Patent Publication Number: US-2023135987-A1

Title: System and Method for Tracking an Expanded State of a Moving Object Using an Online Adapted Compound Measurement Model

Description:
TECHNICAL FIELD 
     The present disclosure relates generally to automotive object tracking, and more specifically to a system and a method for tracking an expanded state of an object using measurements of the object. 
     BACKGROUND 
     Control systems employed by vehicles, such as autonomous vehicles and semi-autonomous vehicles, predict safe motion or path for the vehicles to avoid collision with obstacles, such as other vehicles or pedestrians. In some scenarios, a vehicle is also configured for sensing its surroundings, such as road edges, pedestrians, and other vehicles, with the help of one or more sensors of the vehicle. Some of these sensors include ultrasonic sensors, cameras, and LIDAR sensors, which are used in existing advanced driver assistance systems (ADAS). 
     The control system of the vehicle tracks an object state of the other vehicles (where the object state includes kinematic states) based on automotive radar measurements, to control the vehicle. Extended object tracking (EOT) with multiple measurements per scan has shown improved object tracking than the traditional point object tracking which includes only one measurement per scan, by augmenting the object state from kinematic-only state to both kinematic and extended states. The extended state provides the dimension and orientation of the objects under tracking. To achieve this, spatial distribution (i.e. how automotive radar measurements are spatially distributed around the object) needs to be captured along with sensor noise. Current methods include a framework of a fixed set of points on a rigid body that requires a non-scalable data association between the fixed set of points and automotive radar detections even for a single object tracking. Spatial models, such as the contour model and surface model, bypass the cumbersome data association step. 
     For automotive radar measurements, the contour model reflects the measurement distribution along the contour of an object (e.g., the rigid body), and the surface model assumes the radar measurements are generated from the inner surface of a two-dimensional shape. Examples of the contour model include a simple rectangular shape and a more general star-convex shape modeled by either a random hypersurface model or a Gaussian process model. Some surface models such as Gaussian-based ellipse and hierarchical Gaussian-based ellipse models are computationally much simpler than the contour model that requires much more degrees of freedom to describe a more complex shape. However, the measurements of the object are subject to noise, and reflections are received only from the surface of the object. Therefore, the aforementioned models don&#39;t capture real-world automotive radar measurements. 
     Accordingly, there is a need for a system and a method for tracking both the kinematic and extended states of the object by capturing real-world automotive radar measurements. 
     SUMMARY 
     It is an object of some embodiments to provide a system and a method for tracking an expanded state of an object. The expanded state of the object includes a kinematic state indicative of one or a combination of a position and a velocity of a center of the object, and an extended state indicative of one or a combination of a dimension and an orientation of the object. The center of the object is one or a combination of an arbitrarily selected point, a geometrical center of the object, a center of gravity of the object, a center of a rear axis of wheels of a vehicle, and the like. A sensor, for example, automotive radar, is used to track objects (such as a vehicle). In an embodiment, the automotive radar may provide direct measurements of radial velocities, long operating ranges, small sizes at millimeter or sub-terahertz frequency bands, and high spatial resolutions. 
     In point object tracking, a single measurement per scan is received from the vehicle. The point object tracking provides only the kinematic state (position) of the vehicle. Further, a probabilistic filter with a measurement model having distribution of kinematic states is utilized to track the vehicle. In extended object tracking (EOT), multiple measurements per scan are received. The multiple measurements are spatially structured around the vehicle. The extended object tracking provides both the kinematic and the extended state of the vehicle. The probabilistic filter with a measurement model having distribution of extended states is utilized to track the vehicle. 
     However, real-world automotive radar measurement distributions show that multiple reflections from the vehicle are complex. Due to this complexity, designing a proper measurement model becomes complex. Therefore, regular measurement models are applicable only for the kinematic state and not for the expanded state. 
     To that end, in some embodiments, spatial models such as a contour model and a surface model are used to capture the real-world automotive radar measurements. In particular, in an embodiment, based on principles of the contour and surface models, a compound measurement model (which is a type of surface volume model) is determined. The compound measurement model includes multiple probabilistic distributions constrained to lie on a contour of the object with a predetermined relative geometrical mapping to the center of the object. The multiple probabilistic distributions are used to cover a measurement spread along the contour of the object. 
     The compound measurement model is compound in multiple ways. For example, the compound measurement model has a compound structure, i.e., the multiple probabilistic distributions. Also, the compound measurement model has a compound composition, i.e., functions of the multiple probabilistic distributions, a function of the contour, and their relationship. Further, the compound measurement model has a compound nature, i.e., the multiple probabilistic distributions are based on the measurements and thus represent data-driven approaches of model generation, whereas the contour is based on modeling a shape of the object, e.g., a shape of a vehicle, using principles of physics-based modeling. 
     Additionally, the compound measurement model takes advantage of different principles of modeling the expanded state, i.e., the compound measurement model joins the principles of the contour model and the surface model. As a result, the compound measurement model better represents the physical nature of tracking the object while simplifying measurement assignment. In addition, the multiple probabilistic distributions of the compound measurement model are more flexible over a single distribution of the surface model, and can better describe the contour of the object, and are more flexible to explain the measurements coming from different angles or views of the object. 
     The compound measurement model is learned offline, i.e., in advance. The compound measurement model may be learned in a unit coordinate system or a global coordinate system. Some embodiments are based on the recognition that it is beneficial to learn the compound measurement model in the unit coordinate system, because it simplifies the calculation and makes the compound measurement model agnostic to the dimensions of the object. Each of the multiple probabilistic distributions (represented as ellipses) can be assigned with measurements in a probabilistic manner. The measurements associated with the ellipse may be referred to as ellipse-assigned measurements. 
     According to some embodiments, the offline learned compound measurement model is used for online tracking of the expanded state of the object, i.e., real time tracking of the expanded state of the object. However, there might be mismatches on automotive radar specifications between onboard automotive radar used by the vehicle to obtain measurements and those used for offline data collection, where the offline data collection (also referred to as “offline training data”) is used to train the compound measurement model. 
     Further, the offline training data comprises coarse vehicle labels. Therefore, training the compound measurement model using only the offline training data may lead to an over-smoothed offline learned compound measurement model that averages over different vehicle models. For instance, a coarsely labeled dataset may include sedan and SUV in the same class. Therefore, it is possible that the compound measurement model may not correctly classify between different types of objects (in this case, vehicles such as trucks, cars, tractors, and the likes). 
     To that end, the present disclosure proposes an online adaptation of the compound measurement model that refines the offline learned compound measurement model and further improves the online state estimation performance with a more customized compound measurement model that fits for onboard automotive radar measurements. 
     According to some embodiments, the offline learned compound measurement model is executed for the predetermined time period and the updated expanded state of the object, the predicted expanded state of the object, and measurements performed by the automotive radar of the object are accumulated to form an online batch of training data. 
     Some embodiments are based on the realization that the online batch of training data comprises data accumulated only within the predetermined time period, where the predetermined time period may be of few seconds or few minutes. Thus, the online batch of training data comprises very less training data compared to training data used to train the compound measurement model offline. To increase the accuracy of the compound measurement model trained using the online batch of training data it is important to obtain relationship amongst data within the accumulated data and use the relationship to update parameters of the compound measurement model. 
     To that end, the accumulated updated belief is smoothed using a covariance between the accumulated updated belief and the predicted belief, and a backward recursion and a forward recursion. The smoothed updated belief is used to generate the online batch of training data. In the backward recursion, the accumulated updated belief is smoothed backward from a specific time within the predetermined time period based on measurements at the specific time. Alternatively, in the forward recursion, the accumulated updated belief is smoothed forward from the specific time within the predetermined time period based on measurements at the specific time. 
     In some embodiments, Bayesian smoothing that is customized to the offline learned compound measurement model is applied to the measurements to obtain the smoothed states. 
     Further, the online batch of training data comprises state-decoupled measurements. To state-decouple the online batch of training data, the measurements in the global coordinate system are transformed into a unit-coordinate system that is positioned in the center of the object and oriented such that the x-axis of the unit coordinate system points to the object front using the orientation angle and object center. Finally, the measurements transformed into the unit-coordinate system are normalized by the extent states, i.e., length and width. 
     The state-decoupled online batch of training data is then used for the online learning of the compound measurement model, where the online learning updates the compound measurement model by updating one or more parameters of the offline learned compound measurement model. The parameters of the compound measurement model comprise the number of probabilistic distributions in the compound measurement model, control points that determine the centers of probabilistic distributions, and covariances of each probabilistic distribution. 
     However, the one or more parameters of the compound measurement model are updated such that a predetermined relative geometrical mapping of the multiple probabilistic distributions to a center of an object being tracked is preserved. To that end, control points corresponding to the multiple probabilistic distributions are preserved while updating the parameters of the compound measurement model, where to preserve the control points of the multiple probabilistic distributions a penalty function such as a log-likelihood function is used that enforces the maximal allowable change on the control points. 
     Accordingly, one embodiment discloses a tracking system for tracking an expanded state of an object including a kinematic state indicative of a combination of a position and a velocity of a center of the object and an extended state indicative of a combination of a dimension and an orientation of the object. The tracking system comprises: at least one processor; and memory having instructions stored thereon that, when executed by the at least one processor, cause the tracking system to receive measurements associated with at least one sensor, wherein at least one sensor is configured to probe a scene including the object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission; execute a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie around a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements obtained within the predetermined time period fitting each of the multiple probabilistic distributions, and mapped to the expanded state of the object based on the corresponding geometrical mapping, and wherein the compound measurement model is pretrained offline using an offline training data; accumulate updated belief, predicted belief, and measurements for the predetermined time period to generate an online batch of training data comprising state-decoupled measurements; update the compound measurement model by updating parameters of the compound measurement model based on the online batch of training data; and track the expanded state of the object based on the updated compound measurement model. 
