Tracking of Extended Object or Target Group Using Random Matrix Part I: New Model and Approach

Transcription

1 Tracing of Extended Object or Target Group Using Random Matrix Part I: New Model and Approach Jian Lan Center for Information Engineering Science Research School of Electronics and Information Engineering Xi an Jiaotong University, Xi an, , P. R. China X. Rong Li Department of Electrical Engineering University of New Orleans New Orleans, LA 70148, U.S.A. Abstract The approach of using a random matrix for extended object and group target tracing (EOT and GTT) is efficient. Designing and effectively applying this approach rely on modeling the extended object and group targets accurately. To describe complex dynamical variation and practical observation distortion of the extension in size, shape and orientation, two random matrix based models are proposed. True measurement noise can also be incorporated into the measurement model easily. Facilitated by special properties of the models, an approximate Bayesian approach is proposed to estimate the inematic state and the extension jointly. For maneuvering EOT and GTT, a multiple-model approach is derived by moment matching. To evaluate what is proposed, simulation results of a scenario for maneuvering EOT are also given. The results illustrate the effectiveness of the proposed models and approach. Index Terms Extended Object, Group Target, Random Matrix, Bayesian Approach, Multiple-Model Approach. I. INTRODUCTION Target tracing has been researched over the past several decades. Many theories and techniques have been proposed and developed [1]. Due to limited sensor resolution and large sensor error relative to the target size, most tracing applications consider a target as a point source of measurements [3] [9] [10] [18] [13]. Kinematic state (i.e., position, velocity, acceleration, etc.) of a target is estimated, ignoring details including size, shape and orientation, which simplifies the problem due to practical limitations and theoretical considerations. Nevertheless, target tracing is still complicated because of the uncertainties in target motions and measurements [0]. With increased resolution capabilities of modern sensors (e.g., video cameras, phased array radar), treating an object as a point mass becomes less valid [18] [13]. Thus, traditional methods can not be applied directly. Moreover, modern applications require more and more detailed physical information about the objects for detection, tracing, classification, identification, etc. In this case, an object is treated as an extended object (EO) with size, shape and orientation. Also, for a group of closely spaced targets in formation, tracing applications are facing similar challenges [18] []. Knowing size, shape and Research supported in part by ARO through grant W911NF , ONR-DEPSCoR through grant N , Doctoral Program of Higher Education of China through grant and the Fundamental Research Funds for the Central Universities of China. orientation of group formation can be beneficial for practical applications, e.g., recognition and classification. Thus, both the inematic and extension information of an extended object or group targets (GT) may be needed in tracing applications. Such an extended object or target group may also originate a strongly fluctuating number of measurements [9] [18], due to limited sensor capability, physical properties of EO/GT, and sensor-to-target geometry. Further, one-to-one correspondence between a measurement and an originating point can not be observed directly and easily. In this case, scattering centers of an object and individual targets within a group are also only partially resolvable, which maes the tracing problem even more challenging. Several approaches have been proposed for extended object tracing (EOT) and group target tracing (GTT) [10] [19] [9] [18] [13], e.g., multiple hypothesis tracing (MHT) [17], probability hypothesis density (PHD) filters [8] [] [1], and particle filters [6] [7] [14] [15]. A so-called random hypersurface model was also proposed to describe an extended object [] [3] [4], where varying measurement sources are assumed to lie on scaled versions of the shape boundaries [4]. Based on this nonlinear model, a nonlinear filter is used for state and shape estimation. An approach of using a random matrix initiated by Koch [18] seems promising because it tries to estimate the inematic state and physical extension in such a way that the extended object or target group can