Tracking Multiple Vehicles Using a Variational Radar Model
High-resolution radar sensors are able to resolve multiple detections per object and therefore provide valuable information for vehicle environment perception. For instance, multiple detections allow to infer the size of an object or to more precisel…
Authors: Alex, er Scheel, Klaus Dietmayer
This is a preprint (i.e. the accepted version) of A. Scheel and K. Dietmayer , “Tracking Multiple V ehicles Using a V ariational Radar Model, ” in IEEE Transactions on Intelligent Transportation Systems, vol. 20, no. 10, pp. 3721–3736, 2019. Digital Object Identifier 10.1109/TITS.2018.2879041. c 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collectiv e works, for resale or redistribution to serv ers or lists, or reuse of any copyrighted component of this work in other works. 1 T racking Multiple V ehicles Using a V ariational Radar Model Alexander Scheel and Klaus Dietmayer , Member , IEEE Abstract —High-resolution radar sensors are able to resolv e multiple detections per object and therefor e provide valuable information for vehicle en vironment perception. For instance, multiple detections allow to infer the size of an object or to more precisely measur e the object’ s motion. Y et, the increased amount of data raises the demands on tracking modules: measurement models that are able to process multiple detections for an object are necessary and measurement-to-object associations become more complex. This paper presents a new variational radar model for tracking vehicles using radar detections and demonstrates how this model can be incorporated into a Random-Finite-Set- based multi-object filter . The measurement model is learned from actual data using variational Gaussian mixtures and a voids excessive manual engineering. In combination with the multi- object tracker , the entire process chain fr om the raw mea- surements to the resulting tracks is f ormulated probabilistically . The presented approach is evaluated on experimental data and it is demonstrated that the data-driven measurement model outperforms a manually designed model. Index T erms —radar , tracking, variational methods, au- tonomous vehicles, sensor fusion, machine learning I . I N T RO D U C T I O N R AD AR sensors play an important role for vehicle en- vironment perception due to their ability to directly measure the relativ e radial velocity of an object, their robust- ness to adverse weather conditions, and their low price. In particular , radar data is widely used to track other vehicles in an ego-v ehicle’ s surrounding. Adv ances in automotiv e radar technology hav e led to increased sensor resolution and modern high-resolution radar is able to resolve multiple reflection centers of an object. Thus, each sensor may yield multiple measurements (i.e. detections) per object in a single scan. This additional data is v aluable as it provides more information on the shape, extent, or motion of an object and facilitates tracking objects more precisely and in complex maneuvers. Howe ver , tracking vehicles based on high-resolution radar data poses some challenges. First, one is faced with an ex- tended object problem as the vehicle extent is—at least in the near field—not negligible in comparison to sensor resolution and multiple radar measurements from a vehicle need to be correctly processed to a single estimate. Thus, many classical filters, such as the Kalman filter, which suppose exactly one measurement per cycle, are not directly applicable. Radar data additionally exhibits some peculiarities which further A. Scheel and K. Dietmayer are with the Institute of Measurement, Control, and Microtechnology , Ulm University , 89081 Ulm, Germany , e-mail: alexander .scheel@alumni.uni-ulm.de, klaus.dietmayer@uni-ulm.de complicate data processing: The detections may not always exhibit a clear shape and their number strongly depends on the sensor-to-object constellation. Also, the Doppler measure- ments introduce considerable ambiguity as they only provide the radial portion of an object’ s velocity and the superposition of forward motion and yaw rate causes dif ferent v elocity vectors at dif ferent locations on the vehicle. Some Doppler measurements may ev en originate from rotating wheels and thus do not match the motion o f the rigid body . In addition to data processing, the increased amount of detections further complicates the measurement-to-object association problem which is crucial in multi-object settings. One solution to the extended object problem is to include preprocessing routines that reduce multiple measurements to a single meta-measurement. Several of such approaches hav e been proposed for radar -based tracking. These include clustering and extraction of reference points as in [1], [2], or fitting bounding boxes and L-shapes [3], [4], reflection center models [5], or velocity profiles [6]–[8] to the data. While pre- processing routines are oftentimes effecti ve, computationally fast, and lead to clearly separable system architectures, the y face dif ficulties if the data from a single time step is ambiguous and the correct meta-measurement cannot be easily extracted. An alternativ e approach is to design extended object mea- surement models and filter algorithms which explicitly take all measurements into account. According to [9], which pro- vides an elaborate ov ervie w of extended object tracking, the approaches can be grouped into different modeling paradigms. The first paradigm models objects as a set of measurement sources with a specific spatial structure. An early version of this principle was presented in [10] and variations of it have been applied to radar-based vehicle tracking in [11]–[15]. A second variant of extended object models defines spatial distributions for the location of the measurement as initially proposed in [16] and [17]. A prominent example is the elliptical random matrix model [18] which has been extended for incorporating Doppler measurements in [19]. Also, a polynomial object model for tracking stationary objects such as guard rails [20] and a V olcanormal density for modeling vehicles [21] were proposed for radar applications. The third paradigm is to choose a physics-based approach [9]. Although many of the aforementioned approaches (e.g. [13]) may as well be assigned to this cate gory , it is used here to introduce [22] and [23] which use ray tracing to predict radar measurements. While [22] only considers the rear surface, the direct scattering model from [23] uses a full rectangular 2 description of the vehicle which allows for tracking arbitrary maneuvers with varying aspect angles. Lastly , some extended object models such as the random hypersurface model [24] or the Gaussian process model [25] use a parametric description of the object contour and estimate free form shapes. The Gaussian process approach has, for instance, been used for radar-based vehicle tracking in [26]. One of the major advantages of extended object measure- ment models is that they work on the raw data directly . Thus, they use the entire av ailable information and can resolve ambiguous situations by filtering over time. Still, some approaches such as the random matrix approach rely on restrictive assumptions that are not suitable for v ehicle tracking. Others require a certain amount of modeling and implementation effort such as the approaches based on ray tracing or on sets of reflection centers. Y et, all share the drawback that expert knowledge and manual adaption are necessary for including a certain sensor effect such as the spurious measurements from rotating wheels. Apart from tracking, sensor models are also important for sensor analysis and in simulation applications. Interestingly , there has been a recent development from radar models based on expert kno wledge or physical calculations (e.g. [27] and [28]) towards data-driv en approaches. For instance, [29] uses deep neural networks to simulate a radar power grid from an object list and a grid-based description of the environment. A statistical study on radar measurements from vehicles in dependence on the aspect angle was conducted in [30]. In [31] and [32], kernel density estimation methods are employed to learn a probabilistic measurement model for simulation. In this paper , the idea of leaving the modeling task to machine learning tools is transferred to tracking. A variational radar model for vehicles is learned directly from actual radar detection data. Thus, the shortcomings of existing extended object measurement models are overcome: The engineering effort is diminished and different sensor effects are captured automatically . The process in volv es finding a conditional density function that relates the measurements and v ehicle state. This is similar to the simulation model from [32]. In this work, howe ver , a variational Gaussian mixture (VGM) approach [33]–[35] is used. F or automotiv e radar applica- tions, VGMs have, for example, previously been employed for batch estimation of maps in [36]. In contrast to kernel density estimation, they do not require storing all training data points and instead yield a compact analytical mixture density which can be easily incorporated into a tracking framew ork. As a Bayesian inference technique, VGMs fur- thermore integrate nicely with tracking