A confidence measure for vehicle tracking based on a generalization of Bayes estimation Richard Altendorfer and Stephan Matzka Abstract— In safety-critical driver assistance systems such as automatic emergency braking that require the estimation of the vehicle’s environment usually a measure of confidence or probability of existence for tracked objects is required. We review and assess existing approaches of obtaining such measures. We propose a new method of computing a probability of existence by relaxing the underlying assumption of a Bayes estimator. The benefits of this approach compared to a standard Bayes estimator are demonstrated and illustrated by experimental results.
I. INTRODUCTION Automotive driver assistance systems (DAS) such as adaptive cruise control (ACC) or lane departure warning (LDW) need to perceive the environment using exteroceptive sensors. As DAS become more sophisticated and move from comfort applications such as ACC to safety-critical applications such as automatic emergency braking (AEB), the requirements with respect to their failure modes become more stringent. In particular an erroneous deployment of an automatic emergency braking maneuver, e. g. the automatic braking of a car with a maximum deceleration of up to −1g = −9.8m/s2 going at top speed on an empty highway is considered unacceptable to the driver. In addition to the discomfort and annoyance of the driver, such erroneous deployment can be dangerous and even lethal to the occupants of the braking as well as the following vehicle in case of a rearend collision. Hence qualifying conditions must be imposed such that erroneous deployments or false alarms are kept at an absolute minimum - a typical requirement is 1 false alarm per 10 million driven kilometers. Those qualifying conditions can be applied at every component of the driver assistance system: sensors, perception, controller, and actuation. In this paper we will focus on the perception module. There, those qualifying conditions can deal with the estimated relative dynamics of a driving situation, i. e. how the perceived objects move with respect to the own vehicle and whether any of those poses a threat to the own vehicle. Common criteria are an estimated timeto-collision (TTC), see e. g. [1] or a collision probability see e. g. [2]. On the other hand, estimated objects need not be what they seem to be, e. g. an object could be classified by an identity estimation module as a truck whereas in reality this object corresponds to a passenger car. Even worse, an estimated object might correspond to a real object irrelevant to the vehicle threat assessment, e. g. a soda can, or it can R. Altendorfer is with Driver Assistance Systems, TRW Automotive, 56070 Koblenz, Germany
[email protected] S. Matzka is with Heriot-Watt University, Edinburgh, United Kingdom
[email protected]
even not correspond to any real object at all - a non-existing object based for example on spurious radar reflections. In this paper we will focus on the last case, namely on quantities that estimate the confidence in an estimated object or quality of an estimated object. They should allow the distinction whether an estimated object corresponds to a real object or whether it corresponds to random noise or clutter. Sometimes those quantities can also be interpreted as a probability of existence. In the first part of this paper we will review existing approaches to a confidence measure and assess their ability to discern existing and non-existing objects. In the second part we propose a new approach to computing a probability of existence based on an extension of a Bayes estimator and illustrate some of its features by experimental results. II. REVIEW OF OBJECT CONFIDENCE MEASURES In this section we review existing approaches to computing a confidence measure for an object, in particular the track score function detailed in [3], Integrated Probabilistic Data Association (IPDA) [4] and its extension Joint Integrated Probabilistic Data Association (JIPDA) [5], the Probability Hypothesis Density (PHD) filter [6] as well as the track quality measure introduced in [7]. The derivation of a confidence measure from a cascade of classifiers particularly relevant for image based object detection is discussed in the last subsection. A. Track Score The track score formalism discussed in [8], [3] which is based upon the stochastic modeling in [9] was originally developed as a means to evaluate different hypotheses of associating measurements to tracks. However, it is commonly used for track management, i. e. for track initiation, confirmation, and deletion. The track score computed for every track is defined by the so-called logarithmic likelihood ratio (LLR) LLRk = ln