     Accordingly, another embodiment discloses a tracking method for tracking an expanded state of an object including a kinematic state indicative of one or a combination of a position and a velocity of a center of the object and an extended state indicative of one or a combination of a dimension and an orientation of the object. The tracking method comprises receiving measurements associated with at least one sensor, wherein at least one sensor is configured to probe a scene including the object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission; executing a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie around a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements obtained within the predetermined time period fitting each of the multiple probabilistic distributions, and mapped to the expanded state of the object based on the corresponding geometrical mapping, and wherein the compound measurement model is pretrained offline using an offline training data; accumulating updated belief, predicted belief, and measurements for the predetermined time period to generate an online batch of training data comprising state-decoupled measurements; updating the compound measurement model by updating parameters of the compound measurement model based on the online batch of training data; and tracking the expanded state of the object based on the updated compound measurement model. 
     A non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method for tracking an expanded state of an object, wherein the expanded state includes a kinematic state indicative of one or a combination of a position and a velocity of a center of the object and an extended state indicative of one or a combination of a dimension and an orientation of the object. The method comprises receiving measurements associated with at least one sensor, wherein at least one sensor is configured to probe a scene including the object with one or multiple signal transmissions to produce one or multiple measurements of the object per the transmission; executing a probabilistic filter iteratively tracking a belief on the expanded state of the object, wherein the belief is predicted using a motion model of the object and is updated using a compound measurement model of the object, wherein the compound measurement model includes multiple probabilistic distributions constrained to lie around a contour of the object with a predetermined relative geometrical mapping to the center of the object, wherein in each iteration of the iterative tracking, the belief on the expanded state is updated based on a difference between a predicted belief and an updated belief, wherein the updated belief is estimated based on probabilities of the measurements obtained within the predetermined time period fitting each of the multiple probabilistic distributions, and mapped to the expanded state of the object based on the corresponding geometrical mapping, and wherein the compound measurement model is pretrained offline using an offline training data; accumulating updated belief, predicted belief, and measurements for the predetermined time period to generate an online batch of training data comprising state-decoupled measurements; updating the compound measurement model by updating parameters of the compound measurement model based on the online batch of training data; and tracking the expanded state of the object based on the updated compound measurement model. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The presently disclosed embodiments will be further explained with reference to the attached drawings. The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. 
         FIGS.  1 A,  1 B, and  1 C  collectively show a schematic overview of principles for tracking an expanded state of an object, according to some embodiments. 
         FIG.  2    shows a block diagram of a tracking system for tracking the expanded state of the object, according to some embodiments. 
         FIGS.  3 A,  3 B, and  3 C  collectively show various schematics for tracking a belief on the expanded state of the object using a compound measurement model, according to some embodiments. 
         FIG.  4    shows a workflow for tracking the expanded state of the object by using the compound measurement model, according to some embodiments 
         FIG.  5    shows a flow chart of a method for offline learning of the parameters of the compound measurement model, according to some embodiments 
         FIG.  6    shows a flow chart of a method for online learning of the parameters of the compound measurement model, according to some embodiments. 
         FIGS.  7 A and  7 B  show schematics of transformation of training data collected from different motions of different objects into a common unit coordinate system, according to some embodiments. 
         FIG.  8    shows a block diagram of an expectation-maximization (EM) method for offline learning of the parameters of the compound measurement model, according to some embodiments. 
         FIG.  9    shows a block diagram of the EM method for online learning of the parameters of the compound measurement model, according to some embodiments. 
         FIG.  10 A  shows a flowchart of an unscented Kalman filter-probabilistic multi-hypothesis tracking (UKF-PMHT) algorithm, according to some embodiments. 
         FIG.  10 B  shows a block diagram of steps performed to compute predicted measurements and a covariance matrix, according to some embodiments. 
         FIG.  10 C  shows a block diagram of steps performed to compute synthetic measurements and a synthetic covariance matrix, according to some embodiments. 
         FIG.  10 D  shows a block diagram of steps performed to update the expanded state and a covariance matrix, according to some embodiments. 
         FIG.  10 E  shows a block diagram of steps performed for online learning of the compound measurement model, according to some embodiments. 
         FIG.  10 F  shows a schematic illustrating a backward recursion and a forward recursion, according to some embodiments. 
         FIG.  11 A  shows a schematic of a vehicle including a controller in communication with the system employing principles of some embodiments. 
         FIG.  11 B  shows a schematic of interaction between the controller of the system of  FIG.  11 A  and controllers of the vehicle, according to some embodiments. 
         FIG.  11 C  shows a schematic of an autonomous or semi-autonomous controlled vehicle for which control inputs are generated by using some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. It will be apparent, however, to one skilled in the art that the present disclosure may be practiced without these specific details. In other instances, apparatuses and methods are shown in block diagram form only in order to avoid obscuring the present disclosure. 
     As used in this specification and claims, the terms “for example,” “for instance,” and “such as,” and the verbs “comprising,” “having,” “including,” and their other verb forms, when used in conjunction with a listing of one or more components or other items, are each to be construed as open ended, meaning that that the listing is not to be considered as excluding other, additional components or items. The term “based on” means at least partially based on. Further, it is to be understood that the phraseology and terminology employed herein are for the purpose of the description and should not be regarded as limiting. Any heading utilized within this description is for convenience only and has no legal or limiting effect. 
       FIGS.  1 A,  1 B and  1 C  show a schematic overview of some principles used by some embodiments for tracking an expanded state of an object. The expanded state of the object includes a kinematic state indicative of one or a combination of a position and a velocity of a center of the object, and an extended state indicative of one or a combination of a dimension and an orientation of the object. The center of the object is one or a combination of an arbitrarily selected point, a geometrical center of the object, a center of gravity of the object, a center of a rear axis of wheels of a vehicle, and the like. A sensor  104  (for example, automotive radar) is used to track objects (such as vehicle  106 ). In point object tracking  100 , a single measurement  108  per scan is received from the vehicle  106 . The point object tracking  100  provides only the kinematic state (position) of the vehicle  106 . Further, a probabilistic filter with a measurement model having distribution of kinematic states is utilized to track the vehicle  106 . In extended object tracking (EOT)  102 , multiple measurements  110  per scan are received. The multiple measurements  110  are spatially structured around the vehicle  106 . The EOT  102  provides both the kinematic and the extended state of the vehicle  106 . The probabilistic filter with a measurement model having distribution of extended states is utilized to track the vehicle  106 . 
     However, real-world automotive radar measurement  112  distributions, as illustrated in  FIG.  1 B , show that multiple reflections from the vehicle  106  are complex. Due to this complexity, designing of the measurement model becomes complex. Therefore, regular measurement models are applicable only for the kinematic state and not for the expanded state. 
     To that end, in some embodiments, spatial models such as a contour model  114 , as illustrated in  FIG.  1 C , and a surface model  116  are used to capture the real-world automotive radar measurements  112 . However, the aforesaid spatial models are inaccurate. Some embodiments are based on the recognition that real-world automotive radar measurements  112  are distributed around edges of the objects (the vehicle  106 ) with a certain volume, which gives rise to a surface volume model. To that end, some embodiments are based on objective of formulating a surface volume model  118  that resembles and captures the real-world automotive radar measurements  112 . The surface volume model  118  balances between the contour model  114  and the surface model  116  with more realistic features while keeping the EOT accurate. 
     In particular, in an embodiment, based on principles of the contour model  114  and the surface model  116 , a compound measurement model  120  (which is a type of surface volume model) is determined. The compound measurement model  120  includes multiple probabilistic distributions  122  that are geometrically constrained to a contour  124  of the object. In  FIG.  1 C , this geometrical constraint is that the centers of multiple probabilistic distributions lies on the contour. The compound measurement model then has a predetermined relative geometrical mapping to the center of the object. The multiple probabilistic distributions  122  are used to cover a measurement spread along the contour  124  of the object. 
     The compound measurement model  120  is compound in multiple ways. For example, the compound measurement model  120  has a compound structure, i.e., the multiple probabilistic distributions  122 . Also, the compound measurement model  120  has a compound composition, i.e., functions of the multiple probabilistic distributions  122 , a function of the contour  124 , and their relationship. Further, the compound measurement model  120  has a compound nature, i.e., the multiple probabilistic distributions  122  are based on measurements and thus represent data-driven approaches of model generation, whereas the contour  124  is based on modeling a shape of the object, e.g., a shape of a vehicle, using principles of physics-based modeling. 
     Additionally, the compound measurement model  120  takes advantage of different principles of modeling the expanded state, i.e., the compound measurement model  120  joins the principles of the contour model  114  and the surface model  116 . As a result, the compound measurement model  120  better represents a physical nature of tracking of the object while simplifying measurement assignment. In addition, the multiple probabilistic distributions  122  of the compound measurement model  120  are more flexible over a single distribution of the surface model  116  and may be configured to better describe the contour  124  and are furthermore flexible to explain the measurements coming from different angles or views of the object. 
     Some embodiments are based on understanding that, in theory, the multiple probabilistic distributions  122  can lie on the contour  124 , assuming that a shape of the contour  124  has no restrictions. However, in practice, such assumptions are incorrect and useless for tracking the expanded state. In contrast, the contour  124  of the object is predetermined and the multiple probabilistic distributions  122  are fit to the contour  124  rather than the contour  124  is fit to the multiple probabilistic distributions  122 . This allows reflecting a physical structure of the object during an update stage of the probabilistic filter. 
     The compound measurement model  120  is learned offline, i.e., in advance. The compound measurement model  120  may be learned in a unit coordinate system or a global coordinate system. Some embodiments are based on recognition that it is beneficial to learn the compound measurement model  120  in the unit coordinate system, because it simplifies calculation and makes the compound measurement model  120  agnostic to the dimensions of the object. Each of the multiple probabilistic distributions  122  (represented as ellipses) can be assigned with measurements in a probabilistic manner. The measurements associated with the ellipse may be referred to as ellipse-assigned measurements. 
     Some embodiments are based on a recognition that the expanded state of the object can be tracked online, i.e., in real-time, using the compound measurement model  120 . Specifically, various embodiments track the expanded state of the object using the probabilistic filter that tracks a belief on the expanded state of the object, wherein the belief on the expanded state of the object is predicted using a motion model of the object and is updated using the compound measurement model  120  of the object. 
     Some embodiments are based on the realization that there might be mismatches on radar sensor specifications between onboard sensors used by the vehicle  106  to obtain measurements and those used for offline data collection, where the offline data collection (also referred to as “offline training data”) is used to train the compound measurement model  120 . 