filters and facilitate an integral Bayesian view of the entire problem. Finally , VGMs av oid well-known singularity issues of alternative expectation maximization (EM) approaches and concurrently determine the number of required mixture components [35]. T o av oid the excessi ve manual labeling effort that oftentimes comes with machine learning, the training data set is automatically generated using a reference vehicle. The variational radar model is additionally incorporated into a multi-object framew ork to track multiple vehicles and to tackle measurement-to-object associations, clutter measure- y R x R Ego-V ehicle ω v ϕ b a x VC y VC y SC x SC v D α d y OC x OC Fig. 1. Schematic of the state vector , the radar measurements, and the vehicle (VC), sensor (SC), and object (OC) coordinate systems; adapted from [40] ments, and the fusion of radar data from multiple sensors in a principled way . In particular , an e xtended object labeled multi-Bernoulli (LMB) filter [37] based on finite set statis- tics (FISST) is chosen. FISST [38], [39] is a rather recent theoretical framework which provides a rigorous Bayesian for- mulation of the multi-object problem and mathematical tools for deri ving dif ferent filter algorithms. Thus, it allows for a consistent probabilistic end-to-end formulation of the problem. Nonetheless, an adaption of other tracking approaches (multi- object or single object) to accommodate the variational radar model should be possible. See [9] for a more detailed overvie w of multi-object methods for tracking extended objects. In the remainder of the paper, the tracking problem is first formulated in Section II. The variational radar model and the multi-object measurement likelihood are then de veloped in Section III and Section IV discusses the multi-object tracking approach. The application of the variational radar model to experimental data is shown in Section V and tracking results are ev aluated in Section VI. Section VII concludes the paper . I I . P RO B L E M F O R M U L A T I O N A. V ehicle and Measurement Representation The goal is to recursiv ely provide state estimates for all vehicles in the field of view (FO V) of the radar sensors based on the av ailable measurements. As illustrated in Fig. 1, each vehicle’ s state is described by the composed state vector x k = [ ξ T k , ζ T k ] T ∈ X where X is the state space and ξ k and ζ k are the kinematic and extent portion, respectiv ely . Moreover , the subscript k denotes the time step index. The kinematic state ξ k = [ x R,k , y R,k , ϕ k , v k , ω k ] T combines the position of the rear axle center given by x R,k and y R,k , the yaw angle ϕ k , the vehicle speed v k , and the yaw rate ω k . The extent portion ζ k = [ a k , b k ] T comprises the vehicle width a k and length b k . The position of the rear axle is fixed at 77% of the vehicle length as this value has empirically shown to be suitable for many vehicle types. T o identify the different objects and to extract trajectories ov er time, each state vector is augmented with a unique label ` ∈ L from the label space L . This yields the labeled state vector x k = [ x T k , ` ] T . All present vehicles are combined in the multi-object state which is modeled as the random finite set 3 (RFS) X k = { x (1) k , . . . , x ( n ) k } ⊂ X × L where the cardinality of the set | X | = n equals the number of vehicles. In each measurement cycle, a radar sensor provides a set of detections Z k = { z (1) k , ..., z ( m ) k } ⊂ Z from the measurement space Z which either originate from actual vehicles, sensor noise, or other objects that are not rele vant to the v ehicle track- ing task. Their number m may change from cycle to c ycle. Each detection z k = [ d k , α k , v D,k ] T yields the measured range d k , azimuth angle α k , and Doppler velocity v D,k . While the vehicle state is defined in the ego-vehicle co- ordinate system and the measurements are received in the sensor coordinate system using a polar representation, trans- formations to other coordinate system will be necessary . For instance, learning the vehicle model and computing the likelihood functions requires transforming the object states to the respectiv e sensor coordinate system. Such transformations are indicated by the subscripts SC or OC for the sensor or object coordinate system when learning the variational model. T o av oid cluttered notation, howe ver , the subscripts are omitted in the measurement likelihood and filter update equations. B. The Multi-Object Bayes F ilter The multi-object Bayes filter [39] is used to recursi vely compute the posterior density of the multi-object state π k | k ( X k | Z 1: k ) . This density captures the uncertainty in both the number of set elements as well as their values and can hence be used to obtain estimates of the number of vehicles and their states. It is conditioned on all measurement sets from the first to the k -th time step as denoted by Z 1: k . As in the classical Bayes filter , the estimation procedure is split into a prediction and update step. In the prediction step, the prior multi-object density is computed using the Chapman- K olmogorov equation π k | k − 1 ( X k | Z 1: k − 1 ) = Z f k | k − 1 ( X k | X k − 1 ) π k − 1 | k − 1 ( X k − 1 | Z 1: k − 1 ) δ X k − 1 , (1) where the multi-object transition density f k | k − 1 ( X k | X k − 1 ) gov erns the ev olution of the multi-object state including object motion as well as appearance and disappearance. Information from new measurements is incorporated in the update step π k | k ( X k | Z 1: k ) = g k ( Z k | X k ) π k | k − 1 ( X k | Z 1: k − 1 ) R g k ( Z k | X k ) π k | k − 1 ( X k | Z 1: k − 1 ) δ X k . (2) using the multi-object likelihood function g k ( Z k | X k ) which captures the measurement process and determines how likely the receiv ed measurements are for a specific multi-object state. As the computations in volv e set-v alued random variables and their densities, the integrals in (1) and (2) are set integrals as defined in [39]. Note that the time subscript is dropped in the remainder of the paper to a void cluttered notation. Prior quantities are indicated using the subscript + . I I I . V A R I A T I O NA L R A DA R M O D E L Before the multi-object filter from (1) and (2) can be formulated in detail, the variational radar model and the multi- object likelihood function, which are required during filter update, are dev eloped in this section. First, the basic concept of VGMs is outlined. The approach is then applied to learning a model for a single vehicle. Finally , the model is incorporated into the multi-object likelihood. A. V ariational Gaussian Mixtures VGMs for learning probabilistic models from data were initially presented in [33]. The basic assumption is that the data at hand is generated by an underlying Gaussian mixture model. Howe ver , the parameters of the model are unknown and the goal is thus to estimate the parameter values given the av ailable data. This is done in a Bayesian fashion which in v olves computing posterior densities over the parameter values and allows for including a-priori knowledge about the parameters through prior densities. The estimated posterior parameter densities and the underlying Gaussian mixture then form a probabilistic model of the data which can be used to make predictions on future data points. In the following, the mathematical concepts are briefly outlined. The explanations closely follow [35] to which the reader is referred to for a more detailed and accessible description. Mathematically , the data for learning the model, the training data, is a set of m data points Z D = { z (1) D , . . . , z ( m ) D } . Here, the letter z is reused to emphasize that the training data is measured information ev en though it will ha ve a dif ferent form than the presented radar measurements. The training data was created by a Gaussian mixture model with c components. Each Gaussian distribution N ( ·| µ ( j ) , H − 1 ( j ) ) in the mixture is defined by its mean µ ( j ) , its precision matrix H ( j ) , and is assigned a mixing coefficient w j which measures the contribution of the j -th component to the density . For bre vity , the mean vectors and precision matrices of all components are combined in the parameter sets M and H , respectiv ely , and the weights in the weight vector w . The latent variable l ( i ) is introduced to denote which component created the data point z ( i ) D . Hence, these latent variables are 1-of-K binary vectors where one of the elements l ( i ) j is one and the remaining elements are zero. Again, all vectors are combined in the set of latent variables L . For gi ven latent variables and parameter values, the likelihood of the training data is thus p ( Z D | L, M , H ) = m Y i =1 c Y j =1 N ( z ( i ) D | µ ( j ) , H − 1 ( j ) ) l ( i ) j . (3) Since the Bayesian treatment assumes the unknown latent variables and parameters to be random variables, the full probabilistic model is giv en by the joint distribution of the training data, latent variables, and parameters p ( Z D , L,w , M , H ) = p ( Z D | L, M , H ) p ( L | w ) p ( w ) p ( M | H ) p ( H ) . (4) The factorization follo ws from the Gaussian mixture structure and the Bayesian formulation. Its factors are the data likeli- hood, the distribution of the latent variables for given mixing