The Adaptive Labeled Multi-Bernoulli Filter

Transcription

1 The Adaptive Labeled Multi-ernoulli Filter Andreas Danzer, tephan Reuter, Klaus Dietmayer Institute of Measurement, Control, and Microtechnology, Ulm University Ulm, Germany andreas.danzer, stephan.reuter, arxiv: v1 [cs.y] 2 Dec 218 Abstract This paper proposes a new multi-ernoulli filter called the Adaptive Labeled Multi-ernoulli filter. It combines the relative strengths of the known δ-generalized Labeled Multi- ernoulli and the Labeled Multi-ernoulli filter. The proposed filter provides a more precise target tracking in critical situations, the Labeled Multi-ernoulli filter looses information through the approximation error in the update step. In noncritical situations it inherits the advantage of the Labeled Multi- ernoulli filter to reduce the computational complexity by using the LM approximation. I. INTRODUCTION The aim of multi-object tracking is the estimation of the number of objects as well as their individual states based on noisy measurements, missed detections and false alarms lead to ambiguities in the track-to-measurement association. Further, adequate models for the appearance and disappearance of objects are required. Approaches to tackle multi-object tracking are Joint Probabilistic Data Association (JPDA) [1], Multiple Hypotheses Tracking (MHT) [2], and the Random Finite et (RF) based multi-object ayes filter [3]. ased on the mathematical tools of finite set statistics (FIT) [3], several approximations of the multi-object ayes filter have been proposed during the last decade. The Probability Hypothesis Density (PHD) filter [4], [5], [6], [7] and the Cardinalized PHD (CPHD) filter [8], [9] approximate the multi-object posterior density by the first statistical moment and, in case of the CPHD filter, the cardinality distribution. The Cardinality alanced Multi-Target Multi-ernoulli (C- MeMer) filter [1]) approximates the multi-object posterior using parameters of a multi-ernoulli distribution and only propagates these parameters. In [11], the class of labeled RFs as well as the first analytic implementation of the multiobject ayes filter in form of the Generalized Labeled Multi- ernoulli (GLM) and the δ-glm filter are proposed. The δ-glm filter is shown to outperform the approximations of the multi-object ayes filter and the incorporation of the track labels in the filtering step significantly improves the track extraction in sequential Monte-Carlo (MC) implementations [11]. The labeled Multi-ernoulli (LM) filter [12] efficiently approximates the δ-glm filter by approximating the posterior after each update step using an LM distribution. In [13], the LM filter is shown to outperform PHD, CPHD, and multi-ernoulli filters and to achieve almost the same accuracy as the δ-glm filter. Further, the LM filter is successfully used for the real-time environment perception system of the autonomous car of Ulm University [14], [15], [16] based on radar, lidar, and video sensors. In [12], the approximation error of the LM filter with respect to the cardinality distribution is illustrated in detail. Due to the assumption of statistically independent objects, the LM representation does not facilitate multi-modal cardinality distributions. In contrast to the δ-glm filter which uses multiple hypotheses to represent the data association uncertainty, the LM filter represents the uncertainty within the spatial distribution of each track. Hence, depending on merging and pruning thresholds applied for the spatial distributions of the LM filter, this representation may also lead to a loss of information. Consequently, an adaptive multi-object tracking algorithm, which represents the tracks in δ-glm representation in challenging scenarios (e.g. objects are close by or track-to-measurement association is ambiguous) and uses the computationally efficient LM representation in all other scenarios, is expected to outperform the LM filter and to adjust the computational complexity to the complexity of the scenario. In this contribution, the Adaptive Labeled Multi-ernoulli (ALM) filter is proposed which automatically switches between an LM and δ-glm representation based on Kullback Leibler divergence [17] and entropy [18]. The ALM filter represents the multi-object posterior at each time step using a set of LM and δ-glm distributions. Thus, only