- Email: [email protected]
MULTI CLASSIFIER FUSION TO AIR TARGET CLASSIFICATOIN
MULTI CLASSIFIER FUSION TO AIR TARGET CLASSIFICATOIN
Hassan Golmohammad, Hossein Bolandi and Farhad Fani Saberi
MS student of Iran University of Science and Technology (IUST), Associated Professor of IUST, PhD student of IUST
Abstract: Results of two classifiers are combined to make a more reliable decision about the type of airplanes. Two classifiers are maximum acceleration classifier, which is implemented as an IMM filter, and maximum speed classifier which is a classical Bayesian classifier. Since TBM is a cautious algorithm in data fusion applications, it is adopted for feature fusion. Finally, Monte Carlo simulation is carried out to show the efficiency of the proposed approach. Copyright © 2007 IFAC Keywords: manoeuvring target, tracking filter, sensor fusion, classifier.
1. INTRODUCTION Detection, classification and tracking of targets are three fundamental components of Tactical Surveillance Systems (TSS) which have been based on multi sensor data fusion. Multi sensor data fusion is an emerging technology applied to Department of Defense (DoD) areas such as automatic target recognition (ATR), battle field surveillance, and guidance and control of automatic vehicles and non-DoD applications such as monitoring of complex machinery, medical diagnosis and smart buildings (Hall and Llinas, 1997). In principle, fusion of multi sensor data provides significant advantages over single source data. In a target tracking applications, for example, multi sensor data converted to observations of angular direction, range and range-rate may be converted in turn into an estimate of the target's position and velocity. Similarly observations of the target's attributes, such as radar cross section, infrared spectra, and visual image may be used to classify the target and allow a feature-based classifier to declare an assignment of a label specifying target identity(e.g. F-16 aircraft). Finally, understanding the motion of the target and its relative motion with
respect to the observer may allow a determination of the intent of the target (Hall and Llinas, 1997). Observational data may be combined, or fused, at a variety of levels from the raw data(or observation) level to a feature level or at the decision level. Fig.1 represents the hierarchy of these three fusion levels. The output of each classifier are a decision and finally a single decision about the target is the output of the fusion system (Waltz and Llinas, 1990). Among three fundamental tasks of TSS, classification is the most critical one. An unknown target may be classified from four aspects: · · · ·
Target category: for example, fighter or bomber. Type of the target: for example, F-5 or MIG. Target nationality: friend or foe. Intent of the target: threat or non-threat.
The paper will consider the classification problem in the first level. In other words, the proposed algorithm will identify the target category. There are three basic approaches for information fusion. Feature-level fusion is selected since a feature extraction is possible in this special case. However, in many tactical applications, results achieved by
applying basic approaches are not satisfactory. Two main approaches may be suggested to make a more reliable decision. They are multi level fusion and multi classifier fusion. The implementation of multi level fusion algorithm will be too expensive (McCullough et al., 1996). So, the paper will take a multi classifier fusion approach. Although, the use of kinematic features for target classification is not a new subject, the basic idea of the approach used in this paper is that each target type has its own motion constraints such as a maximum value of speed and a maximum value of acceleration which are a prior known. The more the flight envelopes are different, the easier it is to distinguish between the classes. Ristic, Gordon and Bessell (Ristic et al., 2004) have used the maximum observed acceleration to classify air targets in three classes: commercial, large military aircraft and fighter planes. Their approach has a general weakness. If a fighter plane does not show a high acceleration motion, it will be assumed a large military aircraft or a commercial one. In this paper, we use the same numerical example, and show that by adding a maximum speed classifier to the maximum acceleration classifier used in (Ristic et al., 2004), the classification result will become more reliable. Remainder sections of the paper is organized as follows. In section two, the numerical example is introduced. Section three contains a brief review of the maximum acceleration classifier. In section four, details of the maximum speed classifier will be discussed and finally in section five, an algorithm has been selected for fusing the results of two classifiers. 2. PROBLEM FORMOLATION This section demonstrates the numerical example used for simulation as a motivation. Suppose the aim is to classify air targets into one of the three classes of targets: · Class 1: Commercial planes with very modest maneuvering capabilities. · Class 2: Large military aircrafts such as bombers with medium level of maneuvering capabilities. · Class 3: A class of light and agile military aircrafts such as fighters with extreme maneuvering capabilities. The available target features for classification are the maximum normal acceleration and the maximum tangential speed of the target, during a certain interval of time. These features can be extracted from positional measurements provided by a radar or a visual sensor. The class motion envelopes (the normal acceleration and the tangential speed) for each class are shown in Fig.2. In this case each envelope is an ellipse. For Class1 targets, the motion with small acceleration ( a < 1g , where g=9.81m/s2 is