coefficients p ( L | w ) = m Y i =1 c Y j =1 w l ( i ) j j , (5) 4 and the prior distributions over the mixture model parameters p ( w ) , p ( M | H ) , and p ( H ) . These priors are modeled in conju- gate forms to (3) and (5). The prior of the mixing coefficients is a Dirichlet distribution p ( w ) = Dir( w | ρ 0 ) = C ( ρ 0 ) c Y j =1 w ρ 0 − 1 j (6) with parameter ρ 0 and normalization constant C ( ρ 0 ) . The prior of the mean vectors and precision matrices is a Gaussian- W ishart distribution with independent elements for each com- ponent. It is giv en by p ( M , H ) = p ( M | H ) p ( H ) = c Y j =1 N ( µ ( j ) | γ 0 , β − 1 0 H − 1 ( j ) ) W ( H ( j ) | V 0 , ν 0 ) , (7) with W ( ·| V 0 , ν 0 ) denoting a Wishart density and the param- eters γ 0 , β 0 , V 0 , and ν 0 . T ogether with ρ 0 , these are the hyperparameters of the model which govern the shape of the prior distributions and ho w informativ e they are. The y are used to initialize all mixture components with the same v alue. T o compute the posterior densities ov er the latent variables and model parameters, a v ariational approach is used. It allows for an optimization-based approximation of the true posterior density and is based on maximizing the functional Z q (Φ) ln p ( Z D , Φ) q (Φ) dΦ . (8) Here, the latent variables and model parameters were com- bined in Φ for brevity . The maximum of the functional occurs if the proposal distribution q (Φ) equals the true posterior dis- tribution of the latent v ariables and model parameters p (Φ | Z D ) [35]. Thus, the posterior distrib utions o ver parameters and latent variables are obtained by choosing a certain class of distributions for q (Φ) and maximizing (8) with respect to q (Φ) . For VGMs, the factorized distribution q (Φ) = q ( L, w , M , H ) = q ( L ) q ( w , M , H ) (9) is chosen. An optimal solution can then be found by iterativ ely maximizing (8) with respect to q ( L ) and q ( w , M , H ) . It can be shown that the optimal solution has the structure q ∗ ( L, w , M , H ) = q ∗ ( L ) q ∗ ( w ) q ∗ ( M | H ) q ∗ ( H ) (10) where the different factors take the same form as the distri- butions from (5) to (7) with updated hyperparameters γ ( j ) , β j , V ( j ) , ν j , and ρ j ; see [35] for the full equations. In contrast to the initial v alues of the hyperparameters, these values depend on the training data that is associated to the respectiv e component and hence differ for each component. T o obtain the predictiv e density p ( ˜ z D | Z D ) which measures how likely a new data point ˜ z D is, given the model that was obtained from the training data, the optimized distributions are inserted into (4) and the latent v ariables as well as model parameters are marginalized by integration. This yields a mixture of Student’ s t distributions [35] p ( ˜ z D | Z D ) = 1 P c j =1 ρ j c X j =1 ρ j St( ˜ z D | γ ( j ) , ˜ H ( j ) , ν j + 1 − | ˜ z D | ) , (11) where St( ·|· , · , · ) is a Student’ s t density , ˜ H ( j ) is the precision matrix of the j -th component giv en by ˜ H ( j ) = ( ν j + 1 − | ˜ z D | ) β j 1 + β j V ( j ) , (12) and | ˜ z D | is the dimension of ˜ z D . Note that the reason for obtaining a Student’ s t mixture as predictiv e density instead of Gaussian mixture lies in the inclusion of the uncertainty in the parameter estimates. B. Learning a V ariational Radar Model for V ehicles T o obtain a vehicle measurement model, the VGM approach is applied to actual radar measurements from vehicles and used to find a predictive density for radar detections giv en a particular vehicle state. Ev en though VGMs are able to generalize to a certain extent, it is important to collect data samples from all relev ant areas of the training data space to enable the VGM to detect the structure and basic relationships in the data. For the presented state and measurement vectors, this would imply that data has to be collected in a ten- dimensional space. For instance, samples would be needed for vehicles of different size, with different poses, speeds, and yaw rates. Also, the complex relationship between Doppler measurements and object state as well as the representation of the measurements in polar coordinates may require many mixture components to be able to capture the nonlinearities. T o av oid these issues, the problem is simplified by apply- ing dimension reduction. In particular , the measurements are transformed using the nonlinear transformation function z 0 = z 0 x z 0 y z 0 d = f z ( x S C , z ) = z x,OC /b z y ,OC /a v D − (cos( α ) s 1 + sin( α ) s 2 ) , (13) where z x,OC and z y ,OC are the position of the radar detections in the object coordinate system, s 1 = v cos( ϕ S C ) + ω y R,S C , (14) and s 2 = v sin( ϕ S C ) − ω x R,S C . (15) Thus, the position of all vehicle measurements is transformed to a normalized object coordinate system that is independent of the vehicle dimensions. This results in the coordinates z 0 x and z 0 y . Additionally , the expected Doppler velocity is computed from the vehicle state using (14) and (15). It is subtracted from the measured Doppler velocity and the model therefore only learns the Doppler error z 0 d . Here, information from the vehicle state is used and implicitly enters the measurement model. The object state itself is transformed using x 0 = f x ( x S C ) = ϕ S C − atan2( y R,S C , x R,S C ) . (16) and is hence reduced to a single deriv ed quantity which is approximately the aspect angle under which the sensor sees the vehicle. Concatenating z 0 and x 0 yields the training data representation z D = [ z 0 T , x 0 ] T . 5 Note that this manually designed dimension reduction re- quires some expert knowledge. Other techniques which in- clude the distance to the v ehicle as additional variable or which automatically detect a suitable representation (e.g. [41]) could hav e been used instead. Howe ver , the chosen variant can be interpreted to incorporate the most dominant and well-known properties where the additional ef fort to learn them does not appear to be beneficial. These properties are the basic Doppler measurement principle or the insight that the relative location of measurements will be approximately similar irrespectiv e of the vehicle size or its position in the field of view and will mostly depend on the aspect angle. By computing the predictiv e density (11), the VGM model provides a joint distribution over the transformed measure- ments and state p ( z D ) = p ( z 0 , x 0 ) , where the dependency on Z D is omitted for brevity . Then, the likelihood for the relativ e position of the measurements and the Doppler error for a given aspect angle g z 0 ( z 0 | x 0 ) is obtained through g z 0 ( z 0 | x 0 ) = p ( z 0 , x 0 ) p ( x 0 ) , (17) where p ( x 0 ) is determined from marginalization. See [42] for the corresponding equations. By using the V GM technique, it is assumed that radar detections are generated by an underlying Gaussian mixture structure in which each measurement originates from one of the components. Intuitively , each mixture component can be interpreted to be a particular reflection center of the vehicle with associated position and measurement uncertainty . By including the aspect angle, the model not only learns the number and location but also the rele vance of each reflection center for a particular line of sight. Y et, the mixture density is directly used as a spatial distribution model in this work to av oid an explicit association of detections to reflection centers during tracking. C. Multi-Object Likelihood Function So far , the presented approach allows learning a mea- surement model for a single vehicle which defines where radar detections are expected and how large the deviations from the expected Doppler velocity may be. For updating the multi-object state using (2), howe ver , the formulation of the entire multi-object likelihood g ( Z | X ) , which relates all measurements to all objects, is necessary . 1) Detection-T ype Likelihood: T o this end, the single-object model is incorporated into the multi-object likelihood function from [37] which is designed for detection-type measurements. It is based on several assumptions that hav e also been pre- viously used in other extended object models (see e.g. [17], [43]). These assumptions are: 1) An object is detected with the probability of detection p D ( x, ` ) or misdetected with the complementary prob- ability q D ( x, ` ) = 1 − p D ( x, ` ) . 2) If an object is detected, it giv es rise to a set of measure- ments Z O which follows the single object likelihood function g ( Z O | x, ` ) . The number of receiv ed measure- ments is Poisson distributed with expected value λ T . 3) The measurement set Z is a union of object and clutter measurement sets. The object measurement sets are independently generated by each object. 