a subset of tracks is required to be represented in δ-glm form. The ALM filter is compared to the δ-glm and the LM filter using two simulations. This paper is organized as follows: First, the basics of (labeled) random finite sets are outlined and the LM filter as well as the δ-glm filter are introduced. ection V introduces the switching criteria and the scheme of the ALM filter is detailed in ection VI. Finally, simulation results are shown in ection VII. II. ACKGROUND This sections summarizes multi-object tracking using random finite sets (RF) and introduces the labeled multi- ernoulli and the generalized labeled multi-ernoulli RF. A. Random Finite ets An RF is a finite-set-valued random variable with a random number of points, which are also random and unordered. The RF X = x (1),..., x (N) } X represents the multi-object state X with finite single-target state vectors x (i) X, X c 218 IEEE. Personal use of this material is permitted. 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2 is the state space. Further, the RF Z = z (1),..., z (M) } Z represents the multi-object observations with a random measurement z (i) out of the measurement space Z. The finite set statistics introduced in [3] are a powerful mathematical tool for dealing with RFs.. Multi-ernoulli RF A ernoulli RF is empty with a probability 1 r and has probability r of being a singleton with a distribution p defined on X. Its probability density is given by (see [3]) 1 r X =, π(x) = (1) r p(x) X = x}. The cardinality distribution is a ernoulli distribution with parameter r. A multi-ernoulli RF is the union of M independent ernoulli RFs X (i), thus, X = M i=1 X(i) and is completely described by the parameter set (r (i), p (i) )} M i=1, r(i) is the existence probability and p (i) the spatial distribution. C. Labeled Multi-ernoulli RF In a multi-object scenario, it is often required to estimate the identity of an object in addition to its current state. For that reason, the class of labeled RFs [11] appends a label l L to each state state vector x X. Thus, a labeled RF is an RF on X L with state space X and finite label space L. In the following, labeled state vectors x = (x, l) as well as labeled RFs X are represented by bold letters. Using the projection L : X L L defined by L((x, l)) = l, the distinct label indicator (X) = δ X ( L(X) ), (2) L(X) = L(x) : x X} is the set of labels, ensures that labels l of a realization are distinct. imilar to the multi-ernoulli RF, a labeled multi- ernoulli (LM) RF is completely defined by the parameter set (r (l), p (l) )} l L and its density is given by (see [12]) π(x) = (X)w(L(X))p X, (3) w(l) = i L (1 r (i)) l L 1 L (l)r (l), (4) 1 r (l) p(x, l) = p (l) (x). (5) The cardinality distribution of an LM RF is identical to the one of its unlabeled version and is given by ρ LM (n) = i L (1 r (i)) I F n(l) l I 1 L (l)r (l), (6) 1 r (l) F n (L) denotes all subsets of L with exactly n elements. D. δ-generalized Labeled Multi-ernoulli RF A generalized labeled multi-ernoulli (GLM) RF [11] is a labeled RF with state space X and (discrete) label space L distributed according to π(x) = (X) [ w (c) (L(X)) p (c)] X, (7) c C C is a discrete index set and w (c) (L) = 1, (8) L L c C p (c) (x, l)dx = 1. (9) A δ-generalized labeled multi-ernoulli (δ-glm) RF [11] with state space X and (discrete) label space L is a special case of a generalized labeled multi-ernoulli RF with C = F(L) Ξ, (1) w (c) (L) = w (I,ξ) δ I (L), (11) p (c) = p (I,ξ) = p (ξ) (12) F n (L) denotes all subsets of L, the discrete space Ξ represents the history of track to measurement associations with realizations ξ Ξ and I is a set of track labels. The density of a δ-glm RF is given by π(x) = (X) (I,ξ) F(L) Ξ and its cardinality distribution follows ρ δ-glm (n) = w (I,ξ) δ I (L(X)) [p (ξ)] X (I,ξ) F n(l) Ξ (13) w (I,ξ). (14) Obviously, an LM RF is a special case of a δ-glm RF with only one single component, i.e. p (ξ) (x, l) = p (l) (x): π(x) = (X) I F(L) w (I) δ I (L(X))p X, (15) the weights of the components follow (4). III. δ-generalized LAELED MULTI-ERNOULLI FILTER The δ-glm filter was introduced in [11], it is shown that GLMs and δ-glms are conjugate priors 1 with respect to the prediction and update equations of the multiobject ayes filter [3]. 1 Note: the number of components increases due to the association uncertainty.

