DE102020121064A1 - Method for operating an assistance system in a motor vehicle, computer program product, computer-readable storage medium and assistance system - Google Patents

Abstract

The invention relates to a method for operating an assistance system (2), in which an environment (6) is recorded by means of a recording device (5) by generating recording points (8), and in which the point cloud is evaluated by means of an electronic computing device (3). , wherein the evaluation is carried out on the basis of a mathematical model (9), and a tracking of an object (7) in the environment (6) is carried out by means of the mathematical model (9), the object (7) being defined as a polygonal shape (12 ) is approximated in the mathematical model (9) and at least one of the detection points (8) is assigned to the polygonal shape (9) by means of a Kalman filter, which is set up for processing implied measurement functions, with at least one of the detection points (9) being at least associated with a specific part of the polygonal shape (12) and the detection is performed based on this predetermined condition. The invention also relates to a computer program product, a computer-readable storage medium (4) and an assistance system (2).

Description

The invention relates to a method for operating an assistance system of a motor vehicle, in which an area surrounding the motor vehicle is recorded using a recording device of the assistance system, the area being recorded by generating a large number of recording points as a point cloud using the recording device, and in which the point cloud is recorded using an electronic computing device of the assistance system is evaluated, with the evaluation being carried out on the basis of a mathematical model of the electronic computing device, and using the mathematical model to track an object in the area, with the object being approximated as a polygonal shape in the mathematical model and using a Kalman filter arranged to process implied measurement functions, at least one of the detection points is associated with the polygonal shape. Furthermore, the invention relates to a computer program product, a computer-readable storage medium and an assistance system.

It is already known from the prior art that, for object tracking, the objects are often predefined as a polygonal shape, with the assumption of the closest neighbor often being used here. Furthermore, a so-called probabilistic multi-hypothesis model (PMHT) is already known from the prior art, by means of which improved tracking of a dynamic object can be implemented.

the JP 2019152575 A discloses an object tracking apparatus composed of an object detection unit for detecting at least one distance measurement point representing the position of a tracking object present around the vehicle from distance measurement data obtained from a distance measurement unit mounted on a vehicle. Further, the object tracking device has an object tracking unit, so that the sum of the probabilities that the positions of distance measuring points in a target coordinate system represent those on a tracking object becomes maximum, the target coordinate system based on a tracking object as a reference point, which is based on the location of a tracking object is found in a prescribed period for each of the ranging points obtained in a last prescribed period, in accordance with a shape model representing the probability that a ranging point per position on the tracking object represents a ranging point, the most likely estimate of the probability on the site and the shape model by

the CN 109509207A discloses a method for seamlessly tracking a point target and an extended target belonging to the technical field of high resolution sensor target tracking. The method uses a variable dimension Gaussian process GP to model a variable contour of a target and each sensor resolution unit assumed by the target contour is recorded as a measurement source. The smaller the observed target is in relation to the sensor, the fewer the number of measurement sources on the target profiles and vice versa. The radius number of the GP model is adjusted online using the estimated value of the number of measurement sources. The method can adapt to the change in the appearance of the target and the mutual conversion between extended target (ET) and point target (PT), track a variety of extended targets and point targets. If the target is an ET, the ET-GP-PMHT tracks the position and appearance of the target; if the target is a PT, the ET-GP-PMHT only tracks the target's position and out.

the CN 108363054A belongs to the field of passive radar target tracking (PR) technology and specifically refers to a PR multi-target tracking method used for a single frequency network (SFN) and multipath propagation. The proposed tracking method takes a variety of measurements reaching a receptor through different propagation paths of different external radiation sources as possible target measurements in handling the correlation between measurements and targets, paths and external radiation sources, and correlates the measurements with multipath measurement functions of different bistatic pairs, so that a collection of target information is captured. The target tracking is then carried out in a kind of sliding window.

the DE 10 2018 104 311 A1 relates to a method for estimating the shape of a dynamic object and for tracking this object based on provided object data and an expectation maximization algorithm. It is provided that the shape of the object is approximated as a polygonal shape and each corner and/or each side of the polygonal shape by means of a Kalman filter, which is set up to process the implicit measurement function, a coordinate point required to describe the polygonal shape from the object data is assigned.

The object of the present invention is to create a method, a computer program product, a computer-readable storage medium and an assistance system, by means of which improved object tracking of a dynamic object can be implemented.

This object is achieved by a method, a computer program product, a computer-readable storage medium and an assistance system according to the independent patent claims. Advantageous embodiments are specified in the dependent claims.

One aspect of the invention relates to a method for operating an assistance system of a motor vehicle, in which an environment of the motor vehicle is detected by means of a detection device of the assistance system, the environment being detected by generating a large number of detection points as a point cloud by means of the detection device, and in which the point cloud is evaluated by means of an electronic computing device of the assistance system, with the evaluation being carried out on the basis of a mathematical model of the electronic computing device, and using the mathematical model to track an object in the area, with the object being approximated to the mathematical model as a polygonal shape and at least one of the detection points is assigned to the polygonal shape by means of a Kalman filter, which is set up for processing implicit measurement functions.

Provision is made for at least one of the detection points to be predefined as being assigned to at least one specific part of the polygonal shape and for the detection to be carried out on the basis of this predefined condition using the electronic computing device.

This makes it possible for improved object tracking of the dynamic object to be implemented in the area surrounding the motor vehicle.

In particular, a hypothesis of independent associations is thus removed from the mathematical model and replaced by associating at least one of the detected detection points with a specific part of the polygonal shape. In particular, the mathematical model thus makes use of the fact that the specific part of the polygonal shape of the object has a minimum number of measuring points.

In particular, it can thus be prevented that the hypothesis that states that all associations are independent of one another can lead to undesirable local maxima arising within the mathematical model. This hypothesis implies that all measurements, which correspond in particular to the detection points. can start from the same model, which leads to the behavior that all associations are assumed to be independent of each other. This hypothesis could result in all measurements being assigned to the wrong side of the object when a target is acquired from, say, the rear. Measurements from inside the object are also possible with laser scanners or lidar sensors, and the inside model can be used. The front corners of the object can also be detected from behind. For example, automobiles are not perfectly square, so echoes can bounce off the side surface, so the front corner models must also be used. If the target is accelerating and has just been initialized at a slightly slower speed, the associations for the first and last points will drift to the left and right line models and for the other points to the interior model. Finally, after a few time steps, the first and last points are associated with the front corners and the other points are associated either with the front line or, if the front line is obscured, with the interior model

Furthermore, a similar problem can also occur with radar when a target is detected from the side. The radar device can in principle acquire measurements anywhere on the target, and all models including the theoretically hidden sides must be used. Since radar generates a lot of clutter, i.e. noise, in the vicinity of the target, the probability of the clutter model for correct target tracking must be set quite high. Therefore, even if spots are detected inside the target, they can be associated with clutter and the target will remain "stuck on the wrong side".

Furthermore, each rectangle enclosing all discoveries can have a valid local maximum. All measurements can be assigned to the inner model and none or very few to the contour. This can be particularly problematic with radar since no prior association probabilities of the given sides and corners are set to zero. Estimated target may be too big.

Furthermore, a target can be evaluated as having grown too much, in which case it cannot shrink again. This can happen during initialization when the length uncertainty is large. It can also happen when some clutter measurements cause the target to grow too much.

All of these problems presented are caused by an incorrect choice of posterior association probabilities. In the first example, the problem is that the Kalman filter doesn't know that at least one measurement must come from each corner since the target was acquired from behind, or that at least one measurement must come from the back. The problem in the second example is the same, as some measurements must come from either the front and back left corners, or from the back. In the third and fourth examples, the mathematical model would converge to the correct estimate if it knew that at least one measurement must come from the front left and back right corners.

Therefore, according to the invention, it is now provided that the assumption that all associations are independent of one another is dropped and the knowledge that at least certain models receive one or more detection points is included. This constraint on the association can be active or inactive depending on the viewing angle at which the target is detected or depending on a distance of the object from the sensor. The constraint can also be associated with a certain probability of being valid or not.

According to an advantageous embodiment, the specific part of the polygonal shape is specified as a side of the polygonal shape and/or as a corner of the polygonal shape. In other words, the specific part can be both a corner of the polygonal shape and/or a side of the polygonal shape. For example, if the polygonal shape is specified as a rectangle, a side can be a front side, a left side, a right side and a rear side, for example. Furthermore, the polygonal shape then has four corners. By specifying that at least one of the detection points is assigned to the side and/or the corner, the dynamic object can be reliably tracked.

It is also advantageous if the specific part of the polygonal shape is specified as a function of a detection angle of the object. If, for example, it is determined that a specific side or corner of the polygonal shape is not “visible” using the detection device, a corresponding assignment can be suppressed. If it is then determined, for example, that only one specific side can currently be seen, the specification that one of the detection points is assigned to this side can result in the tracking of the object being able to be carried out in an improved manner.

Furthermore, it has proven to be advantageous if the polygonal shape is specified as a rectangular shape of the object. In particular, object tracking in motor vehicle construction is based on the fact that objects are often rectangular. For example, another motor vehicle in the area can be specified as a rectangular shape. This makes it possible for a large number of different objects, in particular motor vehicles, to be able to be tracked in an improved manner. Furthermore, a simple polygonal shape can thereby be used, as a result of which an assignment to a specific part can be designed very simply. In particular, object tracking can be carried out in a computationally efficient manner. In particular, the rectangular shape then has four sides and four corners. A detection point can then be assigned to at least one of the sides and/or one of the corners, as a result of which improved object tracking can be implemented.

Different detection points can preferably also be assigned to different sides and/or corners. Alternatively or additionally, multiple detection points can also be assigned to a single side and/or a single corner.

It is also advantageous if a probabilistic multi-hypothesis tracker is used as the mathematical model, taking into account the default conditions. The Probabilistic Multi-Hypothesis Tracker is also known as the Probabilistic Multi-Hypothesis Tracker (PMHT). In particular, the PMHT is very computationally efficient, with computational capacity increasing only linearly with the number of targets, while most prior art tracking algorithms increase exponentially with increasing number of targets. The tracking of the dynamic object can thus be implemented in a computationally efficient manner.

It is also advantageous if the recorded detection points are independent of one another in the probabilistic multi-hypothesis tracker, but at least one of the large number of detection points is taken into account with the default condition. In other words, it is provided in particular that a single detection point, for example, fulfills the default condition that it is formed, for example, at a corner or at a side as a specific part of the object. However, the other detection points then in turn have the property that they form independent associations from one another. It is thus possible for the object tracking to be carried out with a reduced computing effort and yet very reliably.

According to a further advantageous embodiment, a Murty algorithm is used by means of the probabilistic multi-hypothesis tracker for evaluating the point cloud, taking into account the default condition. The Murty algorithm is a generalization of the Hungarian algorithm. The Murty algorithm can be used to track the dynamic object with a reduced computing effort using the probabilistic multi-hypothesis tracker.

Alternatively, in a further preferred embodiment, it can be provided that a Gibbs random sample is used by means of the probabilistic multi-hypothesis tracker for evaluating the point cloud, taking into account the default condition. This is an algorithm for generating a sequence for samples of the joint probability distribution of two or more random variables. The goal is to approximate an unknown distribution. Gibbs sampling is particularly useful when the joint distribution of a random vector is unknown, but the conditional distribution of each random variable is known. The basic principle is to select a variable in a repetitive manner and, according to its conditional distribution, produce a value depending on the values of the other variables. The values of the other variables remain unchanged in this iteration step. A Markov chain, for example, can then be derived from the resulting sequence of sample vectors. It can then be shown that the stationary distribution of this Markov chain is precisely the sought-after joint distribution of the random vector.

Furthermore, it has proven to be advantageous if the probability of occurrence for the default condition is taken into account in the mathematical model. In particular, the default condition is introduced as a "soft" default condition, which can be referred to as a "soft constraint". This means that, for example, it can be provided that the probabilistic multi-hypothesis tracking model basically works with associations that are independent of one another if, for example, the probability of occurrence for a corresponding default condition is very low. However, if the probability of occurrence is high enough for the default condition, the default condition is introduced and the probabilistic multi-hypothesis tracker model is used using the default condition.

According to a further advantageous embodiment, the default condition is taken into account or not taken into account depending on a detection angle of the object. In other words, the use of the default condition can be activated or deactivated depending on the detection angle. If, for example, a corresponding detection and an assignment of a detection point to the specific part of the object should not be possible, the default condition can remain unconsidered. Reliable object tracking can thus be carried out.

In a further advantageous embodiment, the default condition is taken into account or not taken into account depending on a detected distance from the object. If, for example, the distance to the object is very large, then the default condition can remain unconsidered, for example. The background to this is that an assignment of detection points in the case of objects that are far away involves a great deal of effort and could possibly fail. Thus, wrong tracking can be prevented from being performed under the default condition. A robust mathematical model can thus be provided, by means of which reliable object tracking can be implemented.

Furthermore, it has proven to be advantageous if the environment is detected by means of a detection device designed as a lidar sensor device and/or by means of a radar sensor device and/or by means of an ultrasonic sensor device. It is thus possible for reliable object tracking to be implemented with different detection devices.

A further aspect of the invention relates to a computer program product with program code means, which are stored on a computer-readable medium, in order to carry out the method for generating a threshold value curve according to the preceding aspect, when the computer program product is processed on a processor of an electronic computing device.