     Some embodiments are based on the realization that offline training data with coarse vehicle labels may lead to an over-smoothed offline learned compound measurement model  120  that averages over different vehicle models. For instance, a coarsely labeled dataset may include sedan and SUV in the same class. For instance, a coarsely labeled dataset may include sedan and SUV in the same class. 
     To that end, the present disclosure proposes an online adaptation of the compound measurement model  120  (also referred to as “online compound measurement model adaptation”) that refines the offline learned compound measurement model  120  and further improves the online state estimation performance with a more customized compound measurement model  120  that fits for onboard automotive radar measurements. 
       FIG.  2    shows a block diagram of a tracking system  200  for tracking the expanded state of the object by using the compound measurement model  120  (shown in previous figures), according to some embodiments. The object may be a vehicle, such as, but not limited to, a car, bike, bus, or truck. Also, the vehicle may be an autonomous or a semi-autonomous vehicle. The expanded state includes the kinematic state and the extended state of the object. The compound measurement model  120  is learned offline using offline training data ( FIG.  5   ). Initially, the learned compound measurement model  120  is executed to track the expanded state of the object for a predetermined time period T. Further, based on measurements obtained within the predetermined time period T, corresponding expanded states are used to update/refine the learned compound measurement model  120 . 
     According to some embodiments, the kinematic state corresponds to motion parameters of the object, such as velocity, acceleration, heading, and turn-rate. In some other embodiments, the kinematic state corresponds to the position of the object with its motion parameters. The tracking system  200  may include a sensor  202  or be operatively connected to a set of sensors to probe a scene with one or multiple signal transmissions. The one or multiple signal transmissions in turn are configured to produce one or multiple measurements of the object per transmission. According to some embodiments, the sensor  202  may be the automotive radar. In some embodiments, the scene includes a moving object. In some other embodiments, the scene may include one or more objects that include both moving objects and stationary objects. 
     The tracking system  200  can have a number of interfaces connecting the tracking system  200  with other systems and devices. For example, a network interface controller (NIC)  214  is adapted to connect the tracking system  200  through a bus  212  to a network  216  connecting the tracking system  200  with a set of sensors. Through the network  216 , either wirelessly or through wires, the tracking system  200  receives data of reflections of the one or multiple signal transmissions to produce the one or multiple measurements of the object per transmission. Additionally or alternatively, the tracking system  200  includes an output interface  220  configured to submit control inputs to a controller  222 . 
     The tracking system  200  also includes a processor  204  configured to execute stored instructions, as well as a memory  206  that stores instructions that are executable by the processor  204 . The processor  204  can be a single core processor, a multi-core processor, a computing cluster, or any number of other configurations. The memory  206  can include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory systems. The processor  204  is connected through the bus  212  to one or more input and output devices. Further the tracking system  200  includes a storage device  208  adapted to store different modules including instructions executable by the processor  204 . The storage device  208  can be implemented using a hard drive, an optical drive, a thumb drive, an array of drives, or any combinations thereof. 
     The storage device  208  is configured to store a motion model  210   a  of the object, a compound measurement model  210   b  of the object (e.g., the compound measurement model  120 ), and a refinement module  210   c . The processor  204  is configured to execute iteratively, for the predetermined time period T, a probabilistic filter, for iteratively tracking a belief on the expanded state of the object predicted using the motion model  210   a  of the object and updated using the compound measurement model  210   b  of the object. After the predetermined time period T, the refinement module  210   c  refines/updates the compound measurement model  210   b  based on measurements obtained during the predetermined time period T and the corresponding predicted belief and the updated belief. The tracking of the belief on the expanded state of the object based on offline learning of the compound measurement model  120  is described in detail below with reference to  FIGS.  3 A,  3 B, and  3 C . 
       FIG.  3 A  shows a schematic for computing predicted measurements and a covariance matrix, according to some embodiments. Given an expanded state  300  of the object and a covariance matrix corresponding to a previous time step, and a motion model of the object, an expanded state  302  of the object for a current time step and a covariance matrix corresponding to the expanded state  302  are predicted by the processor  204 . The expanded state  300  of the object corresponding to the previous time step is denoted as x k−1|k−1 . The predicted expanded state  302  of the object is denoted as x k|k−1  (or x k+1|k ). The expanded state  300  includes various kinematic states, for example, x=[x m , y m , v, ψ, ω] T , where (x m , y m ) T  is the center of the object, v is a polar velocity of the vehicle, ψ is an orientation angle, and ω is a turning rate. In an alternate embodiment, the expanded state  300  includes the extended state in addition to the kinematic states, for example, x=[x m , y m , v, ψ, ω, l, w] T , where l and w are length and width of the object, respectively. Likewise, the predicted expanded state  302  includes predicted kinematic states and/or predicted extended states. In an embodiment, the motion model may be a coordinated turn (CT) motion model with polar velocity. In some other embodiments, for the kinematic states, the CT motion model with the polar velocity is used and, for the extended state, i.e., the length and width, a constant model is used with a process noise with a small covariance as the length and width are unlikely changed over time. 
     The predicted expanded state  302  of the object may be referred to as a predicted belief of the expanded state because this prediction is probabilistic. Some embodiments are based on a recognition that the predicted expanded state  302  of the object may be inaccurate to generate predicted measurement for the expanded state as it requires an accurate spatial model of automotive radar measurements. To this end, in some embodiments, the compound measurement model  304  in a unit coordinate system that is learned offline is used. To align the compound measurement model  304  in the unit coordinate system with the predicted expanded state  302 , the compound measurement model  304  needs to be transformed from the unit coordinate system to the global coordinate system with respect to the predicted expanded state  302 . In particular, the ellipse-assigned measurements in the unit coordinate system need to be transformed into the global coordinate system. 
     Some embodiments are based on realization that such a transformation can be achieved using an unscented transform function  308 . To that end, in an embodiment, the processor  204  generates sigma points for an ellipse  306  (i.e., for a probabilistic distribution of the compound measurement model  304 ). The “ellipse” and “probabilistic distribution” may be used interchangeably and would mean the same. Further, the sigma points are propagated into the unscented transform function  308  which is a function of the predicted state  302  and, consequently, predicted measurements in the global coordinate system corresponding to the ellipse-assigned measurements of the ellipse  306  in the unit coordinate system is determined. Additionally, a covariance corresponding to the predicted measurements is determined based on the predicted measurements. Likewise, the measurements in the global coordinate system corresponding to the ellipse-assigned measurements associated with the rest of the ellipses are determined. To that end, a predicted expanded state model  310 , where the compound measurement model  304  is aligned according to the predicted expanded state  302 , is obtained. Further, synthetic measurements are determined for each probabilistic distribution of the predicted expanded state model  310  as described below with reference to  FIG.  3 B . 
       FIG.  3 B  shows a schematic for determining the synthetic measurements for each probabilistic distribution of the predicted expanded state model  310 , according to some embodiments. The processor  204  receives measurements  312  (represented by cross marks) at the current time step. Some embodiments are based on the realization that the multiple probabilistic distributions  314   a - 314   h  (ellipses) can be treated independently, e.g., parallelly to each other. Such an independent treatment allows considering different view-angles of probing the expanded state of the object. To consider such an independent treatment, some embodiments treat different probabilistic distributions of the multiple probabilistic distributions  314   a - 314   h  as belonging to different objects. Additionally, some embodiments are based on the realization that a soft probabilistic assignment, i.e., probabilistic assignment of the measurements  312  to different probabilistic distributions, is more advantageous than a hard deterministic assignment. Soft probabilistic assignment may avoid catastrophic assignment of hard assignment while keeping the association dimension linear with respect to the number of ellipses and measurements. 
     To that end, processor  204  assigns the measurements  312  to the probabilistic distribution  314  with an association probability. Likewise, processor  204  assigns the measurements  312  to each of the probabilistic distributions  314   a - 314   h  with a corresponding association probability. The measurements with the corresponding association probability associated with each of multiple probabilistic distributions  314   a - 314   h  is referred to as the ‘synthetic measurements’. 
     Further, for the probabilistic distribution  314   a , the processor  204  determines, based on the synthetic measurements associated with the probabilistic distribution  314   a , a synthetic centroid  316   a  and a synthetic covariance matrix defining a spread  316   b . Likewise, for the probabilistic distribution  314   e , the processor  204  determines, based on the synthetic measurements associated with the probabilistic distribution  314   e , a synthetic centroid  318   a  and a synthetic covariance matrix defining a spread  318   b . Likewise, for the probabilistic distribution  314   h , the processor  204  determines, based on the synthetic measurements associated with the probabilistic distribution  314   h , a synthetic centroid  320   a  and a synthetic covariance matrix defining a spread  320   b . In such a way, the synthetic centroid and the synthetic covariance matrix are determined for each probabilistic distribution. Further, using the synthetic measurements associated with each probabilistic distribution, the predicted belief on the expanded state is updated as described below with reference to  FIG.  3 C . 
       FIG.  3 C  shows a schematic for updating the predicted belief on the expanded state  302 , according to some embodiments. The processor  204  updates the predicted belief on the expanded state  302  using the probabilistic filter, such as a Kalman filter, with the synthetic measurements associated with each probabilistic distribution to produce an updated expanded state x k|k    322  of the object. The updated expanded state x k|k    322  of the object may be referred to as an updated belief on the expanded state. Further, the updated belief on the expanded state is used to update the tracked belief. In an embodiment, the tracked belief is updated based on a difference between the predicted belief and the updated belief. Further, the processor  204  tracks the expanded state of the object based on the updated tracked belief on the expanded state. 
     The processor  204  is configured to execute the compound measurement model  210   b  only for the predetermined time period T to predict the belief on the expanded state  302  and further update the predicted belief on the expanded state  302 . After the predetermined time period T, the processor  204  is further configured to accumulate the predicted belief, the updated belief, and the measurements obtained within the predetermined time period to create an online batch of state-decoupled training data and update the compound measurement model  210   b  based on the online batch of the state-decoupled training data. 
     Thus, initially the compound measurement model  304  used for tracking the expanded state of the object, as described above, is learned offline. The offline learning and characteristics of the compound measurement model  304  are described below. 
     The compound measurement model  304  includes, for instance, L Gaussian components (i.e., ellipses) with their component means located on the contour. In an embodiment, the contour may be a B-spline curve. The B-spline curve is advantageous because the B-spline curve provides more control flexibility for enclosed contours. Also, since the B-spline curves satisfy a strong convex hull property, they have a finer shape control. For each ellipse centered at μ l  with an extent Σ l , N k  measurements may be assigned with an association probability ρ i   l . Given a measurement-to-ellipse assignment, a likelihood function is given as 
     