3 A. Prediction The prediction of a δ-generalized labeled multi-ernoulli of the form (13) to the time of the next measurement is given by [ ] X π + (X) = (X) w (I+,ξ) + δ I+ (L(X)) p (ξ) +, (I +,ξ) F(L +) Ξ (16) w (I+,ξ) + = w (I + )w (ξ) (I + L), (17) p (ξ) + (x, l) = 1 L (l)p (ξ) (x, l) + 1 (l)p (x, l), (18) p (ξ) (x, l) = p (, l)f(x, l), p (ξ) (, l) η (ξ) (l), (19) η (ξ) (l) = p (, l)f(x, l), p (ξ) (, l) dx, (2) [ ] L w (ξ) (L) = [ ] I L η (ξ) 1 I (L) w (I,ξ), (21) I L q (ξ) q (ξ) q = (, l), p (ξ) (, l). (22) In (16)-(22), w ( ) is the weight of the birth labels I + and w (ξ) ( ) of the surviving labels I + L. Further, p (, ) is the density of new-born objects and p (ξ) (, ) of surviving objects, depending on the transition density f(x, l) weighted by the probability of survival p (, l) and the prior density p (ξ) (, l). esides, f, g = f(x)g(x)dx denotes the inner product, η (ξ) (l) is a normalization constant and q (, l) = 1 p (, l) the probability that a track disappears.. Update The posterior density after the measurement update of (16) is again a δ-glm RF given by π(x Z) = (X) w (I+,ξ,θ) (Z) (I +,ξ) F(L +) Ξ θ Θ δ I+ (L(X)) w (I+,ξ,θ) (Z) δ θ 1 (: Z })(I + )w (I+,ξ) + [ p (ξ,θ) ( Z)] X (23) [ η (ξ,θ) Z ] I+, (24) p (ξ,θ) (x, l Z) = p(ξ) + (x, l)ψ Z (x, l; θ), (25) η (ξ,θ) Z (l) η (ξ,θ) Z (l) = p (ξ) + (, l), ψ Z (, l; θ), (26) ψ Z (x, l; θ) = δ (θ(l))q D (x, l) + (1 δ (θ(l))) p D(x, l)g(z θ(l) x, l). (27) κ(z θ(l) ) In (23)-(27), θ Θ : I +, 1,..., Z } associates track labels to measurements, θ(i) = represents a missing detection and θ(i) = θ(j) > implies i j. Note, the posterior sets of track labels correspond to the predicted sets of track labels, i.e. I = I +. Here, w (I+,ξ,θ) is the updated weight of a hypothesis (I +, ξ, θ). Further, η (ξ,θ) Z (l) is a normalization constant and ψ Z (x, l; θ) is the measurement likelihood. The likelihood depends on the probability of a missing detection q D (x, l) = 1 p D (x, l) at (x, l) and the spatial likelihood g(z θ(l) x, l) weighted by the detection probability p D (x, l) at (x, l). κ(z θ(l) ) = λ c c(z) models the intensity of Poisson clutter. IV. LAELED MULTI-ERNOULLI FILTER The Labeled Multi-ernoulli (LM) filter was proposed in [12] and is intended as a fast and accurate approximation of the δ-glm filter. While an LM RF is a conjugate prior with respect to the prediction equations, the filter update requires a transformation to δ-glm form and a subsequent approximation. A. Prediction For a multi-object posterior LM RF with parameter set π = (r (l), p (l) )} l L on X L and a multi-object LM birth density π = (r (l), p(l) )} l on X, the multi-object prediction is also an LM RF with state space X and finite label space L + = L and is given by ( )} ( )} π + (X) = r (l) +,, p(l) +, r (l), p(l), (28) l L l r (l) +, = η (l)r (l), (29) p (l) +, = p (, l)f(x, l), p(, l), η (l) (3) η (l) = p (, l)f(x, l), p(, l) dx. (31) In (28)-(31), p (, l) denotes the state dependent survival probability and f(x, l) the single target transition density for track l. Further, η (l) is a normalization constant.. Update ince an LM RF is not a conjugate prior with respect to the measurement update of the multi-object ayes filter, the LM filter update transforms the predicted LM RF to a corresponding δ-glm RF using (15) and subsequently applies the δ-glm update. In order to reduce the computational complexity, the LM components and the measurements are partitioned into approximately statistically independent groups (see [12] for a detailed explanation). In order to close the LM filter recursion, an approximation of the updated δ-glm distribution using an LM RF is required, i.e. ( π (X) π (X) = r (l), p (l))}. (32) l L The parameters of the LM RF are obtained from the updated δ-glm components using r (l) = w (I+,θ) (Z)1 I+ (l), (33) (I +,θ) F(L +) Θ I+ p (l) (x) = 1 r (l) (I +,θ) F(L +) Θ I+ w (I+,θ) (Z)1 I+ (l)p (θ) (x, l Z), (34)