Yet another aspect of the invention relates to a computer-readable storage medium with a computer program product according to the preceding aspect. The computer-readable storage medium can in particular be designed as part of an electronic computing device.

A further aspect of the invention relates to an assistance system for a motor vehicle, with at least one detection device and with an electronic computing device, the assistance system being designed to carry out a method according to the preceding aspect. In particular, the method is carried out using the assistance system.

Yet another aspect of the invention relates to a motor vehicle with an assistance system. The motor vehicle is designed in particular as a passenger car.

Advantageous configurations of the method are to be regarded as advantageous configurations of the computer program product, the computer-readable storage medium, the assistance system and the motor vehicle. For this purpose, the assistance system and the motor vehicle have specific features which enable the method to be carried out or an advantageous embodiment thereof.

Further features of the invention result from the claims, the figures and the description of the figures. The features and combinations of features mentioned above in the description, as well as the features and combinations of features mentioned below in the description of the figures and/or shown alone in the figures, can be used not only in the combination specified in each case, but also in other combinations, without going beyond the scope of the invention leaving. The invention is therefore also to be considered to include and disclose embodiments that are not explicitly shown and explained in the figures, but that result from the explained embodiments and can be generated by separate combinations of features. Versions and combinations of features are also to be regarded as disclosed which therefore do not have all the features of an originally formulated independent claim. Furthermore, embodiments and combinations of features, in particular through the embodiments presented above, are to be regarded as disclosed which go beyond or deviate from the combinations of features presented in the back references of the claims.

The invention will now be explained in more detail using preferred exemplary embodiments and with reference to the accompanying drawings.

• 1 eine schematische Draufsicht auf eine Ausführungsform eines Kraftfahrzeugs mit einer Ausführungsform eines Assistenzsystems;
• 2 eine schematische Draufsicht auf ein erfasstes Objekt mittels des erfindungsgemäßen Verfahrens;
• 3 eine weitere schematische Draufsicht auf ein erfasstes Objekt gemäß dem Verfahren;
• 4 eine weitere schematische Draufsicht auf ein erfasstes Objekts mittels des Verfahrens; und
• 5 eine weitere schematische Draufsicht auf ein erfasstes Objekt mittels des Verfahrens.

show:

• 1 a schematic plan view of an embodiment of a motor vehicle with an embodiment of an assistance system;
• 2 a schematic plan view of a detected object by means of the method according to the invention;
• 3 a further schematic plan view of a detected object according to the method;
• 4 a further schematic plan view of a detected object by means of the method; and
• 5 a further schematic plan view of an object detected by means of the method.

Elements that are the same or have the same function are provided with the same reference symbols in the figures.

1 1 shows an embodiment of a motor vehicle 1 with an embodiment of an assistance system 2 in a schematic top view. The assistance system 2 has at least one electronic computing device 3 . The electronic computing device 3 in turn has, in particular, a computer-readable storage medium 4 . The assistance system 2 also has a detection device 5 . The detection device 5 can be embodied, for example, as a lidar sensor device and/or as a radar sensor device and/or as an ultrasonic sensor device. The detection device 5 is designed to detect an environment 6 of the motor vehicle 1 . In particular, an object 7 , which is dynamic in particular, can be detected in the environment 6 by means of the detection device 5 . At the Dyna mix object 7 can be, for example, another motor vehicle, which is in particular in motion.

Motor vehicle 1 is, in particular, an at least partially autonomously operated motor vehicle, assistance system 2 being designed at least to support at least partially autonomous operation. The motor vehicle 1 can also be designed as a fully autonomous motor vehicle 1 .

In a method for operating the assistance system 2 of the motor vehicle 1, the surroundings 6 of the motor vehicle 1 are detected by means of the detection device 5 of the assistance system 2, the surroundings 6 being detected by generating a large number of detection points 8 ( 2 ) a cloud of points is recorded by means of the detection device 5, and in which the cloud of points is evaluated by means of the electronic computing device 3 of the assistance system 2, the evaluation being carried out on the basis of a mathematical model 9 of the electronic computing device 3, and tracking using the mathematical model 9 of the object 7 in the environment 6 is performed, the object 7 as a polygonal shape 12 ( 2 ) is approximated in the mathematical model 9 and at least one of the detection points 8 is assigned to the polygonal shape 12 by means of a Kalman filter which is set up for processing implicit measurement functions.

Provision is made for at least one of the detection points 8 to be predefined as assigned to at least one specific part of the polygonal shape 12 and for the detection to be carried out on the basis of this predefined condition using the electronic computing device 3 .

In particular, it can be provided that the specific part of the polygonal shape 12 is defined as a side 10 ( 2 ) and/or as a corner 11 ( 2 ) of the polygonal shape 12 is specified.

Furthermore, it can be provided that the specific part of the polygonal shape 12 is specified as a function of a detection angle of the object 7 . Provision can also be made for the polygonal shape 12 to be specified as a rectangular shape of the object 7, which is particularly the case in FIGS 2 until 5 is shown.

Provision can furthermore be made for the default condition to be taken into account or not taken into account depending on a detection angle of the object 7 and/or for the default condition to be taken into account or not taken into account depending on a detected distance to the object 7 .

The Probabilistic Multi-Hypotheses Tracker (PMHT) relies on the following assumptions. The PMHT is used to find out from which part/side 10 of the object 7 a measurement, i.e. a detection point 8, originated, although it was originally developed to find out from which target, i.e. object 7, a measurement came from. Both problems are actually identical since the association information required to build the measurement function or innovation function is sought. The notion of "which goal" can therefore be interchanged with the notion of "which side of a goal" or even with "which side of which goal".

It is assumed that the measurements and innovation vectors are independent of each other and do not depend on the state from another time step. It is further assumed that the object motion model is a Markov process, which means that _{xk +1} depends only on xk. Furthermore, it is assumed that the associations of measurements on a side 10 of the object 7 are independent of one another. In other words, it means that each side 10 of the object 7 can have any number of measurements generated. Another assumption is that the associations are state independent. If they depend on the measurements (or the innovation), this is no longer the case, but knowledge of the state alone does not tell about the association probabilities. The measurements should also be normally distributed and the innovation model should be linear (or linearized). The motion model is also assumed to be linear (or linearized) with additive normal noise uncorrelated with state. The previous PDF (probability density function) of the state of the first time step should also be normally distributed.

Our state X contains the position of the tracked object over all time steps of the batch: $$X={\left[\begin{array}{cc}{x}_1& x_T\end{array}\right]}^T$$

The measuring vector Z contains the measured values of the recording device 5 over all time steps of the batch: $$Z={\left[\begin{array}{cccccc}{\mathrm{e.g}}_1^1& {\mathrm{e.g}}_1^{n\left(1\right)}& {\mathrm{e.g}}_2^1& {\mathrm{e.g}}_2^{n\left(2\right)}& {\mathrm{e.g}}_1^Z& {\mathrm{e.g}}_T^{n\left(T\right)}\end{array}\right]}^T$$

The unobserved random variable K is the assignment of each measurement to a specific side 10 of the object 7, since it is not known from which side 11 of the object 7 each measurement originates. For a specific side i of the polygon, the innovation model is: $$v_{i,t}^{\mathrm{right}}=H_i\left(x_t,{\mathrm{e.g}}_t^{\mathrm{right}}\right)$$

The innovation model 1.3 is a generic representation for each instance of innovation to a vertex or side of the polygon. An instance of K represents an association of each point with a side or vertex of the polygon at each measurement time step, noted as: $$\forall t\in \mathrm{1,}T,\forall \mathrm{right}\in \mathrm{1,}n\left(t\right),k_{\mathrm{right}}\left(t\right)$$

If the polygon has m sides or vertices, the sum is defined over all possible associations: $${\displaystyle {\sum }_K(\cdot )}={\displaystyle {\sum }_{k_1\left(1\right)=1}^m{\displaystyle {\sum }_{k_2\left(1\right)=1}^m{\displaystyle {\sum }_{k_{n\left(t\right)}\left(1\right)=1}^m{\displaystyle {\sum }_{k_1\left(2\right)=1}^m{\displaystyle {\sum }_{k_{n\left(t\right)}\left(2\right)=1}^m{\displaystyle {\sum }_{k_1\left(T\right)=1}^m{\displaystyle {\sum }_{k_{n\left(t\right)}\left(T\right)=1}^m(\cdot )}}}}}}}$$

Expectation maximization as previously derived is used to find the MAP (maximum a posteriori). The index of the iteration is noted in the exponent since the lower index is already used for the time step index. The posterior for the argument X is maximized: $$X_{\text{Max}}=\underset{X}{\text{argmax}}\left\{P\left(X/N=0\right)\right\}=\underset{X}{\text{argmax}}\left\{P\left(N=0/X\right)P\left(X\right)\right\}$$

, womit sich notwendigerweise 1.6 erhöht.The PMHT uses expectation maximization: $$Q\left(X,X^n\right)={\displaystyle {\sum }_{K}P\left(K/X^n,N=0\right)\text{log}P\left(X,K,N=0\right)}$$

, which necessarily increases 1.6.

mit π_i(t) die früheren Assoziationswahrscheinlichkeiten zu jedem Modell i (das bekannt sein soll oder mit Hilfe der Erwartungsmaximierung geschätzt werden soll, siehe unten) und: $$\begin{array}{l}{\omega }_{i,r}^n\left(t\right)=P\left(k_r\left(t\right)=i/x_t^n,N=\mathrm{0,}n\left(t\right)\right)\\ \propto P\left(v_{i,t}^r=0/x_t^n,k_r\left(t\right)=i,n\left(t\right)\right)P\left(k_r\left(t\right)=i/x_t^n,n\left(t\right)\right)={\lambda }_t^{r,n}\left(i\right){\pi }_i\left(t\right)\\ {\omega }_{i,r}^n\left(t\right)=\frac{{\lambda }_t^{r,n}\left(i\right){\pi }_i\left(t\right)}{{\displaystyle {\sum }_{j=1}^m{\lambda }_t^{r,n}\left(j\right){\pi }_j\left(t\right)}}\end{array}$$

The standard derivation of the PMHT shows that 1.7 is equivalent to: $$\begin{array}{l}Q\left(X,X^n\right)=\text{log}\left(P\left(x_1\right){\displaystyle {\prod }_{k=1}^{T-1}P\left(x_{k+1}/x_k\right)}\right)+{\displaystyle {\sum }_{t=1}^T{\displaystyle {\sum }_{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle {\sum }_{i=1}^m{\omega }_{i,\mathrm{right}}^n}\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)}}\\ +{\displaystyle {\sum }_{t=1}^T{\displaystyle {\sum }_{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle {\sum }_{i=1}^m{\omega }_{i,\mathrm{right}}^n}\left(t\right)\text{log}{\pi }_i\left(t\right)}}\end{array}$$

with π _i (t) the previous association probabilities for each model i (which should be known or should be estimated using expectation maximization, see below) and: $$\begin{array}{l}{\omega }_{i,\mathrm{right}}^n\left(t\right)=P\left(k_{\mathrm{right}}\left(t\right)=i/x_t^n,N=\mathrm{0,}n\left(t\right)\right)\\ \propto P\left(v_{i,t}^{\mathrm{right}}=0/x_t^n,k_{\mathrm{right}}\left(t\right)=i,n\left(t\right)\right)P\left(k_{\mathrm{right}}\left(t\right)=i/x_t^n,n\left(t\right)\right)={\lambda }_t^{\mathrm{right},n}\left(i\right){\pi }_i\left(t\right)\\ {\omega }_{i,\mathrm{right}}^n\left(t\right)=\frac{{\lambda }_t^{\mathrm{right},n}\left(i\right){\pi }_i\left(t\right)}{{\displaystyle {\sum }_{j=1}^m{\lambda }_t^{\mathrm{right},n}\left(j\right){\pi }_j\left(t\right)}}\end{array}$$

können für jede Art von Wahrscheinlichkeitsfunktionen ${\lambda }_t^{r,n}\left(i\right)$

berechnet werden. Im Sonderfall der linearisierten Innovationsfunktionen um die aktuelle iterierte Schätzung herum, wie diese mit Xⁿ bedingen, sind die Wahrscheinlichkeiten normalerweise wie folgt verteilt: $$\begin{array}{l}{\lambda }_t^{r,n}\left(i\right)=P\left(v_{i,t}^r=0/x_t^n,k_r\left(t\right)=i,n\left(t\right)\right)\\ =N\left\{0;h_{k_r\left(t\right)}\left(x_t^n,{\widehat{z}}_t^r\right)+H_{t,k_r\left(t\right)}^{r,n}\left(x_t^n-x_t^n\right);Q_{t,k_r\left(t\right)}^{r,n}\right\}\\ =N\left\{0;h_{k_r\left(t\right)}\left(x_t^n,{\widehat{z}}_t^r\right);Q_{t,k_r\left(t\right)}^{r,n}\right\}\\ Q_{t,k_r\left(t\right)}^{r,n}=J_{t,k_r\left(t\right)}^{r,n}{R}_t^r{J}_{t,k_r\left(t\right)}^{r,n^T}\\ J_{t,k_r\left(t\right)}^{r,n}=\frac{\partial h_{k_r\left(t\right)}}{\partial z}\left(x_t^n,{\widehat{z}}_t^r\right)\end{array}$$