Fig. 1. Hierarchy of data fusion different levels(McCullough et al., 1996), the dashedline area shows the focus of the paper’s work. acceleration due to gravity) is known as the nearly constant velocity (CV) motion because the higher acceleration causes sickness in passengers. In this class the maximum speed of targets is slightly less than one Mach(330m/s the speed of sound in free space). For example, Boeing 747 nominal speed is 250m/s which is about 0.85 Mach. Targets of Class2 can move with acceleration up to 3g but not more, due to their size. Maximum speed of this class targets is about 1.5 Mach. For example, the nominal speed of B-58 bomber is 1.3 Mach. Class3 targets are light and agile, so they can accelerate more than 5g and their speed can be more than 2 Mach. For example, maximum speed of F-16, the famous fighter, is 2.4 Mach. Moreover, it is assumed that the steady state acceleration of all three classes of targets can be considered zero because of minimal fuel consumption, minimal stress for pilots, etc. Most target manoeuvres are coupled across different coordinates. For simplicity, however, many maneuver models developed assume that this coordinate coupling is weak and can be neglected. As a consequence, we need to consider only a generic coordinate direction (Li and Jilkov, 2004). Specifically x(t ) = a (t ) + a (t )
(1)
a (t ) = -aa (t ) + w(t )
(2)
0
and 0
0
where a (t ) is the zero mean colored acceleration with a uniform distribution probability density function, a (t ) is the mean of maneuvering acceleration considered as a constant in each sampling period and will be used for classification, w(t ) is a zero mean white noise, and a is the reciprocal of the maneuver acceleration time constant. 0
Letting a (t ) = a (t ) + a (t ) Eqs.(1) and (2) can be rewritten as 0
(3)
x(t ) = a (t )
(4)
and a (t ) = -aa(t ) + aa (t ) + w(t ) The state equation of a filter is
(5)
x (t ) = Ax (t ) + Ba (t ) + B w(t )
(6)
wt
where x (t ) = [x(t ) x (t ) x(t )] is the state vector of the target, and
Fig. 2. Illustration of flight envelopes for different classes in an acceleration-speed (a-v) plane.
é0 1 0 ù é0ù é0 ù A = êê0 0 1 úú, B = êê 0 úú, Bw = êê0úú . êë0 0 - a úû êëa úû êë1 úû This is the famous Singer model and when the sampling period is T , the discrete state equation is
3. IMM FILTER AS AN ACCELERATION ESTIMATOR
T
t
x (k + 1) = Fx (k ) + Ga (k ) + Bw w(k ) where é1 T ê F = ê0 1 ê0 0 ë
(aT - 1 + e ) a (1 - e ) a - aT
-aT
e -aT
éT 2 2ù é1 ê ú ê G = Bw = ê T ú - ê0 ê 1 ú ê0 ë û ë and w(k ) is a zero mean
2
(7)
ù ú ú, ú û
(aT - 1 + e ) a (1 - e ) a
ù ú 1 ú ú e -aT 0 û white noise sequence with T
-aT
2
-aT
variance s w2 = 2as a2 . Only the noisy position data of the target are available, so the measurement equation is y ( k ) = Hx ( k ) + n ( k )
(8)
and
H = [1 0 0] where n (k ) is a zero mean Gaussian measurement noise with variance R, independent from process noise. For a system described by Eq.(7), the Kalman filtering equations will be applied to estimate the state. Fig.3 represents the Singer model block diagram. The filter will be introduced in the next section, not only estimates the speed of the target as a state but also measures the mean of the input acceleration as a distinctive target feature.
As mentioned in previous section, one of the two features used for classification is the maximum normal acceleration. In this section, we use an interacting multiple model(IMM) to classify the target by this feature. This section is a review of the acceleration estimator considered in (Ristic et al., 2004). The proposed solution consists of a bank of classmotion matched (CMM) filters, which are implemented as a single IMM filter. Corresponding to each class, a CMM filter is needed. The three CMM filters are implemented as follows. A Kalman filter excited by input a=0, is used for Class1 targets. An IMM filter with three modes corresponding to inputs a Î {-2 g ,0,2 g} is used for Class2 targets and an IMM filter with five modes corresponding to inputs a Î {-4 g ,-2 g ,0,2 g ,4 g} is used for Class3 targets. Finally, all three CMM filters is merged in a single IMM filter to avoid unnecessary computations. This can be done due to the fact that (Fig.2) Class1 Ì Class 2 Ì Class 3 . The entire span of acceleration values is digitalized so that covers each class maneuver modes. Fig4 represents a graphical meaning of maneuver modes. The value of the acceleration of the target is a key used to classify it, in this figure acceleration modes is shown by circles and arrows indicated possible switching between models. Monte Carlo simulations have been carried out to classify an air target and the results are averaged over 25 Monte Carlo runs. This target is assumed a fighter which does not show any high maneuver motion.
w a
a
j
1 s
a
1 s
v
1 s
x
-a
Fig. 3. Block diagram of the Singer model, a is a reciprocal of the maneuver time constant.
Fig. 4. The digraph used in the design of classmatched IMM filters.