p(Z k |∃)p(∃) p(Z k |@)p(@)
(1)
where Z k = {zk , zk−1 , . . . , z0 } denotes a set of measurements as defined in app. A. Here, we have used ∃ (object exists) and @ (object does not exist) instead of the original notation H1 (true target) and H0 (false alarm). After 0 |∃)p(∃) initialization LLR0 = ln p(z p(z0 |@)p(@) at the first measurement, the LLR is updated additively upon the reception of a new
measurement.1 No dynamical model for transitions between ∃ and @ is assumed. This simple addition is based upon the assumption of the conditional independence of the measurement likelihoods p(Z k |∃) = p(zk |∃) · · · p(z0 |∃)
(2)
which in turn is a consequence of the assumption underlying a Bayesian estimator, see app. A. Indeed, the track score function is simply the logarithm of the odds of a Bayesian estimator for a binary hypothesis, see app. D, as the LLR can be rewritten using Bayes’ theorem LLRk
= =
p(Z k |∃)p(∃) p(Z k |@)p(@) p(∃|Z k ) ln 1 − p(∃|Z k )
ln
(3)
In [3] the measurement likelihood ratio is then factored into a kinematic and a signal-related contribution where the signalrelated contribution can depend upon the amplitude of a radar measurement, for example, whereas the kinematic contribution contains the measurement likelihood of a tracked object assuming a Gaussian distribution p(zk |ξk ) = N (zk ; h(ξk ), R)
p(zk |∃) p(zk |∃, ξk ) → ln p(zk |@) p(zk |@, ξk )
C. Probability Hypothesis Density filter
(5)
hence the track score function deviates from the simple Bayes estimator in (26) as its formula is only valid for a binary hypothesis and not for a continuous state such as ξk . On the other hand, the measurement likelihood is usually already used in the estimation of the kinematic state of a tracked object (e. g. in a Kalman filter) and we will discuss in subsection III-B.3 how to combine the estimation of a probability of existence and the estimation of an object’s state to determine a (position-dependent) probability of existence. In [3] the track score is used for track confirmation and deletion using the sequential probability ratio test. The track deletion for established tracks is proposed to occur when the difference between the current track score and the maximum track score falls below a certain threshold. This means that the absolute value of the true target probability p(∃|Z k ) is not considered for track deletion. An extension of this simple Bayes estimation to probabilistic data association (PDA) is reviewed in the next subsection. 1 Strictly
In [10], [4] an approach is presented that integrally combines the PDA technique (see e. g. [11]) with a probability of track existence. In the track initiation phase the usual binary hypothesis 1. track exists (∃), 2. track does not exist (@) is considered; for an established track three hypotheses are considered: 1. track exists and is observable (∃o ), 2. track exists but is not observable (∃n ), 3. track does not exist (@). The temporal dynamics of those two phases are modeled by separate Markov chains, the measurement update for track initiation is based upon a Bayes estimator as in (25), the measurement update for established tracks is based upon a Bayes estimator with the three states ∃o , ∃n , @ and an assumption about the measurement likelihoods: p(zk |∃n ) = p(zk |@). Those Markov chains have also been used in the integrated track splitting algorithm [12]. In the IPDA framework the association probabilities βk,i are not just the normalized innovations of the kinematic state but also contain the track’s probability of existence as a conditional parameter. This framework is extended in [5] to the joint probabilistic data association (JPDA) technique (see e. g. [11]) where simultaneous associations of one measurement with several tracks are properly taken into account. In [13] an implementation of the two-state Markov chain was proposed for an automotive setting.
(4)
where h is the output function and R is the measurement covariance matrix. This, however, requires the extension of the state in the LLR from the binary hypothesis (∃) to (∃, ξ) ∆LLRk = ln
B. (Joint) Integrated Probabilistic Data Association
speaking, only the update can be called LLR, because the entire p(∃) expression contains not only likelihoods but also p(@) stemming from the initialization.