The posterior association probabilities ${\omega }_{i,\mathrm{right}}^n\left(t\right)$

can for any kind of probability functions ${\lambda }_t^{\mathrm{right},n}\left(i\right)$

be calculated. In the special case of linearized innovation functions around the current iterated estimate, such as those conditional with X ⁿ , the probabilities are usually distributed as follows: $$\begin{array}{l}{\lambda }_t^{\mathrm{right},n}\left(i\right)=P\left(v_{i,t}^{\mathrm{right}}=0/x_t^n,k_{\mathrm{right}}\left(t\right)=i,n\left(t\right)\right)\\ =N\left\{0;H_{k_{\mathrm{right}}\left(t\right)}\left(x_t^n,{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}\left(x_t^n-x_t^n\right);Q_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}\right\}\\ =N\left\{0;H_{k_{\mathrm{right}}\left(t\right)}\left(x_t^n,{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right);Q_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}\right\}\\ Q_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}=J_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}{R}_t^{\mathrm{right}}{J}_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n^T}\\ J_{t,k_{\mathrm{right}}\left(t\right)}^{\mathrm{right},n}=\frac{\partial H_{k_{\mathrm{right}}\left(t\right)}}{\partial \mathrm{e.g}}\left(x_t^n,{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)\end{array}$$

The formulas for the probabilities are now rewritten: $$\begin{array}{l}{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)\\ ={\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}N\left\{0;H_i\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{t/t-1}\left(n\right)\right);Q_{t,i}^{\mathrm{right}}\left(n\right)\right\}\\ =-\frac{{\omega }_{i,\mathrm{right}}^n\left(t\right)}{2}{\left(H_i\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{t/t-1}\left(n\right)\right)\right)}^T{\left(Q_{t,i}^{\mathrm{right}}\left(n\right)\right)}^{-1}\\ \times \left(H_i\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{t/t-1}\left(n\right)\right)\right)\\ =-\frac{1}{2}{\left(H_i\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{t/t-1}\left(n\right)\right)\right)}^T{\left({\tilde{Q}}_{t,i}^{\mathrm{right},n}\left(n\right)\right)}^{-1}\\ \times \left(H_i\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{t/t-1}\left(n\right)\right)\right)\\ {\tilde{Q}}_{t,i}^{\mathrm{right},n}\left(n\right)=\frac{Q_{t,i}^{\mathrm{right}}\left(n\right)}{{\omega }_{i,\mathrm{right}}^n\left(t\right)}=J_{t,i}^{\mathrm{right}}\left(n\right)\frac{R_t^{\mathrm{right}}}{{\omega }_{i,\mathrm{right}}^n\left(t\right)}{J}_{t,i}^{{\mathrm{right}}^T}\left(n\right)=J_{t,i}^{\mathrm{right}}\left(n\right){\tilde{R}}_{t,i}^{\mathrm{right},n}{J}_{t,i}^{{\mathrm{right}}^T}\left(n\right)\end{array}$$

By combining 1.11 and 1.8, Q can be further simplified: $$\begin{array}{l}Q\left(X,X^n\right)=\text{log}\left(P\left(x_1\right){\displaystyle \prod _{k=1}^{T-1}P\left(\frac{x_{k+1}}{x_k}\right)}\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m\text{log}{\tilde{\lambda }}_{t,i}^{\mathrm{right},n}\left(i,n\right)}}}\\ +{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}\\ {\tilde{\lambda }}_{t,i}^{\mathrm{right},n}\left(i,n\right)=\text{log}N\left\{0;H_i\left({\widehat{x}}_{\frac{t}{t}-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}\right)+H_{t,i}^{\mathrm{right}}\left(n\right)\left(x_t-{\widehat{x}}_{\frac{t}{t}-1}\left(n\right)\right);{\tilde{Q}}_{t,i}^{\mathrm{right},n}\left(n\right)\right\}\end{array}$$

This is equivalent to: $$Q\left(X,X^n\right)=\text{log}\left(P\left(X,\tilde{N}\left(n\right)=0\right)\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}$$

Where Ñ(n) is a synthetic innovation vector without association uncertainty, such as: $$\tilde{N}\left(n\right)={\left[\begin{array}{cccc}{v}_{1.1}^{\mathrm{1,}n}\left(n\right)& {\mathrm{<figure-callout\; id="1"\; label="v"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">v</figure-callout>}}_{\mathrm{1,}m}^{n\left(1\right),n}\left(n\right)& v_{T\mathrm{,1}}^{\mathrm{1,}n}\left(n\right)& v_{T,m}^{n\left(T\right),n}\left(n\right)\end{array}\right]}^T$$

um die Zustandsvorhersageerwartung ${\widehat{x}}_{t/t-1}\left(n\right)$

linearisiert wird und die synthetische Messung ${\tilde{z}}_{t,i}^{r,n}$

der Kovarianz ${\tilde{R}}_{t,i}^{r,n}$

verwendet (diese Messung wird synthetisch genannt, da ihre Kovarianz aufgeblasen wurde). Diese Messungen sind natürlich nicht miteinander korreliert (Annahme 1), so wie angenommen wird, dass die gemeinsame Kovarianz der synthetischen Messungen diagonal ist: $${\tilde{R}}^n=\text{diag}\left({\tilde{R}}_{t,i}^{r,n}\right){}_{\begin{array}{c}t=1T\\ i=1m\\ r=1n\left(t\right)\end{array}}$$

Where every innovation $v_{t,i}^{\mathrm{right},n}\left(n\right)=H_i\left(x_t,{\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}\right)$

about the state prediction expectation ${\widehat{x}}_{t/t-1}\left(n\right)$

is linearized and the synthetic measurement ${\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}$

the covariance ${\tilde{R}}_{t,i}^{\mathrm{right},n}$

used (this measurement is called synthetic because its covariance has been inflated). These measurements are of course uncorrelated with each other (Assumption 1), as the common covariance of the synthetic measurements is assumed to be diagonal: $${\tilde{R}}^n=\text{diag}\left({\tilde{R}}_{t,i}^{\mathrm{right},n}\right){}_{\begin{array}{c}t=1T\\ i=1m\\ \mathrm{right}=1n\left(t\right)\end{array}}$$

If the synthetic measurement vector for each time step is defined as: $${\widetilde{\mathrm{e.g}}}_t^n={\left[\begin{array}{cccc}{\widetilde{\mathrm{e.g}}}_{t\mathrm{,1}}^{\mathrm{1,}n}& {\widetilde{\mathrm{e.g}}}_{t,\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{1,}n}& {\widetilde{\mathrm{e.g}}}_{t\mathrm{,1}}^{n\left(t\right),n}& {\widetilde{\mathrm{e.g}}}_{t,m}^{n\left(t\right),n}\end{array}\right]}^T$$

Then the innovation function for this synthetic measurement vector (and for the sake of simplicity Ñ(n)=Ñ is noted): $$\tilde{N}=\tilde{H}\left(X,\tilde{Z}\right)=\left[\begin{array}{l}\begin{array}{l}{H}_1\left(x_1,{\widetilde{\mathrm{e.g}}}_{1.1}^{\mathrm{1,}n}\right)\hfill \\ ⋮\hfill \\ H_m\left(x_1,{\widetilde{\mathrm{e.g}}}_{\mathrm{1,}\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{1,}n}\right)\hfill \\ ⋮\hfill \\ H_1\left(x_1,{\widetilde{\mathrm{e.g}}}_{1.1}^{n\left(1\right),n}\right)\hfill \\ ⋮\hfill \\ H_m\left(x_1,{\widetilde{\mathrm{e.g}}}_{\mathrm{1,}m}^{n\left(1\right),n}\right)\hfill \end{array}\}m\times n\left(1\right)\hfill \\ ⋮\hfill \\ \begin{array}{l}{H}_1\left(x_T,{\widetilde{\mathrm{e.g}}}_{T\mathrm{,1}}^{\mathrm{1,}n}\right)\hfill \\ ⋮\hfill \\ H_m\left(x_T,{\widetilde{\mathrm{e.g}}}_{T,\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{1,}n}\right)\hfill \\ ⋮\hfill \\ H_1\left(x_T,{\widetilde{\mathrm{e.g}}}_{T\mathrm{,1}}^{n\left(T\right),n}\right)\hfill \\ ⋮\hfill \\ H_m\left(x_T,{\widetilde{\mathrm{e.g}}}_{T,m}^{n\left(T\right),n}\right)\hfill \end{array}\}m\times n\left(T\right)\hfill \end{array}\right]=\left[\begin{array}{l}{\tilde{H}}_1\left(x_1,{\widetilde{\mathrm{e.g}}}_1^n\right)\hfill \\ ⋮\hfill \\ {\tilde{H}}_T\left(x_T,{\widetilde{\mathrm{e.g}}}_T^n\right)\hfill \end{array}\right]$$

Optimizing Q is therefore equivalent to optimizing log(P(X, Ñ = 0)), which is equivalent to optimizing P(X, Ñ = 0), which is equivalent to optimizing P(X/Ñ = 0) is since the previous P(Ñ = 0) is independent of X.

ist ebenfalls normalverteilt, da die vorherige Transformation gemäß Annahme 6 linear verläuft. Daher ist pro Induktion der des Messvektors X = [x₁ x_T]^T normalverteilt. Gemäß Annahme 5 ist P(Ñ/X) ebenfalls normalverteilt, und zwar nach der Bayes-Regel: $$P\left(\frac{X}{\tilde{N}}=0\right)\propto P\left(\tilde{N}=\frac{0}{X}\right)P\left(X\right)$$

This is exactly what a Kalman filter would do since it is able to estimate the mean and covariance of any normal distribution P(x _t /Ñ = 0). P(x _t /Ñ = 0) is a normal distribution since, by Assumption 7, x ₁ is assumed to be normally distributed, and therefore: $$x_1\mapsto \left[\begin{array}{l}{x}_1\hfill \\ x_2=f_1{x}_1+q_1\hfill \end{array}\right]$$

is also normally distributed since the previous transformation is linear according to Assumption 6. Therefore, per induction, that of the measurement vector X = [x ₁ x _T ] ^T is normally distributed. According to Assumption 5, P(Ñ/X) is also normally distributed, according to Bayes' rule: $$P\left(\frac{X}{\tilde{N}}=0\right)\propto P\left(\tilde{N}=\frac{0}{X}\right)P\left(X\right)$$

This is also a normal distribution, as shown in the derivation of the Kalman filter.

The maximum of P(X/Ñ=0) is therefore the mean of P(X/Ñ=0), which is the collection of the mean of each P(x _t /Ñ=0), which is the result of Kalman smoothing . The result is: $$\begin{array}{l}{X}^{n+1}={\left[\begin{array}{cc}{x}_1^{n+1}& x_T^{n+1}\end{array}\right]}^T=\underset{X}{\text{argmax}Q}\left(X,X^n\right)\\ =\mathbb{E}\left[X/\tilde{N}=0\right]={\left[\begin{array}{cc}{\widehat{x}}_1\left(n\right)& {\widehat{x}}_T\left(n\right)\end{array}\right]}^T=X{\left(n\right)}^T\end{array}$$

gleich der Erwartung der nicht synthetischen Messung ${\widehat{z}}_t^r$

ist, da nur die Kovarianz aufgeblasen wird: $$\begin{array}{l}{P}_t\left(n\right)={\left({\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)+P_{t/t-1}^{-1}\left(n\right)\right)}^{-1}\\ K_t\left(n\right)=-P_t\left(n\right){\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}\\ {\tilde{R}}_t^n=\text{Diag}{\left({\tilde{R}}_{t,i}^{r,n}\right)}_{t=1mr=1n\left(t\right)}\\ {\tilde{H}}_t\left(n\right)=\frac{\partial {\tilde{h}}_t}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\tilde{z}}}_t^n\right)=\left[\begin{array}{l}\frac{\partial h_1}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_m}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_1}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^{n\left(t\right)}\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_m}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^{n\left(t\right)}\right)\hfill \end{array}\right]=\left[\begin{array}{l}{H}_{t\mathrm{,1}}^1\left(n\right)\hfill \\ ⋮\hfill \\ H_{t,m}^1\left(n\right)\hfill \\ ⋮\hfill \\ H_{t\mathrm{,1}}^{n\left(t\right)}\left(n\right)\hfill \\ ⋮\hfill \\ H_{t,m}^{n\left(t\right)}\left(n\right)\hfill \end{array}\right]\\ {\tilde{J}}_t\left(n\right)=\frac{\partial {\tilde{h}}_t}{\partial {\tilde{z}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\tilde{z}}}_t^n\right)=\left[\begin{array}{l}\frac{\partial h_1}{\partial {\tilde{z}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_m}{\partial {\tilde{z}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_1}{\partial {\tilde{z}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^{n\left(t\right)}\right)\hfill \\ ⋮\hfill \\ \frac{\partial h_m}{\partial {\tilde{z}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{z}}_t^{n\left(t\right)}\right)\hfill \end{array}\right]=\left[\begin{array}{l}{\tilde{J}}_{t\mathrm{,1}}^1\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t,m}^1\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t\mathrm{,1}}^{n\left(t\right)}\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t,m}^{n\left(t\right)}\left(n\right)\hfill \end{array}\right]\end{array}$$

