Modeling for Tracking of Complex Extended Object ... - IEEE Xplore
Modeling for Tracking of Complex Extended Object Using Minkowski Addition Lifan Sun
Jian Lan
X. Rong Li
Center for Information Engineering Science Research (CIESR) School of Electronics and Information Engineering Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P. R. China Email: [email protected], [email protected]
Abstract—This paper considers modeling for tracking of extended objects using measurements of down-range and crossrange extent. Modeling smooth/non-smooth objects of complex geometric shapes using support functions and extended Gaussian images, although efficient, may not be easily obtained. In view of this, the approaches based on support functions and extended Gaussian images are extended and further developed. In this paper, we attempt to model a complex extended object as a Minkowski addition of multiple simple sub-objects, each represented by a support function or extended Gaussian image. Due to favorable properties of the Minkowski addition, the new approach can make full use of common and individual properties of sub-objects and capture more detailed shape information. This facilitates not only object modeling, but also derivation of tracking algorithms. Thus the proposed approach can be extended to the case with a variety of complex objects. Simulation results demonstrated the benefits of the proposed approach. Keywords—Extended object, down-range and cross-range extent, Minkowski addition, support function, extended Gaussian image.
I.
I NTRODUCTION
Target tracking techniques play an important role in many practical applications. They have been developed extensively with fruitful results in the past several decades [1]. Due to limited sensor resolution capabilities, it is assumed that only a point kinematic measurement of a target (if detectable) is available at any time. Although classical target tracking approaches have been extensively used and well developed, they are no longer suitable for many current tracking scenarios. Unlike the conventional point target tracking, extended object tracking (EOT) uses not only the kinematic measurements (position, velocity, acceleration, etc.) of the object centroid but also high resolution sensor measurements, which can provide extra information to improve target identification and classification. For example, modern high resolution sensors are able to resolve multiple point features on an extended object. In recent years, modeling and tracking of extended objects have received more and more attention. Several models and approaches have been proposed for EOT (see, e.g., [2] [3] Research supported in part by grant for State Key Program for Basic Research of China (973) (2013CB329405), the National Natural Science Foundation of China (61203120), NASA/LEQSF(2013-15)-Phase3-06 through grant NNX13AD29A, Fund for the Doctoral Program of Higher Education of China (20110201120007), and the Fundamental Research Funds for the Central Universities of China.
Department of Electrical Engineering University of New Orleans New Orleans, LA 70148, U.S.A Email: [email protected]
[4] [5]). In [6] [7], the true shape of an extended object was modeled as a so-called spatial probability distribution. In [8], a random hypersurface model (RHM) was used to describe an extended object. RHM was extended by approximating the object shape with a star-convex one [9]. Apart from this, a new approach of using a random matrix was proposed to estimate the kinematic state and physical extension of extended objects in a Bayesian framework [10]. The random-matrix-based EOT approach was furthered in [11] [12] [13], and in [14] an new random-matrix approach to tracking non-ellipsoidal extended objects was proposed. In addition, modern surveillance sensors are able to provide one or more dimensions of the range extent measurements of an extended object along the line of sight (LOS) [15]. For instance, a high range resolution (HRR) radar can generate an HRR profile of the target and provides a useful measure of down-range extent [16], and an imaging infrared sensor can provide a measure of cross-range extent. In this context, several existing approaches differ mainly in extended object modeling for approximating the true object shape [15] [17] [18] [19]. However, they can only describe basic geometric shapes such as ellipses or rectangles under the assumption that the major axis of object is parallel to its velocity vector. This is not necessarily the case in many practical tracking scenarios. In view of this, two approaches based on support functions and extended Gaussian images (EGI) respectively were proposed in [20]. They can not only model a wide spectrum of smooth and non-smooth extended objects, but also remove the assumption that the orientation of the object is parallel to its velocity vector. These two approaches have been applied to EOT using measurements of down-range and cross-range extent, and a large range of extended object problems can be handled. It is, however, not feasible, effective, or convenient to describe a complex object by using the above approaches directly. So we consider modeling a complex object as a Minkowski addition of multiple simple (smooth/non-smooth) sub-objects without losing important and useful shape information. As illustrated in Fig. 1, such a complex object is modeled as a Minkowski addition of two sub-objects (i.e., an elliptical sub-object A and a rectangular sub-object B). The Minkowski addition “⊕” is necessarily convex. It plays a fundamental role in shape description [21] and fits well with our problem. Smooth sub-objects are represented by support functions. The Minkowski addition of these sub-objects can be
using measurements of down-range and cross-range extent. A. System Model Consider the following system model: xk = f (xk−1 , wk−1 ) zk = h(xk , vk )
Fig. 1.
