A Multi Scan Clutter Density Estimator

Transcription

1 A Multi Scan Clutter ensity Estimator Woo Chan Kim, aro Mušici, Tae Lyul Song and Jong Sue Bae epartment of Electronic Systems Engineering Hanyang University, Ansan Republic of Korea and Hanwha Institute of core technology team Hanwha Corp. efence R& center Sensor Systems epartment Republic of Korea Abstract ata association attempts to discriminate between the target and the clutter measurements, usually calculating the posterior probabilities of measurement origins. The clutter (spurious) measurements are random and (we presume) follow the Poisson distribution. The Poisson distribution is nonhomogeneous and is parametrized by intensity (the clutter measurement density). The clutter measurement density is almost always a priori unnown, and is often non stationary. Here we propose a measurement oriented clutter density estimator with probability hypothesis density (PH) filtering which integrates information from single-scan spatial clutter density estimator, and can follow and smooth non-stationary clutter information. Keywords Target tracing, false trac discrimination, PH, clutter measurement density estimation, data association. I. INTROUCTION Target tracing attempts to establish the presence and the trajectories of targets in the surveillance space, in the presence of uncertain target measurements (detections) and in the presence of spurious (clutter) measurements. The clutter measurements are random with an (assumed) non homogeneous Poisson distribution [1] in the measurement space. The Poisson distribution is parametrised by the non uniform intensity, aptly termed clutter measurement density by the target tracing community. ata association is required in situations where target tracing is being attempted using measurements of uncertain origin [2]. Automatic target tracing includes trac initiation and the false trac discrimination (FT) [2] [5]. The FT requires a trac quality measure such as the trac score [6] or target existence probability [4], [7] [9]. The data association probabilities, as well as both measures of trac quality mentioned above, depend on the measurement lielihood ratio. For each measurement the measurement lielihood ratio is calculated by dividing the the measurement lielihood and the clutter measurement density. Thus the clutter measurement density is an important parameter. A clear majority of target tracing publications assumes a nown, stationary and uniform clutter measurement density. However, the clutter measurement density is almost always a This wor was supported by efense Acquisition Program Administration and Agency for efense evelopment under the contract U priori unnown, non-uniform, and often non stationary; i.e. it changes with time and propagates in space. Improving the clutter measurement density information translates into improved target tracing performance. Single scan clutter measurement density estimators estimate the clutter measurement density based on the current set of measurements. We may group them into trac oriented [4], [10] and measurement oriented [11]. The trac oriented methods estimate the clutter measurement density for each trac separately using the selected measurements within the trac selection gate. The problems with this approach are that the clutter measurement density within a selection gate is assumed uniform, and that different tracs obtain different clutter measurement estimates for the same measurements. The measurement oriented method [11] estimates clutter density using measurement information only. It is trac independent and each measurement has its own density value. The probability hypothesis density (PH) filter [12] [14] is often used for the target state density estimation. The PH filter is designed to estimate densities, so its use in clutter measurement density estimation is only natural. The PH filter integrates information over multiple scans, thus this is multi scan clutter measurement density estimation. A recursive Maximum Lielihood clutter measurement density estimator using the PH filter is proposed in [15]. It presents clutter measurement density by a dynamic set of Normal-Wishart components. In this paper we also propose the clutter density map estimator based on the PH filter. At each scan the input to the PH filter is a single scan clutter density measurement represented by a Gaussian Mixture, with parameters obtained from the single scan measurement oriented clutter density estimator proposed in [11]. The applicable clutter models are presented in Section II. The interaction between the clutter density estimator and the target tracing is illustrated in Section III. Section IV presents the single scan spatial clutter measurement density estimator (SCME). Proposed clutter map estimator obtained by adapting the PH filter is shown in Section V. The approach is vindicated by simulation studies in Section VI followed by concluding remars in Section VII.

