. To greatly simplify the relations for weight computation (14, 15 and 16), which can be demanding especially for large number of measurement cells, it is possible to utilize the approximation (12) of p(z x (i) = 1) as follows: All, E(i) the measurement pdf s in (14, 15 and 16) are divided by p(z E (i) = ) (11), i.e. they are replaced by the lielihood ratio l(z x (i), E(i) ) = p(z x (i), E(i) )/p(z E (i) = ). This implies the following simplifications. For survived and newborn particles (see [1]): l(z x (i), E(i) = 1) j) N {z(l, : h (l, j) (x (i) ), R} l C(x ) j C(y ) l C(x ) j C(y ) l C(x ) j C(y ) exp ( For dead particles: N {z(l, j) :, R} h(l, j) (x (i) )[h(l, j) (x (i) 2R = ) 2z(l, j) ] ). (17) l(z x (i), E(i) = ) = 1. (18) The estimate of probability of target existence is then given by Ni=1 E (i) ˆP = (19) N and the target state estimate by Ni=1 E (i) ˆx = x(i) Ni=1 E (i). (2) Choices of the proposal densities π and π b greatly influence performance of the. A survey of proposal densities can be found in [6]. The next subsection will discuss choice of proposal densities π(x x (i) 1, E(i) 1 = 1, z ) and π b (x (i) E(i) 1 =, z ) for the TBD approach. Choice of the proposal density In the TBD approach there are two proposal densities, π(x x (i) 1, E(i) 1 = 1, z ) for the surviving particles further denoted as transition proposal and π b (x (i) E(i) 1 =, z ) for the newborn particles, further denoted as TI proposal. As far as the transition proposal is concerned, the simplest choice of the proposal is the transition pdf (9) which does not involve the last available measurement z. In [5] the Rao-Blacwellisation (RB) technique was used to increase efficiency of the proposal density for slightly different sensor model. Further, to reduce the computational demands of the RB technique, which are high in this case as for each particle the proposal density must be calculated, gating techniques were utilized. In [7] it was argued that although the measurement can be used to improve the proposal, due to expected low target SNR s there is little to gain. As far as the TI proposal is concerned, a comparison of three proposal densities was presented in [7]. The first proposal distributes the particle uniformly in the state space, i.e. the TI proposal is equal to the prior TI pdf p b (x ). The second proposal uses available measurement so that particle positions are distributed uniformly within the highest intensity cells. The remaining particle components, i.e. velocities in both directions and intensity are sampled from the prior TI pdf. The third proposal represents a combination of the first two proposals, i.e. half of the particles are distributed uniformly using the prior TI pdf and the other half are places in the highest intensity cells. This paper deals with the TI proposal density only and the goal is to propose a sophisticated TI proposal which achieves high estimation quality with computational demands comparable to the three above mentioned proposals. As was mentioned in Introduction, to use more information from the measurement in the proposal, the composite approach has been chosen. The approach is based on choosing another nonlinear filtering technique and use it as a generator of the proposal density. Based on the comparison published in [6] and the nature of the TBD approach, the GM proposal [1] based on the GM method [11] has been chosen. 4 Gaussian mixture proposal density To design the GM proposal, it is necessary to specify the prior TI pdf p b (x ) in terms of the GM. The state part of each particle consists of position, velocity and intensity components which are considered to be independent for trac initiation. Therefore the prior TI pdf p b (x ) can be written as p b (x ) = p b (x, y )p b (ẋ, ẏ )p b (I ). As far as the velocity components are concerned, in the stage of trac initiation they do not influence the measurement and therefore will be drawn from the prior TI pdf p b (ẋ, ẏ ) and 273

5 omitted in the GM proposal. The other components, i.e. position and intensity, directly influence the measurement and therefore will be present in the GM proposal. The prior TI pdf p b (x, y ) for position components is considered to be uniform within the whole area covered by the measurement, i.e. [, x n x ] in the x-direction and [, y n y ] in the y-direction. The GM approximation p b (x, y ) of this prior TI pdf is given by a rectangular and equally spaced grid of N x N y points, which are distributed within the whole rectangular area covered by the measurement, representing predictive means and by corresponding diagonal predictive covariance matrices which are equal for all grid points. The uniform pdf p b (x, y ) may be approximated by p b (x, y ) up to arbitrary accuracy. The grid may either