4. AN EXTRA FEATURE AXIS This section uses classical Bayesian theory to make a maximum speed classifier. The maximum speed classifier is a classifier which uses the maximum tangential speed of the target to classify targets. The speed is computed from IMM filters’ states. A sensor can be seen as a piece of an equipment that observers some data x Í X and transmits some "opinion" about the actual value of a parameter of interest h Í H , where X and H are called the observation and hypothesis domains, respectively. After that, the sensor communicates its opinion on the value of H under the form of a "likelihood" vector, l (hi x ) for each hi Î H . Target class is a time-invariant attribute which takes values from a discrete set c Î {ci : i = 1,..., n} where n is the number of classes. In order to make a decision about the target class, we need to compute the class probabilities. For this purpose a recursive equation is usually used in optimal Bayesian framework. This equation is Pk +1 (ci ) =
Lik Pk (ci ) å j =1 L jk Pk (c j ) n
(9)
where Lik is the likelihood of class i at time k and Pk (ci ) is the probability of class i at that time and recursive equation is initialized using a prior class probabilities P0 (ci ) which usually assumed to be the same for all different classes. The selection of likelihood functions plays a main role in the classification results. These functions will determine the time at witch the classifier should change its decision about the class of the target. In this research, Rayleigh distribution is adopted as likelihood functions for all three classes. The choice of Rayleigh distribution for this purpose is due to the
an Comparison: "Actual and Estimated Acceleration" 1 Estimated Normal Acc. Actual Normal Acc.
0.5
Normal Acc./g
0 -0.5 -1 -1.5 -2 -2.5 -3
0
10
20
30 40 Sampling Time
50
60
70
Fig. 5. Comparison of the actual and estimated acceleration of the target. Class Probability: "Max. Acceleration Calssifier" 0.7 0.6 0.5 Probability
The trajectory consists of two CV segments and a maneuver duty. The sampling time T is a second and simulation lasts 64 seconds which consists 63 samples. The manoeuvre is between sampling times 20 and 35 with maximum acceleration 2.5g. The motion of the target in remainder sampling times is CV (this kind of movement is usually performed by fighter pilots to hide themselves). Fig.5 compares the real and estimated normal accelerations provided by IMM tracker. It is clear that the real and estimated acceleration of the target is completely in the Class2 acceleration limit. Finally, Fig.6 shows the output of the maximum acceleration classifier. At first, the target is classified in Class1 and when the maneuver is performed, classifier changes its opinion and claims that the target is in the Class2 category. Although the classifier claims that the target is a Class2 airplane but in this situation, this classifier cannot distinguish Class2 and Class3 target, since the probability of Class2 and Class3 is approximately the same. This classification result is the final result achieved later in (Ristic et al., 2004).
0.4 0.3 0.2 0.1 0
0
10
20
30 40 Sampling Time
50
60
70
Fig. 6. Classification results, the output of the Max. acceleration classifier. fact that this distribution, in low speeds, gives the largest likelihood to Class1 and for high speed motions all three likelihood functions approach zero. Rayleigh distribution parameters are 0.7, 1.2 and 1.8 for Class1, Class2 and Class3, respectively. Fig.7 represents these likelihood functions with respect to the normalized speed of the target. Intersections of curves represent the speed at which the classifier should switch between classes. This simple classifier is not ready to use right now, because if a target shows a high speed motion and after then goes back to the low speed class limit, the output of the classifier will be accordingly changed. But we expect that if a target shows a high speed motion in any interval of observation, then after classification result never goes back to a lower speed class. In order to remove this defect in classifier’s work, normalized likelihoods should be renormalized as follows:
L2k
=
(
(
L1k = l1k
l1k L1k
+ l2k L2k
)/ (
L1k
+
)(
L2k
)
(10)
)
(11)
L3k = l1k L1k + l2k L2k + l3k L3k / L1k + L2k + L3k (12)
Rayleigh distributions for three classes 0.7 Class1 Likelihood Class2 Likelihood Class3 Likelihood
0.6 0.5 Probability
where Lik is the normalized likelihood and Lik is the renormalized likelihood for class i used in Bayesian formula. In each iteration, the estimated speed of the target is normalized by Mach number and then the likelihoods are calculated and renormalized likelihoods are used to update class probabilities using Bayesian recursive formula (Eq.9). Fig.8 compares the target speed and its estimated value. In 20th sampling time, the speed of the target becomes more than the Class1 limit and after 19 sampling times, the Class2 limit breaks, too. Fig.9 shows class probabilities of the maximum speed classifier. As it was expected, the decision made by this classifier, contains three parts. In the beginning of the test, Class1 has the biggest probability. In next 35 sampling times, Class2 probability becomes bigger than others and ultimately, when the Class2 limit is exceeded, Class3 is selected by the maximum speed classifier.
0.4 0.3 0.2 0.1 0 0
1
2
hi ÎA
(13)
where m H [ x]( A) denotes the bba of A induced on H for measurement x and A is the complement of A with respect to set H . Let E and F be two "distinct" pieces of evidence and let m H [E ] and m H [F ] be the bbas they induce on H (in this study E and F are corresponding to speed and acceleration classifiers' outputs). We want
7
8
V t Comparison: "Actual and Estimated Speed"
Estimated Speed Actual Speed
Target Speed/Mach
1.6
hi ÎA
6
2
5. TBM FOR FEATURE FUSION
m H [ x]( A) = Õ l (hi x) Õ (1 - l (hi x) )
4 5 Speed/Mach
Fig. 7. Rayleigh distributions used as class likelihood functions.