be treated as one object. Another merit is that the final form of this approach is much simpler than the methods mentioned above. This approach estimates at time both the random vector x representing the inematic state of the object centroid and a symmetric positive definite (SPD) random matrix X representing the object extension (simplified as an ellipsoid with size, shape and orientation). The random-matrix approach is a Bayesian solution for the tracing problem, with the underlying assumption that the sensor error (noise) is negligible compared with the object extension, as pointed out in [13]. Feldmann et al. [11] [1] [13] proposed an extension for the case when the sensor error is not negligible. However, if the measurement noise is considered, the Bayesian solution cannot be derived easily [13], which maes the approach of [13] comparatively more heuristic and non-optimal to some extent. 177

2 Furthermore, the above approach did not consider the following problems adequately: a) Time Variation: The extension of the object can also change in shape, size (for groups) and/or orientation over time. For a practical extended object, its extension changes during maneuver, e.g., adjusting orientation to the direction of velocity especially when it performs a turn. For group targets, the situation is more complicated since they can adjust their positions without much restriction to change the overall extension largely. Considering this phenomenon for EOT and GTT can improve estimation performance. b) Distortion: In practice, observations of an extended object may be distorted from the true extension in shape, size and orientation, depending on the sensor-to-target geometry. In this case, without calibration the estimated extension based on the observations can be far from the true one. However, the existing random-matrix-based approach may not handle these two problems directly. In [18], measurements with small errors which are negligible compared with extension are processed [13], without considering observation distortion directly. Although [13] considers the actual measurement errors and the uncertainties caused by the extension separately, time-variant observation distortion is not considered. As such, the approach of [13] cannot be derived based on Bayes formula and thus it was proposed based on approximations [13]. Extension changes on object sizes are considered in the approach, but changes on object shape and orientation are not modeled explicitly. Thus, some further information about the latter may not be utilized effectively. In general, to handle the distortion problem, practical observation scheme needs to be incorporated into the measurement model for estimation. That is, the distortion of X caused by the current sensor-to-target geometry which relate to x (including the current object position) needs to be considered. For the time-variation problem, it is much better to model extension changes accurately for estimation, given more information about extended objects and group targets. But this information cannot be incorporated easily. Actually, due to the properties of the SPD matrix X and its connections to the ellipsoid, distortion and changes of extension can be generally described by a matrix B, with the required extension as B X B T. Changes or distortion of extension can be described as accurately as possible in terms of entries of B. Unlie [13], estimation can be done approximately based on Bayes formula using this modeling for either maneuvering or non-maneuvering extended object and group targets. In view of this, we propose two models describing dynamic changes and distortion of extension. Then we propose a Bayesian approach for EOT and GTT. For maneuvering EOT and GTT, a multiple-model (MM) Bayesian approach is derived. The effectiveness of our models and approach is demonstrated by some illustrative simulations, compared with the existing random-matrix-based approach. This paper is organized as follows. Section II proposes two models describing extended objects and group targets and a Bayesian approach for EOT and GTT based on them, considering actual measurement noise and the distortion and timevariance problems. Section III proposes an MM approach for maneuvering EOT and GTT. Section IV presents simulation results and Section V concludes the paper. II. A NEW BAYESIAN APPROACH FOR EOT AND GTT A. EOT and GTT Using Random Matrix For EOT/GTT, using a random matrix to describe object extension was first proposed in [18]. It