4) The number of clutter measurements is Poisson dis- tributed with expected value λ C and the values follo w the density p C ( z ) . Hence, they are distributed according to the Poisson RFS [39] g C with intensity function κ ( z ) = λ C p C ( z ) . Using these assumptions, the likelihood of obtaining a set of measurements from a giv en multi-object state is [37] g ( Z | X ) = g C ( Z ) | X | +1 X i =1 X U ( Z ) ∈P i ( Z ) θ ∈ Θ( U ( Z )) ψ U ( Z ) ( ·| θ ) X (18) with ψ U ( Z ) ( x, ` | θ ) = ( p D ( x,` ) g ( U θ ( ` ) ( Z ) | x,` ) [ κ ( · )] U θ ( ` ) ( Z ) , θ ( ` ) > 0 q D ( x, ` ) , θ ( ` ) = 0 , (19) g C ( Z ) = e − λ C [ κ ( · )] Z . (20) Here, the short notation h X , Y x ∈ X h ( x ) , h ∅ = 1 , (21) is used to denote products of a real-valued function h ( · ) applied to all elements of a set. In a nutshell, the function ev aluates the dif ferent possibilities of ho w the measurements could be composed and computes their likelihood. For this purpose, the two sums in (18) are used to ev aluate dif ferent partitions U ( Z ) of the measurement set and dif ferent asso- ciation mappings θ . P i ( Z ) denotes the set of all partitions that contain i mutually exclusi ve clusters. Each association mapping θ : L ( X ) → { 0 , 1 , . . . , |U ( Z ) |} assigns the labels from the multi-object state to the clusters in a partition. The labels are retriev ed using the label projection function L ( X ) = { ` | [ x T , ` ] T ∈ X } . A cluster in a partition may only be assigned to one track, i.e. θ ( ` ) = θ ( ` 0 ) > 0 implies ` = ` 0 while several tracks may be assigned to the index 0 which stands for a misdetection. Θ( U ( Z )) is the space of all possible association mappings and the cluster assigned to track ` is identified by U θ ( ` ) ( Z ) . For the case θ ( ` ) > 0 , (19) computes the single object likelihood g ( Z O | x, ` ) = e − λ T [ λ T g z ( ·| x )] Z O , (22) where Z O = U θ ( ` ) ( Z ) for a specific track-to-cluster associ- ation, and cancels the measurements from the overall clutter term g C ( Z ) . Reformulating the ratio g ( Z O | x, ` ) [ κ ] Z O = e − λ T λ | Z O | T λ | Z O | C Y z ∈ Z O g z ( z | x ) p C ( z ) (23) from (19) separates it into a factor which considers the number of measurements and a factor which compares how well measurements fit to the object and clutter likelihoods. 6 2) Incorporating the V ariational Model: The multi-object likelihood is a density over an RFS that is a subset of the measurement space Z . Hence, the object and clutter likeli- hoods are densities with Z as sample space. In contrast, the conditional density (17) from the v ariational radar model is a density ov er a normalized space where the scaling depends on the object state. Simply inserting (17) into (22) is thus mathematically incorrect and would prohibit a meaningful comparison between different tracks as well as clutter . Y et, the identity g z 0 ( z 0 | x ) p C ( z 0 ) = g z ( f − 1 z ( z 0 , x ) | x ) ∂ f − 1 z ( z 0 ,x ) ∂ z 0 x ∂ z 0 y ∂ z 0 d p C ( f − 1 z ( z 0 , x )) ∂ f − 1 z ( z 0 ,x ) ∂ z 0 x ∂ z 0 y ∂ z 0 d = g z ( f − 1 z ( z 0 , x ) | x ) p C ( f − 1 z ( z 0 , x )) = g z ( z | x ) p C ( z ) (24) states that the ratio between the object and clutter likelihood remains identical if both likelihoods are transformed using the same transformation function. Thus, it can be used to replace the likelihood ratio from (23) with a ne w ratio between the conditional density from the variational radar model and the transformed clutter density . The identity follows from comput- ing the distributions of z 0 as derived distributions from z ; see e.g. [44]. Note that the in verse of the transformation function f − 1 z ( z 0 , x ) , which transforms the measurements from a Carte- sian representation back to the original polar measurement space, exists. Y et, it is not defined at the location of the sensor origin. As this pathological case is not relev ant in practical scenarios, it is neglected here. Also, g z 0 ( z 0 | x ) = g z 0 ( z 0 | x 0 ) as the information of x 0 is fully contained in x . One way to obtain the clutter density for the transformed measurements p C ( z 0 ) is to fully transform the original clutter density o ver the polar measurement space. Here, ho wev er , an alternati ve approach is chosen: It is assumed that clutter is uniformly distributed over the Cartesian sensor coordinate system. Moreover , a mixture between a uniform density and a Gaussian distribution is used for modeling the Doppler values. The Gaussian distribution is centered at the Doppler value of stationary objects and emphasizes that they are the most frequent clutter source. Using the Cartesian representation, the transformation of the density to the space of z 0 is considerably simplified and mostly inv olves scaling by the factor a · b to account for the vehicle size. The corresponding clutter density in the original measurement space which is required in (18) could be determined by transformation. Y et, this factor cancels in the update step and is not required; see Section IV -B. In summary , the ratio (24), which is inserted into the multi- object likelihood, is computed by first transforming the current measurements and state and then ev aluating the densities. Note that information from the state enters through both steps. I V . M U L T I - O B J E C T T R A C K I N G For tracking multiple vehicles, the multi-object measure- ment likelihood from the previous section is used in an extended object LMB filter [37]. This filter has, for instance, also been used in [40] in conjunction with the direct scat- tering model. By modeling the multi-object state using LMB Update T rack Initialization Approximation Measurements ∪ LMB GLMB GLMB Prediction LMB LMB LMB Independent Prediction Feasability Check Fig. 2. Overview of the filtering procedure and generalized labeled multi-Bernoulli (GLMB) distributions [45], it facilitates an analytical solution to (1) and (2). Please refer to [37] for a detailed description including pseudo code. In this paper, the original version is slightly modified to av oid ov erlapping objects as initially proposed in [46]. A schematic ov erview of the filtering procedure is shown in Fig. 2. A. Initialization and Prediction At the end of the last filter recursion and before predic- tion, the distribution ov er the current multi-object state is represented using an LMB distrib ution. It consists of sev eral independent object hypotheses which are described by the existence probability r ( ` ) and the single object state density p ( x, ` ) . The labels of all present object hypotheses define the label space L . The multi-object density is thus gi ven by π ( X ) = ∆( X ) w ( L ( X ))[ p ( · )] X , (25) where w ( I ) = Y i ∈ L 1 − r ( i ) Y ` ∈ I 1 L ( ` ) r ( ` ) 1 − r ( ` ) (26) is the probability that all tracks in the multi-object state X exist and the remaining hypotheses do not. The inclusion function 1 L ( ` ) ensures that only labels from existing hypotheses are used and is 1 if and only if ` ∈ L . Moreov er the distinct label indicator ∆ ( X ) = δ ( |L ( X ) | − | X | ) where δ ( · ) denotes the Kronecker -delta function is used to ensure that each object in a labeled set has a unique label. The cardinality distrib ution can be obtained by marginalizing over the states which yields a Poisson binomial distribution. The expected value of this distribution serves as cardinality estimate and is the sum over the existence probabilities of all hypotheses. In the track initialization stage, new track hypotheses are generated for measurements that have not considerably con- tributed to updating existing tracks and exhibit a significant Doppler velocity . The new hypotheses are labeled with ne w labels from label space B and are assigned an initial existence probability r ( ` ) B as well as a prior state density p B ( x, ` ) . Afterwards, they are appended to the existing hypotheses which yields the new and augmented label space L + = L ∪ B . In the first prediction step, the existing and new tracks are predicted using the standard multi-object transition model. Each object surviv es to the next time step with a probability 7 of persistence p S ( x, ` ) or disappears with complementary probability . If an object surviv es, its states ev olve according to the single object transition density f + ( x + | x, ` ) . Hence [37], r ( ` ) + = η ( ` ) r ( ` ) , (27) p + ( x + , ` ) = R p S ( x, ` ) f + ( x + | x, ` ) p ( x, ` )d x η ( ` ) , (28) η ( ` ) = Z Z p S ( x, ` ) f + ( x + | x, ` ) p ( x, ` )d x d x + . (29) Here, x + is the predicted state, r ( ` ) + the predicted existence probability and p + ( x + , ` ) the predicted state density of hy- pothesis ` . The ne w densities and existence probabilities then constitute the parameters of the prior LMB distribution from the first prediction step. For the application to vehicle tracking, f + ( x + | x, ` ) consists of a constant turn rate and velocity (CTR V) motion model [47] with additiv e noise for the kinematic state and pseudo noise is added to the extent portion. Subsequently , a second prediction step eliminates hypothe- ses with ov erlapping objects by conditioning the predicted multi-object density on the ev ent of physical feasibility F . This yields [46] π + ( X + |F ) = ∆ ( X + ) w + ( L ( X + ))[ p + ( · )] X + , (30) with w + ( I ) = p ( F | I ) ˜ w + ( I ) P J ⊆ L + p ( F | J ) ˜ w + ( J ) , (31) and where ˜ w + ( I ) is obtained by inserting the existence probabilities from the first prediction step (27) into (26). The likelihood for physical feasibility p ( F | I ) is chosen to be 1 if and only if none of the objects in the label set I ov erlap. Here, the predicted mean v alues of the vehicle positions and extents are used to determine possible overlaps. Y et, the prior multi-object density from (30) and (31) is not in LMB form and objects are not independent