Korea

A Probabilistic Strongest Neighbor Filter Algorithm for m Validated Measurements Taek Lyul Song Kye Jin Rhee Dong Gwan Lee Hanyang university Dep. of Control and Instrumentation Engineering 1271 Sa-1-dong, Sangnok-gu, Ansan-si, Gyoenggi-do, 425-791 Korea tsong@hanyang.ac.kr LG Innotek 148-1, Mabuk-ri, Gusung-eup, Yongin-si, Gyoenggi –do, 449-910 Korea Hanyang university Dep. of Control and Instrumentation Engineering 1271 Sa-1-dong, Sangnok-gu, Ansan-si, Gyoenggi-do, 425-791 Korea jbvq@ihanyang.ac.kr efg016@hanmail.net Abstract - The measurement with the strongest signal amplitude in the validation gate is known as the strongest neighbor (SN) measurement. A standard Kalman filter that utilizes the SN at any time as if it is originated from the true target is called the strongest neighbor filter(SNF). Inconsistency of handling the SN as if it is true target is corrected in the existing probabilistic strongest neighbor filter(PSNF) which accounts the probability that the SN is from the true target. It is known that performance of the PSNF is superior to the SNF at a cost of increased computational load. In this paper, we propose a new probabilistic strongest neighbor filter that takes into account the current number of validated measurements in the derivation of probability density functions for the SN which are needed to establish probability weightings and estimation error covariance. The proposed algorithm does not involve infinite summation while the existing PSNF algorithm contains infinite summation that requires approximation for practical usage. Performance of the proposed filter is compared with the existing filters such as the SNF and the PSNF through a series of Monte Carlo simulation runs for aerial target tracking in clutter. The advantages of the new filter in practical applications are studied via analysis and simulation. Keywords: SNF, PSNF, PSNF-m, data association, target tracking, clutter, performance analysis. 1 Introduction Target tracking in a clutter environment requires accurate association of target with a measurement for track maintenance. The existing probabilistic strongest neighbor filter (PSNF)[1] accounts for the probability that the strongest neighbor (SN) in the validation gate is originated from the target while the strongest neighbor filter (SNF)[2] assumes at any time that the SN measurement is target-originated. It is known that the existing PSNF is superior to the SNF in performance at a cost of increased computational load. However, the computational complexity is lower than the probabilistic data association filter (PDAF)[3], which accounts for the probability that each measurement in the validation gate is target-originated. This paper proposes a new form of the PSNF, called the PSNF-m, which takes into account the probability that the SN measurement selected among the m validated measurements in the validation gate is targetoriginated. The concept is similar to [9] which incorporates the number of validated measurements into design of the probabilistic nearest neighbor filter. The PSNF-m considers the current number of measurements in the validation gate whereas the probability in the PSNF is calculated by applying the order statistic and total probability theorem so that it is an averaged value considering all the possible events related to the number of measurements. The proposed PSNF-m is shown to have a similar computational load to an approximated or of the PSNF established by truncating the infinite summation term, and they have similar target tracking performance in clutter. Moreover, performance of the PSNF-m is less sensitive to the spatial clutter density. This fact provides substantial benefit in practice since the clutter density is either time-variant or space-variant and not known exactly beforehand. The performance of the PSNF-m is compared with the PSNF as well as the SNF based on the results of a series of Monte Carlo simulation runs. Sensitivities to the spatial clutter density are tested for the PSNF and the PSNF-m by employing mismatched values for the true and guessed densities. 