2 II. CLUTTER MOEL The vast majority of the target tracing publications assume The point targets: Each target may create zero or one detection per scan, and The infinite sensor resolution: Each measurement has only one source, either clutter or one target. A direct consequence of the infinite resolution sensor assumption is the independence of clutter measurements in disjoint volumes of surveillance space, from which the Poisson distribution of clutter measurements follows directly [16]. The clutter measurement densityρ(y) is non uniform; i.e. it is a function of the coordinate y in the measurement space. The clutter measurement density is non stationary; i.e. it changes its value in time, and may also move in the measurement space. As an example, consider the effects of wind on the clouds, or on the surface of the sea, or vegetation movement when subjected to the gusts of wind. To account for the clutter measurement density dynamics, and following the approach of [15], we model the clutter measurement density analogous to the target state density. We model the clutter as a field of clutter measurements sources, which we term scatterers. Each scatterer is detected, i.e. creates a detection at each scan, with a probability of clutter detection. Each scatterer has a state with both position and velocity components (and even more complex models if deemed necessary), and ρ(x) becomes a function of clutter state space x. Each scatterer propagates by x = Fx 1 +ν 1 (1) where F is the propagation matrix and the scatterer plant noise sequence ν is assumed zero mean, white and Gaussian with covariance matrix Q, independent between scatterers. Without loss of generality, we assume here that the projection from the state space x to the measurement space y is linear, with the projection matrix H. It is important to clarify that these scatterers may or may not be attached to constant physical objects. Consider two examples. In one, the source of clutter is a cloud, which moves as directed by the current wind vector at the cloud altitude. The individual scatterers can not actually be defined; however we can define the density of the scatterers, and the density of the scatterers is being estimated. In effect, the cloud is approximated by a large number of scatterers, each propagating according to (1). Each scatterer is part of the same cloud. In the other example, consider the clutter caused by the vigorous movement of vegetation (foliage) subjected to a gust of wind. In that case the vegetation is stationary, but the clutter measurement density propagates through space following the leading edge of the wind front. In effect, the clutter scatterers jump from tree to tree carried by the wind. The birth process of scatterers is modeled by the birth density, presented here as a Gaussian mixture ρ (0) (x ) = c 0 w c0 N(x ;x c0,p c0 ) (2) Fig. 1. Interaction between clutter measurement density estimator and target tracing where c 0 denotes the birth process component, and x c0 and P c0 denote the mean and the covariance of the component c 0 distribution with N(x;m,Σ) denoting the Gaussian pdf of random variable x, with mean m and covariance Σ. The birth density must cover all clutter state space x with non trivial densities. However, having satisfied the coverage constraint, the proposed clutter measurement density estimator is not overly sensitive to chosen values defining ρ (0). The death process of scatterers is modeled by the probability α that each scatterer will not disappear between scans. The clutter measurement density estimator is again not overly sensitive to this value. enote by Z the set of measurements (detections) received at scan, with random cardinality m, and let z,i Z denote the i-th measurement of Z. enote by Z the sequence of measurement sets Z up to time. III. CLUTTER MEASUREMENT ENSITY AN TARGET TRACKING We anticipate interaction between the target tracing and the clutter measurement density estimator as depicted on Figure 1. Target tracs are updated by measurements, using the predicted clutter measurement density estimates as the parameters. The clutter measurement density estimates are also updated using the measurements, with the feedbac from the target tracer. For reasons of simplicity, we ignore this feedbac in this paper. The clutter measurement density is used in data association and trac quality expressions in the trac update. Here we use the Integrated Probabilistic ata Association [5], [7] as an example, although similar or identical expressions are found in majority of published target tracing algorithms, including the Multi Hypotheses Tracing (MHT) [6] and Integrated Trac Splitting (ITS) [5], [8], [9]. At scan, each trac τ calculates lielihood p τ,i of each received measurement z,i. enote by ˆρ,i the clutter measurement density estimate at time for coordinate z,i in measurement space. Measurement z,i lielihood ratio is calculated by p τ,i /ˆρ,i. The lielihood ratio Λ τ of measurement