anymore. That is, the weight does no longer factorize ov er the set elements. Instead, the multi-object prior is now a variant of the more general GLMB distribution which allows for arbitrary weights and superposition of sev eral multi-object hypotheses [45]. B. Update Substituting the multi-object likelihood equations from Sec- tion III-C and the multi-object prior from the prediction into (2) yields the parameters of the posterior multi-object distribution (cf. [37]) π ( X | Z ) =∆ ( X ) | X | +1 X i =1 X U ( Z ) ∈P i ( Z ) θ ∈ Θ( U ( Z )) w U ( Z ) ( L ( X ) | θ ) × [ p ( ·|U ( Z ) , θ )] X (32) with w U ( Z ) ( I | θ ) = w + ( I ) η U ( Z ) ( ·| θ ) I P J ⊆ L | J | +1 P i =1 X U ( Z ) ∈P i ( Z ) θ ∈ Θ( U ( Z )) w + ( J ) η U ( Z ) ( ·| θ ) J , (33) p ( x, ` |U ( Z ) , θ ) = p + ( x + , ` ) ψ U ( Z ) ( x + , ` | θ ) η U ( Z ) ( ` | θ ) , (34) and η U ( Z ) ( ` | θ ) = Z p + ( x + , ` ) ψ U ( Z ) ( x + , ` | θ )d x + . (35) Again, the distribution is in GLMB form. Y et, the update step introduces additional dependencies among objects which arise from the claim that each cluster in a measurement partition may only be assigned to one object. As observable from the sums in (32), the posterior multi-object distribution is hence composed of sev eral hypotheses that have been updated using different partitioning and clustering possibilities. C. Appr oximation T o avoid a steady increase of multi-object hypotheses in the GLMB posterior ov er time, the posterior GLMB density is approximated by an LMB density at the end of each filter recursion. This procedure has been proposed by [48] and results in a posterior LMB density with parameters [37] r ( ` ) = X I ⊆ L + | I | +1 X i =1 X U ( Z ) ∈P i ( Z ) θ ∈ Θ( U ( Z )) w U ( Z ) ( I | θ )1 L ( ` ) , (36) and p ( x, ` ) = 1 r ( ` ) X I ⊆ L + | I | +1 X i =1 X U ( Z ) ∈P i ( Z ) θ ∈ Θ( U ( Z )) w U ( Z ) ( I | θ )1 L ( ` ) × p ( x, ` |U ( Z ) , θ ) . (37) V ehicle state estimates are then extracted from this result. D. Estimating the Single Object Densities The extended LMB filter internally holds and processes the state densities of the different object hypotheses. In particular , (28) and (34) predict the single object state and update it with the associated measurements, respectiv ely . T o solve these equations, standard Bayesian filtering techniques can be applied. This work uses a particle filter approach as both the transition density and the measurement model are nonlinear . T o reduce the amount of required particles, a simplified approach which is based on the Rao-Blackwellized particle filter (RBPF) technique [49] is applied. Only the kinematic portion ξ is fully represented by particles while the estimation of the extent portion ζ is approximated by employing discrete distributions. At the beginning of the filter procedure, each particle holds a single hypothesis for the v ehicle extent. During prediction, a discrete transition density is applied to each particle. It creates new extent hypotheses by varying the width and length. Thus, a discrete distribution with up to nine elements is generated. The lik elihood is e v aluated for all e xtent hypotheses and the resulting posterior extent distribution of each particle is again reduced to a single extent hypothesis by computing its mean. Note, howe ver , that this step discards information about the extent estimate and that the particles hence do not capture the full extent uncertainty . Y et, the entire procedure introduces a local search for best fitting extent and allows an easy adaption of each particle’ s extent estimate. 8 V . R A DA R M O D E L F R O M E X P E R I M E N T A L D A TA Now that the measurement model and multi-object filter are formulated, they are applied to experimental radar data. This section first describes the process of learning a variational radar model for vehicles. As a supplement, the resulting model is made av ailable online 1 . The application to vehicle tracking is then demonstrated in the following section. A. Experimental Set-Up and Data Set T o generate the measurement data, two vehicles were used. The ego-v ehicle is equipped with four short-range radar sen- sors that are mounted in the corners of the front and rear bumper . The sensors have an opening angle of about 170 ◦ , a range of 43 m, and the sensor axes are rotated by 45 ◦ with respect to the vehicle axis. Thus, an almost complete 360 ◦ cov erage of the close-up range is gi ven. All sensors run at a frequency of 20 Hz and are not synchronized among themselves. Apart from the radar sensors, an IBEO Lux lidar, which serves as reference sensor , is mounted in the center of the front bumper . The second vehicle, a Mercedes E- Class station w agon (S212), serv es as tar get vehicle. Both vehicles are equipped with a GeneSys ADMA which combines a precise dif ferential global positioning system (DGPS) and inertial measurement unit (IMU). It provides the pose of the vehicles in a global coordinate system and the object motion. This allows computing the ground truth position of the target vehicle in both the ego-vehicle coordinate system and the four sensor coordinate systems. The measurement data was collected on a closed test site and on public roads. It includes typical longitudinal and cross traffic situations as well as artificial maneuvers which were designed to achieve a good cov erage of the measurement and state space. These maneuvers, for instance, include circling the stationary ego-v ehicle in dif ferent distances, driving small circles in different parts of the FO V, or driving straight lines at different distances and angles. In case of a stationary vehicle, measuring the orientation in global coordinates is challenging for the DGPS/IMU system. From eye inspection, a mismatch between the vehicle ground truth and the laser measurements was observ able in some sequences. In these cases, the orientation error was manually corrected using the precise measurements from the lidar . The data set was then generated from the recorded measure- ments by computing the ground truth position of the target ve- hicle in sensor coordinates and determining the measurements that originate from the vehicle by gating. That is, only radar detections in a bounding box that exceeds the actual vehicle dimensions by 0.5 m in all directions were paired with the respectiv e ground truth vehicle state. The remaining clutter and measurements from other traffic participants were discarded. Subsequently , the transformation functions (13) and (16) were applied to the extracted detection and vehicle state pairs. The entire data set comprises 336,287 data points from approximately 123 minutes of recorded sensor data. T wo vie ws of the data set are shown in Fig. 3. In particular, a top view 1 The variational radar model is a vailable at https://github .com/A-Scheel/V ariational-Radar-Model − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 0 . 5 0 0 . 5 z 0 x z 0 y (a) Measurement positions in normalized object coordinates − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 10 − 5 0 5 10 z 0 x z 0 d in m/s (b) Doppler error over the length axis of the vehicle Fig. 3. T wo views of the training data points of the measurements in normalized coordinates is sho wn in Fig. 3a and the Doppler error ov er the longitudinal axis in Fig. 3b. It is observable that most measurements originate from the vehicle surface and that deviations from the expected Doppler velocity mostly occur close to the front and rear axles. There is an imbalance in the data set in terms of the number of measurements for dif ferent aspect angles. For example, it contains roughly three times more measurements from the rear perspectiv e than from the front perspectiv e. Also, there are about 20,000 data points in a 5 ◦ interval around the rear perspectiv e x 0 = 0 ◦ , whereas the neighboring 5 ◦ interval around x 0 = − 5 ◦ only contains 4736 data points. While the imbalance over distant aspect angles is mostly eliminated when computing the conditional density , local imbalances can introduce small biases. If only measurements from the rear surface are available for sev eral time steps, for example, it was observed that a model that was learned from the entire data set tends to fav or aspect angles around x 0 = 0 ◦ . T o av oid such issues, a balanced subset of data points was used as training set. It contains 95,688 data points that result in an ev en aspect angle histogram with 5 ◦ bins. B. Resulting V ariational Radar Model A modified MA TLAB implementation 2 of the VGM was used to fit the mixture model to the training data. The number of components was set to c = 70 and the hyperparameter of the Dirichlet prior ov er the mixture weights was set to ρ 0 = 1 . For the Gaussian-Wishart prior, the hyperparameters were set to β 0 = 1 , ν 0 = | z D | + 1 , γ 0 was set to the mean of all training points, and V 0 was initialized as identity matrix. This results in a non-informati ve prior which does not assume a certain form of the VGM parameters. 2 original implementation by Mo Chen, https://de.mathworks.com/matlabcentral/fileexchange/35362-v ariational- bayesian-inference-for-gaussian-mixture-model 9 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 0 . 5 0 0 . 5 z 0 x z 0 y 0 . 5 1 1 . 