2 The existing PSNF The SNF assumes that the strongest neighbor (SN) measurement in the validation gate is originated from the target of interest and the SNF utilizes the SN in the update step of a standard Kalman filter (SKF). The SNF is widely used along with the nearest neighbor filter (NNF) [4], due to computational simplicity in spite of its inconsistency of handling the SN as if it is the true target. The PSNF utilizes the SN in the update step however, it accounts for the probability that the SN is target-oriented such that the target state estimate as well as estimation error covariance is updated with a probabilistic weighting factor. It is known that the PSNF is superior in performance to the SNF at a cost of more involved computation. The validation gate used is the ellipsoid: Rγ (k ) = {ν k ;ν kT S k−1ν k ≤ γ } (1) where ν k is a zero-mean Gaussian residual with covariance of S k for the true measurement, and γ is called the gate size. The volume of the n-dimensional gate Rγ (k ) satisfies 1 n VG = Cn S k 2 γ 2 A8) The target is existing and can be detectable, i.e., it is perceivable [5]. For the SNF and the PSNF, there exist the following three events related to data association with the SN measurement. (2) M 0 : There is no validated measurement; where C 1 = 2 , C 2 = π , C 3 = 43 π etc. The validated measurements consist of n-dimensional location information z and signal amplitude information a . The following assumptions are used in this paper. A1) The true target signal amplitude is the magnitudesquare output of a matched filter so that the signal is χ 2 -distributed with probability density function (pdf) a 1 − 1+ ρ (3) f1 (a ) = e 1+ ρ where ρ is the signal-to-noise ratio. The clutter signal amplitude satisfies f 0 (a) = e − a . M T : The SN measurement is originated from the target; M F : The SN measurement is from a false target. The algorithm of the PSNF derived in [1] is summarized for reference. The PSNF algorithm 1) Prediction step identical to the SKF 2) Update step (a) For the case of M 0 Xˆ k = X k PD PG (1 − CTg ) Pˆk = Pk , M 0 = Pk + KSK T 1 − PD PG (4) A2) The number of validated true measurement is denoted by mT , and mT is at most 1. The probability that mT = 1 is P( m T = 1) = PD PG where PD is the probability of target detection indicating that the target signal amplitude exceeds a threshold τ ; and PG is the probability that the target falls inside the − τ validation gate. PD satisfies PD = e 1+ ρ from A1), and the probability that the false measurement signal exceeds the threshold τ is Pfa = e −τ . A3) The number of validated false measurements in the validation gate, denoted by m F , is Poisson distributed with a spatial density λ such that µ F ( m) = P ( m F = m) = A7) Amplitude is independent of the location. (λVG ) m −λVG e . m! (5) A4) The state prediction error ek = xk − xk for any given time k is a zero-mean Gaussian process with a covariance Pk such that ek ~ N (ek ;0, Pk ) . (b) For the case of M 0 Xˆ k = X k + Kβ 1ν  PD PG PA (1 − CTg )  β 0 − β1  KSK T + β1 β 0 Kνν T K T Pˆk = Pk +   1 − PD PG PA  1 PA = ( I A − PD e −λVG ) PD (1 − e −λVG ) IA = ∫ ∞ τ e a 1 − 1+ ρ e da = PD 1+ ρ ∞ ∑ j =0 (−λVG ) j j !