       
         
           
             
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     (1) and (2) correspond to a sample mean and spread of l-th ellipse.   denotes a Gaussian distribution and   is a Wishart distribution. 
     Some embodiments are based on recognition that the probabilistic distributions of the compound measurement model  304  can be represented using Gaussian distribution to better align with probabilistic filters. For example, in some embodiments, the probabilistic distributions are defined as a random matrix model (RMM) in a probability space (Ω, P, F), where the sample space Ω is a set of matrices. The random matrices are advantageous to represent multi-dimensional probabilistic distributions and parameters of the probabilistic distributions represented as RMMs can be illustrated using oval shapes. According to an embodiment, with all L ellipses and given the measurement-to-ellipse assignment, L random matrices model is defined as 
         p ( Z |θ,ρ)=Σ i=1   L π l ϕ( Z   k   |N   l ,μ l ,Σ l ,ρ),  (3)
 
     where mixture weights π l  are assumed to equal π l =1/L. 
     Further, it is assumed that the ellipse centers are located on a B-spline curve defined by c(r)∈   2×1  of degree d 
         c ( r )=Σ j=0   m   p   j   B   j,d(r) ,0≤ r≤m−d+ 1,  (4)
 
     where p j ∈   2×1  is a j-th control point, m+1 is a number of control points, and B j,d (r) is a basis function with a parameter r. By enforcing μ l =c(r l ) with r l  denoting a corresponding parameter of the l-th ellipse center μ l , a B-spline chained ellipses model (i.e., the compound measurement model  304 ) is defined as 
         p ( Z |θ,ρ)=Σ l=1   L π l ϕ( Z   k   |N   l   ,c ( r   l ),Σ l ,ρ)  (5)
 