3 enote byγ(y) = 1/ρ(y) the clutter sparsity. The SCME estimator of order n finds the sparsity at point y of the measurement space in two steps: Find the n-th smallest distance r(y) between point y and the measurements in Z. If the point y coincides with a measurement, the measurement is ignored. The sparsity estimate is calculated by ˆγ(y) = 1 V (r(y)) (6) n Fig. 2. Single scan spatial clutter measurement density estimation set Z with respect to trac τ at time equals m Λ τ p τ,i = 1 P P G + P P G (3) ˆρ,i i=1 where P and P G denote the probability of target detection, and the target measurement gating (selection) probabilities respectively. The Integrated Probabilistic ata Association (IPA) [4], [7] updates the probability of target existence P{χ τ Z } using the propagated probability of target existence P{χ τ Z 1 } and the measurement lielihood ratio Λ τ by P{χ τ Z } = Λ τ P{χτ Z 1 } 1 (1 Λ τ )P{χτ Z 1 } The probability of target existence is used as the trac quality measure for the false trac discrimination. The data association probabilities β,i τ,i 0 are the posterior probabilities that measurement z,i is the detection of target τ, conditioned on the existence of target τ, and β,0 τ denotes the posterior probability that none of the selected measurements is the target detection. The data association probabilities are used to stochastically distinguish between the target detection and the clutter measurements. { β,i τ = 1 1 P P G, i = 0 Λ τ P P G p τ,i /ˆρ (5),i, i > 0 Thus improving the clutter measurement density estimates will result in improved false trac discrimination and better target trajectory estimates. IV. SINGLE SCAN SPATIAL CLUTTER MEASUREMENT ENSITY ESTIMATOR The spatial clutter measurement density estimator (SCME) [11] is a measurement oriented single scan clutter measurement density estimator which estimates the (reciprocal of) clutter measurement density at the point of each measurement z,i using only measurement set Z. We include a brief overview here, as it is used to generate clutter measurement density measurements to be filtered by the PH multi scan estimator. (4) where V(r) is the volume of the hypersphere with radius r in measurement space V(r) = C L r L = π L/2 Γ(1+L/2) rl (7) with L denoting the dimensionality of the measurement space and C L denoting the volume of L dimensional unit hypersphere; C 1 = 2, C 2 = π and C 3 = 4π/3. In a homogeneous cluttered environment, the bigger n the more precise estimates [11]. In the non-homogeneous environments, the bigger values of n, the worse resolution properties of the SCME. Values of n = 1 or n = 2 are recommended. In reality, the target tracing uses the clutter measurement density only at the selected measurement points, z,i. Thus, if the SCME is used to provide the tracers with the clutter measurement density estimation, the procedure outlined above is repeated for the measurement points only. An illustration of the SCME estimation is provided in Figure 2, forl = 2 andn = 1. Six measurements are presented denoted by the + mars. Around each measurement a circle is drawn ( the two dimensional hypersphere) which touches the nearest (n = 1) measurement. The estimated sparsity for a measurement simply equals the volume of the corresponding circle. The selection gates of two tracs are also drawn (the dashed ellipses) which both include one measurement. The SCME provides the same sparsity (and thus the clutter measurement density) estimate to both tracs. V. MULTI SCAN CLUTTER MEASUREMENT ENSITY ESTIMATOR WITH THE PROBABILITY HYPOTHESIS ENSITY (PH) FILTER The multi scan estimators average information provided by measurements over a number of scans. Each measurement z,i Z is treated as a scatterer detection (measurement) with assumed zero mean and white Gaussian measurement error with covariance R (c),i = r2 (z,i )I L (8) where I L denotes the L-dimensional identity matrix and r(z,i ) is the (first) smallest distance between z,i and another measurement in Z r(z,i ) = min i j z,i z,j (9) with denoting the vector norm. Please recall that each scatterer is detected with probability of clutter detection.