cover the whole area or, based on the idea used in [7], cover only the highest intensity cells. The choice of will be discussed in Section 5. As far as the intensity component I is concerned, its prior TI distribution is also uniform within the interval [I min, I max ] and therefore for the purpose of the GM proposal it will be approximated by a grid of N I equally spaced points representing predictive means and by corresponding variances, which are equal for all intensity grid points. The complete grid of predictive means is constructed by the Cartesian product of the position grid points and intensity grid points. The weights of all terms of the GM approximation are equal and the predictive pdf p(x, y, I ) is given as p(x, y, I ) = N N x N y N I i=1 x y I 1 N x N y N I ˆx i, : ŷ i,, Î i, P x, P y, P I,, (21) where [ˆx i, ŷ i, ]T is a position grid point with predictive covariance matrix x, [ P ] P y,, and Î i, is an intensity grid point with predictive variance P I,. For notational purposes the predictive covariance matrix in (21) will be denoted as P Ṫo compute the GM proposal density in the form N x N y N I p(x, y, I z ) = N x y I : N x N y N I ˆx i, ŷ i, Î i, i=1 α i,, P i,, (22) EKF s are used to compute the filtering means [ ˆx i,, ŷ i,, Î i, ] T and covariance matrices Pi, as ˆx i, ˆx i, ˆx ŷ i, = ŷ i, i, + K i, (z h ŷ Î i, Î i, i, ) (23) Î i, P i, =(I K i, H i, )P, (24) where K i, is the Kalman gain given by K i, = P HT i, (R n x n y + H i, P HT i, ) 1, (25) with R nx n y being n x n y n x n y dimensional diagonal matrix with variance R on its diagonal and H i, being the Jacobian of h ( ) at [ˆx i,, ŷ i,, Î i, ]T. The weights α i, are computed as ˆx i, α i, = N z : h ( ŷ i, ), R nx n y + H P Î i, i, HT. (26) and normalized so that N x N y N I i=1 α i, = 1. It is clear that running N x N y N I extended Kalman filters for n x n y dimensional measurements, even if they may be run in parallel, requires a large amount of computational resources. The following section discusses possibilities of reduction of the requirements. The GM (22) is used as the proposal for the newborn particles in the sampling stage of the. As the proposal (22) corresponds to the filtering pdf, the weights of the samples drawn from (22) have to evaluate only the discrepancy between the uniform prior p b (x ) and the predictive pdf (21), which is negligible for sufficiently large N x, N y and N I. This fact saves computational effort of the weighting step within the. 5 Computational efficiency There are several causes of high computational requirements of the. First, the number of measurements processed by each EKF is n x n y and consequently there is the same number of rows in the Jacobian H. Second, the number of predictive means in (21) is quite high when considering the whole position grid. Remedies to both causes of high computational requirements are discussed in the following subsections. 5.1 Reducing number of measurements processed As has been already mentioned, the target, if present, influences only a few cells within its vicinity. This means that the Jacobian H is sparse and so is the Kalman gain (25). Therefore the influence of the cells that are not in the vicinity of the target is negligible. Note that this fact was confirmed by experiments. The indices (l, j) of the cells at the vicinity of each predictive mean at (x i,, y i, ) position can 274

6 be specified as l C(x) = { x i, s,..., x i, + s}, j C(y) = { y i, s,..., y i, + s} where is the ceiling function, s is the user-specified parameter implying that each EKF now uses (2s + 1) 2 measurements instead of n x n y with (2s + 1) 2 n x n y. If the predictive mean lies close to the borders of monitored area, i.e. closer than s cells in arbitrary direction, the number of the measurements utilized by the corresponding EKF is even lower than (2s + 1) 2. This poses a problem when computing the weight α i, using (26) as its value strongly depends on the actual number of measurements used. To eliminate this dependency, the modified relation for weight α i, computation is used: α i, = Nz l C(x) j C(y) N : h (l, j) ( j) z(l, ˆx i, ŷ i, Î i, ), P z,i,, (27) where P z,i, = R nx n y + H i, P HT i, and N z [(s + 1) 2, (2s + 1) 2 ] is the actual number of measurement used for computation of this weight. 