(1 + j (1 + ρ )) f ( D , a, M T ) β1 = , β 0 = 1 − β1 f ( D , a, M F ) + f ( D , a, M T )  e− a    a − λVG  Pfa  nV 1 − 1+ ρ f ( D, a , M T ) = D N ( D )e   e 1(a − τ ) 2D 1+ ρ a −  e− a nV f ( D, a, M F ) = D (1 − PG e 1+ ρ )λ   Pfa 2D   e− a   − λVG  Pfa  1(a − τ )  e  In the above, K is the filter gain and CTg satisfies CTg = A5) The validated false measurements at any time are i.i.d. uniformly distributed over the gate. A6) The location and amplitude of a validated false measurement are independent of the true measurement at any time and other validated false measurements at any other time.  e− a   − λVG   Pfa    ∫ n∫ γ n − q q 2 e 2 dq 0 γ (6) n −1 − q q 2 e 2 dq 0 − γ γ − γ hence, CTg = (1 − e 2 (1 + )) /(1 − e 2 ) for n = 2 . D is the 2 normalized distance squared (NDS) of the SN measurement with location information z * , so that D = ν S ν , and VD = Cn S *T −1 ellipsoid * 1 2 D n 2 is the volume of the with the gate size D such as T −1 RD = {ν ;ν S ν ≤ D} , 1(x) is the unit step function defined as 1 if x ≥ 0 , 0 for elsewhere, and N (D) is the Gaussian pdf of ν * expressed by using the NDS D such that N ( D) = e 3 RD −D 2 / 2πS . One of the drawbacks of the PSNF is in the calculation of I A the unconditional probability that the SN is targetoriginated, and it is recommended to use interpolation from a tabulated I A versus λVG for various signal-tonoise ratios or to use approximation with the first L terms. In the section a PSNF algorithm which accounts for the number of validated measurements in the update step is derived, and it turns out that the new algorithm is computationally cheaper than the exact from of the existing PSNF with a similar performance in a clutter environment. Under the conditions that the number of validated measurements is m and the SN measurement is targetoriented, the conditional pdf (cpdf) of the signal amplitude a satisfies (conditioning on the sequence of sets of the past measurements is omitted for brevity) f ( a | M T , m) = 1 f ( a, M T , m) . P ( M T , m) (7) Theorem 1: With Assumptions A1) ~ A3), f (a | M T , m) is given by −a e m−1 1 PG (1 − ) f1 (a) µ F (m − 1) (8) P ( M T , m) Pfa and the joint probability P ( M T , m) can be obtained as P ( M T , m ) = PD PG PA µ F ( m − 1) 1 PD ∫ τ ∞ f1 (a )(1 − (9) −a e m−1 ) da Pfa (10) where PA is the probability that the validated target amplitude a T = a is the strongest among the m validated measurements under the assumptions of m T = 1 and m F = m −1 . Proof: See Appendix. Note that for m = 1, PA P ( M T , m) = P (m = 1) = PD PG . T f ( a | M F , m) = 1 f ( a, M F , m) ( P M F , m)  (1 − PD PG ) f cl (a | m) µ F (m) 1   a P ( M F , m) + P ( P − e − 1+ ρ ) f (a | m − 1) µ (m − 1) cl F   G D (11) where f cl (a | m) denotes the cpdf of amplitude a of the clutter-originated SN measurement under the assumption that the number of validated false measurements m F = m , and f cl (a | m) satisfies f cl ( a | m ) = m e −a P fa  e −a 1 −  P fa      m −1 is 1 such (12) , and P ( M F , m) of (11) is expressed as P ( M F , m) = (1 − PD PG ) µ F ( m) + PD PG (1 − PA ) µ F ( m − 1) . (13) Proof: See Appendix. Remark 1: The probability that the number of validated measurements in the validation gate at any time can be obtained by utilizing (9) and (13) as P ( m) = P ( M T , m) + P ( M F , m ) Consider the following Theorem. PA = Theorem 2: With Assumptions A1) ~ A3), f (a | M F , m) is given by = The PSNF-m f ( a | M T , m) = For f (a | M F , m) , one can state the following Theorem. = (1 − PD PG ) µ F (m) + PD PG µ F (m − 1) . (14) The updated error covariance matrices for M 0 and M T of the PSNF-m are identical to the ones for the PSNF however, the error covariance for M F is different. It is required to derive the cpdf of Dt , the NDS of the target, under the assumptions of M F and m validated measurements for the error covariance calculation. Since the target under consideration is perceivable by Assumption A8), the target is not temporarily obscured and it is not proper to use merely the predicted error covariance for M F . Under the perceivability assumption, the following events may occur for M F : 1) The target may be located in the validation gate and detected but the signal amplitude a T is not the strongest among the m validated measurements. 2) The target may be detected but it may not be in the validation gate. 3) The target may not be detected. that Theorem 3: With Assumptions A1) ~ A4), A6), and A8) the cpdf f ( Dt | M F , m) is obtained by f ( D t | M F , m) =  nVDt  N ( D t )((1 − PD 1(γ − D t ))µ F (m)  t  2D  + P (1 − P )1(γ − D t )µ (m − 1)  A F  D  (b) For the case of M 0 Xˆ k = X k + K β1ν * (1 − PD PG )µ F (m) + PD PG (1 − PA )µ F (m − 1) (15) Pk , M F = Pk − KSK T + Proof: See Appendix. Theorem 4: With Assumption A1) ~ A4), A6) and A8), the updated error covariance for M F is given by Pk ,M F = Pk − KSK T + (1 − PD PG CTg )λVG + PD PG CTg (1 − PA )m (1 − PD PG )λVG + PD PG (1 − PA )m m −1 PA = 1 + Remark 2: PA can be evaluated from (14) as a function of m , the number of validated measurements, such as i =1 m −1 i 1 . (i + 1) + i ρ (17) PA does not involve infinite summation as seen in the PSNF algorithm so that the PSNF-m is computationally cheaper without approximation. The probability weighting for M T is evaluated from the a posteriori probabilities as β1 = P( M T | D, a, m) = f ( D, a, M T , m) f ( D, a, M T , m) + f ( D, a, M F , m) (18) where by Assumptions A5) and A7) f ( D , a, M T , m ) = f ( D | a , M T , m) f ( a , M T , m) = f ( D | M T ) f ( a, M T , m ) Note that f ( D | M T ) = N ( D) / PG , f ( D | M F ) = 1/ VG , and f (a, M T , m) and f (a, M F , m) are expressed in (8) and (11) respectively. The proposed algorithm of the PSNF- m is summarized below. 1) Prediction step identical to the SKF 2) Update step (a) For the case of M 0 Xˆ k = X k PD PG (1 − CTg ) Pˆk = Pk ,M0 = Pk + KSK T 1 − PD PG 1 ( PA = 1 for m = 1) (i + 1) + i ρ  e −a PD N ( D) f 1τ (a)1 −  P fa      4 Simulation results Monte Carlo simulation results of 2-dimensional aerial target tracking in clutter are presented to demonstrate the performance of the proposed PSNF-m by comparison with the SNF and the PSNF. The initial position of the target is (7Km, 4Km) and the target is initially moving in a straight line with a speed of 380m/s with 60 o of heading angle from the Y-axis. The target is susceptible to lateral maneuver with an acceleration of AT during tracking. The Singer model [6] is employed for target acceleration dynamic equation for the filters such as (19) f ( D , a, M F , m ) = f ( D | M F ) f ( a, M F , m ) The PSNF-m algorithm m−1 i   −a   λ (1 − PD PG ) f 0τ (a)1 − e    P fa        a − 1+ ρ 1   τ )(m − 1) f 0 (a)  + PG ( PD − e VG    −a     e   + PD N ( D) f 1τ (a )1 −   P fa      f (a) f (a) β 0 = 1 − β1 , f1τ (a ) = 1 , f 0τ (a) = 0 . PD Pfa expected. i i β1 = Note that in the case of m = 0 , Pk ,M F becomes Pk ,M 0 as ∑ (−1) C ∑ (−1) C i =1 KSK T Proof: See Appendix. m −1 (1 − PD PG )λVG + PD PG (1 − PA )m