4 With this measurement model, we use the standard Gaussian Mixture PH filter for multi scan clutter measurement density estimator. It is a recursive estimator, where each recursion cycle consists of The birth process and the density propagation, Measurement selection, ensity update, and ensity component management The start of time recursion is the posterior state at time 1, which is a Gaussian Mixture density ρ 1 1 (x 1 ) = c 1ŵ c 1 N(x 1 ; ˆx c 1, ˆP c 1 ) (10) where c 1 denotes a Gaussian Mixture density component, and each component is parametrized by its weightŵ c 1, mean value ˆx c 1 and covariance ˆP c 1. The filter is initialized at time = 1 with the empty posterior state (no components) at time = 0. A. Prior density: birth process and the density propagation The prior density at time consists of the birth density (2) and the propagated time 1 posterior density (10) ρ 1 (x ) = ρ (0) (x )+ c 1 w c 1 N(x ; x c 1, P c 1 ) (11) where the propagated weight w c 1 = α ŵ c 1 is decreased due to the death process, and the propagated density component mean and covariance [ x c 1 Pc 1 ] = KFP (ˆx c 1, ˆP c 1,F,Q) (12) where KF P denotes the standard Kalman filter propagation equations, and the scatterer propagation and plant noise covariance matrices, F and the Q respectively, are defined in Section II. Thus the prior density is a Gaussian Mixture ρ 1 (x ) = c B. Measurement selection w c N(x ; x c, Pc ) (13) The measurement selection is generally used to limit the computational load of the algorithm. Each density component c selects a subset Zc Z for update. Measurement z,i Z c if it passes test ( ) S ( ) z,i H x c c i z,i H x c ζ (s) (14) where denotes the matrix transpose operator, and the projection matrix H defined in Section II, and S c i = H P c H +R (c),i (15) The selection threshold constant ζ (s) of (14) should be high enough for the scatterer selection probability G to be nominally one, and the G is subsequently ignored in this paper for reasons of clarity. C. ensity update Each density component c generates new density components by pairing with feasible measurement outcomes i c 0, where i = 0 indicates scatterer non-detection possibility. Other feasible values of i c > 0 satisfy z,i c The posterior density equals ρ (x ) = c + c Z c (16) w c N(x ; x c, Pc ) i:z,i Z c ŵ c i N(x ; ˆx c i, ˆP c i ) where the propagated density component weight (17) w c = (1 P (c) )wc (18) and the updated density component weight ŵ c i = ρ 0 + c :z,i Z c i ) wc N(z,i ;H x c,s c (19) wc N(z,i ;H x c c,s i ) with ρ 0 denoting the residual clutter measurement density. Finally, [ ] ˆx c i ˆPc = KF i U (z,i,r (c),i, xc, Pc,H) (20) where KF U denotes the standard Kalman filter update. Thus, the updated estimate of clutter density becomes a Gaussian Mixture ρ (x ) = c ŵ c N(x ; ˆx c, ˆP c ) (21). ensity component management The number of density components grows exponentially in time, and their number should be managed to prevent the saturation of computational resources. Standard component management techniques of pruning and merging [2], [3], [17] [19] are also applicable for PHclutter density map. In Section VI, the authors employ [20] technique which merges density components with identical measurement history in last scans, followed by component pruning. E. Output Output of PH-clutter density estimators is used by the target tracers which need to now the prior clutter measurement density at measurement locations and ρ 1 (z,i ) = w c 1 N(z,i ;H x c 1,H P c 1 H ) c 1 (22) where the birth process is excluded. This operation is not part of (does not feed bac into) the clutter measurement density estimator recursion loop.

5 Fig. 3. Homogeneous clutter experiment situation VI. SIMULATION STUY Here we consider two dimensional clutter measurement density estimation. For simplicity and clarity, a linear measurement system is assumed. We compare two clutter measurement density estimators: the single scan spatial clutter measurement density estimator (SCME) (Section IV) and proposed multi scan PH filter based clutter measurement density map estimator (Section V). Two stationary clutter situations are considered; a homogeneous and a non-homogeneous clutter environment. The surveillance area is 1000 m long and 500 m wide. The sensor measures the target position with the sampling time T = 1s. A number of clutter measurements are also detected in each scan. A single target moves with a uniform motion of 4 meter per second. In this simulation study we follow the clutter measurement density estimates at the target position. Furthermore, for the purpose of simplicity and clarity, we do not include the target measurements in the clutter measurement density estimators. Each simulation run simulates an 80 scans interval, and the statistics are accumulated over 500 simulation runs. The sparsity order for the single scan clutter measurement density estimator SCME n = 1, Section IV. The birth process of scatterers (2) has four components. The surveillance space is divided into four equal segments. The position term of birth components means x c0 are equal to the centers of these segment areas, and the velocity terms x c0 of are set to zero. The position terms of birth components covariance matrices P