Multi-target Bayes filtering where association and the estimation of the number of objects is completely integrated in a Bayesian manner is currently computationally infeasible. However, the multi-target posterior distribution can be approximately characterized by its first order moment – the probability hypothesis density [6]. Its integral over a region of state space yields the expected number of objects in that region. In [14] an automotive implementation was described where individual probabilities of existence for distinct objects were extracted from the multi-modal probability hypothesis density. D. Track Quality In [7] a track quality measure for assignment based trackers is proposed where only one-to-one assignment techniques for measurement to track association are considered. For the temporal dynamics a two-state (∃, @) Markov model is proposed that does not allow the transition @ → ∃. For the measurement update step the likelihoods of events like ”the target exists but the associated measurement is a false alarm” are estimated using implicit assumptions about the statistical independence of the underlying conditional probabilities. In a numerical study the advantages of using this track quality measure for track management over standard fixed logic based algorithms is demonstrated. E. Existence measures from cascaded classifiers In video-based object recognition systems, descriptors such as Haar-like features (cf. [15]) or SIFT [16] are frequently combined with a machine learning system such as
AdaBoost [17]. There, cascaded classifier structures have been shown to be computationally efficient. As an alternative to these cascade approaches, the use of a probabilistic boosting tree (PBT, see fig. 1) for object recognition is investigated in [18]. A PBT has the advantage of providing a probability instead of a binary decision for each object candidate which makes it possible to assign a probability of existence to every candidate. p(Ů|z) q(l1=Ů/ |z)
q(l1=Ů|z)
It is hard to interpret existence without position. The fact that an estimated object exists with a probability of 95% without reference to a position has little practical meaning as its existence at a faraway galaxy is generally not relevant. For the decision to execute an automatic emergency braking maneuver, however, the information that an object exists with 95% probability in a certain region in front of the car is highly relevant. This is achieved by “marginalization” over a certain volume V0 of the kinematic state: Z p(∃, ξ ∈ V0 |Z k ) = p(∃, ξ|Z k )dξ (6) V0 k
If in the estimation of p(∃, ξ|Z ) the initialization as well as the prediction and update step do not mix ∃ and ξ then the q(l2=Ů/ |l1=Ů/,z) q(l2=Ů|l1=Ů/,z) q(l2=Ů/ |l1=Ů,z) q(l2=Ů|l1=Ů,z) joint pdf factorizes: p(∃, ξ|Z k ) = p(∃|Z k )p(ξ|Z k ). While the approaches discussed above model the probability of existence hypotheses and the association uncertainties p(Ů|l2=Ů,l1=Ů,z) p(Ů|l2=Ů/,l1=Ů/,z) p(Ů|l2=Ů,l1=Ů/,z) p(Ů|l2=Ů/,l1=Ů,z) in different ways and to varying degrees of complexity, the update steps for the track score and (J)IPDA are based on a Fig. 1. Probabilistic boosting tree (PBT) with N = 2 levels for the simple Bayes estimator with two or three hypotheses as in computation of a probability of existence p(∃|z) conditioned on the image app. A. However, as discussed in [20], the only assumption in sample z. Cascaded classifiers (gray circles) do not contain left sub-trees a Bayes estimator (13) - namely that the information from the (white circles). new measurement given the underlying state is independent Given a usually rectangular subset z of a video frame, of the previous measurements - should be critically reviewed every link of the PBT corresponds to the outcome of an Ad- on a case-by-case basis. In particular, this assumption reaBoost classifier returning a probability q(ln |ln−1 , . . . , l1 , z) quires the state to be complete, i. e. “large” enough to capture at level n with ln ∈ {∃, @}. A single frame probability of all influencing factors for the measurement. On the other existence p(∃|z) can then be computed by hand, the state for the probability of existence is the smallest X possible - just a binary hypothesis. The limitations of such a p(∃|z) = q(l1 |z)·q(l2 |l1 , z)·. . .