4 Θ I+ denotes the space of mappings. The weights w (I+,θ) and the densities p (θ) are computed using (23)-(27) by setting ξ =. The LM approximation does not affect the spatial distributions of the individual tracks but due to the assumption of statistically independent tracks within an LM RF, the cardinality distribution of the approximation may differ as the mean cardinality is identical (see [12] for additional details). V. WITCHING CRITERIA The aim of the ALM filter is to switch automatically between the LM approximation, facilitating a fast propagation of the density, and the more accurate δ-glm density. In the following, two switching criteria are introduced the Kullback-Leibler criterion evaluates the approximation error of the LM approximation and the Entropy criterion considers the data association uncertainty. A. Kullback-Leibler Criterion In [12], it was shown that the LM approximation loses information about the cardinality distribution. Thus, an intuitive way to detect the information loss, is to examine the difference between the posterior cardinality distribution of the δ-glm RF and its LM approximation. The Kullback- Leibler (KL) divergence [17] is a measure to compare two (discrete) probability distributions P and Q: D KL (P Q) = i P (i) log P (i) Q(i), (35) Q is the approximation of P and the i-th term is zero, if P (i) = since lim x log(x) =. (36) x The cardinality distributions of the δ-glm RF ρ δ-glm (n) and its LM approximation ρ LM (n) are given by (14) and (6), hence the Kullback-Leibler criterion calculates D KL (π(x)) = D KL (ρ δ-glm ρ LM ), (37) π(x) denotes the updated δ-glm RF which facilitates the calculation of its LM. For D KL =, the cardinality distributions are identical, i.e. the LM approximation causes no information loss in the cardinality distribution. D KL > implies that ρ LM differs from ρ δ-glm, a large value of D KL can be interpreted as big difference between the cardinality distributions, i.e., a large approximation error.. Entropy Criterion The δ-glm RF comprises several hypotheses to capture the data association uncertainties, as the LM RF captures the association uncertainty within the spatial distributions of a individual track. Depending on the parameters and the post-processing of the spatial distributions, the δ-glm representation is in general more accurate in challenging situations. For example, two Gaussian components obtained by different track-to-measurement associations may be merged in a Gaussian Mixture (GM) LM filter which results in a loss of information compared to the δ-glm representation holding the two associations in different hypotheses. The Kullback-Leibler criterion does not detect challenging situations with ambiguous data association if the cardinality distributions are identical. Hence, a measure for the data association uncertainty is required which enables a switching to the more accurate δ-glm representation in these situations. The entropy [18] is a measure of unpredictability of information content and is widely used in information theory. In this contribution, the entropy is used to evaluate the trackto-measurement association. Following [18], the entropy is H(P ) = i P (x i ) log P (x i ), (38) P (x i ) is the probability that the event x i occurs. Obviously, for small or large values P (x i ), the entropy is small. This fact can be used to detect ambiguous data associations. The assocation matrix of tracks to measurements is given by r (l1,z1) r (l1,z2) r (l1,zm) r (l2,z1) r (l2,z2) r (l2,zm) A =......, (39) r (ln,z1) r (ln,z2) r (ln,zm) r (li,zj) = (I +,θ) F(L +) Θ I+ w (I+,θ) (Z)1 I+ (l i )δ θ(li)(j) (4) is the probability that track l i is assigned to measurement z j. Further, θ Θ I+ : I +, 1,..., Z } is a mapping of labels to measurements in such a way that θ(i) = θ(j) > implies i j. An unambiguous assignment of measurement z j is characterized by the column vector a j with one value r (li,zj) 1 and all other values r (l k,z j), k = 1, 2,..., i 1, i + 1,..., n. With the above-mentioned property of entropy, such a column vector results in a small value H(a j ). Hence, the entropy for a distribution π(x) is m H(π(X)) = H(a