1.8
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
10
20
30 40 Sampling Time
50
60
70
Fig. 8. Comparison of the actual and estimated speeds of the target. Class Probability: "Max. Speed Calssifer" 1 0.9 0.8 0.7 Probability
The inherent philosophy of the introduced algorithm is illustrated in Fig.10. The available target features for classification, the maximum normal acceleration and maximum along-track speed of the target, are extracted from positional measurements provided by IMM tracker. The target state and maneuver estimates are monitored. At each filtering step, the estimated speed from CMM filters converts to speed likelihoods and together with the acceleration likelihoods both are trying to improve the classification process. After then, class probabilities are updated by renormalized likelihoods and Bayesian recursive formula. The results are two feature-based classifications that should be fused. Among many approaches available for feature fusion, like Bayesian and Dempster-Shefer theories, transferable belief model(TBM) is adopted for this purpose. The TBM is a model to represent quantified uncertainties based on belief functions and unrelated to any underlying probability model (Delmotte and Smets, 2004). The central element of the TBM is the basic belief assignment(bba), denoted m . For A Í H , m( A) is the part of belief that supports A i.e. the fact that the actual value h of H is in A , and due to a lack of information, does not support any strict subset of A . Given the likelihoods l (hi x) for every hi Î H , for x Í X and for every A Í H , Smets (Smets, 1978) has proved
3
0.6
Class1 Class2 Class3
0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 40 Sampling Time
50
60
70
Fig. 9. Classification results, the output of the Max. speed classifier. to build the bba m H [E , F ] that results from the combination of the two pieces of evidence. In terminology of TBM, this operation is called conjunctive rule of combinations and is defined by m H [E , F ]( A) =
å m H [E ]( B)m H [F ](C ) , "A Í H (14)
B ,C Í H , B ÇC = A
Fig. 10. The inherent philosophy of the paper classification approach. Finally, for decision making, we should transfer to probability functions. This transformation is done using the so called pignistic transformation given by: BetP H (h) = å
A:hÎ AÍ H
m H ( A) , A (1 - m H (Æ))
"h Î H
(15)
where A is the number of elements in A and Æ is the empty subset of H . Now, we are ready to apply the TBM theory to the feature fusion problem mentioned in the paper. The output values of two distinct classification approaches introduced in previous sections, will be fed as an input of the TBM algorithm. In each iteration the probability of different classes are assumed as a likelihood of that class. Then for each classification algorithm, the amount of the belief of every subset of H are calculated using Eq.13. For fusion of two bbas( m H [E ] , m H [F ] ), the conjunctive rule of combinations is used to calculate m H [E , F ] (Eq.14). Ultimately, Eq.15 (pignistic transformation) is applied to make the final decision. Fig.11 presents the results of the maximum speed and the maximum acceleration classifiers’ fusion. It is clear that as time elapses, the probability of Class3 becomes greater than two others which means that the target is a fighter. This result is different from the result of the maximum acceleration classifier at the end of section 3. Class Probability: "TBM output" 0.9 0.8 0.7
Probability
0.6 Class1 Class2 Class3
0.5 0.4 0.3 0.2 0.1 0
0
10
20
30 40 Sampling Time
50
60
70
Fig. 11. Classification results, after fusion of two classifiers using TBM.
6. CONCLUSIONS The paper studies the classification problem in airborne applications. The feature-level fusion and multi-classifier fusion approaches are selected from the basic and advanced classification techniques, respectively. Two classifiers are the maximum speed and the maximum acceleration classifiers. In view of the fact that each classifier probes the detected target with a different point of view, two classification results may be different, as in the proposed simulation. So, two probable different decisions could be combined using belief functions. In fact, while using the TBM algorithm, class probabilities are transferred to the belief space. Then, associated beliefs are fused to a single belief and this belief is transferred back to the probability space. Results of the simulation confirms that the final decision made by the proposed approach is reliable enough. REFERENCES Delmotte, F. and P. Smets (2004), Target identification based on the transferable belief model interpretation of Dempster-Shefer model, IEEE Trans. On Systems, Man and Cybernetics, vol. 34, pp. 457-471. Hall, D.L. and J. Llinas (1997), An introduction to multisensor data fusion, Proceedings of the IEEE, vol. 85, pp. 6-23. Li, X.R. and V.P. Jilkov (2003), Survey of maneuvering target tracking. PartI: dynamic models, IEEE Trans. On Aerospace and Electronic Systems, vol. 39, pp. 1333-1364. McCullough, C.L., B.V. Dasarathy and P.C. Lindberg (1996), Multi-level sensor fusion for improved target discrimination, Proceedings of the IEEE, vol. 85, pp. 3674-3675. Ristic, B., N. Gordon and A. Bessell (2004), On target classification using kinematic data, Information fusion, vol. 5, pp. 15-21. Smets, P. (1978), “Un modele mathematicostatistique stimulant le processus du diagnostic medical,” PhD dissertation , Univ. Libre Bruxelles, Bruxelles, Belgium. Waltz, E. and J. Linas (1990), Multisensor Data Fusion ,Artech House, Norwood, Ma.