·p(∃|lN , . . . , l1 , z) small state were tried to be overcome in e. g. [3] by the inclul1 ,...,lN ∈{∃,@} sion of the state in the LLR measurement update (5). Here we where p(∃|lN , . . . , l1 , z) denote the probabilities of the leaf propose a different approach, namely the incorporation of the measurement history into the measurement likelihood. This nodes of the PBT with N levels. In [19] the PBT method is extended towards use with cas- is not possible in ordinary Bayes estimation because by the caded classifiers. There, determining the probability p(∃|z) completeness of the state the measurements are conditionally cannot be performed in the same manner as for PBTs. First, independent, see (2). By relaxing the assumption underlying cascaded classifiers do not contain left sub-trees and thus a Bayes estimator that the estimated state be complete, we cannot be expanded completely. Second, the strong learner’s will use a “generalized” Bayes estimator that allows such a threshold Θn at each level n is adapted to guarantee a certain dependence of the measurement history within a Bayesian context. The salient distinction between the two estimators detection rate. The absence of left sub-trees can be remedied by the is summarized in the following box: use of empirical probabilities, whereas the adaptation of • Bayes estimation: the strong learner’s threshold Θn influences the calculation state x complete of q(ln |ln−1 , . . . , l1 , z). An adaptation of the method to p(zk |Z k−1 , xk ) = p(zk |xk ) determine the probabilities q(ln |ln−1 , . . . , l1 , z) for a strong • Generalized Bayes estimation: learner’s threshold Θn 6= 0 is presented in [19]. state x incomplete k−1 III. P ROBABILITY OF EXISTENCE USING A GENERALIZED p(zk |Z k−1 , xk ) = p(zk |Zk−l , xk ) BAYESIAN ESTIMATOR (no simplification or restriction to a finite measurement history) A. Concept p(Ů|l1=Ů/,z)
p(Ů|l1=Ů,z)
All of the previously introduced confidence measures serve as an indicator of how much confidence we have that the estimated track actually corresponds to an existing object. Those confidence measures are either the probability of a binary hypothesis p(∃|Z k ) or a joint probability density function (pdf) p(∃, ξ|Z k ) where ξ is a state vector containing e. g. position, speed, etc, of the object.
For details of the derivation of a standard Bayes estimator and the generalized Bayes estimator we refer to appendices A and C. B. Algorithm The proposed algorithm consists of three parts: 1. the prediction step using a Markov model, 2. the update step using
a generalized Bayesian estimator (see app. C) where the measurement likelihoods take into account the measurement history, and 3. the marginalization of the joint pdf to arrive at a position dependent probability of existence. Contrary to (J)IPDA, we do not consider probabilistic data association techniques but focus on unique neighbor association methods. In this setting it will be shown below that the joint pdf factorizes p(∃, ξ|Z k ) = p(∃|Z k )p(ξ|Z k ). The estimation of kinematic state ξ and probability of existence ∃ is schematically depicted in fig. 2.
(
p ξk Z k
(
p ξ k +1 Z
)
k
)
(
p ∃k Z
(
)
p ∃k +1 Z prediction
(
p ξ k +1 Z k +1
association, update
prediction
k
sensor measurements
k
)
(
p ∃k +1 , ξ k +1 ∈ V0 Z localized probability of ex.
) update
(
p ∃k +1 Z k +1
k +1
)
)
clues about existence: detection, radar amplitude, classifier score, etc
Fig. 2. Separate estimation of object state ξ and existence ∃ under the factorization assumption p(∃, ξ|Z k ) = p(∃|Z k )p(ξ|Z k ). For a localized probability of existence to be used for an automatic emergency brake trigger, for example, the joint pdf is “marginalized” over a certain volume V0 of the kinematic state.