Illustration of Minkowski addition
directly transformed into a simple addition of their respective support functions. Non-smooth sub-objects are represented by EGI directly. Their Minkowski addition can also be easily obtained by merging their EGI (to be demonstrated later). Some symmetric non-smooth objects can also be modeled in the framework of support functions. For example, we can obtain the support function representations of rectangular objects by utilizing the connection between the EGI and the support function indirectly [20]. In this case, a complex object (e.g., Fig. 1) can also be modeled as a Minkowski addition of smooth and non-smooth sub-objects represented by support functions. This largely facilitates object modeling. Moreover, the down-range and cross-range extent measurements have natural ties with support functions [20]. Tracking algorithms based on the proposed models can be easily developed to estimate the kinematic state and object extension jointly. The proposed modeling for tracking of complex extended objects using Minkowski addition has the following advantages. i) It can make full use of common and individual properties of the sub-objects and capture more detailed shape information of complex extended objects, which facilitates object modeling considerably. ii) Due to the concise mathematical forms and favorable properties of the Minkowski addition, the state vector dimension needed for modeling and estimation can be greatly reduced. Thus the complex object modeling is largely simplified. iii) The approaches based on support functions and EGI are extended and further developed, and the proposed models can be easily implemented for EOT by using Minkowski addition. This paper is organized as follows. Section II reviews the existing work on EOT based on support functions and extended Gaussian images using measurements of down-range and cross-range extent. Section III proposes a model describing a complex extended object as a Minkowski addition of multiple simple sub-objects, each represented by a support function or EGI. Also, we develop a tracking algorithm based on the proposed model. In Section IV, simulation results are presented to demonstrate the effectiveness of the proposed modeling and estimation. The last section concludes the paper. II.
EOT BASED
S UPPORT F UNCTIONS G AUSSIAN IMAGES
ON
It describes the target dynamics and sensor measurements, where xk is the state vector with transition function f , zk is the measurement vector with the measurement function h, k is the time index, and wk ∼ N (0, Qk ) and vk ∼ N (0, Rk ) are independent Gaussian noise. Here it is assumed that a high resolution sensor provides measurements of down-range extent D and cross-range extent C along LOS, as well as the range r and bearing β measurements of the object centroid, given by zk = [rk , βk , Dk , Ck ]′ . Unlike a point target, the state vector ′ ′ ′ of an extended object is given by xk = [(xm k ) , (ek ) ] , which and the vector consists of the centroid kinematic state xm k ek characterizing the shape. Consider xm = [x , x ˙ , y ˙ k ]′ , k k k, y k where (x, y) and (˙x, y˙ ) are the position and velocity in the Cartesian plane, respectively. B. Modeling Based on Support Function In mathematics, the support function H of a non-empty closed convex set K in Euclidean space RN describes the (signed) distance from a fixed reference point to the supporting hyperplane of K [21]. In general, it is practical to assume that the reference point lies in the interior of K (usually taken to be the centroid for simplicity), which makes H always positive for every unit vector [20]. Support functions are frequently used to describe smooth convex figures, since such figures are uniquely determined by H. If K is a convex body in the plane R2 and v = [cos θ, sin θ]′ is a unit vector denoting the viewing direction, the support function H(θ) is H(θ) = sup x′ v
(2)
x∈K
Then it is the distance from the origin O to the supporting line (see Fig. 2) Ls (θ) = {x ∈ R2 |x′ v = H(θ), θ ∈ [0, 2π]} of K perpendicular to v. The support function not only can describe object shapes, but more importantly has natural ties with the down-range and cross-range extent measurements of extended objects. In [20],
AND EXTENDED
In this section, we review briefly the existing work on EOT based on support functions and extended Gaussian images
(1)
Fig. 2.