5.2 Reducing number of Kalman filters and precomputation When specifying the number of predictive GM terms in (21), it must be noted that use of too many terms is as performance-wise bad as that of too few terms [2] due to excessive competition from the unnecessary GM terms. In this case it is possible to reduce the number of position grid points using the idea borrowed from [7] based on choosing a specified number of the highest intensity cells. Similarly, in the case of GM proposal it is possible to consider only the position grid points located within the cells with highest intensity. A discussion of the choice of is a part of the numerical example. Another way of reducing the computational requirements of the GM proposal is to precompute as much terms as possible. The Kalman gain (25), filtering covariance matrix P i,, and the term R nx n y + H i, P HT i, are among the terms that can be computed in advance as they are independent of the measurement z. The filtering mean [ˆx i,, ŷ i,, Î i, ] T and the weight α i, must be calculated at each time step using (23) and (27), respectively. 6 Numerical example To assess performance of the and compare it with the with bootstrap TI proposal density based on distributing particles uniformly within the highest intensity cells, further denoted as the, both filters processed the same measurements. The comparison is based on 1 Monte Carlo experiments. As the and differ only in the TI proposal, the comparison is based on performance at the TI stage, i.e. when the target appears in the scene. For both the and N = 1 particles were used. The parameters of the measurement equation (5) defined as sizes of the cells are x = y = 1, the deviation of the target spread function =.7, the variance of the measurement noise R = 3. Each frame of data consisted of n x = n y = 64 cells in the x and y directions, the intensity I was considered within range [1, 3] which corresponds to the pea target strengths between approximately.69 and 1.23dB in SNR. Note that the whole range of intensity considered belongs to the TBD with low SNR. This is illustrated by the fact that even for the highest intensity considered (I = 3) due to the measurement noise in average 71 measurement cells have higher intensity than the cell containing the target. The true position of the target is drawn from the prior TI pdf p b (x, y ), i.e. uniformly within the whole region. As the velocity components of the state do not influence the measurement and are generated from their prior TI pdf for both and, they will not be considered in the experiments. The considered 4 position grid points within each measurement cell that was taen into account with their variance P x, = P y, =.5 and 6 intensity grid points with their variance equal to P I, = 4. They constitute the predictive GM. Experiment I This experiment deals with performance of the and for different SNR s. The performance was measured in terms of root mean square error (RMSE) of the position components and intensity component RM SE(x) = 1 1 (ˆx (m) x (m)) 1 2 (28) m=1 RM SE(I) = 1 1 (Î (m) I (m)) 1 2, (29) m=1 where ˆx (m) and Î (m) are estimates provided by the or according to (2) at the mth simulation. Note that error of the position in the y direction was not evaluated as it was equal to the error in the x direction. Both the and considered = 1 cells of the highest measurement intensities. The results are depicted in Fig. 1. As far as the position estimate is considered, the clearly outperforms the by approximately 1% for all SNR s. The intensity estimate of the achieves smaller error than that of the for larger SNR. For small SNR the intensity estimate is better. The experiments indicated that a significant increase of intensity estimate quality for low SNR can be achieved by pruning the insignificant terms from the GM proposal but this issue requires further analysis. 275

7 RMSE(I) = 1 RMSE(I) SNR = dB RMSE(x) SNR [db] SNR [db] Figure 1: RMSE of and for varying SNR Experiment II This experiment deals with performance of the and for different choices of. The results are depicted in Fig. 2 for 1.52dB SNR (I = 11), in Fig. 3 for 6.71dB (I = 2)and in Fig. 4 for 9.94dB SNR (I = 29). The shaded area highlights a usually considered range of. As far as quality of the intensity estimates of both and is considered, it can be seen that for low SNR (1.52dB) with increasing number of measurements ( ) the RMSE decreases for both and. According to the figure, the intensity estimate quality of is worse than that of the in this case but if the is very low (up to = 1), the outperforms the. For medium (6.71dB) and high (9,94dB) SNR s increasing brings decrease of the RMSE only for the first few measurements. Further increase of causes on contrary increase of the RMSE. This behavior of the intensity estimates is in accord with [2]. For high SNR only few of the highest intensity cell measurements brings useful information for both the and. The remaining measurements are unnecessary and pose excessive competition for the terms corresponding to the quality measurements. For low SNR each measurement cell brings only little information and no competition exists in this case. As far as position estimates are considered, for approximately < 2 RMSE