j ), (41) j=1 a small value indicates an unambiguous data association and a large value represents an uncertain track-to-measurement assignment. VI. THE ADAPTIVE LAELED MULTI-ERNOULLI FILTER The δ-glm filter of [11] is shown in [13], [19], [2] to outperform the LM filter [12] in challenging scenarios, e.g. containing closely spaced objects in combination with missed detections and false alarms, at the cost of a significantly higher computational complexity. The main idea of the Adaptive Labeled Multi-ernoulli (ALM) filter proposed in this section is to combine the advantages of the δ-glm filter in critical situations with the efficiency of the LM filter. Thus, the ALM filter uses the δ-glm distribution to represent tracks

5 irth Model Track Extraction Prediction plitting Measurements Pruning Fig. 1. cheme of the proposed ALM filter. Merging Update in critical situations as accurate as possible and uses LM distributions for the representation of all other tracks. Using the assumption, that well separated objects are statistically independent of each other, the ALM filter uses several independent multi-object distributions in LM and δ-glm form to represent the environment, i.e. π δ-glm (X (δ) ) = π LM (X (L) ) = } π (i) nδ δ-glm (X(i) ) i=1 } nl π (i) LM (X(i) ) i=1, (42), (43) n δ is the number of δ-glm distributions and n L denotes the number of LM distributions. Consequently, the multi-object posterior density is given by the set } π(x) = π δ-glm (X (δ) ), π LM (X (L) ). (44) Fig. 1 illustrates the scheme of the ALM filter which propagates the density (44) over time. Obviously, each component of the ALM filter is required to be able to handle both representations, LM and δ-glm. In the following, the individual components of the ALM filter are presented in detail. A. irth Model The birth model is responsible for initializing new tracks. The ALM filter may use a static birth model [11] requiring known birth locations or an adaptive birth model [12], [13] which facilitates the appearance of objects any in the state space. Due to the structure of the ALM filter, several new-born objects may not be represented using a single LM distribution since this would require an additional splitting of distributions before the measurement update. Hence, each new-born object l is represented by an individual LM distribution consisting of a single component. Prediction π (l ) = r (l ) (z), p(l ) (x) }. (45) In the prediction step, the ALM filter predicts each distribution π (i) δ-glm (X(i) ), i = 1,..., n δ, using the standard δ-glm prediction equations. Further, the standard LM prediction is applied for each π (i) LM (X(i) ), i = 1,..., n L. C. Measurements ince the ALM filter holds multiple multi-object distributions at the same time, not every received measurement affects each multi-object density. Hence, the measurements module performs a gating procedure for the distributions and the set of measurements. Obviously, this gating procedure is important for a parallel execution of the update step in the manner of [12]. The measurements module always performs the assignment of observations z (i) Z based on the LM distribution. Consequently, a δ-glm RF π (i) δ-glm (X(i) ) has to be approximated by an LM RF π (i) LM (X(i) ) according to (32). This temporary approximation facilitates a faster observation to distribution association. ince the spatial distribution of the LM approximation is equivalent [12], the temporary approximation does not influence the gating result. After that, the gating examines if a received observation z (i) affects the track l of the LM distribution with d MHD (ẑ (l) +, z (i) ) < γ z, (46) by ẑ (l) + is the predicted measurement of track l and γ z is the gating distance threshold. The value of γ z depends on the desired σ-gate of the confidence interval and can be calculated using the inverse Chi-squared cumulative distribution. D. Merging Due to the new-born objects represented by individual LM RFs and the prediction of the existing multi-object densities, it is possible that distributions influence each other in the update step. Hence the merging combines all multi-object densities with common measurements. The measurements module assigned a set of measurements Z (i) to the predicted multi-object density π (i) +. Consequently, two predicted densities π (i) + and π (j) + with common measurements Z (i) Z (j) (47) have to be merged into a single multi-object distribution π (i,j) +. The merging itself depends on the representations of π (i) + and π (j) + and is introduced in the following. 