Nevertheless, running a Kalman filter on the synthetic measurement Z will be cumbersome since the number of measurements has been increased by a factor of m. A method is therefore needed to efficiently perform Kalman measurement updating for a large number of measurements. In fact, updating the Kalman measurement usually needs to reverse the covariance of the innovation. It is possible to write the Kalman gain and the updated state covariance matrix in the information form for each measured value update in each time step t using the inversion Lemma. We also use the fact that the expectation is a synthetic measurement ${\widehat{\widetilde{\mathrm{e.g}}}}_{t,i}^{\mathrm{right},n}$

equal to the expectation of non-synthetic measurement ${\widehat{\mathrm{e.g}}}_t^{\mathrm{right}}$

is, since only the covariance is inflated: $$\begin{array}{l}{P}_t\left(n\right)={\left({\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)+P_{t/t-1}^{-1}\left(n\right)\right)}^{-1}\\ K_t\left(n\right)=-P_t\left(n\right){\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}\\ {\tilde{R}}_t^n=\text{Diag}{\left({\tilde{R}}_{t,i}^{\mathrm{right},n}\right)}_{t=1m\mathrm{right}=1n\left(t\right)}\\ {\tilde{H}}_t\left(n\right)=\frac{\partial {\tilde{H}}_t}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\widetilde{\mathrm{e.g}}}}_t^n\right)=\left[\begin{array}{l}\frac{\partial H_1}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_m}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_1}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{n\left(t\right)}\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_m}{\partial x_t}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{n\left(t\right)}\right)\hfill \end{array}\right]=\left[\begin{array}{l}{H}_{t\mathrm{,1}}^1\left(n\right)\hfill \\ ⋮\hfill \\ H_{t,m}^1\left(n\right)\hfill \\ ⋮\hfill \\ H_{t\mathrm{,1}}^{n\left(t\right)}\left(n\right)\hfill \\ ⋮\hfill \\ H_{t,m}^{n\left(t\right)}\left(n\right)\hfill \end{array}\right]\\ {\tilde{J}}_t\left(n\right)=\frac{\partial {\tilde{H}}_t}{\partial {\widetilde{\mathrm{e.g}}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\widetilde{\mathrm{e.g}}}}_t^n\right)=\left[\begin{array}{l}\frac{\partial H_1}{\partial {\widetilde{\mathrm{e.g}}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_m}{\partial {\widetilde{\mathrm{e.g}}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^1\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_1}{\partial {\widetilde{\mathrm{e.g}}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{n\left(t\right)}\right)\hfill \\ ⋮\hfill \\ \frac{\partial H_m}{\partial {\widetilde{\mathrm{e.g}}}_t^n}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widehat{\mathrm{e.g}}}_t^{n\left(t\right)}\right)\hfill \end{array}\right]=\left[\begin{array}{l}{\tilde{J}}_{t\mathrm{,1}}^1\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t,m}^1\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t\mathrm{,1}}^{n\left(t\right)}\left(n\right)\hfill \\ ⋮\hfill \\ {\tilde{J}}_{t,m}^{n\left(t\right)}\left(n\right)\hfill \end{array}\right]\end{array}$$

nur von ${\tilde{z}}_{t,i}^{r,n}$

und nicht von den anderen Komponenten von ${\tilde{z}}_t^n$

abhängt, hat jede Submatrix ${\tilde{J}}_{t,i}^r\left(n\right)$

die Form: $$\begin{array}{l}{\tilde{J}}_{t,i}^r\left(n\right)=\left[\begin{array}{ccc}\underset{p_{t,i}^r}{\underbrace{\begin{array}{cc}0& 0\end{array}}}& \frac{\partial h_i}{\partial {\tilde{z}}_{t,i}^{r,n}}\left({\widehat{x}}_{t/t-1}\left(n\right),{\tilde{z}}_{t,i}^{r,n}\right)& \underset{q_{t,i}^r}{\underbrace{\begin{array}{cc}0& 0\end{array}}}\end{array}\right]\\ =\left[\begin{array}{ccc}\underset{p_{t,i}^r}{\underbrace{\begin{array}{cc}0& 0\end{array}}}& J_{t,i}^r\left(n\right)& \underset{q_{t,i}^r}{\underbrace{\begin{array}{cc}0& 0\end{array}}}\end{array}\right]\end{array}$$

wobei $p_{t,i}^r$

und $q_{t,i}^r$

die Anzahl der Variablen vor bzw. nach ${\tilde{z}}_{t,i}^{r,n}$

in dem synthetischen Meßvektor ${\tilde{z}}_t^r$

sind. Auch g ${\tilde{J}}_{t,i}^r\left(n\right)$

hat eine Anzahl von Zeilen, die der Dimension der Innovation h_i entspricht (dies hängt von i ab).The inversion necessary to calculate P _t (n) has the dimension of state, which is small compared to the dimension of innovation. Also due to the fact that $H_i\left(x_t,{\widetilde{\mathrm{e.g}}}_t^n\right)$

only from ${\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}$

and not by the other components of ${\widetilde{\mathrm{e.g}}}_t^n$

depends, each submatrix has ${\tilde{J}}_{t,i}^{\mathrm{right}}\left(n\right)$

form: $$\begin{array}{l}{\tilde{J}}_{t,i}^{\mathrm{right}}\left(n\right)=\left[\begin{array}{ccc}\underset{p_{t,i}^{\mathrm{right}}}{\underbrace{\begin{array}{cc}0& 0\end{array}}}& \frac{\partial H_i}{\partial {\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}}\left({\widehat{x}}_{t/t-1}\left(n\right),{\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}\right)& \underset{q_{t,i}^{\mathrm{right}}}{\underbrace{\begin{array}{cc}0& 0\end{array}}}\end{array}\right]\\ =\left[\begin{array}{ccc}\underset{p_{t,i}^{\mathrm{right}}}{\underbrace{\begin{array}{cc}0& 0\end{array}}}& J_{t,i}^{\mathrm{right}}\left(n\right)& \underset{q_{t,i}^{\mathrm{right}}}{\underbrace{\begin{array}{cc}0& 0\end{array}}}\end{array}\right]\end{array}$$

whereby $p_{t,i}^{\mathrm{right}}$

and $q_{t,i}^{\mathrm{right}}$

the number of variables before or after ${\widetilde{\mathrm{e.g}}}_{t,i}^{\mathrm{right},n}$

in the synthetic measurement vector ${\widetilde{\mathrm{e.g}}}_t^{\mathrm{right}}$

are. also g ${\tilde{J}}_{t,i}^{\mathrm{right}}\left(n\right)$

has a number of rows equal to the innovation dimension h _i (this depends on i).

ist folglich blockdiagonal und ihre Umkehrung ist kostengünstig und der Begriff ${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)$

ist ebenfalls kostengünstig zu berechnen, da es sich um die einfache Summe handelt: $${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)={\displaystyle \sum _{r=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{H}_{t,i}^{r^T}\left(n\right){\left(J_{t,i}^r\left(n\right){\tilde{R}}_{t,i}^{r,n}{J}_{t,i}^{r^T}\left(n\right)\right)}^{-1}{H}_{t,i}^r\left(n\right)}}$$

The Matrix ${\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)$

is consequently block diagonal and its inverse is cost effective and the term ${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)$

can also be calculated inexpensively, since it is the simple sum: $${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}{\tilde{H}}_t\left(n\right)={\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{H}_{t,i}^{{\mathrm{right}}^T}\left(n\right){\left(J_{t,i}^{\mathrm{right}}\left(n\right){\tilde{R}}_{t,i}^{\mathrm{right},n}{J}_{t,i}^{{\mathrm{right}}^T}\left(n\right)\right)}^{-1}{H}_{t,i}^{\mathrm{right}}\left(n\right)}}$$

blockdiagonal ist, ergibt sich: $${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}={\left[H_{t,i}^{r^T}\left(n\right){\left(J_{t,i}^r\left(n\right){\tilde{R}}_{t,i}^{r,n}{J}_{t,i}^{rT}\left(n\right)\right)}^{-1}\right]}_{\begin{array}{c}t=1T\\ i=1m\\ r=1n\left(t\right)\end{array}}$$

Similarly, due to the fact that ${\tilde{J}}_T\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)$

is block diagonal, we get: $${\tilde{H}}_t^T\left(n\right){\left({\tilde{J}}_t\left(n\right){\tilde{R}}_t^n{\tilde{J}}_t^T\left(n\right)\right)}^{-1}={\left[H_{t,i}^{{\mathrm{right}}^T}\left(n\right){\left(J_{t,i}^{\mathrm{right}}\left(n\right){\tilde{R}}_{t,i}^{\mathrm{right},n}{J}_{t,i}^{\mathrm{right}T}\left(n\right)\right)}^{-1}\right]}_{\begin{array}{c}t=1T\\ i=1m\\ \mathrm{right}=1n\left(t\right)\end{array}}$$

It is also possible to update the previous probabilities using expectation maximization if they are not exactly known. An actual estimate of π is assumed at the nth iteration, denoted π ⁿ . The function can therefore be rewritten in such a way that it is optimized: $$Q\left(X,\pi ,X^n\right){\displaystyle \sum _{k}P\left(\frac{K}{X^n},{\pi }^n,N=0\right)}\text{log}P\left(X,K,N=0\right)$$

wobei ${\omega }_{i,r}^n\left(t\right)$

unter Verwendung von πⁿ berechnet wird. Es kann als 1.26 aufgeteilt werden in: $$\begin{array}{l}Q\left(X,\pi ,X^n,{\pi }^n\right)=Q_X\left(X,X^n,{\pi }^n\right)+{\displaystyle \sum _{t=1}^T{Q}_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)}\hfill \\ =Q_X\left(X,X^n,{\pi }^n\right)+Q_{\pi }\left(\pi ,N^n,{\pi }^n\right)\hfill \\ Q_X\left(X,X^n,{\pi }^n\right)=\text{log}\left(P\left(X,\tilde{N}=0\right)\right)\hfill \\ Q_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)={\displaystyle \sum _{r=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,r}^n}\left(t\right)\text{log}{\pi }_i\left(i\right)}\hfill \\ Q_{\pi }\left(\pi ,X^n,{\pi }^n\right)={\displaystyle \sum _{t=1}^T{Q}_{t,\pi }\left(\pi \left(t\right),X^n\right)}\hfill \end{array}$$

Q is a function of π because it is implicit in P(X, K, N = 0). Identical to what has already been done, Q(X, π, X ⁿ ) can be rewritten such that: $$Q\left(X,\pi \text{,}{X}^n,{\pi }^n\right)=\text{log}\left(P\left(X,\tilde{N}=0\right)\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}$$

whereby ${\omega }_{i,\mathrm{right}}^n\left(t\right)$

is calculated using π ⁿ . It can be split as 1.26 into: $$\begin{array}{l}Q\left(X,\pi ,X^n,{\pi }^n\right)=Q_X\left(X,X^n,{\pi }^n\right)+{\displaystyle \sum _{t=1}^T{Q}_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)}\hfill \\ =Q_X\left(X,X^n,{\pi }^n\right)+Q_{\pi }\left(\pi ,N^n,{\pi }^n\right)\hfill \\ Q_X\left(X,X^n,{\pi }^n\right)=\text{log}\left(P\left(X,\tilde{N}=0\right)\right)\hfill \\ Q_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)={\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n}\left(t\right)\text{log}{\pi }_i\left(i\right)}\hfill \\ Q_{\pi }\left(\pi ,X^n,{\pi }^n\right)={\displaystyle \sum _{t=1}^T{Q}_{t,\pi }\left(\pi \left(t\right),X^n\right)}\hfill \end{array}$$

Q _x (X,X ⁿ , π ⁿ ) has already been maximized with the Kalman filter, now Q _π (π,X ⁿ , π ⁿ ) can be maximized. Q _t,π (π(t),X ⁿ ,π ⁿ ) is maximized for each time step t with respect to π(t) and thus Q _π (π,X ⁿ ,π ⁿ ) is maximized. It is maximized as: $${\pi }^{n+1}\left(t\right)=\text{bad}\underset{\pi \left(t\right)}{{\displaystyle \text{Max}}}{Q}_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)=\text{bad}\underset{\pi \left(t\right)}{{\displaystyle \text{Max}}}{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}$$

Under the condition: $${\displaystyle \sum _{i=1}^m{\pi }_i^{n+1}\left(t\right)=1}$$

To get the Lagrangian: $$L_t=Q_{t,\pi }\left(\pi \left(t\right),X^n,{\pi }^n\right)+{\lambda }_t\left(1-{\displaystyle \sum _{i=1}^m{\pi }_i^{n+1}\left(t\right)}\right)$$

The bounded maximum of the Lagrangian satisfies the condition for any fixed (i',t'): $$\begin{array}{l}\frac{\partial L_t}{\partial {\pi }_i,\left(t\text{'}\right)}=0\\ \iff {\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}\frac{{\omega }_{i\text{'},\mathrm{right}}^n\left(t\text{'}\right)}{{\pi }_{i\text{'}}^{n+1}\left(t\text{'}\right)}-{\lambda }_t=0}\\ \iff {\pi }_{i\text{'}}^{n+1}\left(t\text{'}\right)=\frac{1}{{\lambda }_t}{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\omega }_{i\text{'},\mathrm{right}}^n\left(t\text{'}\right)}\end{array}$$

With the restriction that all priors should be summed to 1, we get: $$\begin{array}{l}{\displaystyle \sum _{i\text{'}=1}^m{\pi }_{i\text{'}}^{n+1}\left(t\text{'}\right)}=1\\ {\displaystyle \sum _{i\text{'}=1}^m{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\omega }_{i\text{'},\mathrm{right}}^n\left(t\text{'}\right)={\lambda }_t={\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i\text{'}=1}^m{\omega }_{i\text{'},\mathrm{right}}^n\left(t\text{'}\right)}}}}={\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}1=n\left(t\right)}\end{array}$$

Finally, the formula for updating the previous probabilities is generated: $${\pi }_i^{n+1}\left(t\right)=\frac{1}{n\left(t\right)}{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\omega }_{i,\mathrm{right}}^n\left(t\right)}$$

Equation 1.33 allows the PMHT to figure out by itself which sides 10 of the object 7 are more likely to be detected and increases the performance of the filter. However, if a side 10 was not detected at all during a time step, the updated prior probabilities are 0 and the PMHT is therefore no longer able to associate that side 10 of object 7 even if that side 10 is suddenly detected, typically due to a change in the position of the tracked object 7 relative to the detection device. To prevent this effect, 1.33 is used, ensuring that the updated prior probabilities do not become smaller than a defined threshold.