of both and decrease with increasing with the achieving smaller error than the, whereas for approximately > 2 the RMSE of both and increases with increasing. This fact is also a manifestation of too many predictive terms. Computational requirements To compare computational requirements of the and, computational time was calculated for varying number of measurement cells with the highest intensity. The RMSE(x) Figure 2: RMSE of the and for varying and 1.52 db SNR RMSE(I) RMSE(x) SNR = 6.717dB Figure 3: RMSE of the and for varying and 6.71 db SNR experiments were executed within the MATLAB environment on a 2GHz PC. The results consisting in computational time of sampling 1 samples from the proposal densities of the (blue solid line), (red dashed line) and time spent within the for actual drawing samples and weighting, i.e. without calculating the GM terms by (22), (27), (green dotted line) are depicted on Fig. 5. The results show that for < 2 the achieves even lower computational complexity than the. For > 2 the is computationally more demanding than the due to increasing number of terms of the filtering pdf (22) that are computed. It must be pointed out that in this case the effect of the precomputation and the reduction of the number of EKF s proposed in Section 5 is reduction of computational demands by more than 87%. Also it should be noted that for the considering < 1 is usually sufficient for achieving high estimate quality. 276

8 RMSE(I) RMSE(x) SNR = 9.938dB Figure 4: RMSE of the and for varying and 9.94 db SNR time [s] SNR = 9.938dB sampling and weighting part of Figure 5: Computational time of the and for varying and 9.94dB SNR 7 Conclusion The paper focused on TI proposal density of the in the TBD approach to estimation of the target state. The aim was to utilize more information from the available measurement to increase quality of the particles drawn from the TI proposal in comparison with the usual bootstrap TI proposal. To compute the new proposal, the GM method based on a ban of EKF s was used. The quality of the target position estimate in terms of the RMSE achieved by the is better than that of the. Also quality of the target intensity estimate achieved by the is better than that of the (with the exception of SNR lower than 6dB which can be prevented by substantial pruning of the GM proposal). The computational efficiency of the was increased by choosing only small number of measurements with high intensity, limiting number of GM terms and precomputing as many terms of the EKF s as possible. The achieved results showed that with a careful choice of parameters the achieves higher quality of estimates with lower computational complexity than the. The future wor will be devoted to analysis of possibilities of GM pruning techniques which, according to experiments carried out, achieves better intensity estimates for very low SNR. 8 Acnowledgments The wor was supported by the Ministry of Education, Youth and Sports of the Czech Republic, project No. 1M572, by the Czech Science Foundation, project GA12/8/442 and by the University of West Bohemia, project POSTDOC-9. References [1] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracing Applications. Artech House, 24. [2] X. R. Li and Y. Bar-Shalom, Multiple-model estimation with variable structure, IEEE Transactions on Automatic Control, vol. 41, no. 4, pp , [3] A. Doucet, N. De Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice. Springer, 21, ch. An Introduction to Sequential Monte Carlo Methods, (Ed. Doucet A., de Freitas N., and Gordon N.). [4] Y. Boers and J. Driessen, Particle filter based detection for tracing, in Proceedings of the American Control Conference, 21, vol. 6, 21. [5] A. Ooi, A. Doucet, B. N. Vo, and B. Ristic, Particle filter for tracing linear Gaussian target with nonlinear observations, in Proc. SPIE, vol. 596, 23. [6] O. Straa and M. Simandl, Sampling densities of particle filter: a survey and comparison, in Proceedings of the 26th American Control Conference (ACC). New Yor: AACC, 27, pp [7] M. G. Rutten, B. Ristic, and N. Gordon, A comparison of particle filters for recursive trac-before-detect, in 7th Internatinal Conference on Information Fusion (FUSION), 25, pp [8] D. Musici, R. Evans, and S. Stanovic, Integrated probabilistic data association, IEEE Trans. on Automatic Control, vol. 39, no. 6, pp , [9] A. Doucet, N. De Freitas, and N. Gordon, Eds., Sequential Monte Carlo Methods in Practice. Springer, 21, (Ed. Doucet A., de Freitas N., and Gordon N.). [1] J. H. Kotecha and P. M. Djuric, Gaussian sum particle filtering, in IEEE Transactions on Signal Processing, ser. 51, no. 1, 23. [11] K. Ito and K. Xiong, Gaussian filters for nonlinear filtering problems, IEEE Trans. on Automatic Control, vol. 45, no. 5, pp ,