estimates jointly the inematic state represented by a random vector x and the object extension (size, shape and orientation) represented by an SPD d d random matrix X where d is the dimension of the space for EOT/GTT (e.g., d =,3 means two- or three-dimensional space). The extension is thus simplified as an ellipsoid surface defined by (y H x ) T X 1 (y H x ) = 1 (1) where y is the variable representing the points on the surface of the ellipsoid, H is the measurement matrix converting the object state to the object location. The dynamic model for state x is assumed as [18] x = Φ x 1 +w, w N(0,D X ) () where Φ = F I d with F being the dynamic matrix in the one-dimensional space (e.g., the Singer model used in [18]), I d R d d is the identity matrix, stands for the Kronecer product, N(µ, Σ) denotes the normal distribution with mean µ and covariance matrix Σ, D = σ D is the covariance matrix of the independent Gaussian process noise w in the one-dimensional model, σ is the variance of the acceleration noise, and D is a coefficient matrix. This model indicates that the centroid state x is affected by the extension X. Let Z = {z r }n r=1 denote a set of n vector-valued position measurements at time. In [18] and [13], the measurement model is assumed as z r = H x +v r (3) where H = H I d with H = [1 0 0] being the measurement matrix in the one-dimensional space (assuming that only positions are measured and the state in one-dimensional space is [position, velocity, acceleration] T ), v r is the Gaussian white noise independent of v t (r,t = 1,...,n,r t). For other types of measurements, see [18] and [13] for more details. It is further assumed [18] that the Gaussian measurement noise has v r N(0,X ) (4) Under the assumption that the random number n of measurements is independent of x and X, the lielihood function of x and X given Z can be obtained as [18] p[z n,x,x ] N( z ; H x,x /n )W( Z ;n 1,X ) (5) where W(Y;a,C) is the density of the Wishart distribution for the SPD random matrix Y R d d, defined as W(Y;a,C) = 1 c C 1 a Y 1 (a d 1) etr( 1 C 1 Y) (6) 178

3 with a > d 1. Here etr( ) stands for exp(trace( )), c is the normalization factor (c and c will always be normalization factors in the sequel), and z = 1 n n r=1 z r, n Z = (z r z )(z r z ) T (7) r=1 Equation (5) with a separated form for x and X is the ey to deriving an Bayesian estimator for EOT and GTT in [18]. Actually, (4) indicates that the true measurement noise is negligible compared with X. [13] discovered this and pointed out that it is better to consider both the true measurement error and the uncertainties in X simultaneously. It assumed the measurement noise as v r N(0,λX +R ) (8) where λ is a scalar describing the effect of X, R is the covariance matrix of the true measurement error. Equation (8) indicates that the measurements have two sources of uncertainties: extension and the true measurement noise. Although (8) seems more practical than (4), it is not as handy as (5) for estimation. Under assumption (8), a recursive Bayesian estimator lie that of [18] cannot be derived directly [13]. Thus, [13] proposed approximations leading to the loss of optimality. Assuming the extension tends to be time invariant, [13] and [18] adopt some direct equations with justification to describe extension evolution (see equations (8), (9) in [13] and (45), (46) in [18] for details). [18] further points out that the following prior model can also be used to describe extension evolution: p[x X 1 ] = W ( X ;δ 1,X 1 /δ 1 ) (9) These methods can be easily adopted for EOT and GTT. However, they actually consider only the evolution of the object size, not shape nor orientation (e.g., δ 1 in (9) is a scalar). As analyzed above, the two models differ from each other mainly in their basic assumptions, especially of the measurement models. Model (4) is simple and maes the estimation easy. Model (8) is more practical but much harder to handle, leading to a comparatively more heuristic approach. B. New Models for EOT and GTT Using Random Matrix Different extension models may go along with different estimation methods. The above analysis illustrates that for EOT/GTT, it is better for the models developed to have a) a general form capable of describing practical phenomena and dynamics of EO and GT easily; b) the potential to develop Bayesian estimation conveniently. Specifically, the models need to describe a) the dynamics of extension evolution and b) the observations distorted from the true extension in size, shape and orientation. For these two purposes, we propose the following two models, respectively. 