1) Merging of LMs: The merging of two LM RFs is used as in [12] after the parallel group update step. Two LM densities π (i) (X (i) ) and π (j) (X (j) ) are merged to the distribution π (i,j) (X (i,j) ) = π (i) (X (i) ) π (j) (X (j) ). (48) 2) Merging of δ-glms: The merging of two δ-glm RF is not as simple as the LM merging, because all combinations of the hypotheses of the two RFs have to be considered. To calculate the combined δ-glm RF, each component of π (i) δ-glm (X(i) ) has to be multiplied with each component of π (j) δ-glm (X(j) ) resulting in π (i,j) (X (i,j) ) = (X (i,j) ) w (I,ξ) δ I (L(X (i) ) [p (ξ)] X (i) (49) (I,ξ) F(L (i) ) Ξ (i) w (Ĩ, ξ) δĩ(l(x (j) )) [p ] ( ξ) X (j). (Ĩ, ξ) F(L (j) ) Ξ (j) (5) Hence, the merged number of components is given by the product of the individual number of components.

6 3) Merging of an LM with a δ-glm: The merging of an LM RF with a δ-glm RF always results in a δ-glm. First, the LM π (i) LM (X) is transformed into a corresponding δ-glm π (i) δ-glm (X) using (15). Due to the fact that both densities are now in δ-glm form, the δ-glm merging according to (5) is used to merge the densities. E. Update According to the representation of the predicted multi-object density, the ALM filter performs either a δ-glm or a LM update, which are given by (23) and (32), respectively. Observe: in case of the LM update it is required to store the resulting δ-glm density of the update step in addition for the following steps. After the update, the ALM filter uses the criteria presented in ection V to decide whether a switching of the multi-object representation is necessary or not. An updated LM RF indicates a noncritical situation before the update, but it is possible that the correction step changed this fact. Therefore, the filter examines the cardinality distributions of the LM approximation π(x) and the δ-glm posterior π(x) using the Kullback-Leibler criterion (see ection V-A) to detect an information loss and Entropy criterion (see V-) to handle ambiguous track-to-measurement associations. If one of the criteria detects a critical situation, i.e. the KL divergence or the entropy exceed an application-specific threshold, the filter replaces the LM approximation by the δ-glm RF obtained during the filter update and propagates the incorporated tracks using the δ-glm filter in the next filter cycle. If a loss of information caused a switching, the filter uses the KL divergence to examine whether the critical situation is solved. Otherwise, the Entropy criterion is used. This implies that only the criterion which detected the critical situation, can trigger switching back to propagating a set of tracks using an LM RF. Once the KL divergence or the entropy fall below the threshold, a challenging situation is resolved. F. Pruning oth, the LM update and the δ-glm update, produce components with negligible influence. To reduce the computational cost, the pruning removes these components. The LM pruning removes all tracks l with marginal existence probability r (l), so the resulting LM RF is ( π (i) LM = r (l), p (l)) : r (l) > µ r, (51) }l L(i) µ r represents the application dependent minimum existence probability. The δ-glm pruning removes all hypotheses (I, ξ, θ) with insignificant weight w (I,ξ,θ), which leads to } π (i) δ-glm = (I, ξ, θ) : w (I,ξ,θ) > µ w (52) using the threshold µ w. I F(L (i) ) G. plitting In the course of time, it is possible that tracks in a density move apart, so that the RF can be splitted into multiple smaller