5 2 Density value Fig. 4. Marginal density p ( z 0 x , z 0 y ) A useful feature of the VGM approach is that it internally penalizes the model complexity . Unnecessary components— i.e. components which explain no or only very fe w measurements—automatically receive low weights. From the 70 initially proposed components, 20 recei ved a mixing weight below 10 − 5 . As these components do not contribute to the model and only increase computation time, they are removed. Thus, the number of components of the final mixture is c = 50 . As a visualization of the full, four-dimensional joint density p ( z 0 , x 0 ) is dif ficult, Fig. 4 illustrates the marginal density p ( z 0 x , z 0 y ) . The VGM has identified that most measurements originate from the vehicle surface. Also, it identified the centers of the front and rear surface as well as the four wheels and wheel houses as typical measurement sources. The conditional density p ( z 0 x , z 0 y | x 0 ) shows where measure- ments are expected for a giv en aspect angle x 0 . Fig. 5 depicts examples for different values of x 0 . Figure 5a shows the conditional density when looking at the v ehicle front. The aspect angle is close to the π , − π boundary . Since the VGM does not consider the periodic nature of the aspect angle, an abrupt change in the inv olved mixture components occurs when the sign of the aspect angle changes. This could be further improv ed by adapting the standard VGM to periodic states. As the components on both sides are similar and e xpand ov er the boundary , howe ver , this issue has so far not been noticeable during application. A view from the right side of the vehicle is illustrated in Fig. 5b. Clearly , measurements are expected close to the right vehicle surface ( z 0 y = − 0 . 5 ). Also, the positions of the right wheels are identifiable as frequent measurement sources. This effect is again visible in Fig. 5c where the vehicle is viewed from rear right. In addition to the wheels, the vehicle corner becomes another prominent feature. When viewed from the rear , measurements tend to originate from the center of the rear surface as shown in Fig. 5d. Additionally , relatively low weighted components on the vehicle interior come into play . A possible explanation for the components close to z 0 x = 0 is that the sensor recei ves reflections from the rear axle or the edge of the vehicle roof. The conditional density p ( z 0 x , z 0 d | z y = − 0 . 5 , x 0 = − π 2 ) is depicted in Fig. 6. It shows the density of the Doppler error on the right vehicle surface over the vehicle length when looking from the right. It can be observed that the model expects larger Doppler errors in the vicinity of the wheels. In this case, the VGM has automatically learned the occurence of spurious measurements from rotating wheels. V I . T R AC K I N G U S I N G E X P E R I M E N T A L D A TA In this section, results for the tracking performance of the multi-object tracking approach in combination with the variational radar model are presented. The algorithm was im- plemented in MA TLAB and applied to different experimental scenarios that were recorded using the same ego-v ehicle as in section Section V. The section starts with some practical remarks on the implementation and the tracking accuracy is subsequently assessed for single and multi-vehicle scenarios. The performance is compared to the manually designed direct scattering approach from [40]. It differs in the single object likelihood and uses a clutter density which is defined in polar coordinates. The multi-object filter core is identical. A. Practical Implementation Issues 1) Number of P articles: Upon initialization, the number of particles for representing the birth density p B ( x, ` ) is 900 to cover the wide range of possible states. This number is gradually reduced by 100 in the following update steps until the number of particles reaches 300, which is the steady state value for tracked objects. 2) Constraints on V ehicle Dimensions: The dimensions of a vehicle are restricted to maximum and minimum v alues. These are a min = 1 . 4 m and a max = 2 . 5 m for the width as well as b min = 2 . 5 m and b max = 7 m for the length. Additionally , the ratio between the length and width is restricted to minimum and maximum values of 1.7 and 3.5, respectiv ely . Thus, only extent hypotheses with reasonable proportions are allowed. 3) Pr ocess and Measur ement Model P arameters: During prediction, process noise is added to the kinematic states of the vehicles. The noise is modeled as uniform distributions centered at 0 and sampled for each particle. The maximum values are defined for the interv al of one second and adjusted proportionally to the time difference between consecutiv e prediction steps. The normalized values are 3 m/s for the position, 0.698 rad/s for the angle, 9 m/s 2 for the velocity , and 3 rad/s 2 for the yaw rate. The probability of persistence is made dependent on the time between two consecuti ve updates and determined from an exponential distrib ution which models that an object persists for an average of 10 s in and 0.1 s outside the FO V. In the measurement model, the probability of detection is set to 0.8 and slowly decreased towards the boundaries of the FO V. The e xpected number of object and clutter measurements are set to λ T = 5 and λ C = 30 . 4) P artitioning and Association: Ev aluating all possible measurement partitions and cluster-to-object associations as demanded by the multi-object likelihood function (18) is computationally intractable even for a moderate amount of measurements. Therefore, only meaningful partitions are ev al- uated for obtaining the posterior multi-object density (32). Partitions are generated in two ways. In a first step, DBSCAN [50] with different distance thresholds between 0.5 m and 5 m is applied. Additionally , the predicted tracks are used to generate partitions by combining all measurements that are in the vicinity of an existing track. The resulting clus- ters in the partitions and particularly the contained Doppler measurements are further analyzed. If the measurements do 10 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 0 . 5 0 0 . 5 z 0 x z 0 y (a) x 0 = − 3 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 0 . 5 0 0 . 5 z 0 x z 0 y (b) x 0 = − π 2 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 0 . 5 0 0 . 5 z 0 x z 0 y (c) x 0 = − π 4 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 − 0 . 5 0 0 . 5 z 0 x z 0 y 5 10 15 Density value (d) x 0 = 0 Fig. 5. Marginal density p ( z 0 x , z 0 y | x 0 ) conditioned on the aspect angle x 0 . The line of sight between the sensor and the center of the rear axle is indicated by the arrow and the dashed rectangle depicts the normalized v ehicle dimensions. − 0 . 5 0 0 . 5 1 − 1 0 1 z 0 x z 0 d in m/s 1 2 Density value in s/m Fig. 6. Conditional density p ( z 0 x , z 0 d | z y = − 0 . 5 , x 0 = − π 2 ) not conform to consistent rigid body motion, the clusters are split and additional partitions with the resulting subclusters are added. This allows excluding clutter measurements as for example measurements from rotating wheels. As will be shown, this step is mainly necessary for the direct scattering tracking approach that is used for comparison. For each multi- object state hypothesis, the ten best association variants are determined using Murty’ s algorithm [51] and ev aluated. 5) Initialization and Pruning: New vehicle hypotheses are initialized as soon as there is a measurement cluster with at least tw o measurements that exhibit relev ant Doppler velocities and have not considerably contributed to updating an existing vehicle. The goal is to a void the creation of new hypotheses for stationary objects or from single temporary clutter measure- ments. Ne w vehicle hypotheses are assigned a birth existence probability of r ( ` ) B = 0 . 1 and the birth density p B ( x, ` ) is formed by creating particles with different plausible states. For poses where the length is observable, suitable length hypotheses between 2.5 m and 7 m are initialized. If the length is not observable, random values between 4 m and 5 m, that are closer to typical vehicle lengths, are created. As soon as the existence probability of a vehicle hypothesis falls belo w 0.01, it is pruned from the multi-object density . 6) Ego-Motion Compensation: Motion of the ego-v ehicle affects tracking in two ways and hence, compensation proce- dures are added. In measurement processing, the contribution of e go-vehicle motion is computed and remov ed from the Doppler measurements. Also, the prediction routine needs to account for the moving ego-v ehicle coordinate system in which the vehicles are tracked. Therefore, an additional step transforms the vehicles from the last to the current vehicle coordinate system. 7) Sensor Fusion: The presented tracking approach is used in a centralized fusion architecture to fuse the data from all four radar sensors of the vehicle. That is, a ne w update is triggered each time ne w data from a sensor arri ves and the information is fused into the posterior multi-object density . Measurements arriv e in order of recording time and out-of- sequence problems are not considered. 