c0 are calculated by using the lengths and widths of the segment areas. The velocity terms of P c0 are set to cover the maximum anticipated scatterer velocity of 1m/s and are thus equal to 0.25I 2 (m/s) 2 [21]. The probability of scatterer survival α = 0.98 (Section II). The residual clutter measurement density ρ 0 = 0. The PH filter component management (Section V-) is implemented by first merging components [20] with history length equal to 1, followed by pruning components with weights less than If the number of remaining components is bigger than 1,000, then only 1,000 strongest density components are retained. The multi scan PH estimator is parametrised by various values of, from 0.2 to 1. Fig. 4. Fig. 5. Homogeneous clutter: Performance index mean Homogeneous clutter: Performance index RMS error Two performance measures are used to asses the performance of the clutter measurement density estimators; the performance index and the relative root mean square (RMS) error. These values are measured (and averaged across simulation runs) at the points of the target trajectory over time. For the single scan SCME estimator the performance index is defined by 1 η (y) = (23) ˆγ (y)ρ(y) For the multi scan PH estimator the performance index is defined by η (y) = ρ (y)/ρ(y) (24) These performance indices are compatible, as the clutter measurement sparsity ˆγ (y) is the reciprocal of the clutter measurement density. A. Homogeneous Clutter ensity The simulated environment is depicted in Figure 3. The clutter measurement density is uniform and equals ρ = Figure 4 shows the performance index average of the single scan SCME estimator, as well as the multi scan PH estimator parametrized by the choices for. The Exact curve shows the ideal outcome, which would be obtained if the

6 Fig. 6. Non-homogeneous clutter experiment situation Fig. 8. Non-homogeneous clutter: Performance index RMS error Fig. 7. Non-homogeneous clutter: Performance index mean estimators returned the true value of the clutter measurement density. The averaged relative RMS performance is presented in Figure 5. It is defined as the RMS error divided by the true value of the clutter measurement density. As evidenced in Figure 4 all estimators are essentially unbiased in this environment (at least there is no evidence here for the contrary). The averaged relative RMS SCME equals one, which confirms the conclusion of [11] that this estimator has exponential distribution (the standard deviation equals mean for the exponential pdf). ecreasing the chosen probability of scatterer detection improves averaging, as evidenced by decreasing RMS values shown in Figure 5. The highest averaging tried for = 0.2 decreases the relative RMS errors by almost an order of magnitude. B. Non-homogeneous Clutter ensity The clutter measurement density is very rarely (almost never) homogeneous in real applications. The non homogeneous clutter simulated environment is depicted in Figure 6. The base clutter measurement density is uniform and equals ρ = 10 4, with a heavy clutter patch with ρ = The target is within the heavy clutter patch from = 25 up to and including = 49. Figure 7 shows the performance index average of the single scan SCME estimator, as well as the multi scan PH estimator parametrized by the choices for. The Exact curve again shows the ideal outcome. The averaged relative RMS performance is presented in Figure 8. As evidenced in Figure 7 all estimators here suffer from the edge effect, where they are biased near the boundary of areas with two different clutter measurement densities. The recovery was the worst for the single scan SCME estimator, whereas it depends on the parameter for the multi scan PH estimator. In this environment, the RMS values again decrease with the chosen probability of scatterer detection. However this is somewhat balanced by a bias in the high clutter region. The high clutter region is small enough for the transient effects to persist throughout. On a standard Windows based system with Intel 3.3 GHz CPU, the C++ program taes approximately 15 seconds to simulate one 80 seconds interval. Therefore, this approach seems suitable for real time systems. VII. CONCLUSION This paper proposes a solution to clutter measurement density estimation using the multi scan PH filter. The clutter information is averaged over a number of scans, producing better averaging than the single scan SCME estimator. Additionally, it provides the ability to trac non stationary clutter. This should improve functionality of the target tracing in clutter. The current research (not finalized yet at the time of paper submission) includes dealing with the target measurements, as well as tracing the dynamic clutter in surveillance region. Full interaction with target tracing; i.e. using the integrated target tracing with proposed clutter measurement estimator is also left for future publication once the indicated research is finalized. REFERENCES [1] F. Smith and J. Malin, Models for radar scatterer density in terrain images, IEEE Trans. Aerospace Electronic Systems, vol. 22, no. 5, pp , Sep [2] S. Blacman and R. Popoli, esign and Analysis of Modern Tracing Systems. Artech House, 1999.

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