• Alle Messungen können der falschen Seite 10 des Objekts 7 zugeordnet werden, wenn ein Ziel von hinten erfasst wird: Auch bei Laserscannern sind Messungen aus dem Inneren des Objekts 7 üblich und es muss das Innenmodell verwendet werden. Auch die vorderen Ecken 11 können von hinten detektiert werden (Autos sind nicht perfekt rechteckig, so dass Echos von der Seitenfläche reflektiert werden können), so dass auch die Modelle für die vorderen Ecken 11 verwendet werden müssen. Wenn das Ziel beschleunigt wird und gerade mit einer etwas zu geringen Geschwindigkeit initialisiert wurde, driften die Assoziationen für den ersten und letzten Punkt zum linken und rechten Linienmodell und für die anderen Punkte zum Innenmodell. Schließlich, nach einigen Zeitschritten, werden der erste und der letzte Punkt mit den vorderen Ecken 11 assoziiert und die anderen Punkte entweder mit der vorderen Linie oder, auch wenn die vordere Linie verdeckt ist, mit dem Innenmodell oder dem Clutter.

The hypothesis stating that “all associations are independent of each other” may actually result in the PMHT presented above getting stuck in undesired local maxima. This hypothesis implies that all measurements can start from the same model, which can lead to the following behavior: "All associations are independent of each other":

• All measurements can be assigned to the wrong side 10 of the object 7 when a target is detected from behind: measurements from inside the object 7 are also common with laser scanners and the inside model must be used. Also the front corners 11 can be detected from behind (cars are not perfectly square so echoes can be reflected from the side surface), so the models for the front corners 11 must also be used. If the target is accelerating and has just been initialized at a slightly too low speed, the associations for the first and last points will drift to the left and right line models and for the other points to the interior model. Finally, after some time steps, the first and the last point become associated with the front corners 11 and the other points either with the front line or, even if the front line is obscured, with the interior model or the clutter.

A similar problem can occur with radar technology when a target is detected from the side 10. Radar can in principle acquire measurements anywhere on the target, and all models including the essentially hidden sides 10 must be used. Because radar can generate a lot of clutter near the target, the clutter model's probability of correctly tracking the target must be set quite high. Therefore, even if spots are detected inside the target, they can be associated with clutter and the target will get stuck on the wrong side 10.

Furthermore, each rectangle enclosing all discoveries can have a valid local maximum. All measurements can be assigned to the inner model and none or very few to the contour. This can be particularly problematic with radar since we cannot set the previous association probabilities of the hidden faces 10 and corners 11 to 0. Estimated target may be too big.

Furthermore, once a target has grown too much, it can never shrink again. This can happen during initialization when the length uncertainty is large. It can also happen if some clutter measurements caused the target to grow too much.

All of these problems are caused by an incorrect choice of posterior association probabilities. In the first example, the problem is that the filter doesn't know that at least one measurement must come from each corner 11 since we are capturing the target from behind, or that at least one measurement must come from the back. The problem in the second example is the same, as some measurements must come from either the front and back left corners 11 or from the back. In the third and fourth examples, the PMHT would converge to the correct estimate if it knew that at least one measurement must come from the front left and back right corners 11 .

Therefore, according to the invention, the assumption that all associations are independent of each other is dropped and the knowledge that “at least certain models receive one (or more) measurements” is included. This limitation of the association can be active or inactive depending on the viewing angle at which the object 7 is detected or depending on its distance from the detection device 5 . A certain probability can also be assigned to the constraint as to whether this is valid or not, in order to avoid hard switching on/off of this constraint.

The hypothesis that "the association of measurements to a side 10 of the object 7 are independent" is removed and replaced by the new hypotheses.

For each time step, at least a certain number of measurements, which can also be different for each time step, must be assigned to a specific set of models (which can be different for each time step). The variable κ is generated, which represents a case of such a possible association over all time steps, and κ(t) for a specific time step t. For example, if at time step t at least one measurement from the models i ₁ (t),, i _{m ct} (t) where the number of bounded models at time step tm is _c,t ≤ m, then an instance of (κ(1),, κ(T)) and for each time step κ(t) = ( _{kr 1 (t)} = i ₁ (t),,k _{r m c, t (t)} (t)). It should be noted that for a given κ, not all measurements have a fixed association, only those associated with the constrained models.

Furthermore, all instances of κ are a priori equally probable and state-independent.

Furthermore, for a given κ, the associations for the unconstrained measures, the measures whose association was not fixed, are independent of each other, which corresponds to the same hypothesis as for the standard PMHT.

Furthermore, the prior association probabilities are independent of state, consistent with the same hypothesis as for the standard PMHT, and are identical for all instances of κ.

Möglichkeiten hat, wodurch κ ${\displaystyle {\prod }_{t=1}^T\left(\begin{array}{c}n\left(t\right)\\ m_{c,t}\end{array}\right)}$

Möglichkeiten hat.The number of possible instances for κ is combinatorial. For example, if the constraint is at least one measurement, it must be satisfied with the bounded models i ₁ (t),, i _{m c, t} et), κ(t) are associated, where this $\left(\begin{array}{c}n\left(t\right)\\ m_{c,t}\end{array}\right)$

has opportunities, whereby κ ${\displaystyle {\prod }_{t=1}^T\left(\begin{array}{c}n\left(t\right)\\ m_{c,t}\end{array}\right)}$

has opportunities.

In principle, any kind of boundary condition is possible as long as there are enough measurements, whereby it cannot be specified that there should be more boundary conditions than measurements. Because of the combinatorial number of possibilities, a method is also needed to sample κ only for the most likely cases.

The Q-function 1.7 should therefore be maximized: $${\displaystyle \sum _{K}P\left(\frac{K}{X^n},N=0\right)lOGP\left(X\right)=lOGP\left(X\right){\displaystyle \sum _{K}P\left(\frac{K}{X^n},N=0\right)}=P\left(X\right)}$$

This creates: $$\begin{array}{l}Q\left(X,X^n\right)={\displaystyle \sum _{K}P\left(\frac{K}{X^n},N=0\right)\text{log}P\left(X,K,N=0\right)}\\ =\text{log}P\left(X\right)+{\displaystyle \sum _{K}P\left(\frac{K}{X^n},N=0\right)}\text{log}\left(P\left(N=\frac{0}{X},K\right)P\left(\frac{K}{X}\right)\right)\\ =\text{log}P\left(X\right)+Q_K\left(X,X^n\right)\end{array}$$

Here it is emphasized that Σ _K means a sum over all possible associations that satisfy the boundary conditions, i.e. the required minimum number of measurements is assigned to the models with constraints and the other measurements can be assigned anywhere. It is now intended, in particular in the constrained PMHT, to express this sum as a double sum Σ _K = Σ _K Σ _K(κ) , where K(κ) represents a set of associations of the unconstrained measurements for a given κ. In other words, one instance of κ is first fixed and then the remaining unconstrained measurements are summed before summing over all possible κ.

We therefore first calculate P(K(κ)/X ⁿ , N = 0). From the definition of the conditional probability and using the hypothesis that the associations of the unconstrained measurements are independent of each other, we get: $$\begin{array}{l}P\left(\frac{K\left(k\right)}{X^n},N=0\right)=P\left(\frac{K\left(k\right)}{X^n},N=\mathrm{0,}k\right)P\left(\frac{k}{X^n},N=0\right)\\ ={\displaystyle \prod _{t=1T\mathrm{right}=1n\left(T\right)\forall t\in \mathrm{1,}T,\forall i\in \mathrm{1,}{m}_{c,t}\mathrm{right}\ne {\mathrm{right}}_i\left(t\right)}{\omega }_{k_{\mathrm{right}}\left(t\right),\mathrm{right}}^{n,k}\left(t\right)P\left(\frac{k}{X^n},N=0\right)}\\ {\omega }_{i,\mathrm{right}}^{n,k}\left(t\right)=P\left(k_{\mathrm{right}}\left(t\right)=\frac{i}{X^n},N=\mathrm{0,}k\right)\end{array}$$

For simplicity, the set (t, r)\κ is introduced as the set of indices of all unconstrained measurements, so that the equation above is simplified: $$\begin{array}{l}P\left(\frac{K\left(k\right)}{X^n},N=0\right)=P\left(\frac{K\left(k\right)}{X^n},N=\mathrm{0,}k\right)P\left(\frac{k}{X^n},N\text{=0}\right)\\ ={\displaystyle \prod _{\left(t,\mathrm{right}\right)\setminus k}{\omega }_{k_{\mathrm{right}}\left(t\right),\mathrm{right}}^{n,k}\left(t\right)P=\left(\frac{k}{X^n},N=0\right)}\end{array}$$

Taking Bayes' rule into account, we get: $$\begin{array}{l}{\omega }_{i,\mathrm{right}}^{n,k}\left(t\right)=P\left(k_{\mathrm{right}}\left(t\right)=\frac{i}{X^n},N=\mathrm{0,}k\right)\\ \propto P\left(N=\frac{0}{X^n},k_{\mathrm{right}}\left(t\right)=i,k\right)P\left(k_{\mathrm{right}}\left(t\right)=\frac{i}{X^n},k\right)\end{array}$$

The fourth hypothesis then gives: $$P\left(k_{\mathrm{right}}\left(t\right)=\frac{i}{X^n},k\right)={\pi }_i\left(t\right)$$

Due to the fact that the measurements are independent of each other, and for a given r and t, according to the third hypothesis, the probability of the other measurements is independent of i, and for a given association k _r (t) = i is the value of κ irrelevant to the probability, since the association of this measurement is fixed, we get: $$\begin{array}{l}P\left(N=\frac{0}{X^n},k_{\mathrm{right}}\left(t\right)=i,k\right)\propto P\left(v_{i,t}^{\mathrm{right}}=\frac{0}{x_t^n},k_{\mathrm{right}}\left(t\right)=i,k\right)\\ =P\left(v_{i,t}^{\mathrm{right}}=\frac{0}{x_t^n},k_{\mathrm{right}}\left(t\right)=i\right)={\lambda }_t^{\mathrm{right},n}\left(i\right)\end{array}$$

Therefore, the posterior association probabilities of the unconstrained measurements yield the same result as for the standard PMHT: $${\omega }_{i,\mathrm{right}}^{n,k}\left(t\right)={\omega }_{i,\mathrm{right}}^n\left(t\right)=\frac{{\lambda }_t^{\mathrm{right},n}\left(i\right){\pi }_i\left(t\right)}{{\displaystyle {\sum }_{j=1}^m{\lambda }_t^{\mathrm{right},n}}\left(j\right){\pi }_j\left(t\right)}$$

It now becomes P(κ/X ⁿ ,N = 0) and using Bayes' rule: $$P\left(\frac{k}{X^n},N=0\right)\propto P\left(N=\frac{0}{X^n},k\right)P\left(\frac{k}{X^n}\right)$$

According to the second hypothesis, all P(κ/X ⁿ ) are equally probable and therefore vanish in the normalization factor and can be ignored. It will (t,r)\κ introduced as the set of indices of all constrained measurements for a given κ, and these are determined using the independence between the measurements: $$P\left(N=\frac{0}{X^n},k\right)={\displaystyle \prod _{\overline{\left(t,\mathrm{right}\right)\setminus k}}P\left(v_{k_{\mathrm{right}}\left(t\right),t}^{\mathrm{right}}=\frac{0}{x_t^n},k_{\mathrm{right}}\left(t\right),k\right)}{\displaystyle \prod _{\left(t,\mathrm{right}\right)\setminus k}P\left(v_t^{\mathrm{right}}=\frac{0}{x_t^n},k\right)}$$

The probability of a measurement for an unknown association can be determined by marginalizing across all models: $$P\left(v_t^{\mathrm{right}}=\frac{0}{x_t^n},k\right)={\displaystyle \sum _{i=1}^{M}P\left(v_{i,t}^{\mathrm{right}}=\frac{0}{x_t^n},k,k_{\mathrm{right}}\left(t\right)=i\right)}P\left(k_{\mathrm{right}}\left(t\right)=\frac{i}{x_t^n},k\right)$$