1) Extension dynamic model: The dynamic model for extension evolution is assumed as p[x X 1 ] = W(X ;δ,a X 1 A T ) (10) where δ > d 1 is the degrees of freedom, and matrix A R d d describes the specific evolution mode. Remar 1: (a) From (10) and the properties of the Wishart distribution, it follows that [16] E[X X 1 ] = δ A X 1 A T (11) This illustrates: i) scalar δ can describe the dependence of the extension on size over time; ii) A can describe the dependence of the extension on orientation (if A is a rotation matrix), size (e.g, A = λi d ), or shape (if A is some other matrix). Note that (10) with A = I d / δ and δ = δ 1 reduces to (9). (b) As the degrees of freedom, δ also describes the uncertainty of this evolution [18]. ) Measurement model: To describe the observation distortion of extension, the measurement model in (3) is used with v r N(0,B X B T ) (1) where B R d d is a matrix describing distortion of the observed extension (embodied by the covariance matrix of multiple measurements) from the true one. Remar : (a) Model (1) can also incorporate (8) approximately. As in [13], assume X E[X Z 1 ] = X 1,Z 1 = {Z 1,...,Z 1 }. Then we have λx +R = (λx +R ) 1 1 X X X T (λx +R ) T B X B T (13) where B (λ X 1/ 1 +R ) X 1/ 1. (Note that X 1 = ˆX 1 /(ˆv 1 d ) if we assume p[x Z 1 ] IW(X 1 ;ˆv 1, ˆX 1 ). See (0) (1).) (b) Directly using noise model (8) is difficult to derive a Bayesian estimator in a simple form [13] mainly because the corresponding lielihood function cannot be factorized into terms for x and X as in (14). Using model (1) is much easier. Specifically, the lielihood here can be written as p[z n,x,x ] = Π n r=1 N(zr ; H x,b X B T ) N( z ; H x, B X B T n )W( Z ;n 1,B X B T ) (14) Equation (14) can be used to derive the Bayesian estimator easily, to be demonstrated later. (c) B in (1) can also represent the distortion of the observed extension from the true extension in size, shape and orientation, just lie A in (10). When the distortion is induced by sensor-to-target geometry and so B is a function of x (denoted as B(x )), this model can also be used by letting B (x ) B (ˆx 1 ) with the predicted state ˆx 1 = E[x Z 1 ]. 179

4 C. New Bayesian EOT and GTT The above new models are more general than those of [18] and [13] to some extent. Furthermore, their special forms can facilitate Bayesian EOT and GTT. For example, the lielihood (14) based on (1) can be handled easily for Bayesian estimation. We derive the Bayesian estimator utilizing the special forms of the new models. The purpose of Bayesian estimation is to obtain the conditional probability density function (pdf) p[x,x Z ], which can be written by Bayes formula as p[x,x Z ] = 1 c p[z x,x,z 1 ]p[x,x Z 1 ] (15) Since the lielihood p[z x,x,z 1 ] has been obtained by (14), we discuss only the predicted pdf p[x,x Z 1 ]. Under the assumption that the temporal change of the object extension has no impact on the prediction of inematic states (as given in [18]), i.e., p[x 1 X,Z 1 ] = p[x 1 X 1,Z 1 ], this predicted density p[x,x Z 1 ] = p[x, X,Z 1 ]p[x Z 1 ] (16) can be given by the following independent integrations [18]: p[x X,Z 1 ] = p[x X,x 1 ]p[x 1 X 1,Z 1 ]dx 1 (17) p[x Z 1 ] = p[x X 1 ]p[x 1 Z 1 ]dx 1 (18) Then based on (14), (15) and (16), p[x,x Z ] can be obtained. Details are discussed next based on the derivation process given in [18]. 1) Prediction: Assume [18] p[x 1,X 1 Z 1 ] = p[x 1 X 1,Z 1 ]p[x 1 Z 1 ] = N(x 1 ;ˆx 1,P 1 X 1 )IW(X 1 ;ˆv 1, ˆX 1 ) (19) where IW(Y; a, C) is the inverted Wishart (IW) density, defined as IW(Y;a,C) = 1 c C a d 1 Y a etr( 1 CY 1 ) (0) with a > d being the degrees of freedom and E[Y] = C/(a d ), a d > 0 (1) How to obtain the conditional pdf p[x,x Z 1 ] based on (17) and (18) under assumption (19) is discussed next. a) Kinematic Part: Since assumption () is the same as that in [18], p[x X,Z 1 ] can also be calculated similarly as [18] p[x X,Z 1 ] = N(x ;ˆx 1,P 1 X ) () where ˆx 1 and P 1 are given in Table I. b) Extension Part: Under assumption (10), p[x Z 1 ] can be calculated based on model (17) as p[x Z 1 ] = p[x X 1 ]p[x 1 Z 1 ]dx 1 (3) = GB II d (X ;δ /,(ˆv 1 d 1)/;A ˆX 1 A T,0) where GB II d ( ) is the so-called Generalized Beta Type II (GBII) distribution [16] [18]. To obtain recursive estimation, we consider approximating this GBII distribution with an inverted Wishart distribution by moment matching similarly as in [18]: p[x Z 1 ] IW(X ;ˆv 