Hassan Golmohammad, Hossein Bolandi and Farhad Fani Saberi
MS student of Iran University of Science and Technology (IUST), Associated Professor of IUST, PhD student of IUST
Abstract: Results of two classifiers are combined to make a more reliable decision about the type of airplanes. Two classifiers are maximum acceleration classifier, which is implemented as an IMM filter, and maximum speed classifier which is a classical Bayesian classifier. Since TBM is a cautious algorithm in data fusion applications, it is adopted for feature fusion. Finally, Monte Carlo simulation is carried out to show the efficiency of the proposed approach. Copyright © 2007 IFAC Keywords: manoeuvring target, tracking filter, sensor fusion, classifier.
1. INTRODUCTION Detection, classification and tracking of targets are three fundamental components of Tactical Surveillance Systems (TSS) which have been based on multi sensor data fusion. Multi sensor data fusion is an emerging technology applied to Department of Defense (DoD) areas such as automatic target recognition (ATR), battle field surveillance, and guidance and control of automatic vehicles and non-DoD applications such as monitoring of complex machinery, medical diagnosis and smart buildings (Hall and Llinas, 1997). In principle, fusion of multi sensor data provides significant advantages over single source data. In a target tracking applications, for example, multi sensor data converted to observations of angular direction, range and range-rate may be converted in turn into an estimate of the target's position and velocity. Similarly observations of the target's attributes, such as radar cross section, infrared spectra, and visual image may be used to classify the target and allow a feature-based classifier to declare an assignment of a label specifying target identity(e.g. F-16 aircraft). Finally, understanding the motion of the target and its relative motion with
respect to the observer may allow a determination of the intent of the target (Hall and Llinas, 1997). Observational data may be combined, or fused, at a variety of levels from the raw data(or observation) level to a feature level or at the decision level. Fig.1 represents the hierarchy of these three fusion levels. The output of each classifier are a decision and finally a single decision about the target is the output of the fusion system (Waltz and Llinas, 1990). Among three fundamental tasks of TSS, classification is the most critical one. An unknown target may be classified from four aspects: · · · ·
Target category: for example, fighter or bomber. Type of the target: for example, F-5 or MIG. Target nationality: friend or foe. Intent of the target: threat or non-threat.
The paper will consider the classification problem in the first level. In other words, the proposed algorithm will identify the target category. There are three basic approaches for information fusion. Feature-level fusion is selected since a feature extraction is possible in this special case. However, in many tactical applications, results achieved by
applying basic approaches are not satisfactory. Two main approaches may be suggested to make a more reliable decision. They are multi level fusion and multi classifier fusion. The implementation of multi level fusion algorithm will be too expensive (McCullough et al., 1996). So, the paper will take a multi classifier fusion approach. Although, the use of kinematic features for target classification is not a new subject, the basic idea of the approach used in this paper is that each target type has its own motion constraints such as a maximum value of speed and a maximum value of acceleration which are a prior known. The more the flight envelopes are different, the easier it is to distinguish between the classes. Ristic, Gordon and Bessell (Ristic et al., 2004) have used the maximum observed acceleration to classify air targets in three classes: commercial, large military aircraft and fighter planes. Their approach has a general weakness. If a fighter plane does not show a high acceleration motion, it will be assumed a large military aircraft or a commercial one. In this paper, we use the same numerical example, and show that by adding a maximum speed classifier to the maximum acceleration classifier used in (Ristic et al., 2004), the classification result will become more reliable. Remainder sections of the paper is organized as follows. In section two, the numerical example is introduced. Section three contains a brief review of the maximum acceleration classifier. In section four, details of the maximum speed classifier will be discussed and finally in section five, an algorithm has been selected for fusing the results of two classifiers. 2. PROBLEM FORMOLATION This section demonstrates the numerical example used for simulation as a motivation. Suppose the aim is to classify air targets into one of the three classes of targets: · Class 1: Commercial planes with very modest maneuvering capabilities. · Class 2: Large military aircrafts such as bombers with medium level of maneuvering capabilities. · Class 3: A class of light and agile military aircrafts such as fighters with extreme maneuvering capabilities. The available target features for classification are the maximum normal acceleration and the maximum tangential speed of the target, during a certain interval of time. These features can be extracted from positional measurements provided by a radar or a visual sensor. The class motion envelopes (the normal acceleration and the tangential speed) for each class are shown in Fig.2. In this case each envelope is an ellipse. For Class1 targets, the motion with small acceleration ( a < 1g , where g=9.81m/s2 is