8) Computation T ime: The prototype MA TLAB implemen- tation presented in this paper is not intended for real-time calculations and the runtime is therefore not in the focus of the ev aluations. A rough analysis has, howe ver , revealed that approximately 85% of the computation time is spent on ev aluating the v ariational radar model. Also, first porting to C++ on a single core has indicated potential for object update times in the two-digit milliseconds range. T o achieve real- time capability , the crucial point of future work is thus to find fast implementations for e valuating the densities and to use parallelization for the particle implementation. 11 T ABLE I R M SE V A L UE S F O R T HE FIG U R E E IG H T S C E NA RI O ( 8 ) A N D A LL S I N G LE O B JE C T S C EN A RI O S ( A LL ) U S I NG TH E T H E V A R I A T I O NA L M O DE L ( V M ) A N D T H E D I RE C T S C A T T E R IN G M O D EL (D S M ) States VM 8 DSM 8 VM all DSM all x R in m 0.10 0.25 0.27 0.31 y R in m 0.13 0.19 0.26 0.40 ϕ in ◦ 2.29 3.60 7.28 9.41 v in m 0.25 0.35 0.36 0.55 ω in ◦ /s 3.57 6.15 5.54 8.63 a in m 0.19 0.33 0.26 0.32 b in m 0.16 0.49 0.28 0.50 B. Single-Object Accuracy The tracking accuracy for a single vehicle is ev aluated on experimental data that was recorded using the same ego and target vehicle on a different day . In total, ten scenarios were ev aluated. They comprise situations with oncoming and crossing traffic or passing and turning vehicles. Due to the Monte Carlo implementation, which in volves random genera- tion and propagation of particles, estimation results are subject to random effects. T o diminish these effects, all scenarios were ev aluated 20 times and the results are av eraged o ver these runs. In the following, one scenario in which the target vehicle driv es a figure eight in front of the stationary ego-v ehicle and is visible to the two front sensors is examined in detail. The scenario is challenging for se veral reasons: The aspect angle on the target vehicle changes constantly , it deviates from classical longitudinal traffic scenarios in that it contains a turning vehicle and cross traffic where the Doppler measurements do generally not equal the vehicle speed, and it is highly dynamic with yaw rates up to 60 ◦ /s. Figure 7 sho ws the estimation results, reference values, and resulting estimation errors for all components of the state vector . An excerpt of the scenario from an exemplary run is shown in Fig. 8. T able I lists root mean squared error (RMSE) values for the figure eight scenario as well as combined values over all single object scenarios. For comparison, results for the direct scattering approach are also provided. The variational radar model considerably outperforms the manually designed direct scattering model for all states. Despite the complicated maneuver , it achie ves especially precise estimation results for the figure eight scenario. The accuracy decreases for both ap- proaches when averaging over all scenarios. In contrast to the figure eight scenario, where the target vehicle is visible from all four sides, it is only partially visible over longer periods of time in other scenarios. This deteriorates size estimation and leads to correlated position errors. Also, vehicles in greater distance or vehicles with straight motion trend to yield fewer measurements. Thus, accurate orientation estimation is more difficult. An exemplary case is presented in the next section. C. Multi-Object P erformance The multi-object performance is assessed using nine differ - ent scenarios with three vehicles: the ego-vehicle, the E-Class target vehicle, and an additional Mercedes C-Class station wagon (S205), which is also equipped with a DGPS/IMU system. The nine scenarios comprise different situations such as oncoming traf fic, cross traf fic, overtaking, and occlusions. Again, the results are av eraged over 20 Monte Carlo runs. An ex emplary run of one of the scenarios is sho wn in Fig. 9. Here, two vehicles are approaching the stationary ego-vehicle and pass it on both sides. The target vehicles are continuously tracked and cross the FO Vs of all four radar sensors. As mentioned before, it is observ able that the tracking results are very precise in the direct vicinity of the ego-v ehicle and become less precise to wards FO V boundaries as measurements become more scarce and less accurate. The cardinality estimate is plotted in Fig. 10. As soon as the v ehicles enter the FO V, the true cardinality rises to one and then two. It decreases once the vehicles leav e the FO V. The filter is mostly able to correctly estimate the cardinality . Y et, it takes a considerable amount of time to initialize the second track. This is because the second vehicle only creates single measurements in the far range while the initialization routine expects at least a cluster of two. Thus, the vehicle is not set up before two measurements are created at a distance of approximately 35 m. An adaption of the initialization routine could eliminate this issue. Figure 11 depicts two additional excerpts of the scenario. Here, the a verage estimate of the upper vehicle at 10.54 s and corresponding radar measurements from the front left sensor are shown for direct scattering model (Fig. 11a) and for the variational radar model (Fig. 11b). Additionally , the measurements from the cluster that has contributed the most to updating the vehicle are indicated. The direct scattering model fav ors a cluster which excludes six measurements that originate from the wheels and exhibit an especially large or small Doppler velocity . This due to the fact that measurements from rotating wheels are not considered in the model. In this case, the direct scattering model profits from the multi-object approach which allo ws for dif ferent partitioning hypotheses. In contrast, the variational model has learned the effect of spu- rious measurements of the rotating wheels and uses a cluster which contains all measurements. The additional information helps to locate the position of the vehicle axes and might thus be a cause for the improved length estimation performance. A second multi-object scenario is sho wn in Fig. 12. The ego-v ehicle is first passed by the two target vehicles on both sides. Then, they closely driv e in parallel in front of the ego- vehicle before they depart to the left and the right at around 30 seconds. Since the ego-v ehicle is moving, the trajectories, which are estimated in vehicle coordinates, are dif ficult to visualize and thus not plotted. At 22.06 s, the benefit of using a multiple extended object tracking approach which considers multiple partitions and associations becomes apparent: The target vehicles are driving so close to each other that the distance between measurements from both vehicles is lo wer as for example the distance in between measurements from the upper vehicle at 32.87 s. Using a clustering routine with fixed parameter set would not be effecti ve in both situations and classical preprocessing routines would most likely merge the two close-by vehicles into a single object. By considering different hypotheses, ho wev er , the algorithm is able to find the right associations in both cases. The cardinality estimates of the variational and the direct 12 0 10 20 30 40 50 60 − 0 . 25 0 0 . 25 T ime in s e b in m 0 2 4 6 b in m 0 10 20 30 40 50 60 − 0 . 2 0 0 . 2 T ime in s e a in m 0 0 . 5 1 1 . 5 2 a in m − 0 . 25 0 0 . 25 e ω in rad/s − 2 − 1 0 1 2 ω in rad/s − 0 . 5 0 0 . 5 e v in m/s 0 2 4 6 v in m/s − 0 . 1 0 0 . 1 e ϕ in rad − 4 − 2 0 2 4 ϕ in rad − 0 . 25 0 0 . 25 e x , e y in m − 20 0 20 40 x R , y R in m Fig. 7. Figure eight scenario: Estimates (solid) and ground truth (dashed) as well as errors e averaged ov er 20 runs. The y-position is plotted in gray . 10 5 0 − 5 − 10 15 20 25 30 t = 2.40 s t = 4.40 s t = 6.26 s t = 8.76 s t = 11.90 s t = 13.76 s t = 16.01 s t = 17.51 s y VC in m x VC in m Fig. 8. Excerpt of the figure eight scenario: radar measurements with indicated Doppler velocity from the front left ( ) and front right ( ) sensor, estimated trajectory (solid) and exemplary vehicle poses (solid rectangles), reference trajectory (dashed) and reference poses (dashed rectangles) scattering approach are compared using histograms of cardi- nality errors over all nine multi-object scenarios. T o obtain the ground truth, all vehicles in the sensor FO V with a speed greater than 1 m/s were counted. The histograms are sho wn in Fig. 13. While the direct scattering approach estimates the correct cardinality in 66.8% of the update steps, the variational approach is correct in 73.8% of the time. The direct scattering approach ov erestimates the cardinality in 17.5% of the update steps, whereas the percentage is reduced to 15.2% when using the variational radar model. This suggests that the variational radar model performs better in distinguishing clutter from actual vehicles. False tracks are mostly caused by spurious measurements with non-zero Doppler velocity and may surviv e if there are matching clutter measurements from stationary objects o ver several time steps. Such cardinality errors are caused by a violation of the assumption of inde- pendent and uniform clutter . The cardinality is underestimated in 15.7% of the update steps for the direct scattering model and 11.0% for the variational radar model. In these cases, the filter has either not yet initialized a track, has assigned too low existence probabilities to the vehicle hypotheses, or vehicle tracks are temporarily lost. Delayed