1, ˆX 1 ) (4) with ˆv 1 and ˆX 1 shown in Table I. It thus follows from (16) that p[x,x Z 1 ] =N(x ;ˆx 1,P 1 X ) IW(X ;ˆv 1, ˆX 1 ) (5) ) Update: Given n, (14) and (5), the conditional pdf p[x,x Z ] is given by (15) with p[x,x Z ] = p[z x,x,z 1 ]p[x,x Z 1 ]/c N( z ; H x,b X B T /n )W( Z ;n 1,B X B T ) N(x ;ˆx 1,P 1 X )IW(X ;ˆv 1, ˆX 1 ) (6) The final form can be obtained by factorizing the right side of the above equation, as given next following [18]. a) Kinematic Part: Based on the derivation of the Kalman filter for the Gaussian case, the product of the two Gaussian pdfs in (6) can be factorized as N( z ; H x,b X B T /n )N(x ;ˆx 1,P 1 X ) =N( z ; H x 1,S 1 X )N(x ;ˆx,P X ) (7) where ˆx and P are given next. To simplify the estimation process, we adopt the following approximation: B X B T γ X (8) where γ is a scalar and can be given by setting the determinants of both sides to be identical: B X B T = γ X γ = B B T 1/d = B /d (9) Treat z as the measurement, H = H I d the measurement matrix,b X B T /n the measurement noise, and ˆx 1 the predicted mean and P 1 X the predicted covariance matrix. Then based on (8) and (9), ˆx and P in (7) are given in Table I. 180

5 b) Extension Part: Substituting (7) into (6) yields approximately p[x,x Z ] N(x ;ˆx,P X )N( z ; H x 1,S 1 X ) W( Z ;n 1,B X B T )IW(X ;ˆv 1, ˆX 1 ) (30) The last three pdfs on the right side of (30) can be further combined as IW(X ;ˆv, ˆX ) with ˆv and ˆX given in Table I. Then, p[x,x Z ] =p[x X,Z ]p[x Z ] N(x ;ˆx,P X )IW(X ;ˆv, ˆX ) (31) The detailed derivation of the above results is omitted due to space limitation. The estimation process (for one cycle) is summarized in Table I. State Extension Table I THE NEW BAYESIAN APPROACH The new approach ˆx 1 = (F I d )ˆx 1 P 1 = F P 1 F T +D ˆx = ˆx 1 +(K I d )G P = P 1 K S 1 K T S 1 = H P 1 H T + B /d n K = P 1 H TS 1 1 G z (H I d )x 1 E[x Z ] = ˆx ˆX 1 = δ (ˆv λ 1 1 d )A ˆX 1 A T λ 1 = ˆv 1 d ˆv 1 = δ (λ 1 1)(λ 1 ) +d+4 λ 1 (λ 1 +δ ) ˆX = ˆX 1 +N 1 +B 1 Z B T ˆv = ˆv 1 +n N 1 = S 1 1 G G T E[X Z ] = ˆX /(ˆv d ) Remar 3: (a) Our approach differs from that of [18] in the extension prediction and the update of both centroid state and extension. (b) Consider the formulation of measurement noise in [13]. According to approximation (13), S 1 in Table I can be written as S 1 = H P 1 H T + B /d /n = H P 1 H T + λi d +R X d /n (3) When the true measurement noise is negligible compared with the extension (i.e., R X X 1 ) and thus R X 1 1 0, we have S 1 H P 1 H T + λ/n. If we further let λ = 1, S 1 in (3) reduces to that in [18]. This justifies the validity of the new approach to some extent, when the underlying assumption in [18] is satisfied. (c) WithB = (λ X 1/ 1 +R ) X 1/ 1 as in (13),S 1 of (3) incorporates the true measurement noise represented by R. Utilizing this information about the true noise, the new approach can thus estimate the centroid state accurately. Furthermore, extension X can also be estimated accurately because the true measurement noise (represented by R ) is incorporated into the estimation process for X, since ˆX = ˆX 1 +N 1 +B 1 Z B T. (d) Here, the prediction ( ˆX 1 and ˆv 1 ) of the extension is different from that in [13]. Using this new approach, more prior information about extension dynamics can be incorporated to improve the estimation results (since p[x X 1 ] = W(X ;δ,a X 1 A T ), using appropriate δ and A can describe different cases accurately). III. MULTIPLE-MODEL APPROACH FOR MANEUVERING EOT AND GTT When an extended object (target group) maneuvers, both inematical state and extension may undergo abrupt changes. For maneuvering EOT and GTT (MEOT and MGTT), a multiple-model (MM) approach for state and extension estimation is derived next. In [13], an MM approach for MEOT and MGTT was proposed by integrating the corresponding single-model approach with the interacting multiple-model (IMM) algorithm [5]. The obtained MM approach is simple and direct. However, this MM approach is heuristic to some extent, especially for calculating lielihood functions. In view of this, we propose a new MM approach for MEOT and MGTT based on the models and the estimation approach given in Section II. In practice, motions of maneuvering EO/GT may be complicated. To describe practical motions accurately, a hybrid system framewor is powerful and convenient. For EO/GT, the following hybrid system is considered x =Φ j x 1 +ω j, ωj N(0,Dj X ) z r = H j x +υ r,j, υr,j N(0,B j X (B j )T ) p[x X 1 ] = W(X ;δ j,aj X 1(A j )T ) (33) where Φ j = Fj I d, Hj = Hj I d, j = 1,...,N is the index of model j, N is the number of models, r = 1,...,n, and n is the number of measurements received at time. In the following parts, m j denotes model j or the event that it is in effect at time. Here, the purpose of estimation is to obtain the conditional pdf p[x,x Z ], which can be expanded as p[x,x Z ] = N p[x,x m j,z ]P{m j Z } (34) j=1 where by Bayes formula p[x,x m j,z ] = (c ) 1 p[z x,x,m j,z 1 ]p[x,x m j,z 1 ] (35) Then the MM state estimation based on the hybrid system (33) can be given next. 181