1) Prediction: In [8] and [3] the dynamical model for the track score was considered constant, i. e. it was assumed that a true target does not spontaneously change to clutter from one time step to the next. In the military setting of [9], an object’s birth and destruction was explicitly modeled. In [13] the dynamics for the probability of existence was modeled by a Markov chain and depended upon the sensor ranges and object occlusions such that for example the transition probability from existence to non-existence was higher outside the sensor field of view (FOV). The rationale for this is that the opposite of existence included nonexistence, non-perceivability, and non-relevance. On the other hand, we want to purely model an object’s existence without any occlusion or relevance reasoning. In automotive traffic the existence of a real object does not actually change over time unless it e. g. crashes and disintegrates into pieces; this holds even if it moves outside a sensor’s FOV. However, we still want to decrease the probability of existence from one time step to the next independent of any received measurements - this is a conservative, precautionary measure with the purpose of counteracting unmodeled effects and modeling errors that lead to a too high estimated probability of existence. For a consistent framework where the transition probabilities add up to one we therefore model the prediction stage as a complete Markov chain2 - albeit with very small birth and destruction probabilities: p(∃k+1 |@k ) 1, p(@k+1 |∃k ) 1. This is Markov chain one as in [4]. This Markov chain has the important side-effect of limiting the probability of existence to a value < 1.0. 2 For
B.
a review of how Markov chains arise in Bayes estimation see app.
p(∃k +1 ∃k ) p(∃k +1 ∃k )
∃
p(∃k +1 ∃k )
∃ p(∃k +1 ∃k )
Fig. 3.
Markov model for probability of existence dynamics.
2) Update: Using the generalized Bayesian estimator as derived in app. C the update step using measurement likelihoods for measurement information that gives direct clues about an object’s existence (a measurement has been received and associated, radar amplitude, etc) becomes LLR(∃k |Z k ) = LLR(∃k |Z k−1 ) + ln
k−1 p(zk |Zk−l , ∃k ) k−1 p(zk |Zk−l , @k )
(7)
This is the generalization of the update formula eq. (26) for a binary hypothesis as derived in app. D. If, for example, the measurement information used consists of measurement detected and associated (D) or not associated or not detected (6D) only, and the history of the last l = 3 measurements is taken into account, one could e. g. assume that the measurement likelihood of not having an associated measurement at k given that no measurements have been associated in the past three cycles and given the object does not exist is larger than if a measurement has been associated in the past three cycles: p(6Dk |{6D, D 6 ,D 6 }, @k ) > p(6Dk |{D, D 6 ,D 6 }, @k )
(8)
Unlike in [3] where the kinematic state innovation is taken into account for the update, we do not include it here, as it would require an extension of the state as in (5). However, a large state covariance reduces the localized probability of existence discussed in the next paragraph. 3) Localized probability of existence: In order to obtain the probability that a tracked object exists in a certain state volume V0 , the joint pdf is integrated over V0 . Since both the above described update and prediction steps do not mix ∃ and ξ, the pdf factorizes p(∃k , ξk |Z k ) = p(∃k |Z k )p(ξk |Z k )
(9)
and the “marginalization” simplifies to Z p(∃k , ξk ∈ V0 |Z k ) = p(∃k , ξk |Z k )dξk V0 Z k = p(∃k |Z ) p(ξk |Z k )dξk (10) V0
A large state covariance will reduce the integral in (10). IV. E XPERIMENTAL RESULTS We have evaluated the above algorithm as part of an automotive collision mitigation braking (CMB) system based on a 77GHz radar sensor with a cycle time of 40ms. The system can autonomously brake in case of a perceived
k−1 p(zk |Zk−l , ∃k ) k−1 p(zk |Zk−l , @k )
≈
k−1 X i=k−l
ci ln
p(zi |∃i ) p(zi |@i )
Confidence measure (LLR)
16 14 12 190
100
192
194
196
198 200 time [s]
202
204
206
208
1 missed detection 2 missed detections
many missed detections
80 target amplitude
190
192
194
196
198 200 time [s]
202
204
206
208
Fig. 4. LLR using ordinary Bayes estimation and using generalized Bayes estimation for an existing sample track over its life time.