The support function
the down-range extent D(θ) and cross-range extent C(θ) can be directly expressed in terms of support functions:
and (4), respectively. Thus, the object shape state ek with dimension 2N + 1 is given by
D(θ) = H(θ) + H(θ + π) (3) π π C(θ) = H(θ + ) + H(θ − ) (4) 2 2 and an approach based on support functions was proposed to model smooth objects. As an example of an elliptical object, it is approximated by using a 2 × 2 symmetric positive semidefinite matrix Ek as a suitable parametric representation. Let Ek represent an ellipse centered at Xkc , comprising the set of points given by
ek = [ak , ak , bk , ..., ak , bk ]′
{Xk ∈ R2 |(Xk − Xkc )′ Ek−1 (Xk − Xkc ) = 1} where Xk = (xk , yk ) is a point on the boundary of the ellipse. The support function H(θk ) of an ellipse at viewing angle θk is given by [24] H(θk ) = (vk′ Ek vk )1/2 = ([cos θk , sin θk ]Ek [cos θk , sin θk ]′ )1/2
(5)
′
where vk = [cos θk , sin θk ] . Geometric properties of the ellipse are reflected in the algebraic properties of the corresponding matrix Ek , which carries useful and important information about the ellipse extension (i.e., size, shape and orientation). Thus entries of the matrix Ek can be considered to form shape parameters. However, the estimated shape Ek may lose positive semi-definiteness, leading to unpredictable results. So, the Cholesky decomposition Ek = Lk L′k is utilized to handle this problem, where " # (1) Lk 0 Lk = (6) (2) (3) Lk Lk
(0)
(1)
(1)
(N )
(N )
(11)
This method may be extended to other smooth object shapes by using suitable parametric representations. It should be noted that the approach based on support functions can not be applied directly to an extended object modeled as a non-smooth shape such as a rectangle. In [20], another approach based on extended Gaussian images was proposed to deal with this problem. C. Modeling Based on EGI The extended Gaussian image (EGI) is particularly convenient to represent a convex body K concisely [25]. In particular, if we have the EGI parameters of a convex polygon K, the shape of K is uniquely determined. Moreover, the EGI has two other useful properties: translation invariance and orientation equivariance: it is not affected by translation of the object, and rotation of the object leads to an equal rotation of its EGI.
is a lower triangular matrix with positive diagonal entries. Then Eq. (5) can be rewritten as H(θk ) = ([cos θk , sin θk ]Lk L′k [cos θk , sin θk ]′ )1/2
(7)
Fig. 3.
EGI representation of a convex polygon
According to Eqs. (3) and (4), we have D(θk ) = 2([cos θk , sin θk ]Lk L′k [cos θk , sin θk ]′ )1/2
(8)
C(θk ) = 2([− sin θk , cos θk ]Lk L′k [− sin θk , cos θk ]′ )1/2 (9) The shape state ek is changed to consist of the non-zero entries (1) (2) (3) Lk , Lk and Lk and is included in the state vector xk = (1) (2) (3) [xk , x˙ k , yk , y˙ k , Lk , Lk , Lk ]′ . Note that by selecting a suitable parametric representation for the object shape, it may be extended to a variety of object shapes, that is, different parametric representations will lead to different shapes. In this context, the Fourier series expansion is introduced to object modeling based on support functions. As such, more detailed shape information can be described and utilized. As a real-valued integrable periodic function on [0, 2π] (i.e., H(θ) = H(θ + 2π)), the support function H(θk ) of an extended object can be expanded in a Fourier series of a degree N in θk as X (0) (j) (j) H(θk ) = ak + (ak cos jθk + bk cos jθk ) (10) j=1,2,...N
Correspondingly, the down-range extent D(θk ) and crossrange extent C(θk ) can be easily obtained by using Eqs. (3)
Due to these properties, the EGI representation is widely used for object recognition and reconstruction in computer vision, graphics, and image processing. If K is an N -sided polygon whose jth edge has length lj and outer unit normal vector uj = [cos αj , sin αj ]′ , its EGI can be represented by N vectors lj uj for j = 1, 2, ..., N counterclockwise (see Fig. 3), and its EGI parameters are easily obtained as {l1 , ..., lN , α1 , ..., αN } [26]. If the viewing diection is v = [cos θ, sin θ]′ , the downrange and cross-range extent of the object K can be written as [20] N
D(θ) = H(θ) + H(θ + π) =
1X lj | sin(θ − αj )| 2 j=1
(12)
N
π π 1X C(θ) = H(θ − ) + H(θ + ) = lj | cos(θ − αj )| (13) 2 2 2 j=1 For a rectangular object, its EGI parameters can be easily obtained as {lk,1 , lk,2 , lk,1 , lk,2 , αk , αk + π2 , αk + π, αk + 3π 2 }. Thus lk,1 , lk,2 (i.e., the lengths of minor and major axes) and the angle of orientation αk can be considered to form shape parameters and included in the state vector xk =
′ ′ [(xm k ) , lk,1 , lk,2 , αk ] to be estimated. The EGI-based modeling approach works for other non-smooth shapes besides rectangles. It should be noted that the down-range extent D(θ) and cross-range extent C(θ) can be calculated by using the EGI parameters in Eqs. (12) and (13), respectively. This provides a connection between the EGI and support function. For example, the support function representation of a rectangular object is easily obtained by utilizing its symmetric property, that is,
1 H(θk ) = H(θk + π) = D(θk ) 2 1 = (lk,1 | sin(θk − αk )| + lk,2 | cos(θk − αk )|) (14) 2 As such, some non-smooth objects having symmetry can be represented by support functions indirectly. For more details on EGI-based modeling we refer the reader to [20]. III.