6 A. The Interacting MM approach for EOT 1) Reinitialization: Based on the total probability theorem, the conditional pdf of predicted state and extension becomes p[x,x m j,z 1 ] = p[x,x x 1,X 1,m j,z 1 ] with p[x 1,X 1 m j,z 1 ]dx 1 dx 1 (36) p[x 1,X 1 m j,z 1 ] = i p[x 1,X 1 m j,mi 1,Z 1 ]λ i j p[x 1,X 1 m i 1,Z 1 ]λ i j i N(x 1 ;ˆx j,0 1,Pj,0 1 X 1)IW(X 1 ;ˆv j,0 j,0 1, ˆX 1 ) (37) where p[x 1,X 1 m i 1,Z 1 ] = N(x 1 ;ˆx i 1,Pi 1 X 1 )IW(X 1 ;ˆv 1 i, ˆX 1 i ) calculated at 1, and λi j P{m i 1 mj,z 1 } can be calculated easily as in IMM. The approximation in the last line of (37) follows from moment matching, given in the next subsection. Based on (36) and (37), we have p[x,x m j,z 1 ] p[x,x m j,{ˆxj,0 1,Pj,0 1,ˆvj,0 j,0 1, ˆX 1 }] (38) where ˆx j,0 1,Pj,0 1,ˆvj,0 j,0 1, ˆX 1 are the parameters of the pdf in (37) and are used to replace Z 1 given m j. This approximation reinitializes m j by interacting the previous estimation results, similar to IMM. ) Filtering: Once p[x,x m j,z 1 ] is given by (38), p[x,x m j,z ] can be obtained based on (35) by the approach of Section II, as shown in Table I. 3) Probability Update: µ j P{mj Z } = 1 c Λ j P{mj Z 1 } (39) where P{m j Z 1 } can also be calculated as in IMM, and the lielihood can be given as Λ j p[z m j,z 1 ] = p[z x,x,m j,z 1 ]p[x,x m j,z 1 ]dx dx = p[z m j,x,x ]p[x,x m j,z 1 ]dx dx (40) Substitution of (14) and (38) into (40) yields Λ j = (n (d 1) π nd n ) 1 Γd [ aj +n B j (Bj )T 1 n S j 1 1 ˆXj 1 a j ]Γ 1 d [aj ] ˆX a j n (41) where a j = ˆvj 1 d 1. The proof of (41) is omitted due to space limitation. 4) Fusion: Given µ j and p[x,x m j,z ], the overall conditional pdf p[x,x Z ] can be obtained as in (34). As pointed out in the reinitialization part, this fusion process can also be completed by moment matching, illustrated next. B. Moment Matching Moment matching is important for the MM estimation, e.g., as in (37) for interaction and in (34) for fusion. Moment matching provides a convenient solution to the approximation problem N N(x ;ˆx j,pj X )IW(X ;ˆv j, ˆX j )µj (4) j=1 N(x ;ˆx,P X )IW(X ;ˆv, ˆX ) (43) Because x is a random vector while X is an SPD random matrix, the moment matching is more difficult than in a typical MM solution, e.g. IMM. Here, we consider using the following staced matrix for moment matching [ ] X e [x,0 (sd) (d 1) ] (44) X where s is the dimension of the state in one-dimensional space and thus x R (sd) 1. Then moment matching can be done for X e easily, as summarized in Table II. The derivation is omitted due to space limitation. State Table II MOMENT MATCHING FOR MM ESTIMATION N j=1 N(x ;ˆx j,pj X )IW(X ;ˆv j, ˆX j )µj N(x ;ˆx,P X )IW(X ;ˆv, ˆX ) The moment matching approach ˆx = ˆx m = N j=1 ˆxj µj P = [p i,j ] s s ([p i,j ] s s ) T p i,j = 1 d d l=1 q (i 1)d+l,(j 1)d+l [q l,h ] sd sd = [ v (I s ˆX )P x,m (I s ˆX )] 1 P x,m = N j=1 µj [Px,j +(ˆx j ˆxm )( )T ] P x,j = (P j ˆX j )/(ˆvj +bmse ) v = ˆv +b MSE, b MSE = s d sd 3 Extension ˆv = 1 (A + A 8(B +))+d ˆX = (ˆv d ) X m X m = N X j j=1 µj, X j = ˆX j /(ˆvj d ) P X,m = N j=1 µj [PX,j +( X j X m)( )T ] A = B +C +5 B = [tr( X m)] /tr(p X,m ),C =tr[( X m) ]/tr(p X,m ) Remar 4: (a) The above MM approach can be directly derived based on the total probability theorem, because the single-model (approximately) Bayesian approach proposed before provides the joint distribution of state and extension. (b) In the MM approach, the lielihood function is different from that in [13] because it was generated by Gaussian pdfs in [13] intuitively, but derived here using Bayes formula. (c) Because of the properties of the proposed models, the MM approach can be used for MEOT and MGTT with extension changing in size, shape and orientation, which is