a better classifier. Receiver operating characteristic (ROC)
(11)
Here the single measurement likelihoods are obtained from a CA-CFAR detector as in app. E. In addition, if no more than two detections are missed over the last l updates, their measurement likelihoods are not included in the generalized measurement likelihood computation. This reflects the radar’s characteristic pattern of missed detections, see fig. 4. Likewise, if there are only two or less detections over the last l updates, they are also not included in the likelihood computation. In this manner such heuristic rules can be incorporated in a Bayesian context. p(∃k |Z k ) The following plot depicts LLRk = ln 1−p(∃ during k k |Z ) a track’s life for a Bayesian estimator and for a generalized Bayesian estimator. For better illustration of the effect of the generalized Bayesian estimator the localized probability of existence is not shown here as variations of the position covariance would be superimposed on the variations due to the measurement likelihoods. As a consequence of our heuristic reasoning the probability of existence does not significantly decrease in case of one or two missed detections whereas the fall-off before track deletion is slightly deferred due to the low-pass behavior of the FIR-filter in eq. (11). For a quantitative analysis of the performance of the two estimators we have evaluated and compared their receiver operating characteristics (ROC, see e. g. [21]) in fig. 5 with a combination of highway and city driving scenarios. An estimated track at every time step was classified as “existent” if its LLR was above a certain threshold and “non-existent” otherwise. The classification result was then compared to the object’s true existence. Variation of the threshold for the LLR yielded the ROC curve in fig. 5. It can be seen that the ROC curve of the generalized Bayesian estimator is “northwest” of the ordinary Bayesian estimator and is hence
Bayes generalized Bayes
18
60
1
0.9
0.8 True existence rate
ln
20
120 Amplitude [dB]
impending collision with a deceleration of up to −0.5g. As a qualifying condition for collision mitigation braking in order to meet the stringent limits on the false alarm rate the localized probability of existence p(∃, ξ ∈ V0 |Z k ) must exceed a certain threshold. During the life time of a stable, established track of the radar tracker it can happen - without any apparent reason that for a few consecutive measurement cycles no targets can be associated. On the other hand a few consecutive targets in the same target measurement bin need not indicate the presence of a real object. Instead of trying to understand and model this behavior in terms of the highly complex physical and algorithmic aspects involved (radar transmission, reflection, reception, signal processing, estimation, ...) we will model this behavior for the computation of a probability of existence in terms of generalized measurement likelihoods as in (7). The expression for the generalized measurement likelihood is modeled as a linear superposition of single measurement likelihoods with weights ci decreasing with measurement age
variation of LLR threshold
0.7
0.6
0.5
0.4 Bayes generalized Bayes 0.3
0
1
2
3 4 5 False existence rate
6
7
8 −3
x 10
Fig. 5. Receiver operating characteristic (ROC) curves for ordinary and generalized Bayes estimation. The curve parameter is the LLR threshold.
A couple of remarks are in order: first, the ROC curves depend on the driving scenarios used. For example, as a Bayesian estimator needs to build up its LLR value after initialization over several consecutive associations before it exceeds the threshold, the true positive rate will never reach 1.0. The shorter the life time of a track, the smaller its true positive rate. Hence these results based on the data set used must be considered as preliminary. Second, the true and false positive rates are the classification rates for the (nonlocalized) probability of existence, they are not the rates for a CMB system where a false positive rate of the order of 10−4 would be unacceptably large. For a CMB trigger conditions on the localized probability of existence and on a collision probability coming from the estimated relative trajectory must be met - among others - which will drastically reduce the false alarm rate. Third, while the heuristic model in (11) already yields an improvement over an ordinary Bayes
estimator, for a more descriptive approach machine learning techniques could be used to determine those generalized measurement likelihoods. V. CONCLUSIONS In this paper a new method for computing a probability of existence for target tracking is presented. After a review and assessment of existing approaches we propose an approach where the joint probability distribution function p(∃, ξ|Z k ) factorizes allowing for a decoupled estimation of state and probability of existence. For the update step we propose a generalization of a Bayes estimator that incorporates the measurement history in measurement likelihoods without the need to enlarge the state in order to account for unmodeled effects. The prediction step is implemented by a Markov chain. The joint pdf is then marginalized over a relevant state volume as a qualifying condition for the deployment of a collision mitigation braking system. The advantage of this approach over an ordinary Bayes estimator in the update step is demonstrated by a superior ROC curve generated by varying the threshold of the logarithmic likelihood ratio. VI. ACKNOWLEDGEMENTS Helpful discussions with S. Wirkert are gratefully acknowledged.