Außerdem wir vorgegeben, identisch wie zuvor, dass für eine gegebene Assoziation k_r(t) = i der Wert von κ für die Wahrscheinlichkeit irrelevant ist, da die Assoziation dieser Messung festgelegt ist. So ergibt sich: $$P\left({\nu }_t^r=\frac{0}{x_t^n},\kappa \right)={\displaystyle \sum _{i=1}^M{\lambda }_t^{r,n}\left(i\right){\pi }_i\left(t\right)}$$

was dem Normalisierungsfaktor von 2.8 entspricht. Die Wahrscheinlichkeit der posterioren Assoziation einer beschränkten Zuweisung nach der n^th Iteration kann schließlich durch Normalisierung erhalten werden, um sicherzustellen, dass die Summe über κ gleich 1 ist: $$\begin{array}{l}P\left(\frac{\kappa }{X^n},N=0\right)={\omega }_{\kappa }^n=\\ \frac{{\displaystyle {\prod }_{\overline{\left(t,r\right)\setminus \kappa }}P\left({\nu }_{k_r\left(t\right),t}^r=\frac{0}{x_t^n},k_r\left(t\right),\kappa \right){\displaystyle {\prod }_{\left(t,r\right)\setminus \kappa }{\displaystyle {\sum }_{i=1}^M{\lambda }_t^{r,n}\left(i\right)}}{\pi }_i\left(t\right)}}{{\displaystyle {\sum }_{\kappa \text{'}}{\displaystyle {\prod }_{\overline{\left(t\text{'},r\text{'}\right)\setminus \kappa \text{'}}}P\left({\nu }_{k_{r\text{'}}\left(t\text{'}\right),t\text{'}}^{r\text{'}}=\frac{0}{x_{t\text{'}}^n},k_{r\text{'}}\left(t\text{'}\right),\kappa \text{'}\right){\displaystyle {\prod }_{\left(t\text{'},r\text{'}\right)\setminus \kappa \text{'}}{\displaystyle {\sum }_{i\text{'}=1}^M{\lambda }_{t\text{'}}^{r\text{'},n\text{'}}\left(i\text{'}\right)}}{\pi }_{i\text{'}}\left(t\text{'}\right)}}}\end{array}$$

According to the fourth hypothesis, $P\left(k_{\mathrm{right}}\left(t\right)=i/x_t^n,k\right)={\pi }_i\left(t\right).$

We also assume, identically as before, that for a given association k _r (t) = i, the value of κ is irrelevant to the probability since the association of this measurement is fixed. This is how it turns out: $$P\left(v_t^{\mathrm{right}}=\frac{0}{x_t^n},k\right)={\displaystyle \sum _{i=1}^M{\lambda }_t^{\mathrm{right},n}\left(i\right){\pi }_i\left(t\right)}$$

which corresponds to the normalization factor of 2.8. Finally, the probability of the posterior association of a constrained assignment after the n ^th iteration can be obtained by normalizing to ensure that the sum over κ equals 1: $$\begin{array}{l}P\left(\frac{k}{X^n},N=0\right)={\omega }_k^n=\\ \frac{{\displaystyle {\prod }_{\overline{\left(t,\mathrm{right}\right)\setminus k}}P\left(v_{k_{\mathrm{right}}\left(t\right),t}^{\mathrm{right}}=\frac{0}{x_t^n},k_{\mathrm{right}}\left(t\right),k\right){\displaystyle {\prod }_{\left(t,\mathrm{right}\right)\setminus k}{\displaystyle {\sum }_{i=1}^M{\lambda }_t^{\mathrm{right},n}\left(i\right)}}{\pi }_i\left(t\right)}}{{\displaystyle {\sum }_{k\text{'}}{\displaystyle {\prod }_{\overline{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k\text{'}}}P\left(v_{k_{\mathrm{right}\text{'}}\left(t\text{'}\right),t\text{'}}^{\mathrm{right}\text{'}}=\frac{0}{x_{t\text{'}}^n},k_{\mathrm{right}\text{'}}\left(t\text{'}\right),k\text{'}\right){\displaystyle {\prod }_{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k\text{'}}{\displaystyle {\sum }_{i\text{'}=1}^M{\lambda }_{t\text{'}}^{\mathrm{right}\text{'},n\text{'}}\left(i\text{'}\right)}}{\pi }_{i\text{'}}\left(t\text{'}\right)}}}\end{array}$$

It will now explicitly state the terms under the log from the Q-function. Per independence between the measurements arise: $$\begin{array}{l}P\left(N=\frac{0}{X},K\left(k\right)\right)={\displaystyle \prod _{\overline{\left(t,\mathrm{right}\right)\setminus k}}P\left(v_{k_{\mathrm{right}}\left(t\right),t}^{\mathrm{right}}=\frac{0}{x_t},k_{\mathrm{right}}\left(t\right)\right)}{\displaystyle \prod _{\left(t,\mathrm{right}\right)\setminus k}P}\left(v_{k_{\mathrm{right}}\left(t\right),t}^{\mathrm{right}}=\frac{0}{x_t},k_{\mathrm{right}}\left(t\right)\right)\\ ={\displaystyle \prod _{\overline{\left(t,\mathrm{right}\right)\setminus k}}{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)}{\displaystyle \prod _{\left(t,\mathrm{right}\right)\setminus k}{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)}\end{array}$$

The first product term depends only on κ and not on the others that are later used to factor it. For this reason, P(N = 0/X, K( _K )) is not expressed as a single product of the probabilities of all measurements.

Now, as for the second term within the log, using the definition of conditional probability and hypothesis two, that all conditional hypotheses are a priori equally likely and independent of state: $$P\left(K\left(k\right)\right)/X)=P\left(K\left(k\right)/X,k\right)P\left(k/X\right)\propto P\left(K\left(k\right)/X,k\right)$$

Considering the fourth hypothesis: $$P\left(K\left(k\right)\right)/X)\propto {\displaystyle \prod _{\left(t,\mathrm{right}\right)\setminus k}{\pi }_{k_{\mathrm{right}}\left(t\right)}}\left(t\right)$$

Q _K can now be simplified from 2.2 to 2.4, 2.14 and 2.16, taking into account that the second equality holds up to a constant due to the proportionality factor of 2.16: $$\begin{array}{c}{Q}_K\left(X,X^n\right){\displaystyle \sum _{K}P\left(\frac{K}{X^n},N=0\right)}\text{log}\left(P\left(N=\frac{0}{X},K\right)P\left(\frac{K}{X}\right)\right)\\ ={\displaystyle \sum _k{\displaystyle \sum _{K\left(k\right)}{\displaystyle \prod _{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k}{\omega }_{k_{\mathrm{right}\text{'}}\left(t\right),\mathrm{right}\text{'}}^n\left(t\text{'}\right){\omega }_k^n}}}\\ \times \left[{\displaystyle \sum _{\overline{\left(t,\mathrm{right}\right)\setminus k}}\text{log}{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)+{\displaystyle \sum _{\left(t,\mathrm{right}\right)\setminus k}\left(\text{log}{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)+\text{log}{\pi }_{k_{\mathrm{right}}\left(t\right)}\left(t\right)\right)}}\right]\\ ={\displaystyle \sum _k{\omega }_k^n\left[{\displaystyle \sum _{\overline{\left(t,\mathrm{right}\right)\setminus k}}lOG{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)\times {\displaystyle \sum _{K\left(k\right)}{\displaystyle \prod _{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k}{\omega }_{k_{\mathrm{right}\text{'}}\left(t\text{'}\right),\mathrm{right}\text{'}}^n\left(t\text{'}\right)}}}\right]+}\\ {\displaystyle \sum _k{\omega }_k^n{\displaystyle \sum _{K\left(k\right)}\left[{\displaystyle \prod _{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k}{\omega }_{k_{\mathrm{right}\text{'}}\left(t\text{'}\right),\mathrm{right}\text{'}}^n\left(t\text{'}\right)}{\displaystyle \sum _{\left(t,\mathrm{right}\right)\setminus k}\left(\text{log}{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)+\text{log}{\pi }_{k_{\mathrm{right}}\left(t\right)}\left(t\right)\right)}\right]}}\end{array}$$

If a probability is summed over the entire sample, we get: $${\displaystyle \sum _{K\left(k\right)}{\displaystyle \prod _{\left(t\text{'},\mathrm{right}\text{'}\right)\setminus k}{\omega }_{k_{\mathrm{right}\text{'}}\left(t\text{'}\right),\mathrm{right}\text{'}}^n}\left(t\text{'}\right)}=1$$

A transformation is applied next: $$\begin{array}{l}{Q}_K\left(X,X^n\right)=\\ {\displaystyle \sum _k{\omega }_k^n}{\displaystyle \sum _{\overline{\left(t,\mathrm{right}\right)\setminus k}}lOG{\lambda }_t^{\mathrm{right}}\left(k_{\mathrm{right}}\left(t\right),n\right)+{\displaystyle \sum _k{\omega }_k^n}{\displaystyle \sum _{\left(t,\mathrm{right}\right)\setminus k}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)}}\left(lOG{\lambda }_t^{\mathrm{right}}\left(i,n\right)+lOG{\pi }_i\left(t\right)\right)}\end{array}$$

To determine how the maximum of _QK can be found, we define: $$\begin{array}{l}\forall k,\left(t,\mathrm{right}\right)\setminus k,n_{i,\mathrm{right}}^{n,k}\left(t\right)={\omega }_{i,\mathrm{right}}^n\left(t\right)\hfill \\ \forall k,\overline{\left(t,\mathrm{right}\right)\setminus k},n_{i,\mathrm{right}}^{n,k}\left(t\right)={\delta }_{k_{\mathrm{right}\left(t\right)},i}\hfill \end{array}$$

Where δ is the Kronecker Delta and: $$\begin{array}{l}\forall k,\left(t,\mathrm{right}\right)\setminus k,{\tilde{n}}_{i,\mathrm{right}}^{n,k}\left(t\right)={\omega }_{i,\mathrm{right}}^n\left(t\right)\hfill \\ \forall k,\overline{\left(t,\mathrm{right}\right)\setminus k},{\tilde{n}}_{i,\mathrm{right}}^{n,k}\left(t\right)=0\hfill \end{array}$$

With the equation from 2.19 we get: $$\begin{array}{l}{Q}_K\left(X,X^n\right)\\ \begin{array}{c}={\displaystyle \sum _k{\omega }_k^n{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{n}_{i,\mathrm{right}}^{n,k}\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)+{\displaystyle \sum _k{\omega }_k^n{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\tilde{n}}_{i,\mathrm{right}}^{n,k}\left(t\right)\text{log}{\pi }_i\left(t\right)}}}}}}}}\\ ={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _k{\omega }_k^n{n}_{i,\mathrm{right}}^{n,k}\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)}}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _k{\omega }_k^n{\tilde{n}}_{i,\mathrm{right}}^{n,k}\left(t\right)\text{log}{\pi }_i\left(t\right)}}}}\end{array}\end{array}$$

Ultimately, the result is: $$\begin{array}{c}{\Omega }_{i,\mathrm{right}}^n\left(t\right)={\displaystyle \sum _k{\omega }_k^n{n}_{i,\mathrm{right}}^{n,k}\left(t\right)}\\ {\tilde{\Omega }}_{i,\mathrm{right}}^n\left(t\right)={\displaystyle \sum _k{\omega }_k^n{\tilde{n}}_{i,\mathrm{right}}^{n,k}\left(t\right)}\end{array}$$

Whereby it can be expressed similarly to 1.8: $$Q_K\left(X,X^n\right)={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\Omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\tilde{\Omega }}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i(t)}}}}}}$$

anstelle von ${\omega }_{i,r}^{\text{n}}\left(t\right)$

für die nicht-beschränkten PMHT erhalten werden. Die Aktualisierung der früheren Assoziationswahrscheinlichkeiten erfolgt ebenfalls auf die gleiche Weise wie 1.33, jedoch unter Verwendung von ${\tilde{\Omega }}_{i,r}^{\text{n}}\left(t\right)$

anstelle von ${\omega }_{i,r}^n\left(t\right).$

The solution that maximizes _QK is therefore obtained with a Kalman filter using synthetic measurements associated with the posterior association weights ${\mathrm{\Omega }}_{i,\mathrm{right}}^{\text{n}}\left(t\right)$

instead of ${\omega }_{i,\mathrm{right}}^{\text{n}}\left(t\right)$

can be obtained for the non-constrained PMHT. Updating the previous association probabilities is also done in the same way as 1.33 but using ${\tilde{\Omega }}_{i,\mathrm{right}}^{\text{n}}\left(t\right)$

instead of ${\omega }_{i,\mathrm{right}}^n\left(t\right).$

As shown, the exact calculation of the constrained posterior association probabilities 2.23 is unsolvable due to the combinatorial number of possibilities for the constrained associations κ. The standard PMHT circumvents this problem by assuming independence between the associations, but this cannot be avoided if this hypothesis is dropped. This combinatorial number of associations is a well-known problem in tracking multiple objects 7, and two main techniques are used to perform the calculation. Here, Murty's algorithm or Gibbs sampling or Gibbs spot check is known.

The Murtys algorithm is a generalization of the well-known Hungarian assignment algorithm. It aims to find the k-best joint assignments for a given cost matrix that assigns costs to each possible individual assignment.