     where parameters of the the B-spline chained ellipses model (i.e., the compound measurement model  304 ) are a number of measurements for each component N, the control points of the B-spline curve {p j } j=0   m  and the covariance matrices of each component {Σ l } l=1   L . 
       FIG.  4    shows a workflow for tracking the expanded state of the object by using the compound measurement model  210   b , according to some embodiments.  FIG.  4    is described below in conjunction with  FIG.  2   . The proposed tracking system (in  FIG.  2   ) uses the compound measurement model  210   b  for tracking the expanded state of the object. To that end, the compound measurement model  210   b  is configured to learn offline  400 . The offline learning  400  of the compound measurement model  210   b  is described below in detail with respect to  FIG.  5   . Based on the offline learning, the compound measurement model  210   b  is used by the tracking system to track the expanded state of the object online  402  for a predetermined time period T. To track the expanded state of the object online  402 , the tracking system uses the motion model  210   a  to predict the belief on the expanded state of the object and uses the offline learned compound measurement model  210   b  to update the predicted belief. The online tracking of the expanded state  402  is described in detail with respect to  FIG.  10 A- 10 D . 
     After the predetermined time period T, the predicted belief, the updated predicted belief, and the measurements within the time period T are accumulated to form an online batch of training data comprising state-decoupled measurements. The online batch of training data is then used for an online learning  404  of the compound measurement model  210   b , where the online learning  404  updates/refines the compound measurement model  210   b  by updating one or more parameters of the compound measurement model  210   b . The parameters of the compound measurement model  210   b  comprise a number of measurements for each probabilistic distribution of the multiple probabilistic distributions comprised by the compound measurement model  210   b , control points corresponding to the multiple probabilistic distributions, and covariances between the multiple probabilistic distributions. The online learning  404  of the compound measurement model  210   b  is described later in detail with respect to  FIG.  9   . 
       FIG.  5    shows a flow chart of a method for offline learning of the parameters of the compound measurement model  304 , according to some embodiments. The compound measurement model  304  learns offline using an offline training data. At step  500 , the method includes accepting  500  training data including different measurements of different motions of different objects. At step  502 , the method includes transforming  502  the training data into a common coordinate system. 
     Some embodiments are based on recognition that the parameters of the compound measurement model  304  can be learned offline based on the training data and knowledge of the contour of the object to be tracked using various statistical methods, such as an expectation-maximization (EM) method. To that end, at step  504 , the method includes learning  504  the parameters of the compound measurement model from the training data, using the statistical method, such as the EM method. 
     Some embodiments are based on the realization that offline training data, used for offline learning by the compound measurement model  304 , comprises coarse vehicle labels which may lead to an over-smoothed offline learned compound measurement model  304  that averages over different vehicle models. For instance, a coarsely labeled dataset may include sedan and SUV in the same class. For instance, a coarsely labeled dataset may include sedan and SUV in the same class. 
     Therefore, the present disclosure proposes an online adaptation of the compound measurement model  304  (also referred to as “online compound measurement model adaptation”) that refines the offline learned compound measurement model  304  and improves the online state estimation performance (i.e., real-time tracking of the expanded state of the object). Refining the compound measurement model  304  comprises updating the parameters of the compound measurement model  304 . 
       FIG.  6    shows a flow chart of a method for online learning of the parameters of the compound measurement model  304 , according to some embodiments. The compound measurement model  304  learned offline as shown in  FIG.  5    is executed for a predetermined time period to update predicted belief on the expanded state of the object and track the expanded state of the object based on the updated predicted belief. After the predetermined time period, the method for learning of the parameters of the compound measurement model  304  includes at step  600 , accumulating predicted belief, updated belief, and measurements within the predetermined time period. 
     At step  602 , generating an online batch of state-decouple training data based on the accumulated predicted belief, updated belief, and measurements within the predetermined time period. Some embodiments are based on recognition that the compound measurement model  304  can be refined to better track the expanded state of the object by updating the parameters of the compound measurement model  304 . The parameters of the compound measurement model  301  are updated based on the online batch of training data and knowledge of the contour of the object to be tracked using various statistical methods, such as the EM method. Accordingly, at step  604 , the method for online learning of the compound measurement model  304  includes refining the compound measurement model  304  by updating parameters of the compound measurement model  304 . 
       FIGS.  7 A and  7 B  show schematics of transformation of the training data collected from different motions of different objects into the common unit coordinate system, according to some embodiments. The different measurements collected from tracking different trajectories  700  and  702  are converted into a respective object-centered (OC) coordinate system  704  and  706 . Then, the converted measurements are aggregated  708 . In some implementations, the measurements are collected for motion of similar type of objects, e.g., from motions of similar class of vehicles. For example, the embodiments, for each trajectory, convert the measurements from each time step from global coordinate (GC) to the object-centered (OC) coordinate, and aggregate OC measurements from all trajectories for vehicles with a similar size (e.g., sedan). 
     Next, as shown in  FIG.  7 B , the embodiments convert the aggregated OC  708  measurements to a unit coordinate (UC) system  710 . In some implementations, the conversion to UC system is performed by various normalization techniques that allow using the converted training data for machine learning. Further, the measurements in the unit coordinate system  710  are used as training data for learning the parameters of the compound measurement model  304 . 
       FIG.  8    shows a block diagram of the EM method for offline learning of the parameters of the compound measurement model  304 , according to some embodiments. In some embodiments, the measurements in the unit coordinate system  710  are used as the training data  800 , denoted as Z={z j } i=1   N , for the EM method. In some other embodiments, the aggregated OC measurements  708  are received and converted into a unit coordinate system, which originates at the object center m=[x m , y m ]∈   2×1  and is oriented such that x-axis points towards object&#39;s front, by applying the following coordinate transformation: 
         {tilde over (z)}   i   =S   −1   R   ψ   −1 ( z   i   −m ),  (6)
 
     where R ψ ∈   2×2  is a rotation matrix as a function of the orientation angle ψ, S=diag(l, w) is a scaling matrix. 
     The training data  800  in the unit coordinate system, and initial parameters  802  such as a control point p j   0  and extent Σ l   0  are input data to the EM method. The EM method includes two main steps, namely, an expectation step  804  and a maximization step  806 . 
     The expectation step  804  is to update hidden random variables {ρ l ,  z   l ,  Z   l }. At first, a posterior association probability of each measurement is calculated as 
     
       
         
           
             
               
                 
                   
                     
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     The maximization step  806  is to update the model parameters θ={p j , Σ l } based on the Q-function of (5) as 
     
       
         
           
             
               
                 
                   
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     The B-spline curve in a matrix-vector form can be reformatted as μ l =B l p, where 
     B l =blkdiag(n l   T , n l   T ), n l =[B 0,d (r l ), . . . , B m,d (r l )] T  and p=[p x   T , p y   T ] with p x   T  and
 
p y   T  denoting the control inputs in x and y coordinates, respectively. By setting derivative of Q(θ) (with respect to θ) to 0, the control input can be given as p=H + M, where H +  is Moore-Penrose inverse of
 
         H=Σ   l=1   L (   N     l   B   l   T Σ l   −1   B   l ) and
 
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     Further, iterations are carried out between the estimates of p and Σ l  until a convergence criterion  808  is achieved. The convergence criterion  808  may be a predetermined likelihood in (8), relative changes of the estimated parameters over consecutive iterations is smaller than predefined values, or a predetermined maximum number of iterations. 
     According to some embodiments, the offline learned compound measurement model is used for online tracking of the expanded state of the object, i.e., real time tracking of the expanded state of the object. Some embodiments are based on the realization that the probabilistic nature of the compound measurement model can be beneficially aligned with probabilistic multi-hypothesis tracking (PMHT) methods. For example, such an alignment allows implementing the probabilistic filter using at least a variation of a Kalman filter. For example, one embodiment uses an unscented Kalman filter-probabilistic multi-hypothesis tracking (UKF-PMHT) method. The unscented Kalman filter (UKF) is used for transforming the compound measurement model from the unit coordinate system into the global coordinate system. The probabilistic multi-hypothesis tracking (PMHT) method is then applied to assign the measurements at the current time step to different ellipsis components in a probabilistic fashion and update the expanded state of the object. 
     In an embodiment, given the offline learned compound measurement model and assuming a measurement x μ  in the unit coordinate system is distributed with respect to the l-th ellipse  (μ l , Σ l ), the corresponding measurement h l,k (x k|k−1 ) in the global coordinate system is defined as 
         h   l,k ( x   k|k−1 )= m   k|k−1   +R   ψk|k−1   ·S   k|k−1   ·x   μ   (10)
 
     where m ψk|k−1 , R ψk|k−1  and s=diag(l k|k−1 , w k|k−1 ) are defined the same way as (6) except that all augments are given by the predicted state (k|k−1) with corresponding predictive distributions (e.g., the Gaussian distribution). 
     Some embodiments are based on realization that since the transformation in (10) is nonlinear, particularly with respect to the predictive orientation angle, an unscented transform (UT) can be used to determine a mean  h   l,k (x k|k−1 ) and a covariance matrix X l  of h l,k (x k|k−1 ). To this end, the predicted expanded state is augmented with x μ  as x aug =[x k|k−1 , x μ ] T     n     a     ×1  with n a =9. Then, 2n a +1 weighted samples, i.e., the sigma points, are determined such that they can describe a true mean  x   aug  and a covariance matrix P aug  of x aug : 
         A   0   = x     aug   , W   0 =κ/( n   a +κ),  W   i≥1 =0.5( n   a +κ),
 
         A   i≤n     a     = x     aug +(√{square root over (( n   a +κ) P   aug )}) i ,  (11)
 