7 more general than the approach proposed in [13], which considers only abrupt change of extension size. IV. SIMULATION RESULTS In this part, the new MM approach is compared with the approaches proposed in [18] (Koch) and [13] (Feldmann) by simulation. The scenario of [13] for maneuvering extended object tracing was simulated. The extended object was an ellipse with diameters 340m and 80m (about the size of an aircraft carrier of the Nimitz-class, see [13]). The trajectory is shown in Figure 1, where the speed was assumed constant at 7 nots (about 50 m/h). y direction x direction Figure 1. Trajectory of the extended object It is assumed that scattering centers were uniformly distributed over the extension. And the true measurement noise was distributed as zero-mean Gaussian with variance R. Thus, the assumed measurement noise (used for estimation) is distributed as N(υ r;0,λx + R ) with λ = 1/4 and R = diag([50,50 ]) m. The number of measurements in each scan was Poisson distributed with mean 0. The compared approaches are designed as follows. (a) In Koch s (single-model) approach, τ = 8T (T is the sampling period andτ is a temporal decay constant describing extension evolution, see equations (8) and (9) in [18] for details) and other parameters were the same as in [18]. (b) The parameters of Feldmann s (MM) approach were designed following [13]. (c) In the first version of the proposed MM approach (MM1), 3 models were adopted: m 1 with low inematical process noise and low extension agility (small D, large δ ), m with high inematical process noise and high extension agility (large D, small δ ), m 3 with moderate inematical process noise and high extension agility (median D, small δ ), and A = I d /δ 1/,B = (λ X 1/ 1 +R ) X 1/ 1 withλ = 1/4 for uniformly distributed targets (see [13]). (d) In the second version of the proposed MM approach (MM), 3 models were adopted: m 1 is the same as in MM1; m and m 3 with high inematical process noise and high extension agility (larged, small δ ), and A j is a rotation matrix with A j = 1 [ ] cosθ j sinθ j δ j sinθ j cosθ j (45) where θ = 10π/180 rad and θ 3 = 10π/180 rad. Other parameters of MM are the same as MM1. The comparison results are the RMS errors over N s = 500 Monte Carlo runs. The RMS errors of extension estimation were calculated as follows [13]. RMSE X = ( 1 N s tr[( N X l X ) ]) 1/ (46) s l=1 where l means the l th run, and X l is the estimated mean of X on run l. The comparison results are shown in Figures 5. They show that the proposed MM approach has better performance than Koch s (single-model) approach, especially for extension estimation. The comparison clearly shows that the new approach can utilize the prior information about the true measurement noise effectively. Compared with Feldmann s (MM) approach, the new approach also has better performance for both centroid state and extension estimation. Especially during maneuvers, MM has better performance than the others in position, velocity and extension estimation. The average probabilities of MM in Figure 5 also illustrates the validity of the proposed MM approach. Note that models m and m 3 in MM cannot be used directly in Feldmann s MM approach. position RMSE velocity RMSE Koch s approach Feldmann s approach MM1 MM time 5 Figure. RMS errors of position (m) Koch s approach Feldmann s approach MM1 MM time Figure 3. RMS errors of velocity (m/s) The comparison demonstrated the effectiveness of the proposed models and approach for EOT. 183

8 extension RMSE average probabilities in MM x 10 4 Koch s approach Feldmann s approach MM1 MM time Figure 4. RMS errors of extension (m ) model 1 model model time Figure 5. Average probabilities of MM V. CONCLUSION For EOT and GTT, the paper first proposed two randommatrix-based models describing evolution and observation distortion of object extension, respectively. Based on the models, an approximate Bayesian approach for the estimation of inematic state and extension has been proposed. Compared with the existing random-matrix approach, the new approach can model extension dynamics and measurements more generally and effectively. When true measurement noise is negligible compared with extension, the new approach degenerates to Koch s approach. If it is not negligible, the new approach can still estimate the extension effectively. Based on the proposed Bayesian approach, an MM approach for maneuvering EOT and GTT has also been derived by using moment matching. Simulation for maneuvering EOT and GTT demonstrated the effectiveness of the proposed models and approach, compared with the existing ones. In general, the main advantages of our wor are the flexibility of describing and the effectiveness of estimating an extended object or target group