KSK T T Pˆk = Pk ,M F (1 − β1 ) + ( Pk − KSK T ) β1 + β 0 β1 Kν *ν * K T (16) PA = 1 + (1 − PD PG CTg )λVG + PD PG CTg (1 − PA )m  O2 x& = O2  O 2  I2 O2 O2  O2  O2    w  I 2  x + O2   X    wY  1 − I 2   I 2  τ  (20) where x is composed of target position, velocity, and acceleration components, 1 / τ is a bandwidth of target acceleration and τ = 5 (sec) . I 2 and O2 represent 2 × 2 identity matrix and zero matrix, respectively. The process noise ( wX , wY )T is a zero-mean white Gaussian noise vector with the power spectral density of 1.6 × 10 −4 I 2 m 2 / sec5 . The location information z corrupted by a measurement noise vector can be described by z k = ( I 2 , O2 , O2 ) x + υ k (21) where υ k is a zero-mean white Gaussian noise vector sequence with covariance of (20) 2 m 2 I 2 . The sampling frequency for target tracking is chosen to be 10 Hz . Table 1 is a summary of track loss percentages obtained from 500 runs of Monte Carlo simulation resulting from employing the SNF, the PSNF, and the PSNF-m for the cases of fixed ρ = 10, γ = 9, AT = 0 and varying PD and λ . Track loss is declared if the position estimation error in the X or Y axis exceeds 10 times the standard deviation of the measurement noise. The results indicate that the performance of the SNF gets worse for lower PD and larger λ while the PSNF and the PSNF-m show similar and excellent tracking performance. Simulation results with different parameter sets indicate similar characteristics. Table 1: Track loss percentage PD 0.7 0.8 0.9 λ SNF PSNF PSNF-m 0.00005 10.4 0 0 0.0001 13.4 0 0 0.00015 17.4 0 0 0.0002 22.6 0 0 0.0003 23.0 0 0 0.00005 6.4 0 0 Table 2: Sensitivity to λ in terms of track loss percentages λˆ (×10−4 ) 0.5 1.0 1.5 2.0 4.5 6.5 7.5 PSNF 18.6 17.8 13.4 11.2 13.8 12.4 11.6 PSNF-m 20.6 16.0 14.2 9.0 7.2 5.4 7.4 PSNF 40.6 29.4 25.6 23.4 22.4 22.2 21.8 PSNF-m 39.4 26.0 22.4 22.8 14.0 10.6 10.8 Case I Case II 5 Conclusions A new PSNF, called the PSNF-m, based on the current number of validated measurements in the gate is derived in this paper. Simulation results show the PSNF-m has similar performance to the existing PSNF with less computational complexity. It is found that the PSNF-m is less sensitive to the spatial clutter density due to the fact that the number of measurements in the current validation gate used in the algorithm takes the number of clutters into account and thus it makes the algorithm more adaptable to the current clutter environment. Therefore, the PSNF-m has advantages in practical applications for which the exact density is not known beforehand or it is not fixed in time and space. 0.0001 9.2 0 0 0.00015 13.4 0 0 0.0002 14.8 0 0 0.0003 16.0 0 0 0.00005 2.8 0 0 0.0001 5.4 0 0 Under M T , the number of validated true measurement m should be 1 and m F = m − 1 . From the Bayes’ rule [8], f ( a, M T , m) of (7) becomes 0.00015 9.6 0 0 0.0002 11.2 0 0 0.0003 13.4 0 0 Since it is hard to choose the correct λ for a clutter environment in practice, many on-line estimation algorithms for λ have been suggested [7]. However they may cost an increased computational load, and convergence may take too long for environment changes. The results of sensitivity to λ for the PSNF and the PSNF-m are summarized in Table 2. The true λ is chosen as λ =0.00015 while each filter uses a guessed value λ̂ . Target undergoes lateral 2g-maneuver at t = 3 sec in this case and the zero-mean process noise vector is modeled with the power spectral density of 2.13I 2 m 2 / sec5 to accommodate target maneuver. Table 2 idicates the track loss percentages obtained from 500 