distributions of the same type. The splitting uses the grouping algorithm of [12]. Therefore, a δ-glm RF temporary is approximated by an LM RF according to (32) during the splitting procedure. Then, the partitioning scheme is applied to find a possible splitting of the RF. The module splits an LM distribution π (i) in multiple new densities π (j) such that N π (i) = π (j), (53) j=1 N is the number of identified object groups. The δ-glm splitting uses the labels of tracks in the groups to create several new δ-glm RF. The splitted δ-glm densities π (j) only approximate the original distribution π (i), because during the splitting, hypotheses containing labels of two different groups, are divided in new hypotheses, which only contain the labels according to the new distribution. ince the influence of tracks from different groups is marginal, the occurred approximation error is negligible. H. Track Extraction The track extraction decides whether a track with label l exists or not by using the existence probability r (l). An LM density ( π (j) = r (l), p (l))} (54) l L (j) implicitly contains the existence probability. According to [12], the extraction of the track is } ˆX = (ˆx, l) : r (l) > ϑ, (55) the parameter ϑ is an application specific threshold and ˆx = arg x max p (l) (x). ince a δ-glm density does not contain the existence probability, the module has to calculate this value. Following [12], the existence probability for a track l is given by r (l) = w (I,θ) (Z)1 I (l), (56) (I,θ) F(L) Θ I w (I,θ) is the weight of the corresponding hypothesis (I, θ). Afterwards, the extraction uses (55) to choose existing tracks. VII. REULT This section evaluates the ALM filter and compares it with the LM filter [12] and the δ-glm filter [11]. For the evaluation, a gaussian mixture (GM) implementation is used. The scenario consists of two targets on a two dimensional region [ 1, 1]m [ 1, 1]m. The target state x k = [p x,k, ṗ x,k, p y,k, ṗ y,k ] T comprises the position and velocity in x and y direction. Measurements are noisy vectors z k = [z x,k, z y,k ] T. The clutter measurements are uniformly

7 y [m] y [m] 1, , 1, LM filter ALM filter 1, 1, , x [m] Fig. 2. Ground truth trajectories of the two objects (black lines with endposition marked by a triangle), the estimated trajectories (circles/squares) of the LM filter (above) and the result of the ALM filter (below). distributed over the measurement space and their number follows a Poisson distribution with mean value λ c. The state model is a standard constant velocity model the standard deviation of the process noise for the velocity in x and y direction is given by σv 2 = 5m/s 2. The cycle time of the sensor is T = 1 s and the standard deviation of the sensor measurements consisting of x and y positions is σ ε = 1m. The survival probability of the targets is state independent and given by p,k =.99, the detection probability is p D,k =.98. Furthermore, the birth densities are two multi-ernoulli RF π (i) = r (i) (z), p(i) (x z)}2 i=1, r (1) = r(2) =.5, p(i) = N (x; m (i), P ) with m (1) = [ 1,,, ] T, m (2) = [1,,, ]T and P = diag([1, 1, 1, 1] T ) 2. The thresholds for an automatic switching between an LM and δ-glm representation are 1 4 for the Kullback-Leibler criterion and.5 for the Entropy criterion. In a δ-glm density, the number of components is limited to 5 and the pruning removes all components with a weight below 1 5. In an LM density, all tracks with an existence probability below.1 are pruned. For both densities, the threshold in the track extraction is set to.5 and an extracted track is represented by the gaussian component with the highest weight. Figure 2 shows the true trajectories together with the tracking result of a single run. Obviously, the LM filter can not handle the situation in the region [ 1, 1]m [, 25]m. In OPAT [m] Time k LM δ-glm ALM Fig. 3. OPAT distances of order p = 1 and cut-off c = 3 for GM implementation with λ c = 5 and p D =.98 (averaged over 1 MC runs). Time [s] Time k LM δ-glm ALM Fig. 4. Computation time of