Wenn mehr Messungen als Nebenbedingungen verfügbar sind, können die anderen Messungen jedem Modell zugeordnet werden.µ _i (t) at each time step t is used as the minimum number of measurements to be associated with model i, which can be zero if there is no constraint on that model. The total number of measurements to be assigned to the constrained models is therefore ${\displaystyle {\sum }_{i=1}^m{\mu }_i}\left(t\right)=m_{c,t}.$

If more measurements are available as constraints, the other measurements can be assigned to each model.

ergeben. Der größte Wert von ${\omega }_{\kappa }^n$

ergibt sich für den größten Zähler aus 2.13, der das Produkt der gemeinsamen Wahrscheinlichkeit mit den beschränkten Modellen ist: $${\displaystyle \prod _{\overline{\left(t,r\right)\text{}\kappa }}P}\left(v_{k_r\left(t\right),t}^r=\frac{0}{x_t^n},k_r\left(t\right),\kappa \right)$$

The goal is to find the k-best assignments κ that have the k-largest values of ${\omega }_k^n$

result. The greatest value of ${\omega }_k^n$

results for the largest numerator from 2.13, which is the product of the joint probability with the restricted models: $${\displaystyle \prod _{\overline{\left(t,\mathrm{right}\right)\text{}k}}P}\left(v_{k_{\mathrm{right}}\left(t\right),t}^{\mathrm{right}}=\frac{0}{x_t^n},k_{\mathrm{right}}\left(t\right),k\right)$$

With the probability of each model: $${\displaystyle \prod _{\left(t,\mathrm{right}\right)\text{}k}{\displaystyle \sum _{i=1}^M{\mathrm{\lambda }}_t^{\mathrm{right},n}}}\left(i\right){\pi }_i\left(t\right)$$

It is common to express the cost of the assignment in logarithmic space, since Murty's algorithm finds the k-best assignments whose cost is the sum of the costs of each individual assignment, which is equivalent to a joint probability optimization since the sum of the logarithm is the logarithm of the product is.

und die Kosten für eine Zuordnung zu einem beliebigen Modell: $${\mathrm{\Lambda }}_t^{r,n}=\text{log}{\displaystyle \sum _{i=1}^M{\mathrm{\lambda }}_t^{r,n}}\left(i\right){\pi }_i\left(t\right)$$

The cost of assigning a measurement to a specific model i is (t, r): $${\mathrm{\Lambda }}_{i,t}^{\mathrm{right},n}=\text{log}P\left(v_{i,t}^{\mathrm{right}}=\frac{0}{x_t^n}\right)$$

and the cost of mapping to any model: $${\mathrm{\Lambda }}_t^{\mathrm{right},n}=\text{log}{\displaystyle \sum _{i=1}^M{\mathrm{\lambda }}_t^{\mathrm{right},n}}\left(i\right){\pi }_i\left(t\right)$$

The cost matrix results for each time step t: $$\left[\begin{array}{ccc}\stackrel{{\mu }_1\left(t\right)}{\overbrace{\begin{array}{cc}{\mathrm{<figure-callout\; id="1"\; label="\Lambda "\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{\mathrm{1,}\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{1,}\mathrm{<figure-callout\; id="1"\; label="n\; \Lambda "\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">n\; </figure-callout>}}& & {\mathrm{\Lambda }}_{}^{}{\mathrm{}}_{\mathrm{1,}\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{1,}n}\\ ⋮& \ddots & ⋮\\ {\mathrm{\Lambda }}_{\mathrm{1,}t}^{n\left(t\right),\mathrm{<figure-callout\; id="1"\; label="n\; \Lambda "\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">n\; </figure-callout>}}& & {\mathrm{\Lambda }}_{}^{}{\mathrm{}}_{\mathrm{1,}t}^{n\left(t\right),n}\end{array}}}& \stackrel{{\mu }_m\left(t\right)}{\overbrace{\begin{array}{ccc}{\mathrm{\Lambda }}_{m,\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{1,}n}& & {\mathrm{\Lambda }}_{m,\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">t</figure-callout>}}^{\mathrm{1,}n}\\ ⋮& \ddots & ⋮\\ {\mathrm{\Lambda }}_{m,t}^{n\left(t\right),n}& & {\mathrm{\Lambda }}_{m,t}^{n\left(t\right),n}\end{array}}}& \stackrel{n\left(t\right)-{\displaystyle {\sum }_{i=1}^m{\mu }_i\left(t\right)}}{\overbrace{\begin{array}{cc}{\mathrm{<figure-callout\; id="1"\; label="\Lambda \; t"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_t^{}& & {\mathrm{\Lambda }}_t^{}{}_{\mathrm{1,}n}^{}\\ ⋮& \ddots & ⋮\\ {\mathrm{\Lambda }}_t^{n\left(t\right),n}& & {\mathrm{\Lambda }}_t^{n\left(t\right),n}\end{array}}}\end{array}\right]$$

Since the maximum likelihood is obtained from the maximum logarithm, Murty's algorithm is used to maximize the cost from the cost matrix. Using the k-best assignments returned by Murty's algorithm, for each assignment returned, the numerator of 2.13 can be computed, and the denominator is the sum of all numerators. From there the bounded back weights 2.23 can be computed to run the Kalman filter.

An alternative to Murty's algorithm for simplifying a combinatorial association problem is to use Gibbs sampling.

The idea is that instead of finding the k-best matches, sample from the distribution of matches.

If a sample is taken from a one-dimensional distribution, this can be done using the so-called inverse distribution transformation. For sampling for a random va riable X whose CDF (cumulative distribution function) is F, we start sampling from a uniformly distributed random variable U on [0,1]. It is the CDF of F ^-1 (U)F: $$P\left(f^{-1}\left(u\right)\le x\right)=P(f\left(f^{-1}\left(u\right)\right)\le f\left(x\right))=P\left(u\le f\left(x\right)\right)=f\left(x\right)$$

Sampling from a one-dimensional distribution is therefore straightforward: samples are generated from a uniform distribution on [0,1] and passed through the inverse of the CDF. Samples are taken from a two-dimensional random vector (x ₁ ,x ₂ ), whose joint distribution is known, so that the marginal distribution of one dimension must first be restored by marginalization: $$P\left(x_1\right)={\displaystyle \int P}\left(x_1,x_2\right)\mathrm{<figure-callout\; id="2"\; label="i.e\; x"\; filenames="DE102020121064A1\_0122.png"\; state="\{\{state\}\}">i.e</figure-callout>}{x}_{}{}_2$$

für x₁ unter Verwendung der inversen Verteilungstransformation der CDF von 2.31 erhalten wird, kann x₂ aus der bedingten Wahrscheinlichkeit abgezogen werden: $$P\left(\frac{x_2}{x_1}=x_1^i\right)=\frac{P\left(x_2,x_1=x_1^i\right)}{P\left(x_1=x_1^i\right)}$$

Once a sample $x_1^i$

for x ₁ is obtained using the inverse distribution transformation of the CDF of 2.31, x ₂ can be subtracted from the conditional probability: $$P\left(\frac{x_2}{x_1}=x_1^i\right)=\frac{P\left(x_2,x_1=x_1^i\right)}{P\left(x_1=x_1^i\right)}$$

However, this method is not applicable in this case. The association κ has many dimensions and the integration 2.31 is very complex, since in order to marginalize it, one would have to integrate over dx ₂ dx ₃ . Gibbs sampling is one way to avoid this.

Suppose we want to obtain k samples of X = (x ₁ ,...,x _n ) from a joint distribution P(x ₁ ,...,x _n ). The ith sample is $X^i=\left(x_1^i,...,x_n^i\right).$

ein Vektor ist, wird jede Komponente des Vektors, $x_j^{i+1}$

entnommen, aus der Verteilung dieser Komponente, die auf alle anderen bisher entnommenen Komponenten konditioniert wurde, insbesondere unter Verwendung der inversen Verteilungstransformation. Es werden jedoch die Komponenten von Xⁱ⁺¹ bis zu $x_{j-1}^{i+1}$

konditioniert und danach die Komponenten von Xⁱ, beginnend mit $x_{j+1}^i\text{bis}{x}_n^i.$

Um dies zu erreichen, beproben werden die Komponenten der Reihe nach beprobt, beginnend mit der ersten Komponente. Um $x_j^{i+1}$

formeller zu beproben, wird es gemäß der durch $P\left(x_j^{i+1}\text{/}{x}_1^{i+1},\dots ,x_{j-1}^{i+1},x_{j+1}^i,\dots ,x_n^i\right)$

spezifizierten Verteilung aktualisiert. Es wird den Wert verwendet, den die (j + 1)^th Komponente in der i-ten Stichprobe hatte, nicht die (i + 1)^th Stichprobe. Es wird dann der geschilderte Schritt k-mal wiederholt.The following steps are then carried out. It starts with an initial value X ⁱ . In this case, this value can be the best match obtained with the Hungarian algorithm. The next sample X ⁱ⁺¹ is then taken. There $X^{i+1}=\left(x_1^{i+1},x_2^{i+1},...,x_n^{i+1}\right)$

is a vector, each component of the vector, $x_j^{i+1}$

extracted, from the distribution of that component conditioned on all other components extracted so far, in particular using the inverse distribution transformation. However, the components of X ⁱ⁺¹ up to $x_{j-1}^{i+1}$

conditioned and then the components of X ⁱ , starting with $x_{j+1}^i\text{until}{x}_n^i.$

To achieve this, the components are sampled sequentially, starting with the first component. Around $x_j^{i+1}$

to sample more formally, it will be carried out according to the $P\left(x_j^{i+1}\text{/}{x}_1^{i+1},...,x_{j-1}^{i+1},x_{j+1}^i,...,x_n^i\right)$

specified distribution updated. It uses the value that the (j + 1) ^th component had in the ith sample, not the (i + 1) ^th sample. The described step is then repeated k times.

When such sampling is performed, these important facts apply that the samples approximate the joint distribution of all variables. Furthermore, the marginal distribution of any subset of variables can be approximated by considering the samples for that subset of variables and ignoring the rest. The expected value of each variable can be approximated by averaging across all samples.

Therefore, to perform Gibbs sampling on κ, only the conditional probability mass functions of a given assignment are needed, provided that all other constrained assignments are fixed. For each time step t, the probability that a given measurement r is associated with a constrained model j _i (t) is searched for, knowing that all other constrained models already have a measurement assigned: $$\begin{array}{l}P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n,N=0)\end{array}$$

Applying Bayes' rule gives: $$\begin{array}{l}\begin{array}{l}P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n,N=0)\end{array}\hfill \\ \begin{array}{l}\propto P\left(N=\frac{0}{k_{{\mathrm{right}}_1\left(t\right)}}=j_1\left(t\right),...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n\right)\times \\ P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n)\end{array}\hfill \end{array}$$

Based on the second assumption, all restricted assignments are a priori equally likely, so 2.34 simplifies to: $$\begin{array}{l}P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n,N=0)\\ \propto P\left(N=\frac{0}{k_{{\mathrm{right}}_1\left(t\right)}}=j_1\left(t\right),...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n\right)\end{array}$$

wobei $P\left(v_t^{r\text{'}}=0\text{/}{x}_t^n\right)$

durch 2.12 gegeben ist und $P\left(v_{i,t}^r=0\text{/}{k}_r\left(t\right)=j_i\left(t\right),x_t^n\right)=$

${\lambda }_t^{r,n}\left(j_i,\left(t\right)\right)$

aus 2.7 bereits bekannt ist und verwendet wird, um die hinteren Gewichte 2.8 zu erhalten.The probability of the measurements whose relationship is known is constant for all r and therefore vanishes in the normalization factor. 2.35 is therefore further simplified to: $$\begin{array}{l}\begin{array}{l}\begin{array}{l}P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n,N=0)\end{array}\hfill \\ \propto {\displaystyle \prod _{\mathrm{right}\text{'}=1\mathrm{right}\text{'}\ne {\mathrm{right}}_1,...,{\mathrm{right}}_{i-1},{\mathrm{right}}_{i+1},...,{\mathrm{right}}_{m_{c,t}\left(t\right)}}^{n\left(t\right)}P\left(\frac{v_t^{\mathrm{right}\text{'}}}{k_{\mathrm{right}}\left(t\right)}=j_i\left(t\right),x_t^n\right)}\hfill \end{array}\\ =P\left(v_{i,t}^{\mathrm{right}}=\frac{0}{k_{\mathrm{right}}\left(t\right)}=j_i\left(t\right),x_t^n\right){\displaystyle \prod _{\mathrm{right}\text{'}=1\mathrm{right}\text{'}\ne {\mathrm{right}}_{\mathrm{1,}...,}{\mathrm{right}}_{i-1},{\mathrm{right}}_{i+1},...,{\mathrm{right}}_{m_{c,t}\left(t\right)},\mathrm{right}}^{n\left(t\right)}P\left(v_t^{\mathrm{right}\text{'}}=\frac{0}{x_t^n}\right)}\end{array}$$

whereby $P\left(v_t^{\mathrm{right}\text{'}}=0\text{/}{x}_t^n\right)$

is given by 2.12 and $P\left(v_{i,t}^{\mathrm{right}}=0\text{/}{k}_{\mathrm{right}}\left(t\right)=j_i\left(t\right),x_t^n\right)=$

${\lambda }_t^{\mathrm{right},n}\left(j_i,\left(t\right)\right)$

is already known from 2.7 and is used to obtain the rear weights 2.8.