         A   i&gt;n     a     = x     aug −(√{square root over (( n   a +κ) P   aug )}) i−n     a   ,  (12)
 
     where κ is a scaling parameter such thāt κ+n a ≠0 and (√{square root over (A)}) i  denotes i-th row of matrix square root of A. Each sigma point is then propagated through the nonlinear function of (10), i.e.,    i =h l,k (   i ), and a first two moments of h l,k (x k|k−1 ) are computed as 
           h     l,k =Σ i=0   2n     a     W   i   B   i ,  (13)
 
         X   l =Σ i=0   n     a     W   i ( B   i   − h     l,k )(( B   i   −h   l,k )) T .  (14)
 
     In the global coordinate system, a measurement z i  that is assigned to the l-th ellipse can be defined as 
         z   i   =h   l,k ( x   k|k−1 )+ n   l ,  (15)
 
     where h l,k (x k|k−1 )˜ ( h   l,k , X l ), is a corresponding reflection center, and n l ˜  (0,R) is a measurement noise. 
     Given measurements at time k, Z k ={z i,k } i=1   N     k    and L offline learned {h l,k (x k|k−1 )} i=1   L  obtained at time step k, the PMHT is used to assign the measurements to each ellipse component. Different from a general PMHT algorithm to handle measurement-to-object association and update the kinematic states of multiple objects over consecutive time steps, the PMHT algorithm here is applied to handle the measurement-to-ellipsis association and update both kinematic and extent states of a single object over the current time step k. Specifically, the PMHT employs the EM algorithm for a soft measurement-to-ellipsis assignment that in turn creates a synthetic measurement for each component. Mathematically, the measurement-to-ellipsis association weights ρ i,k   l , synthetic measurements  z   l.k  and corresponding synthetic covariance matrix C zz  are derived as follows 
     
       
         
           
             
               
                 
                   
                     
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     Further, a covariance between the expanded state and measurements C xz  is calculated during the UT procedure (10) and the filter gain is calculated as K=C xz C zz   −1 . The expanded state x k,l  and the covariance matrix C l,n  are updated based on the l-th measurement equation in (15). The PMHT iterates between the expectation and maximization steps until a predefined maximum iteration number N iter  is reached. In each iteration n, the expanded state x k,l  and the covariance matrix C l,n  are updated incrementally by each component (i.e., over l) in order of (10) and (16)-(18). An overall UKF-PMHT tracking algorithm is described below with reference to  FIG.  10 A . 
     According to some embodiments, the offline learned compound measurement model  304  is refined using the refinement module  210   c  that updates or refines the compound measurement model  304  online (i.e., in real-time). To achieve the online adaptation of the compound measurement model  304 , the expanded state of the object tracked online by the compound measurement model  304  within the T time steps is smoothed to remove noise. To that end, at least one of a backward recursion and a forward recursion is performed to smooth the online compound measurements by filtering expanded state with all observed measurements within the T time steps. In some embodiments, the smoothing is performed by applying a Bayesian state smoothing technique. In a preferred embodiment, an unscented Rauch-Tung-Striebel (RTS) smoother is applied to compute smoother gain, smoothed mean, and the smoothed covariance matrix at each time step k of the T time step by recursively computing a posterior of the expanded state conditioned on all observed measurements backward from the filtered expanded state estimate at the last time step. 
     Then, the smoothed expanded states are used to convert all observed measurements, within the T time steps, in the global coordinate system to a batch of state-decoupled training data in a unit coordinate system, where the batch of state-decoupled training data is used for refining the compound measurement model  304  by updating parameters of the compound measurement model  304 . 
     The online learning of the parameters (θ) of the compound measurement model  304  updates the parameters of the offline learned compound measurement model  304  after the predetermined time period T. Thus, the online learning refines the offline learned compound measurement model  304 . Therefore, the online learning is also referred to as online adaptation of the compound measurement model  304 . As the online batch of training data are state-decoupled and only depends on the compound measurement model  304 , a statistical algorithm such as the EM algorithm can be used to update the model parameters within a regularization on the distance to the parameters of the offline learned compound measurement model  304 . Thus, the parameters of the offline learned compound measurement model (also referred to as “pretrained compound measurement model”) are updated based on regularized distances between parameters of the updated compound measurement model and the parameters of the offline learned compound measurement model. 
     However, the parameters of the offline learned compound measurement model are updated such that the predetermined relative geometrical mapping of the multiple probabilistic distributions to the center of the object is preserved. To that end, control points corresponding to the multiple probabilistic distributions are preserved while updating the parameters of the offline learned compound measurement model, where to preserve the control points of the multiple probabilistic distributions a penalty function such as a log-likelihood function is used that enforces the maximal allowable change on the control points. 
     For the online adaptation of the compound measurement model  304 , updated states x k|k , predicted states x k|k−1  and measurements Z k  of the past and the future (with respect to time k within the predetermined time period T) are accumulated. The online adaptation improves tracking the expanded state of the object by generating an online batch of training data and updating the parameters of the compound measurement model  304 , where the online batch of the training data comprises state decoupled measurements. 
     To generate the online batch of training data comprising the state decoupled measurements, the measurements obtained during the T time steps are used to smooth the updated expanded state of the object. The offline learned compound measurement model  304  enables tracking of the expanded state of the object at each time step k, where the expanded state comprises kinematic state elements (i.e., the first 5 elements in equation (19)) and the extent state elements in terms of the length and width in equation (19), 
         x   k =[ x   m,k   ,y   m,k   ,v   k ,ψ k ,ω k   ,l   k   ,w   k ] T   (19)
 
     where [x m,k , y m,k ] T  is the center of the object (for example, vehicle as shown in  FIG.  7   ), v k  is the polar velocity, ψ k  is the heading orientation, ω k  is the turning rate, l k  and w k  are length and width of the object, respectively and Z k ={z i,k } i=1   N     k    with N k  denoting the number of measurements at time k. Further, a standard point-target Bayesian smoothing algorithm can be directly applied on the updated expanded states (also referred to as “augmented kinematic-and-extent state vector”). 
     In some embodiments, the Bayesian smoothing based unscented RTS smoother of the expanded states is used to smooth the expanded states after T time steps. To smooth the expanded states after T time steps, initially sigma points x i  and corresponding weights W i   (m)  and W i   (C)  are generated according to statistics of the updated expanded state x k|k , where the updated expanded state x k|k  at time k follows the Gaussian distribution with the statistics of the updated expanded states x k|k  comprising mean of the Gaussian distribution as  x   k  and covariance C x,k : 
         X   0   = x     k , 
         X   i   = x     k +(√{square root over (( M +λ) C   x,k )}) i   , i= 1, . . . , M  
 
         X   i   = x     k −(√{square root over (( M +λ) C   x,k )}) i   , i=M+ 1, . . . ,2 M  
 
         W   0   (m) =λ/( M +λ),
 
         W   0   (C) =λ/( M +λ)+(1−α 2 +β),
 
         W   i   (m)   =W   i   (C) =1/[2( M +λ)]  i= 1, . . . 2 M   (20)
 
     where M is the dimension of the updated expanded state, λ is a scaling parameter as λ=α 2 (M+K)−M, α is the parameter that determines the spread of the sigma points around the mean  x   k , κ is the secondary scaling parameter usually set to 0, and β is used to incorporate the prior knowledge of the updated state distribution. 
     Each sigma point is then propagated through the nonlinear motion model (for example, coordinated turn (CT) motion model) x k+1|k =g(x k|k ) with polar velocity. The propagated sigma points are given as y i =g(x i ). The propagated sigma points are then used to compute the predicted mean  x     k + 1   and its covariance C   x ,k+ 1   of the predicted state x k+1|k  can be computed using the propagated sigma points: 
           x       k + 1   =Σ i=0   2M   W   i   (m)   y   i   (21)
 
         C     x ,k+ 1   =Σ i=0   2M   W   i   (C) [ y   i   − x       k + 1   ][ y   i   − x       k + 1   ] T .  (22)
 
     Further, to smooth the updated expanded states a cross-covariance matrix between the predicted expanded states x k+1|k  and the updated expanded states x k|k  as: 
         D   k+1 =Σ i=0   2M   W   i   (C) [ X   i   − x     k ][ y   i   − x       k + 1   ] T .  (23)
 
     Finally, the smoothed states x k   s  and smoothed covariance matrix C x,k   s  are calculated using equations (24) and (25), where x k   s  is Gaussian distributed with the smoothed mean  x   k   s : 
           x     k   s   = x     k   +G   k (   x     k+1   s   − x       k + 1   )  (24)
 