the LM, δ-glm and ALM filter (averaged over 1 MC runs). this critical situation, the data association of tracks to measurements is uncertain and the LM filter erroneously switches the track labels. In contrast, the ALM filter successfully detects the critical situation using the criteria from ection V and uses the δ-glm representation until the ambiguity is resolved. As a result, the ALM filter can deal with such situations and does not switch the track labels. The OPAT distances [21] in Fig. 3 illustrate the difference between the LM, δ-glm and ALM filters. In noncritical situations (time k < 55), the performance of the LM and ALM filter is identical since the ALM filter uses the LM representation. However, LM and ALM perform slightly worse than the δ-glm filter which is expected due to the approximations in the update step. In challenging situations with data association uncertaintities (time k > 5 k < 65), the ALM and δ-glm filter outperform the LM filter. Obviously, the ALM filter can handle the situation due to the propagation of multiple hypotheses in most of the Monte Carlo runs. In contrast, the LM filter loses too much information due to the LM approximation and almost always switches the track labels. At time k = 91, the OPAT increases for all filters due to the disappearance of both tracks and the short delay until both tracks are abandoned. Figure 2 obviously illustrates this fact. After a certain time, the estimation of the tracks matches to the ground truth resulting in a declining OPAT distance. Figure 4 shows the computation time of the compared filters. Obviously, the LM and ALM filter significantly outperform the δ-glm. The execution time of the ALM filter is almost the same as of the LM filter. Only in critical situations (k > 5 k < 65), the ALM filter needs more time for the calculation, but nevertheless, it outperforms the δ-glm filter in such situations. In [22], a fast implementation of the δ-glm filter is proposed. This implemenation reduces the execution time of the δ-glm filter, but would also speed up the ALM filter in critical situations. The ALM filter represents tracks by partitioned multi-ernoulli RFs resulting

8 y [m] 1, , 1, , x [m] Fig. 5. Ground truth trajectories of a scenario with up to 16 objects, the start position is marked by an circle and the end position by a triangle. OPAT [m] Time k LM ALM Fig. 6. OPAT distances of order p = 1 and cut-off c = 3 for GM implementation with λ c = 25 and p D =.98 (averaged over 1 MC runs). from the merging and splitting module. ince the groups are assumed to be independent, the prediction, measurements, update, pruning and track extraction modules can be performed in parallel, which also speeds up the algorithm. In a second example, the performance of the ALM filter is evaluated in a scenario with many targets. Figure 5 illustrates the scenario with up to 16 objects involving birth and death of objects and considering missed detections and clutter measurements. The OPAT distances in Figure 6 show that the ALM filter always performs same or better than the LM filter. VIII. CONCLUION This paper has proposed a new efficient multi-target tracking filter based on a Labeled Multi-ernoulli and δ-generalized Multi-ernoulli filter. The proposed Adaptive Labeled Multi- ernoulli filter combines a low computational complexity of the LM filter with the accuracy of the δ-glm filter. With the Kullback-Leibler distance and an intuitive interpretation of the entropy, the filter uses simple mathematical tools to detect challenging situations in a tracking scenario. ince the criteria do not depend on the representation of the spatial distributions, the principles of the ALM filter may also be used in sequential Monte Carlo implementations as well as the recently published Gamma Gaussian Inverse Wishart implementation [2]. The modular structure of the ALM filter further facilitates the replacement of individual components and the extension of the filter with new features. 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