Fig. 1. Hierarchy of data fusion different levels(McCullough et al., 1996), the dashedline area shows the focus of the paper’s work. acceleration due to gravity) is known as the nearly constant velocity (CV) motion because the higher acceleration causes sickness in passengers. In this class the maximum speed of targets is slightly less than one Mach(330m/s the speed of sound in free space). For example, Boeing 747 nominal speed is 250m/s which is about 0.85 Mach. Targets of Class2 can move with acceleration up to 3g but not more, due to their size. Maximum speed of this class targets is about 1.5 Mach. For example, the nominal speed of B-58 bomber is 1.3 Mach. Class3 targets are light and agile, so they can accelerate more than 5g and their speed can be more than 2 Mach. For example, maximum speed of F-16, the famous fighter, is 2.4 Mach. Moreover, it is assumed that the steady state acceleration of all three classes of targets can be considered zero because of minimal fuel consumption, minimal stress for pilots, etc. Most target manoeuvres are coupled across different coordinates. For simplicity, however, many maneuver models developed assume that this coordinate coupling is weak and can be neglected. As a consequence, we need to consider only a generic coordinate direction (Li and Jilkov, 2004). Specifically x(t ) = a (t ) + a (t )
(1)
a (t ) = -aa (t ) + w(t )
(2)
0
and 0
0
where a (t ) is the zero mean colored acceleration with a uniform distribution probability density function, a (t ) is the mean of maneuvering acceleration considered as a constant in each sampling period and will be used for classification, w(t ) is a zero mean white noise, and a is the reciprocal of the maneuver acceleration time constant. 0
Letting a (t ) = a (t ) + a (t ) Eqs.(1) and (2) can be rewritten as 0
(3)
x(t ) = a (t )
(4)
and a (t ) = -aa(t ) + aa (t ) + w(t ) The state equation of a filter is
(5)
x (t ) = Ax (t ) + Ba (t ) + B w(t )
(6)
wt
where x (t ) = [x(t ) x (t ) x(t )] is the state vector of the target, and
Fig. 2. Illustration of flight envelopes for different classes in an acceleration-speed (a-v) plane.
é0 1 0 ù é0ù é0 ù A = êê0 0 1 úú, B = êê 0 úú, Bw = êê0úú . êë0 0 - a úû êëa úû êë1 úû This is the famous Singer model and when the sampling period is T , the discrete state equation is
3. IMM FILTER AS AN ACCELERATION ESTIMATOR
T
t
x (k + 1) = Fx (k ) + Ga (k ) + Bw w(k ) where é1 T ê F = ê0 1 ê0 0 ë
(aT - 1 + e ) a (1 - e ) a - aT
-aT
e -aT
éT 2 2ù é1 ê ú ê G = Bw = ê T ú - ê0 ê 1 ú ê0 ë û ë and w(k ) is a zero mean
2
(7)
ù ú ú, ú û
(aT - 1 + e ) a (1 - e ) a
ù ú 1 ú ú e -aT 0 û white noise sequence with T
-aT
2
-aT
variance s w2 = 2as a2 . Only the noisy position data of the target are available, so the measurement equation is y ( k ) = Hx ( k ) + n ( k )
(8)
and
H = [1 0 0] where n (k ) is a zero mean Gaussian measurement noise with variance R, independent from process noise. For a system described by Eq.(7), the Kalman filtering equations will be applied to estimate the state. Fig.3 represents the Singer model block diagram. The filter will be introduced in the next section, not only estimates the speed of the target as a state but also measures the mean of the input acceleration as a distinctive target feature.
As mentioned in previous section, one of the two features used for classification is the maximum normal acceleration. In this section, we use an interacting multiple model(IMM) to classify the target by this feature. This section is a review of the acceleration estimator considered in (Ristic et al., 2004). The proposed solution consists of a bank of classmotion matched (CMM) filters, which are implemented as a single IMM filter. Corresponding to each class, a CMM filter is needed. The three CMM filters are implemented as follows. A Kalman filter excited by input a=0, is used for Class1 targets. An IMM filter with three modes corresponding to inputs a Î {-2 g ,0,2 g} is used for Class2 targets and an IMM filter with five modes corresponding to inputs a Î {-4 g ,-2 g ,0,2 g ,4 g} is used for Class3 targets. Finally, all three CMM filters is merged in a single IMM filter to avoid unnecessary computations. This can be done due to the fact that (Fig.2) Class1 Ì Class 2 Ì Class 3 . The entire span of acceleration values is digitalized so that covers each class maneuver modes. Fig4 represents a graphical meaning of maneuver modes. The value of the acceleration of the target is a key used to classify it, in this figure acceleration modes is shown by circles and arrows indicated possible switching between models. Monte Carlo simulations have been carried out to classify an air target and the results are averaged over 25 Monte Carlo runs. This target is assumed a fighter which does not show any high maneuver motion.
w a
a
j
1 s
a
1 s
v
1 s
x
-a
Fig. 3. Block diagram of the Singer model, a is a reciprocal of the maneuver time constant.
Fig. 4. The digraph used in the design of classmatched IMM filters.