         C   x,k   s   =C   x,k   +G   k ( C   x,k+1   s   −C     x ,k+ 1   ) G   k   T ,  (25)
 
     where G k =D k+1 [C   x ,k+ 1   ] −1  is the smoother gain. In an example embodiment,  x   T,k   s = x   T  and C x,T   s  at time step T. Further, equations (24) and (25) are iteratively computed backwards in time step k=T−1, . . . , 1 to generate the smoothed states. 
     Further, an online batch of training data that may fit better for onboard radar sensors is generated, where the tracking of the expanded states of the object is improved by using the online batch of training data. The online batch of training data is generated using measurements Z={Z 1 , . . . , Z T } up to time step T and corresponding smoothed states {x 1   s , . . . , x T   s } as x k   x ˜N( x   k   s , C x,k   s ), where  x   k   s  and C x,k   s  are obtained using equations (24) and (25). 
     Further, each element in the online batch of training data is state decoupled by removing underlying unknown states {x 1 , . . . , x T } from the measurements Z in the global coordinate system using the smoothed mean { x   1   s , . . . ,  x   T   s } 
         z   i,k   s =[ S (   l     k   s   , w     k   s )] −1 [ R ( ψ   k   s )] −1 ( z   i,k   − m     k   s ),  (26)
 
     where { m   k   s ,  ψ   k   s ,  l   k   s ,  w   k   s } denote the same state variables as defined equation (10) except that they now correspond to the smoothed state mean  x   k   s  of equation (24). For each time step k, the online training dataset is grouped as Z k   s ={z i,k   s } i=1   N     k   , k=1, . . . , K. 
     To state-decouple the online batch of training data, the measurements in the global coordinate system are transformed into a unit-coordinate system that is positioned in the center of the object and oriented such that the x-axis of the new coordinate system points to the object front using the orientation angle ψ k   s  and object center  m   k   s . Then, the measurements transformed into the unit-coordinate system are further normalized by the extent states, i.e., length  l   k   s  and width  w   k   s  via the scaling matrix S. In an example embodiment, the length and the width used in equation (26) may be choses as the average values of smoothed length and width over all time steps T, i.e., S=1/Tdiag(Σ k   l   k   s , Σ k w k   s ). 
     Further, for simplicity all the smoothed measurements Z s ={Z 1   s , . . . , Z T   s } are grouped as the online batch of training data and the measurement index is re-assigned in Z s  as Z s ={z 1   s , . . . , z N     online     s } with N online =Σ k N k  denoting the total number of measurements over T steps. Based on the online training data N online , the online adaptation of the compound measurement model  304  is conducted by the EM algorithm. 
       FIG.  9    shows a block diagram of the EM method for online learning of the parameters of the compound measurement model  304 , according to some embodiments. The online batch of the training data (Z s )  600  is input to the EM method for online learning. The EM method includes an expectation step  902  and a maximization step  904 . 
     At the expectation step  902 , n-th online measurement z n   s  from the online batch of training data (Z s )  600  is assigned to the l-th ellipsis component via the posterior association probability ω n   l : 
     
       
         
           
             
               
                 
                   
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     In equation (27), μ l  and 4Σ l  is the mean and the covariance matrix of each component, where to uniformly distribute measurements in each ellipsis component a scaling factor of 4 is used, ∈ is the probability of the uniformly distributed outliers. Further, based on the posterior association probability ω n   l , the synthetic measurements  z   l , synthetic measurement spread  Z   l , and the sum of weights for the l-th component are calculated, respectively, as: 
     
       
         
           
             
               
                 
                   
                     
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     At the maximization step  904 , the model parameters θ={p, Σ l } are updated around the offline learned model parameters θ offline  based on a regularized log-likelihood function that enforces the maximal allowable change on the control points p of the B-spline curve: 
     
       
         
           
             
               
                 
                   
                     
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     where p offline ∈   2(m+1)×1  are the control points corresponding to the offline learned model, λ is regularization parameter that controls the spatial model adaptation rate, and ∥.∥ 2  denotes the l 2  norm. λ. 
     The B-spline curve can be reformulated in a matric-vector form as μ l =B l p, B l =blkdiag(n l   T , n l   T ), n l =[B 0,d (r l ), . . . B m,d (r l )] T , and p=[p x   T , p y   T ] T  with p x   T    
     and p y   T  denoting the control points in the x and y-coordinates, respectively. By taking the gradient of  (θ) (with respect to the parameters θ) to 0, the control input can be given as p=H + M, where M=Σ l=1   L ( M   l B l   T Σ l   −1 )+2λp offline  and 
     H +  is Moore-Penrose inverse of H=Σ l=1   L  ( M   l B l   T Σ l   −1 B l )+2λI with I denoting the identity matrix, and 
     
       
         
           
             
               
                 
                   
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     The PMHT iterates between the estimates of p and Σ l  until a convergence criterion  906  is achieved. The convergence criterion  906  may be a predetermined likelihood in (30), relative changes of the estimated parameters θ* online  over consecutive iterations is smaller than predefined values θ online   i , or a predetermined maximum number of iterations N iter . In this way, the model parameters θ online ={p, Σ l } are updated from the iterations in the EM method and the updated compound measurement model  304  p({tilde over (Z)}, Z s |θ online , ρ) replaces the offline learned compound measurement model  304  p({tilde over (Z)}|θ offline , ρ), where the tracking system  200  uses the updated compound measurement model  304  p({tilde over (Z)}, Z s |θ online , ρ) to track the object. An overall UKF-PMHT tracking algorithm is described below with reference to  FIG.  10 A . 
       FIG.  10 A  shows a flowchart of the UKF-PMHT tracking algorithm, according to some embodiments. The UKF-PMHT tracking algorithm is executed by the processor  204 . The UKF-PMHT tracking algorithm includes two stages, namely, a prediction stage  1000  and an update stage  1004 . In the prediction stage  1000 , a belief on the expanded state of the object and a corresponding covariance matrix is predicted using the motion model. Further, at block  1002 , an iteration of the update stage  1004  is initiated by setting an iteration index n=1, and x 1,1 =x k|k−1  and C 1,1 =C k|k−1 . 
     At block  1006  of the update stage  1004 , the predicted measurements and the covariance matrix are computed based on an offline learned measurement model  1012 .  FIG.  10 B  shows a block diagram of steps performed to compute the predicted measurements and the covariance matrix, according to some embodiments. At block  1022 , the sigma points    i=1   n     a    are generated for a first ellipse (i.e., l=1) of the compound measurement model  1012  and the latest updated expanded state, using equations (11) and (12). At block  1024 , the sigma points are propagated through an affine transform given by (10). At block  1026 , the predicted measurements and the covariance matrix are computed according to equations (13) and (14), respectively, for the first ellipse. 
     Referring back to  FIG.  10 A , at block  1008 , given measurements  1014  at time k, the synthetic measurements and the synthetic covariance matrix are computed.  FIG.  10 C  shows a block diagram of steps performed to compute the synthetic measurements and the synthetic covariance matrix, according to some embodiments. At block  1028 , the measurements-to-ellipse association weight is computed according to equation (16) for the first ellipse. At block  1030 , the synthetic measurements and the synthetic covariance matrix are computed using equations (17) and (18), respectively, for the first ellipse. 
     Referring back to  FIG.  10 A , at block  1010 , the expanded state x l,n  and the covariance matrix C l,n  are updated. Here, subscript l is an ellipse index.  FIG.  10 D  shows a block diagram of steps performed to update the expanded state x l,n  and the covariance matrix C l,n  for the first ellipse (i.e., l=1), according to some embodiments. At block  1032 , the cross-covariance matrix is computed as 
     
       
         
           
             
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     At block  1034 , a Kalman filter gain is computed. The Kalman filter gain is given by 
         K   l   =C   xz,l   C   zz,l   −1 . 
     At block  1036 , the expanded state x l,n  and the covariance matrix C l,n  are updated as 
         x   l,n   =x   l−1,n   +K   l (   z     l,k   − h     l,k ) 
     
       
      