with complicated dynamics and observation schemes. REFERENCES [1] Y. Bar-Shalom and X. R. Li, Estimation and Tracing: Principles, Techniques, and Software. Boston, MA: Artech House, (Reprinted by YBS Publishing, 1998). [] M. Baum and U. D. Hanebec, Random Hypersurface Models for Extended Object Tracing, in Proceedings of the 9th IEEE International Symposium on Signal Processing and Information Technology, Ajman, United Arab Emirates, Dec. 009, [3] M. Baum, B. Noac, and U. D. Hanebec, Extended Object and Group Tracing with Elliptic Random Hypersurface Models, in Proceedings of the 13th International Conference on Information Fusion, Edinburgh, United Kingdom, July 010, pp [4] M. Baum and U. D. Hanebec, Shape Tracing of Extended Objects and Group Targets with Star-Convex RHMs, in Proceedings of the 14th International Conference on Information Fusion, Chicago, Illinois, USA, July 011, pp [5] H. A. P. Blom and Y. Bar-Shalom, The Interacting Multiple Model Algorithm for Systems with Marovian Switching Coefficients, IEEE Transactions on Automatic Control, vol. 33, no. 8, pp , [6] A. Carmi, S. J. Godsill, F. Septier, Evolutionary MCMC Particle Filtering for Target Cluster Tracing, in IEEE 13th DSP Worshop and the 5th SPE Worshop, Marco Island, Florida, USA, Jan. 009, pp [7] A. Carmi, F. Septier, and S. Godsill, The Gaussian Mixture MCMC Particle Algorithm for Dynamic Cluster Tracing, to appear in Automatica, 01. [8] D. Clar and S. Godsill, Group Target Tracing with the Gaussian Mixture Probability Hypothesis Density Filter, in Proceedings of the 3rd International Conference on Intelligent Sensors, Sensor Networs and Information, Melbourne, Australia, Dec. 007, pp [9] F. E. Daum and R. J. Fitzgerald, Importance of Resolution in Multiple- Target Tracing, in Proceedings of the SPIE Conference on Signal and Data Processing of Small Targets, Orlando, Florida, USA, July 1994, pp [10] J. Dezert, Tracing Maneuvering and Bending Extended Target in Cluttered Environment, in Proceedings of the SPIE Conference on Signal and Data Processing of Small Targets, Orlando, Florida, USA, Apr. 1998, pp [11] M. Feldmann and D. Franen, Tracing of Extended Objects and Group Targets Using Random Matrices A New Approach, in Proceedings of the 11th International Conference on Information Fusion, Cologne, Germany, June July 008, pp [1] M. Feldmann and D. Franen, Advances on Tracing of Extended Objects and Group Targets Using Random Matrices, in Proceedings of the 1th International Conference on Information Fusion, Seattle, Washington, USA, July 009, pp [13] M. Feldmann, D. Franen, and W. Koch, Tracing of Extended Objects and Group Targets Using Random Matrices, IEEE Transactions on Signal Processing, vol. 59, no. 4, pp , 011. [14] K. Gilholm, S. Godsill, S. Masell, and D. Salmond. Poisson Models for Extended Target and Group Tracing. in Proceedings of the SPIE Conference on Signal and Data Processing of Small Targets, San Diego, USA, Sep. 005, Vol. 5913, pp [15] K. Gilholm and D. Salmond, Spatial Distribution Model for Tracing Extended Objects, IEE Proceedings on Radar, Sonar and Navigation, vol. 15, no. 5, pp , 005. [16] A. K. Gupta, and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC, 000. [17] W. Koch and G. van Keu, Multiple Hypothesis Trac Maintenance with Possibly Unresolved Measurements, IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 3, pp , [18] J. W. Koch, Bayesian Approach to Extended Object and Cluster Tracing Using Random Matrices, IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no. 3, pp , 008. [19] X. R. Li and J. Dezert, Layered Multiple-Model Algorithm with Application to Tracing Maneuvering and Bending Extended Target in Clutter, in Proceedings of the 1st International Conference on Information Fusion, Las Vegas, NV, USA, July 1998, pp [0] X. R. Li and V. P. Jilov, Survey of Maneuvering Target Tracing-Part I: Dynamic Models, IEEE Transactions on Aerospace and Electronic Systems, vol. 39, no. 4, pp , 003. [1] R. Mahler, PHD Filters for Nonstandard Targets, I: Extended Targets, in Proceedings of the 1th International Conference on Information Fusion, Seattle, Washington, USA, July 009, pp [] B.-T. Vo, B.-N. Vo, and A. Cantoni, Bayesian Filtering with Random Finite Set Observations, IEEE Transactions on Signal Processing, vol. 56, no. 4, pp , 008. [3] M. J. Waxman and O. E. Drummond, A Bibliography of Cluster (Group) Tracing, in Proceedings of the SPIE Conference on Signal and Data Processing of Small Targets, Orlando, Florida, USA, Apr. 004, pp