4. AN EXTRA FEATURE AXIS This section uses classical Bayesian theory to make a maximum speed classifier. The maximum speed classifier is a classifier which uses the maximum tangential speed of the target to classify targets. The speed is computed from IMM filters’ states. A sensor can be seen as a piece of an equipment that observers some data x Í X and transmits some "opinion" about the actual value of a parameter of interest h Í H , where X and H are called the observation and hypothesis domains, respectively. After that, the sensor communicates its opinion on the value of H under the form of a "likelihood" vector, l (hi x ) for each hi Î H . Target class is a time-invariant attribute which takes values from a discrete set c Î {ci : i = 1,..., n} where n is the number of classes. In order to make a decision about the target class, we need to compute the class probabilities. For this purpose a recursive equation is usually used in optimal Bayesian framework. This equation is Pk +1 (ci ) =
Lik Pk (ci ) å j =1 L jk Pk (c j ) n
(9)
where Lik is the likelihood of class i at time k and Pk (ci ) is the probability of class i at that time and recursive equation is initialized using a prior class probabilities P0 (ci ) which usually assumed to be the same for all different classes. The selection of likelihood functions plays a main role in the classification results. These functions will determine the time at witch the classifier should change its decision about the class of the target. In this research, Rayleigh distribution is adopted as likelihood functions for all three classes. The choice of Rayleigh distribution for this purpose is due to the
an Comparison: "Actual and Estimated Acceleration" 1 Estimated Normal Acc. Actual Normal Acc.
0.5
Normal Acc./g
0 -0.5 -1 -1.5 -2 -2.5 -3
0
10
20
30 40 Sampling Time
50
60
70
Fig. 5. Comparison of the actual and estimated acceleration of the target. Class Probability: "Max. Acceleration Calssifier" 0.7 0.6 0.5 Probability
The trajectory consists of two CV segments and a maneuver duty. The sampling time T is a second and simulation lasts 64 seconds which consists 63 samples. The manoeuvre is between sampling times 20 and 35 with maximum acceleration 2.5g. The motion of the target in remainder sampling times is CV (this kind of movement is usually performed by fighter pilots to hide themselves). Fig.5 compares the real and estimated normal accelerations provided by IMM tracker. It is clear that the real and estimated acceleration of the target is completely in the Class2 acceleration limit. Finally, Fig.6 shows the output of the maximum acceleration classifier. At first, the target is classified in Class1 and when the maneuver is performed, classifier changes its opinion and claims that the target is in the Class2 category. Although the classifier claims that the target is a Class2 airplane but in this situation, this classifier cannot distinguish Class2 and Class3 target, since the probability of Class2 and Class3 is approximately the same. This classification result is the final result achieved later in (Ristic et al., 2004).
0.4 0.3 0.2 0.1 0
0
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20
30 40 Sampling Time
50
60
70
Fig. 6. Classification results, the output of the Max. acceleration classifier. fact that this distribution, in low speeds, gives the largest likelihood to Class1 and for high speed motions all three likelihood functions approach zero. Rayleigh distribution parameters are 0.7, 1.2 and 1.8 for Class1, Class2 and Class3, respectively. Fig.7 represents these likelihood functions with respect to the normalized speed of the target. Intersections of curves represent the speed at which the classifier should switch between classes. This simple classifier is not ready to use right now, because if a target shows a high speed motion and after then goes back to the low speed class limit, the output of the classifier will be accordingly changed. But we expect that if a target shows a high speed motion in any interval of observation, then after classification result never goes back to a lower speed class. In order to remove this defect in classifier’s work, normalized likelihoods should be renormalized as follows:
L2k
=
(
(
L1k = l1k
l1k L1k
+ l2k L2k
)/ (
L1k
+
)(
L2k
)
(10)
)
(11)
L3k = l1k L1k + l2k L2k + l3k L3k / L1k + L2k + L3k (12)
Rayleigh distributions for three classes 0.7 Class1 Likelihood Class2 Likelihood Class3 Likelihood
0.6 0.5 Probability
where Lik is the normalized likelihood and Lik is the renormalized likelihood for class i used in Bayesian formula. In each iteration, the estimated speed of the target is normalized by Mach number and then the likelihoods are calculated and renormalized likelihoods are used to update class probabilities using Bayesian recursive formula (Eq.9). Fig.8 compares the target speed and its estimated value. In 20th sampling time, the speed of the target becomes more than the Class1 limit and after 19 sampling times, the Class2 limit breaks, too. Fig.9 shows class probabilities of the maximum speed classifier. As it was expected, the decision made by this classifier, contains three parts. In the beginning of the test, Class1 has the biggest probability. In next 35 sampling times, Class2 probability becomes bigger than others and ultimately, when the Class2 limit is exceeded, Class3 is selected by the maximum speed classifier.
0.4 0.3 0.2 0.1 0 0
1
2
hi ÎA
(13)
where m H [ x]( A) denotes the bba of A induced on H for measurement x and A is the complement of A with respect to set H . Let E and F be two "distinct" pieces of evidence and let m H [E ] and m H [F ] be the bbas they induce on H (in this study E and F are corresponding to speed and acceleration classifiers' outputs). We want
7
8
V t Comparison: "Actual and Estimated Speed"
Estimated Speed Actual Speed
Target Speed/Mach
1.6
hi ÎA
6
2
5. TBM FOR FEATURE FUSION
m H [ x]( A) = Õ l (hi x) Õ (1 - l (hi x) )
4 5 Speed/Mach
Fig. 7. Rayleigh distributions used as class likelihood functions.