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     Further, the same functions given at the blocks  1006 ,  1008 , and  1010  are executed for a second ellipse (i.e., l=2) of the compound measurement model  1012 . For the second ellipse, the updated expanded state x l,n  and the covariance matrix C l,n  are used to compute the predicted measurements and the covariance matrix. In other words, the predicted measurements and the covariance matrix are computed using the latest updated expanded state and the covariance matrix. Likewise, the same functions given at the blocks  1006 ,  1008 , and  1010  are executed for the rest of ellipses of the compound measurement model  1012 . To that end, to complete an iteration of the update stage  1004 , a number of internal iterations l=1 . . . L, where L is a number of ellipses of the compound measurement model  1012 , are executed. The complete execution of the iteration (n=1) of the update stage  704  yields an expanded state of the object at time k. Further, in the next iteration (i.e., n=2), the update stage  1004  is executed to yield an updated expanded state. The update stage  1004  is iteratively executed until a convergence criterion  1016  is achieved. The convergence criterion  1010  may be that iterations (N iter ) are carried out for a time t less than a predetermined time period T, till relative changes of the estimated parameters θ* over consecutive iterations is smaller than predefined values. Once the convergence criterion  1016  is achieved, an updated expanded state x k|k =x L,N     iter    and a corresponding covariance matrix C k|k =C L,N     iter    are outputted  1018 . 
     On satisfying the convergence criteria is achieved i.e., after the predetermined time period T, the updated expanded x k|k  states along with corresponding measurements obtained within the time period T are used to for online learning  1020  of the compound measurement model. The online learning  1020  refines the offline learned  1012  compound measurement model by updating the parameters of the compound measurement model. 
       FIG.  10 E  shows a block diagram of steps performed for online learning  1020  of the compound measurement model, according to some embodiments. At step  1038 , the updated expanded states are smoothed using the measurements obtained within the predetermine T time steps using equations (24) and (25) and the covariance (23) between the updated expanded state and the predicted expanded state accumulated within the predetermined T time steps. The updated expanded states are smoothed using the measurements obtained within the predetermined T time steps by using at least one of: the backward recursion and the forward recursion. To that end, some embodiments use Bayesian smoothing based RTS smoother. Further, at step  1040 , the online batch of training data comprising state decoupled measurements is generated based on the smoothed expanded states using equation (26). Finally, the online batch of training data is used to refine the offline learned compound measurement model by updating its parameters using EM method ( FIG.  9   ). 
       FIG.  10 F  shows a schematic illustrating a backward recursion  1044  and a forward recursion  1046 , according to some embodiments.  FIG.  10 F  shows accumulated data  1042 , where the accumulated data  1042  comprises updated belief, predicted belief, and the measurements obtained within a predetermined time period T sec. The accumulated data  1042  is used to generate the online batch of training data, which is used to update the compound measurement model, and where the expanded states of objects are tracked using the updated compound measurement model. The online batch of training data comprises data accumulated only within the predetermined time period T sec, where T may be of few seconds or few minutes. Thus, the online batch of training data comprises very less training data compared to training data used to train the compound measurement model offline. To increase the accuracy of the compound measurement model trained using the online batch of training data it is important to obtain relationship amongst data within the accumulated data  1042  and use the relationship to update parameters of the compound measurement model. 
     To that end, the accumulated updated belief is smoothed using the backward recursion  1044  and the forward recursion  1046  to generate the online batch of training data. In the backward recursion  1044 , the accumulated updated belief is smoothed backward from a specific time t sec (for example, 5 th  sec) within the predetermined time period T sec (for example, 10 sec) based on measurements at the specific time t sec. Alternatively, in the forward recursion  1046 , the accumulated updated belief is smoothed forward from the specific time t sec (for example, 5 th  sec) within the predetermined time period T sec (for example, 10 sec) based on measurements at the specific time t sec. 
       FIG.  11 A  shows a schematic of a vehicle  1100  including a controller  1102  in communication with the tracking system  200  employing principles of some embodiments. The vehicle  1100  may be any type of wheeled vehicle, such as a passenger car, bus, or rover. Also, the vehicle  100  can be an autonomous or semi-autonomous vehicle. For example, some embodiments control the motion of the vehicle  1100 . Examples of the motion include lateral motion of the vehicle controlled by a steering system  1104  of the vehicle  1100 . In one embodiment, the steering system  1104  is controlled by the controller  1102 . Additionally or alternatively, the steering system  1104  may be controlled by a driver of the vehicle  1100 . 
     In some embodiments, the vehicle  1100  may include an engine  1110 , which can be controlled by the controller  1102  or by other components of the vehicle  1100 . In some embodiments, the vehicle may include an electric motor in place of the engine  1110  and can be controlled by the controller  1102  or by other components of the vehicle  1100 . The vehicle  1100  can also include one or more sensors  1106  to sense the surrounding environment. Examples of the sensors  1106  include distance range finders, such as radars. In some embodiments, the vehicle  1100  includes one or more sensors  1108  to sense its current motion parameters and internal status. Examples of the one or more sensors  1108  include global positioning system (GPS), accelerometers, inertial measurement units, gyroscopes, shaft rotational sensors, torque sensors, deflection sensors, pressure sensor, and flow sensors. The sensors provide information to the controller  1102 . The vehicle may be equipped with a transceiver  1112  enabling communication capabilities of the controller  1102  through wired or wireless communication channels with the tracking system  200  of some embodiments. For example, through the transceiver  1112 , the controller  802  receives the control inputs from the tracking system  200 . 
       FIG.  11 B  shows a schematic of interaction between the controller  1102  and controllers  1114  of the vehicle  1100 , according to some embodiments. For example, in some embodiments, the controllers  1114  of the vehicle  1100  are steering control  1116  and brake/throttle controllers  1118  that control rotation and acceleration of the vehicle  1100 . In such a case, the controller  1102  outputs control commands, based on the control inputs, to the controllers  1116  and  1118  to control the kinematic state of the vehicle. In some embodiments, the controllers  1114  also includes high-level controllers, e.g. a lane-keeping assist controller  1120  that further process the control commands of the controller  1102 . In both cases, the controllers  1114  utilize the output of the controller  1102  i.e. control commands to control at least one actuator of the vehicle  1100 , such as the steering wheel and/or the brakes of the vehicle  1100 , in order to control the motion of the vehicle  1100 . 
       FIG.  11 C  shows a schematic of an autonomous or semi-autonomous controlled vehicle  1122  for which the control inputs are generated by using some embodiments. The controlled vehicle  1122  may be equipped with the tracking system  200 . In some embodiments, an expanded state of each of the obstacles  1124  are tracked by the controlled vehicle  1122  and subsequently, the control inputs are generated based on the tracked expanded states of the obstacles. In some embodiments, the control inputs include commands specifying values of one or combination of a steering angle of the wheels of the vehicle and a rotational velocity of the wheels, and the measurements include values of one or combination of a rotation rate of the vehicle and an acceleration of the vehicle. 
     The generated control inputs aim to keep the controlled vehicle  1122  within particular bounds of road  1126 , and aims to avoid other uncontrolled vehicles, i.e., obstacles  1124  for the controlled vehicle  1122 . For example, based on the control inputs, the autonomous or semi-autonomous controlled vehicle  1122  may, for example, pass another vehicle on the left or on the right side or instead to stay behind another vehicle within the current lane of the road  1126 . 
     The following description provides exemplary embodiments only, and is not intended to limit the scope, applicability, or configuration of the disclosure. Rather, the following description of the exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Contemplated are various changes that may be made in the function and arrangement of elements without departing from the spirit and scope of the subject matter disclosed as set forth in the appended claims. 
     Specific details are given in the following description to provide a thorough understanding of the embodiments. However, understood by one of ordinary skill in the art can be that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the subject matter disclosed may be shown as components in block diagram form in order not to obscure the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail in order to avoid obscuring the embodiments. Further, like reference numbers and designations in the various drawings indicate like elements. 
     Also, individual embodiments may be described as a process which is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. Although a flowchart may describe the operations as a sequential process, many of the operations can be performed in parallel or concurrently. In addition, the order of the operations may be re-arranged. A process may be terminated when its operations are completed, but may have additional steps not discussed or included in a figure. Furthermore, not all operations in any particularly described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the function&#39;s termination can correspond to a return of the function to the calling function or the main function. 
     Furthermore, embodiments of the subject matter disclosed may be implemented, at least in part, either manually or automatically. Manual or automatic implementations may be executed, or at least assisted, through the use of machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware or microcode, the program code or code segments to perform the necessary tasks may be stored in a machine readable medium. A processor(s) may perform the necessary tasks. 
     Various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments. 
     Embodiments of the present disclosure may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts concurrently, even though shown as sequential acts in illustrative embodiments. 
     Although the present disclosure has been described with reference to certain preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the present disclosure. Therefore, it is the aspect of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the present disclosure.