The final result is obtained after normalization over r: $$\begin{array}{l}P(k_{\mathrm{right}}\left(t\right)=j_i\left(t\right)\text{/}{k}_{{\mathrm{right}}_1\left(t\right)}=j_1\left(t\right),...,k_{{\mathrm{right}}_{i-1}\left(t\right)}=j_{i-1}\left(t\right),k_{{\mathrm{right}}_{i+1}\left(t\right)}=j_{i+1}\left(t\right),\\ ...,k_{{\mathrm{right}}_{m_{c,t}}}\left(t\right)=j_{m_{c,t}\left(t\right)}\left(t\right),X^n,N=0)\\ =\frac{{\lambda }_t^{\mathrm{right},n}\left(j_i\left(t\right)\right){\displaystyle {\prod }_{\mathrm{right}\text{'}=1\mathrm{right}\text{'}\ne {\mathrm{right}}_1,...,{\mathrm{right}}_{i-1},{\mathrm{right}}_{i+1},...,{\mathrm{right}}_{m_{c,t}\left(t\right)},\mathrm{right}}^{n\left(t\right)}P\left(v_t^{\mathrm{right}\text{'}}=0\text{/}{x}_t^n\right)}}{{\displaystyle {\sum }_{s=1s\ne {\mathrm{right}}_1,...,{\mathrm{right}}_{i-1},{\mathrm{right}}_{i+1},...,{\mathrm{right}}_{m_{c,t}\left(t\right)}}^{n\left(t\right)}{\lambda }_t^{s,n}}\left(j_i\left(t\right)\right){\displaystyle {\prod }_{s\text{'}=1s\text{'}\ne {\mathrm{right}}_1,...,{\mathrm{right}}_{i-1},{\mathrm{right}}_{i+1},...,{\mathrm{right}}_{m_{c,t}\left(t\right)},s}^{n\left(t\right)}P\left(v_t^{S\text{'}}=0\text{/}{x}_t^n\right)}}\end{array}$$

It cannot always be guaranteed that a number of conditions will be met. It is therefore proposed in a preferred embodiment how the probability that a boundary condition is valid can be taken into account. Assuming C were the case that the proposed constraint is valid, and p _c = P(C) its probability, then we can marginalize the MAP of the state PDF, which expectation maximization tries to find from 1.6 as follows: $$\begin{array}{l}{X}_{\text{Max}}=\underset{X}{\text{argmax}}\left\{P\left(\frac{X}{N}=0\right)\right\}=\underset{X}{\text{argmax}}\left\{P\left(N=\frac{0}{X}\right)P\left(X\right)\right\}\\ =\underset{X}{\text{argmax}}\left\{\left(P\left(N=\frac{0}{X},C\right)p_c+P\left(N=\frac{0}{X},\overline{C}\right)\left(1-p_c\right)\right)P\left(X\right)\right\}\end{array}$$

The case where the constraint is invalid is considered here. For example, if the constraint is that at least one measurement is associated with a particular model, it must be taken into account that no measurement is associated with that model. That is, if this constraint is assigned a 30% probability of being valid, the posterior association weight of that model will never go beyond that value, even if the measurement is directly above that model. Ultimately, it will completely prevent more measurements from being associated with this model than defined by the constraint.

Therefore it is preferable to optimize instead of 2.38: $$X_{max}=\underset{X}{\text{argmax}}\left\{\left(P\left(N=\frac{0}{X},C\right)p_c+P\left(N=\frac{0}{X}\right)\left(1-p_c\right)\right)P\left(X\right)\right\}$$

This leads to optimization: $$\begin{array}{c}{Q}_K\left(X,X^n\right)=p_c{Q}_K^C\left(X,X^n\right)+\left(1-p_c\right)Q_K^{\overline{C}}\left(X,X^n\right)\\ Q_K^C\left(X,X^n\right)={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\mathrm{\Omega }}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{{\displaystyle \sum _{i=1}^m\tilde{\Omega }}}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}}}\\ Q_K^{\overline{C}}\left(X,X^n\right)={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\omega }_{i,\mathrm{right}}^n\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{{\displaystyle \sum _{i=1}^m\omega }}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}}}\end{array}$$

Then the easy conditions of the posterior association weights are introduced as follows: $$\begin{array}{c}{\mathrm{\Delta }}_{i,\mathrm{right}}^n\left(t\right)=p_c{\mathrm{\Omega }}_{i,\mathrm{right}}^n\left(t\right)+\left(1-p_c\right){\omega }_{i,\mathrm{right}}^n\left(t\right)\\ {\tilde{\Delta }}_{i,\mathrm{right}}^n\left(t\right)=p_c{\tilde{\Omega }}_{i,\mathrm{right}}^n\left(t\right)+\left(1-p_c\right){\omega }_{i,\mathrm{right}}^n\left(t\right)\end{array}$$

This reduces the optimization to: $$Q_K\left(X,X^n\right)={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{\displaystyle \sum _{i=1}^m{\mathrm{\Delta }}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\lambda }_t^{\mathrm{right}}\left(i,n\right)+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{\mathrm{right}=1}^{n\left(t\right)}{{\displaystyle \sum _{i=1}^m\tilde{\Delta }}}_{i,\mathrm{right}}^n\left(t\right)\text{log}{\pi }_i\left(t\right)}}}}}$$

A Kalman filter is now used and the earlier association probabilities are updated using the soft condition of the posterior association weights 2.41.

die einem bestimmten Modell i zugeordnet ist, mindestens p_cµ_i(t) beträgt. Wenn genügend Messungen in der Nähe dieses Modells liegen, kann es jedoch zu µ_i(t) oder mehr konvergieren.This algorithm ensures that the total pseudo number of the synthetic measurements ${\displaystyle {\sum }_{\mathrm{right}=1}^{n\left(t\right)}{\mathrm{\Delta }}_{i,\mathrm{right}}^n}\left(t\right),$

associated with a particular model i is at least p _c µ _i (t). However, if enough measurements are close to this model, it can converge to µ _i (t) or more.

definiert werden, wie zum Beispiel ${\displaystyle {\sum }_l{p}_c^l}\le 1$

und die Gewichte der beschränkten posterioren Assoziation können erhalten werden: $$\begin{array}{c}{\mathrm{\Delta }}_{i,r}^n\left(t\right)={\displaystyle \sum _l{p}_c^l}{\mathrm{\Omega }}_{i,r}^{n,l}\left(t\right)+\left(1-{\displaystyle \sum _l{p}_c^l}\right){\omega }_{i,r}^n\left(t\right)\\ {\tilde{\Delta }}_{i,r}^n\left(t\right)={\displaystyle \sum _l{p}_c^l}{\tilde{\Omega }}_{i,r}^{n,l}\left(t\right)+\left(1-{\displaystyle \sum _l{p}_c^l}\right){\omega }_{i,r}^n\left(t\right)\end{array}$$

In a generalized way, several constraints can be related to the probability $p_c^l$

be defined, such as ${\displaystyle {\sum }_l{p}_c^l}\le 1$

and the constrained posterior association weights can be obtained: $$\begin{array}{c}{\mathrm{\Delta }}_{i,\mathrm{right}}^n\left(t\right)={\displaystyle \sum _l{p}_c^l}{\mathrm{\Omega }}_{i,\mathrm{right}}^{n,l}\left(t\right)+\left(1-{\displaystyle \sum _l{p}_c^l}\right){\omega }_{i,\mathrm{right}}^n\left(t\right)\\ {\tilde{\Delta }}_{i,\mathrm{right}}^n\left(t\right)={\displaystyle \sum _l{p}_c^l}{\tilde{\Omega }}_{i,\mathrm{right}}^{n,l}\left(t\right)+\left(1-{\displaystyle \sum _l{p}_c^l}\right){\omega }_{i,\mathrm{right}}^n\left(t\right)\end{array}$$

The four following examples in the 2 until 5 describe the limited PMHT as an example. In the first example according to 2 the boundary condition is used that at least one measurement must start from the back. In the second example according to 3 the constraint is used that at least one measurement must start from the left side 10. In the third example according to 4 at least one measurement must start from the rear left and front left corners 11. According to the fourth example 5 at least one measurement must start from the front left corner 11. All constraints are applied softly with p _c =0.3.

An iterative Kalman filter is also used to remove non-linearity problems. In addition, a limited Kalman gain ensures that the length and width of the object 7 does not become smaller than a realistic minimum value.

As shown, the constrained PMHT avoids the unwanted local maxima and converges to the desired result. Some other boundary conditions can lead to identical results, for example the third case shows similar results when at least three measurements should come from the left side 10 and one measurement from the back right corner 11. Both Murty's algorithm and Gibbs sampling give identical results.

2 shows a schematic plan view of the object 7. In particular, in a left part of the 2 shown how the detection points 8 are detected and how the object 7 actually behaves. On the right side of the 2 shows in particular that the polygonal shape 12 is adapted accordingly by means of the mathematical model 9 used, due to the specifications that at least one of the detection points 8 must be assigned to a rear side of the object 7 . It can thus be seen in particular that the detection points 8 are assigned to the polygonal shape 12 of the rear side. The actual object 13 is then shown accordingly, whereby it can be seen that the assignment of the polygonal shape 12 corresponds to the actual object 13 .

3 FIG. 12 shows a further schematic plan view, with the selected default condition being that at least one of the detection points 8 is assigned to a left-hand side 10 of the object 7 . On the left 10 of the 3 shows how the detection points 8 are detected and the polygonal shape 12 is assigned. On the right side of the 3 It is shown how the polygonal shape 12 corresponds to the actual object 13.

4 shows a further schematic top view of the object 7. In the present exemplary embodiment, the default condition is in particular that at least one of the detection points 8 must come from the rear left corner 11 and the front right corner 11. A first estimate 14 is shown. Furthermore, it is shown how the mathematical model 9 is then used to develop that the polygonal shape 12 is adapted in accordance with the actual object 13 .

5 shows a further schematic plan view of the object 7, wherein in the present exemplary embodiment a default condition is that at least one of the detection points 8 must come from the front left corner 11. A first estimate 14 is also shown here, with the polygonal shape 12 then being developed accordingly, which in turn is then associated with the actual object 13 .

Overall, the figures show constrained associations for advanced object tracking.

QUOTES INCLUDED IN DESCRIPTION

This list of documents cited by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent Literature Cited

• JP 2019152575 A [0003]
• CN109509207A [0004]
• CN 108363054A [0005]
• DE 102018104311 A1 [0006]

Claims (15)

Method for operating an assistance system (2) of a motor vehicle (1), in which by means of a detection device (5) of the assistance system (2) an environment (6) of the motor vehicle (1) is detected, the environment (6) by generating a plurality of detection points (8) is detected as a point cloud by the detection device (5), and in which the point cloud is evaluated by an electronic computing device (3) of the assistance system (2), the evaluation being based on a mathematical model (9) of the electronic computing device (3) is carried out, and by means of the mathematical model (9) a tracking of an object (7) in the environment (6) is carried out, the object (7) being approximated as a polygonal shape (12) in the mathematical model (9). and at least one of the detection points (8) is assigned to the polygonal shape (9) by means of a Kalman filter which is set up for processing implied measurement functions d, characterized in that at least one of the detection points (9) is predetermined as being assigned to at least one specific part of the polygonal shape (12) and the detection is carried out on the basis of this predetermined condition by means of the electronic computing device (3). procedure after claim 1 , characterized in that the specific part of the polygonal shape (12) is specified as a side (10) of the polygonal shape (12) and/or as a corner (11) of the polygonal shape (12). procedure after claim 1 or 2 , characterized in that the specific part of the polygonal shape (12) is specified as a function of a detection angle of the object (7). Method according to one of the preceding claims, characterized in that the polygonal shape (12) is specified as a rectangular shape of the object (7). Method according to one of the preceding claims, characterized in that a probabalistic multi-hypothesis tracker is used as the mathematical model (9), taking into account the default condition. procedure after claim 5 , characterized in that in the case of the probabalistic multi-hypothesis tracker the detected detection points (8) are independent of one another, but at least one of the plurality of detection points (8) is taken into account with the default condition. Procedure according to one of Claims 5 or 6 , characterized in that a Murty algorithm is used by means of the probabalistic multi-hypothesis tracker for evaluating the point cloud, taking into account the default condition. Procedure according to one of Claims 5 or 6 , characterized in that a Gibbs random sample is used by means of the probabalistic multi-hypothesis tracker for evaluating the point cloud, taking into account the default condition. Method according to one of the preceding claims, characterized in that a probability of occurrence for the default condition is taken into account in the mathematical model (9). Method according to one of the preceding claims, characterized in that the default condition is taken into account or not taken into account depending on a detection angle of the object (7). Method according to one of the preceding claims, characterized in that the default condition is taken into account or not taken into account depending on a detected distance from the object (7). Method according to one of the preceding claims, characterized in that the environment (6) is detected by means of a detection device (5) designed as a lidar sensor device and/or by means of a radar sensor device and/or by means of an ultrasonic sensor device. Computer program product with program code means, which are stored in a computer-readable storage medium, for the method according to one of the preceding Claims 1 until 12 through result when the computer program product is processed on a processor of an electronic computing device (3). Computer-readable storage medium containing a computer program product Claim 13 . Assistance system (2) for a motor vehicle (1), with at least one detection device (5) and with an electronic computing device (3), wherein the assistance system (2) for carrying out a method according to one of Claims 1 until 12 is trained.

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