1.8
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
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20
30 40 Sampling Time
50
60
70
Fig. 8. Comparison of the actual and estimated speeds of the target. Class Probability: "Max. Speed Calssifer" 1 0.9 0.8 0.7 Probability
The inherent philosophy of the introduced algorithm is illustrated in Fig.10. The available target features for classification, the maximum normal acceleration and maximum along-track speed of the target, are extracted from positional measurements provided by IMM tracker. The target state and maneuver estimates are monitored. At each filtering step, the estimated speed from CMM filters converts to speed likelihoods and together with the acceleration likelihoods both are trying to improve the classification process. After then, class probabilities are updated by renormalized likelihoods and Bayesian recursive formula. The results are two feature-based classifications that should be fused. Among many approaches available for feature fusion, like Bayesian and Dempster-Shefer theories, transferable belief model(TBM) is adopted for this purpose. The TBM is a model to represent quantified uncertainties based on belief functions and unrelated to any underlying probability model (Delmotte and Smets, 2004). The central element of the TBM is the basic belief assignment(bba), denoted m . For A Í H , m( A) is the part of belief that supports A i.e. the fact that the actual value h of H is in A , and due to a lack of information, does not support any strict subset of A . Given the likelihoods l (hi x) for every hi Î H , for x Í X and for every A Í H , Smets (Smets, 1978) has proved
3
0.6
Class1 Class2 Class3
0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 40 Sampling Time
50
60
70
Fig. 9. Classification results, the output of the Max. speed classifier. to build the bba m H [E , F ] that results from the combination of the two pieces of evidence. In terminology of TBM, this operation is called conjunctive rule of combinations and is defined by m H [E , F ]( A) =
å m H [E ]( B)m H [F ](C ) , "A Í H (14)
B ,C Í H , B ÇC = A
Fig. 10. The inherent philosophy of the paper classification approach. Finally, for decision making, we should transfer to probability functions. This transformation is done using the so called pignistic transformation given by: BetP H (h) = å
A:hÎ AÍ H
m H ( A) , A (1 - m H (Æ))
"h Î H
(15)
where A is the number of elements in A and Æ is the empty subset of H . Now, we are ready to apply the TBM theory to the feature fusion problem mentioned in the paper. The output values of two distinct classification approaches introduced in previous sections, will be fed as an input of the TBM algorithm. In each iteration the probability of different classes are assumed as a likelihood of that class. Then for each classification algorithm, the amount of the belief of every subset of H are calculated using Eq.13. For fusion of two bbas( m H [E ] , m H [F ] ), the conjunctive rule of combinations is used to calculate m H [E , F ] (Eq.14). Ultimately, Eq.15 (pignistic transformation) is applied to make the final decision. Fig.11 presents the results of the maximum speed and the maximum acceleration classifiers’ fusion. It is clear that as time elapses, the probability of Class3 becomes greater than two others which means that the target is a fighter. This result is different from the result of the maximum acceleration classifier at the end of section 3. Class Probability: "TBM output" 0.9 0.8 0.7
Probability
0.6 Class1 Class2 Class3
0.5 0.4 0.3 0.2 0.1 0
0
10
20
30 40 Sampling Time
50
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Fig. 11. Classification results, after fusion of two classifiers using TBM.
6. CONCLUSIONS The paper studies the classification problem in airborne applications. The feature-level fusion and multi-classifier fusion approaches are selected from the basic and advanced classification techniques, respectively. Two classifiers are the maximum speed and the maximum acceleration classifiers. In view of the fact that each classifier probes the detected target with a different point of view, two classification results may be different, as in the proposed simulation. So, two probable different decisions could be combined using belief functions. In fact, while using the TBM algorithm, class probabilities are transferred to the belief space. Then, associated beliefs are fused to a single belief and this belief is transferred back to the probability space. Results of the simulation confirms that the final decision made by the proposed approach is reliable enough. REFERENCES Delmotte, F. and P. Smets (2004), Target identification based on the transferable belief model interpretation of Dempster-Shefer model, IEEE Trans. On Systems, Man and Cybernetics, vol. 34, pp. 457-471. Hall, D.L. and J. Llinas (1997), An introduction to multisensor data fusion, Proceedings of the IEEE, vol. 85, pp. 6-23. Li, X.R. and V.P. Jilkov (2003), Survey of maneuvering target tracking. PartI: dynamic models, IEEE Trans. On Aerospace and Electronic Systems, vol. 39, pp. 1333-1364. McCullough, C.L., B.V. Dasarathy and P.C. Lindberg (1996), Multi-level sensor fusion for improved target discrimination, Proceedings of the IEEE, vol. 85, pp. 3674-3675. Ristic, B., N. Gordon and A. Bessell (2004), On target classification using kinematic data, Information fusion, vol. 5, pp. 15-21. Smets, P. (1978), “Un modele mathematicostatistique stimulant le processus du diagnostic medical,” PhD dissertation , Univ. Libre Bruxelles, Bruxelles, Belgium. Waltz, E. and J. Linas (1990), Multisensor Data Fusion ,Artech House, Norwood, Ma.