B. Bayes estimation and Markov chains If a dynamical model is part of the estimator, then the prediction xk−1 → xk must be specified. For discrete valued xk , the law of total probability can be used as in (15) to write X p(xk |Z k−1 ) = p(xk |Z k−1 , xk−1 )p(xk−1 |Z k−1 ) (17) xk−1
Analogous to the assumption (13) we assume that the dynamics of the system can be expressed in terms of the state x only, i. e. p(xk |Z k−1 , xk−1 ) = p(xk |xk−1 )
Hence the prediction step can be implemented in terms of a Markov chain. X p(xk |Z k−1 ) = p(xk |xk−1 )p(xk−1 |Z k−1 ) (19) xk−1
Note however, that Bayes estimation and Markov chains address different aspects: a Bayes estimator specifies the incorporation of new sensory information; a Markov chain specifies the discrete (temporal) dynamics. C. Generalized Bayes estimation Let us relax the underlying assumption (13) of a Bayesian estimator. Instead of
A PPENDIX A. Bayes estimation The recursive formulation of a general Bayes estimator with individual measurements zk , a set of measurements accumulated over time Z k = {zk , zk−1 , . . . , z0 } with temporal index k and estimated state xk reads (see e. g. [20], [22]) p(zk |xk )p(xk |Z k−1 ) (12) p(zk |Z k−1 ) The key (and the only) assumption made in the derivation of a Bayes estimator (12) is (see e. g. [20]) p(xk |Z k ) =
p(zk |Z k−1 , xk ) = p(zk |xk )
(13)
This means that the probability of observing zk given the underlying state xk is independent of the previous measurements Z k−1 or in other words: in p(zk |Z k−1 , xk ) the information about the previous measurements Z k−1 is already completely contained in xk . A consequence of (13) is the conditional independence of the new measurement with respect to the old measurements ⇒ p(zk , Z k−1 |xk ) = p(zk |xk )p(Z k−1 |xk )
Under the assumption (13) this simplifies to Z p(zk |Z k−1 ) = p(zk |xk )p(xk |Z k−1 )dxk
p(zk |Z k−1 , xk ) = p(zk |xk )
(20)
k−1 p(zk |Z k−1 , xk ) = p(zk |Zk−l , xk )
(21)
we assume
k−1 where Zk−l = {zk−1 , zk−2 , . . . , zk−l }. Hence the measurement zk is not only conditionally dependent upon the state xk k−1 but also on a series of previous measurements Zk−l . In this situation the state xk is assumed not to fully contain all the information necessary to predict the new measurement. We will see that despite this generalization the resulting formula looks similar to that of a standard Bayesian estimator. We want to compute
p(xk |Z k ) =
(16)
Formula (12) provides the basis for most standard estimators such as the Kalman filter and the particle filter.