Monte Carlo simulation runs for case I and case II. For case I, PD =0.7, ρ =10 and various values of λ̂ are used while PD =0.7, ρ =5 are used for case II. The results show that the PSNF- m is less sensitive to λ̂ , which indicates that the PSNF-m has substantial advantages in practical usage. Appendix A. Proof of Theorem 1 T f ( a, M T , m) = P ( M T | a T = a, m T = 1, m F = m − 1) × f (a | m T = 1, m F = m − 1) P (m T = 1) P (m F = m − 1) (A-1) where the first term of the RHS implies the probability that all the m − 1 false measurements in the validation gate have signal amplitudes larger than the threshold τ and smaller than a T = a ,  e −a P ( M T | a T = a, m T = 1, m F = m − 1) = 1 −  Pfa      m −1 (A-2) where the probability that a validated false measurement has signal amplitude smaller than a T = a is equal to 1− e −a is used. f (a | mT = 1, m F = m − 1) in (A-1) is equal to Pfa 1 f1 (a ) and P(mT = 1) = PD PG , P(m F = m − 1) = µ F (m − 1) PD from Assumptions A2) and A3). Therefore, inserting (A1) into (7) becomes f ( a | M T , m) =  e− a 1 PG 1 − P( M T , m)  Pfa where P ( M T , m) = ∫    m−1 f1 ( a) µ F (m − 1) (8) ∞ τ Similarly, f cl (a | m F = m) is obtained by replacing m to m − 1 from (12). Note that P( M F | a F = a, mT = 0, m F = m) = 1, P(m T = 0) = 1 − PD PG , P(m T = 1) = PD PG . P( M F | a F = a, m T = 1, m F = m − 1) in the second term of the RHS of (A-5) is rewritten as f (a, M T , m)da . If we denote PA as P ( M F | a F = a, mT = 1, m F = m − 1) = P(aT < a | mT = 1) the probability that the validated true measurement has the largest signal amplitude among the m validated measurements, PA becomes PA = ∫ ∞ τ = 1 PD ∫ τ a f1 (a )da a f (a F < aT = a | mT = 1, m F = m − 1)da = 1− (A-3) 1 −1+ ρ e PD (A-6) F where a is the signal amplitude of any clutter and f ( a F < a T = a | m T = 1, m F = m − 1) = P(a F 〈 a | m F = m − 1) f (a T = a | m T = 1)  e −a = 1 −  Pfa      (A-4) m −1 Therefore, from (A-6), (12) and (A-5), f (a | M F , m) satisfies (11) and P( M F , m) is obtained from f (a, M F , m) of (A-5) by 1 f1 (a ) . PD P ( M F , m) = Inserting (A-4) into (A-3) leads P ( M T , m) in (8) to satisfy P ( M T , m ) = PD PG PA µ F ( m − 1) . (9) ∫ τ ∞ f (a, M F , m)da (A-7) which results in (13). C. Proof of Theorem 3 Under the perceivability assumption A8), the cpdf of D , the NDS of target, conditional on M and m becomes ( Dt is denoted as D for brevity) t B. Proof of Theorem 2 Under M F , m F ≥ 1 and m T = 1 or 0 among the m validated measurements. By the Bayes’ rule, f (a, M F , m) in (11) f (a, M F , m) = P( M F | a F = a, m T = 0, m F = m) f cl (a | m F = m) f ( D | M F , m) = nV D 1 N ( D) P ( M F , m) 2 D  P 1( D − γ ) µ F (m) + PD (1 − PA )1(γ − D) µ F (m − 1)   ×  D   + (1 − PD ) µ F (m)  (A-8) × P(m T = 0)µ F (m) + P(M F | a F = a, m T = 1, m F = m − 1) f cl (a | m F = m − 1) × P(m T = 1)µ F (m − 1) (A-5) where f cl (a | m F = m) represents the cpdf of a , the signal amplitude of SN measurement associated with a clutter conditioned on m , the number of validated measurements. f cl (a | m F = m) is given by, where the three terms of the RHS represent the event 2), the event 1), and the event 3) of Section Ⅲ, respectively. nVD Note that is the Jacobian used to express the pdf 2D of the NDS D associated with the target. By using 1( D − γ ) = 1 − 1(γ − D ) , and inserting P( M F , m) of (13) to (A-8), (15) is obtained. D. Proof of Theorem 4 The cpdf of ek , f (ek | M F , m) for the corresponding covariance Pk ,M F can be expressed by using the residual f cl ( a | m F f 0 (a )  e − a m! = m) = 1− 1! (m − 1)! P fa  P fa     m −1 (12) ν t of the true target measurement ( denoted here as ν for brevity ) f (ek | M F , m) = ∫ f (e k Ων | ν , M F , m) f (ν | M F , m)dν (A-9) Therefore, Pk ,M F satisfies, [ Pk , M F = E ek e | M F , m T k ∫e e = k T k Note that ] f ( e k | M F , m ) de k (A-10) Ω ek ∫ ∫e e = k T k f (ek | ν , M F , m ) f (ν | M F , m ) dek dν nVDt t N ( Dt ) is the pdf of the NDS 2D Dt = ν T S −1ν where ν is a Gaussian process such that nVDt ν ~ N (ν ; 0, S) . N ( Dt )1(γ − Dt ) is equivalent to 2 Dt N (ν ;0, S ) defined inside the validation gate only. From (15), f (ν | M F , m) can be obtained as Ων Ω ek Note that f (ek | ν , M F , m) = N (ek ; Kν , Pk − KSK T ) [2] and the cpdf uses ν only regardless of the underlying event M F and m . Hence f (ν | M F , m) = ( 1 N (ν ;0, S ) P ( M F , m) × (1 − PD 1(ν ; Rγ )) µ F (m) + PD (1 − PA )1(ν ; Rγ ) µ F (m − 1) ) (A-18) ∫e e T k k f (ek | ν , M F , m)dek = Pk − KSK T + Kνν T K T . (A-11) Ωe where 1(ν ; Rγ ) is a multivariate unit step function defined as 1 for ν ∈ Rγ , 0 for elsewhere. Applying (A-18) to the last term of (A-13) and utilizing (A-14) ~ (A-17), one can obtain By inserting (A-11) to (A-10), Pk ,M F becomes, Pk ,M F = ∫ (P − KSK T k + Kνν T K T ) f (ν | M F , m)dν (A-12) Pk,MF = Pk − KSKT Ων where f (ν | M F , m) can be obtained from f ( D | M F , m) of (15) by change of variables. (A-12) id further proceeded to t Pk ,M F = Pk − KSK T + ∫ Kνν T K T f (ν | M F , m)dν (A-13) Ων since integrating f (ν | M F , m) over Ων results in 1, the ν - + (1− PD PGCTg )µ F (m) + (1− PD PG CTg (1− PA )µ F (m −1)) (1− PD PG )µ F (m) + (1− PD PG (1− PA )µ F (m −1)) KSKT (A-19) which is further reduced to (16) by using µ F (m) of (5). References [1] X. R. Li and X. Zhi. PSNF: A refined strongest independent matrix Pk − KSK T of (A-12) can be placed neighbor filter for tracking in clutter. Proceedings of the 35th CDC, pp.2557-2562, Kobe Japan, Dec. 1996. outside the integral. Now, we would like to apply the same method as [2] to evaluate the integral term of (A-13). [2] X. R. Li. Tracking in clutter with strongest neighbor measurements-Part I: Theoretical analysis. IEEE Prerequisite equations are summarized as Trans. on Automatic Control, 43(11):1560-1578, 1998. T T [3] Y. Bar-Shalom and T. E. Fortmann. Tracking and Kνν K N (ν ; 0, S )dν Data Association. Academic Press, New York, 1988. VG [4] S. S. Blackman. Multiple-Target Tracking with 1 ν T S −1ν N (ν ;0, S )dν KSK T = (A-14) Radar Applications. Artech House, 1986. n VG [5] N. Li and X. R. Li. Target perceivability and its applications. IEEE Trans. on Signal Processing, q γ n  1  nC − =  n +1 n n q 2 e 2 dq  KSK T , 49(11):2588-2604, 2001. n  2 2 π 2 0  [6] T. L. Song, J. Y. Ahn, and C. Park. A suboptimal filter design with pseudomeasurements for target Kνν T K T N (ν ;0, S ) dν = KSK T , (A-15) tracking. IEEE Trans. on Aerospace and Electronic Ων Systems, 24(1): 28-39, 1988. γ n −1 − q nCn [7] X. R. Li and N. Li. Integrated real-time estimation of PG = N (ν ;0, S )dν = n +1 n q 2 e 2 dq , (A-16) clutter density for tracking. IEEE Trans. on Signal 2 2 2 π 0 VG Processing, 48(10):2797-2805, 2000. [8] A. Papoulis and S. U. Pillai. Probability, Random Variables, and Stochastic Processes, 4th ed. The and CTg of (6) satisfies McGraw Hill Co., 2002. [9] T. L. Song and S. J. Shin. A probabilistic nearest neighbor filter for m validated measurements. ν T S −1ν N (ν ; 0, S )dν Proceedings of the 6th International Conference on V (A-17) CTg = G . Information Fusion, Cairns, Queensland, Australia, nPG 8-11 July 2003. ∫ ∫ ∫ ∫ ∫ ∫ ∫