initialization makes up 6.2% and 6.9% of the cardinality errors for the direct scattering and variational radar models, respectively . D. Generalization So far , the variational model was tested using the same E-Class target vehicle that generated the training data and a rather similar C-Class vehicle. T o demonstrate that the model is applicable to a wider range of vehicle types, it was applied to an urban scenario. The ego-v ehicle stands at a T -intersection, while elev en dif ferent vehicles pass it. The vehicle types range from compact cars ov er sedans and conv ertibles to vans. Unfortunately , no accurate ground truth is available for these vehicles. Therefore, the vehicle poses and dimensions where manually labeled using the lidar sensor of the ego-vehicle as 13 − 50 − 40 − 30 − 20 − 10 0 10 20 30 40 50 − 10 − 5 0 5 10 t = 5.00 s t = 8.39 s t = 8.39 s t = 9.70 s t = 9.70 s t = 11.25 s t = 11.25 s t = 12.50 s t = 12.50 s t = 13.75 s t = 13.75 s t = 16.24 s x VC in m y VC in m Fig. 9. Scenario with two oncoming vehicles: Estimated (solid) and ground truth (dashed) trajectories, exemplary v ehicle poses (estimates: solid rectangles, ground truth: dashed rectangles), corresponding measurements with Doppler velocity (front left: , front right: , rear left: , rear right: ), and sensor FOVs 0 2 4 6 8 10 12 14 16 18 0 1 2 T ime in s Cardinality Fig. 10. Cardinality estimate (black) and ground truth (dashed) for the scenario with two oncoming vehicles 4 6 8 10 12 0 2 4 y VC in m x VC in m (a) Direct scattering model 4 6 8 10 12 0 2 4 y VC in m x VC in m (b) V ariational radar model Fig. 11. Comparison of the measurement clusters that contributed the most during update: cluster measurements ( ), other measurements ( ), a verage vehicle estimate with center of the rear axle (rectangle and cross), reference vehicle (dashed rectangle) − 10 0 10 20 30 − 10 0 10 t = 6.24 s t = 9.99 s t = 22.06 s t = 32.87 s t = 32.87 s t = 13.74 s t = 13.74 s t = 16.24 s x VC in m y VC in m Fig. 12. T wo vehicles driving closely: Estimated (solid) and ground truth (dashed) vehicle poses, corresponding measurements with Doppler velocity (front left: , front right: , rear left: , rear right: ), and sensor FO Vs reference. The labels are only av ailable in the lidar FO V which approximately covers 100 ◦ in front of the ego-vehicle. T wo ex emplary situations are shown in Fig. 14. Figure 14a shows one of the most challenging situations for the algorithm. Here, two sedan vehicles cross in front of the ego-vehicle and the front sedan temporarily occludes the second vehicle. While the front sedan is tracked continuously , the track of the rear − 3 − 2 − 1 0 1 2 3 0 0 . 25 0 . 5 0 . 75 Cardinality error Percentage Fig. 13. Histogram of cardinality estimation errors for the multi-object scenarios: variational model ( ) and direct scattering model ( ) sedan is lost during occlusion. In this situation, the sensors do not provide measurements from this vehicle over a period of 25 update steps. This causes the probability of existence to drop below the pruning threshold. Once the vehicle is visible again, it is reinitialized. Dropping the assumption of object measurements being generated independently and including an occlusion model as for example used in [52] could alleviate this issue. Figure 14b sho ws a constellation of three vehicles. A v an takes a left turn past the ego-vehicle whereas a conv ertible and a compact van are driving straight. Here, all three vehicles are continuously tracked. All in all, the variational radar model did not show difficul- ties with a particular vehicle type e ven though it was trained using data from a single vehicle. The RMSE v alues where computed with respect to the manually created labels in front of the ego-vehicle and av eraged over the elev en vehicles and 20 Monte Carlo runs. They are 0.21 m and 0.52 m for x R and y R , 4.3 ◦ for ϕ , 0.40 m for the width and 0.54 m for the length. T rack estimates are av ailable for the labeled vehicles in 95.1% of the update steps. The remaining 4.9% are due to delayed initialization of entering vehicles and the track loss during occlusion (cf. Fig. 14a). From visual inspection, an expectable degradation of performance occurs in the far field where the number as well as the accuracy of measurements decreases and the number of misdetections and clutter increases. The indicated ability to generalize to other vehicles is not surprising as ev en high-resolution radar measurements are not yet at the resolution performance of other sensor types such as lidar . Hence, the rough extent of the vehicles is observable b ut the details that distinguish dif ferent vehicles are still concealed. Problems are expected as soon as the vehicle appearance changes drastically , e.g. with additional wheels or truck bodies with distinct reflection characteristics. 14 10 0 − 10 − 20 − 30 − 10 0 10 20 t = 7.50 s t = 8.36 s t = 9.36 s t = 7.50 s t = 8.36 s t = 9.36 s y VC in m x VC in m (a) Occlusion situation 10 0 − 10 − 20 − 30 − 10 0 10 20 t = 22.00 s t = 23.36 s t = 22.00 s t = 23.36 s t = 22.00 s t = 23.36 s y VC in m x VC in m (b) T urning van and two other vehicles Fig. 14. T -intersection scenario: T wo e xcerpts with estimated (solid) and true (dashed) trajectories (only available in the lidar FOV), exemplary vehicle poses (solid rectangles) and true poses (dashed rectangles), corresponding measurements (front left: , front right: , rear left: , rear right: ), and sensor FO Vs V I I . C O N C L U S I O N In this paper, a v ariational radar model for vehicles that is learned from actual radar data was presented and included in a FISST-based multi-object filter . Both measurement mod- eling and multi-object filtering are formulated in an integral Bayesian f ashion. FISST provides a rigorous mathematical formulation of the multi-object problem which allows for a natural incorporation of the variational radar model. The multi- object filter considers object dependencies, e.g. that objects should not ov erlap and that measurements may only originate from one object, and is able to filter ov er sev eral measurement associations and partitions. By learning a vehicle model from actual radar data, the variational radar model is a close approximation of the true measurement likelihood and avoids the need for excessi ve manual engineering. Also, it was shown that it is able to outperform state of the art extended object methods in both the single and multi-object performance. The capability to generalize to objects that are not contained in the training data was shown using a real-world example. There are se veral possible extensions of the approach to ov ercome some limitations. F or e xample, using other nonlinear estimation techniques such as a unscented Kalman filter (UKF) for the single-object densities could simplify the approach and facilitate fast real time implementations. So far , the approach does not learn all parameters that are in volved in the multi- object likelihood. Learning additional parameters such as the expected number of measurements or the clutter densities could further improv e modeling accuracy . 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Murty , “An Aglorithm for Ranking all the Assignments in Order of Increasing Cost, ” Oper . Res. , vol. 16, no. 3, pp. 682–687, May-Jun. 1968. [52] K. Granstr ¨ om, S. Reuter, D. Meißner, and A. Scheel, “A multiple model PHD approach to tracking of cars under an assumed rectangular shape, ” in Pr oc. 17th Int. Conf. Inf . Fusion , 2014. Alexander Scheel was born in Heilbronn, Germany , in 1987. In 2013, he received his Diploma degree (equiv alent to M.Sc. degree) in automotive and en- gine technology from the Univ ersity of Stuttgart. Since December 2013, he has been working as research assistant at the Institute for Measurement, Control, and Microtechnology at Ulm University where he became group leader for autonomous driv- ing in 2016. In 2018, he joined the autonomous driving department of Robert Bosch GmbH. Alexan- der Scheel specializes in en vironment perception for autonomous vehicles. In particular, his research interests include multi-object tracking, sensor data fusion, and measurement models for extended objects. Klaus Dietmayer (M05) was born in Celle, Ger- many in 1962. He receiv ed his Diploma degree (equiv alent to M.Sc. degree) in 1989 electrical en- gineering from the T echnical Univ ersity of Braun- schweig (German y), and the Dr .-Ing. degree (equiv- alent to Ph.D.) in 1994 from the Uni versity of Armed Forces in Hambur g (Germany). In 1994 he joined the Philips Semiconductors Systems Laboratory in Hambur g, Germany as a research engineer . Since 1996 he became a manager in the field of networks and sensors for automoti ve applications. In 2000 he was appointed to a professorship at Ulm University in the field of measurement and control. Currently he is Full Professor and Director of the Institute of Measurement, Control and Microtechnology in the school of Engineering and Computer Science at Ulm Uni versity . His research interests include information fusion, multi-object tracking, en vironment perception for advanced automotive driver assistance, and E-Mobility . Klaus Dietmayer is member of the IEEE and the German society of engineers VDI/VDE.
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