p(Z k |xk )p(xk ) p(Z k )
(22)
Applying the chain rule of conditional probabilities to p(Z k |xk ) where Z k = {zk , Z k−1 } we obtain p(Z k |xk ) = p(zk , Z k−1 |xk ) = p(zk |Z k−1 , xk )p(Z k−1 |xk )
(14)
Using the law of total probability, the denominator of (12) can be expressed in terms of p(xk |Z k−1 ) Z k−1 p(zk |Z ) = p(zk |Z k−1 , xk )p(xk |Z k−1 )dxk (15)
(18)
k−1 = p(zk |Zk−l , xk )p(Z k−1 |xk )
where the last equality is due to assumption (21). Hence eq. (22) reads k−1 p(zk |Zk−l , xk )p(Z k−1 |xk )p(xk ) p(Z k ) k−1 p(zk |Zk−l , xk )p(xk |Z k−1 )p(Z k−1 ) = p(Z k ) k−1 p(zk |Zk−l , xk )p(xk |Z k−1 ) = p(zk |Z k−1 )
p(xk |Z k ) =
which is identical to ordinary Bayes estimation (12) except k−1 that the measurement likelihood p(zk |Zk−l , xk ) can now k−1 depend upon previous measurements Zk−l . The denominator can be written under the assumption (21) using again the law of total probability Z k−1 p(zk |Z k−1 ) = p(zk |Zk−l , xk )p(xk |Z k−1 )dxk (23) D. Bayes estimation of a binary hypothesis If the state xk consists of just a single, binary hypothesis Hk ,3 i. e. either the hypothesis is true Hk or it is false H k , eq. (12) can be simplified. With xk ∈ {Hk , H k } the denominator of (12) using (16) reads p(zk |Z k−1 ) = p(zk |Hk )p(Hk |Z k−1 ) +p(zk |H k )(1 − p(Hk |Z k−1 ))
(24)
Hence the recursive Bayes estimator for a binary hypothesis reads p(Hk |Z k ) =
p(ai |∃i ) = p(ai |@i ) =
1 2σ 2 (1
a
+ ρ)
e
i − 2σ2 (1+ρ)
1 − ai2 e 2σ 2σ 2
(27) (28)
For no detection (zi = D 6 ) the likelihoods are obtained by integrating the amplitude up to the detection threshold using the pdf appropriate for a cell averaging constant false alarm rate (CA-CFAR) detector for single pulse detection and Gaussian clutter and noise [24] p(6D|∃i ) = 1 −
1
τ (1 + m(1+ρ) )m 1 p(6D|@i ) = 1 − τ m ) (1 + m
(29) (30)
Here, τ is the CA-CFAR threshold multiplier and m is the number of cells to be averaged over. R EFERENCES
p(zk |Hk )p(Hk |Z k−1 ) (25) p(zk |Hk )p(Hk |Z k−1 ) + p(zk |H k )(1 − p(Hk |Z k−1 )) This form can be found in eq. (6.15) in [8] as well as in eq. (2.14) in [4]. p(Hk |Z k ) By taking the logarithm of the odds p(H = |Z k ) p(Hk |Z k ) 1−p(Hk |Z k )
variance σ 2 of the assumed Gaussian clutter and noise, and the signal-to-noise ratio (SNR) ρ are given by (see e. g. [23])
k
the Bayes recursion simplifies considerably:
LLR(Hk |Z k ) := ln
p(Hk |Z k ) 1 − p(Hk |Z k )
= LLR(Hk |Z k−1 ) + ln
p(zk |Hk ) (26) p(zk |H k )
By using Bayes’ theorem, the Bayes estimation using hypothesis likelihoods can be reformulated in terms of single measurement hypothesis probabilities: p(Hk |Z k ) = p(Hk |zk )p(Hk |Z k−1 )(1 − p(Hk )) p(Hk |zk )p(Hk |Z k−1 )(1 − p(Hk )) +(1 − p(Hk |zk ))(1 − p(Hk |Z k−1 ))p(Hk ) Here, the “prior” probability p(Hk ) appears. This probability is often tacitly assumed to be 0.5. E. Measurement likelihoods for a CA-CFAR detector For the case of a Swerling I target model the measurement likelihoods for a detected target in terms of the measured amplitude zi = ai (output of a square law detector), the 3 The notation H can be slightly confusing as the hypothesis H does k not change over time, only its probability does. Nevertheless, in order to maintain the previously established notation for general, continuously valued states, we assign a subscript k to H. Hence p(Hk ) for example denotes the probability of hypothesis H at discrete time k; it could have also been written as pk (H).
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