M ODELING FOR T RACKING OF C OMPLEX E XTENDED O BJECT U SING M INKOWSKI A DDITION
In this section, we propose an approach to modeling a complex extended object by using a Minkowski addition of multiple simple sub-objects, each represented by a support function or EGI. With this approach, common and individual properties of the sub-objects can be sufficiently utilized to obtain a suitable parametric representation of a complex extended object. As such, more detailed shape information can be captured. This will largely facilitate target tracking based on the proposed models. A. Minkowski Addition If A and B are two arbitrary sets of points in real Euclidean n-dimensional space Rn , their Minkowski addition A ⊕ B is defined as [21] A ⊕ B = {a + b, a ∈ A, b ∈ B}
HλA (θ) = λHA (θ)
(18)
If λ1 , ..., λm are positive real numbers and K1 , ..., Km are convex bodies, then the support function of their linear combination λ1 K1 ⊕ · · · ⊕ λm Km is given by Hλ1 K1 ⊕···⊕λm Km (θ) = λ1 HK1 (θ) + · · · + λm HKm (θ) (19) Obviously from the above equation, the Minkowski addition of convex bodies K1 , ...Km becomes addition of support functions. C. Down-range and Cross-range Extent Here a complex object K is described as a Minkowski addition of multiple simple sub-objects Ki ∀ i = 1, ..., m (i.e., K = K1 ⊕ · · · ⊕ Km ), represented by addition of support functions: HK (θ) = HK1 ⊕···⊕Km (θ) = HK1 (θ) + · · · + HKm (θ) (20) As shown in Eqs. (3) and (4), the down-range extent D and cross-range extent C of an extended object can be expressed in terms of support functions. Thus down-range extent and cross-range extent of K can be easily obtained as follows. Since K = K1 ⊕ · · · ⊕ Km , DK (θ) = DK1 ⊕···⊕Km (θ) = HK1 ⊕···⊕Km (θ) + HK1 ⊕···⊕Km (θ + π) = HK1 (θ) + HK1 (θ + π) + · · · + HKm (θ) + HKm (θ + π) = DK1 (θ) + · · · + DKm (θ)
(21)
(15)
where “+” denotes the vector addition of two points. The Minkowski addition “⊕” is widely used in computational geometry, and it plays a fundamental role in shape description and analysis. The inverse of the “⊕” operator is “⊖”, denoting the Minkowski decomposition operator. If Ki is a compact convex set in Rn , and λi ≥ 0 ∀ i = 1, ..., m, then the vector sum λ1 K1 ⊕ · · · ⊕ λm Km = {λ1 k1 + · · · + λm km , ki ∈ Ki } (16) is called a Minkowski linear combination. The addition and scalar multiplication in Eq. (16) are called Minkowski addition and Minkowski scalar multiplication, respectively. Note that this Minkowski linear combination turns out to yield a convex set. B. Minkowski Addition and Support Functions In the general convex case, if both bodies A and B are represented by support functions for all viewing direction vectors v = [cos θ, sin θ]′ in Rn , the Minkowski addition A⊕B is a unique convex body with HA⊕B (θ) = HA (θ) + HB (θ)
where HA (θ) + HB (θ) is also a support function. For λ ≥ 0, the scalar multiplication is defined as
(17)
CK (θ) = CK1 ⊕···⊕Km (θ)
π π = HK1 ⊕···⊕Km (θ − ) + HK1 ⊕···⊕Km (θ + ) 2 2 π π = HK1 (θ − ) + HK1 (θ + ) + · · · 2 2 π π + HKm (θ − ) + HKm (θ + ) 2 2 = CK1 (θ) + · · · + CKm (θ) (22)
where DKi (θ) = HKi (θ) + HKi (θ + π), i = 1, ..., m π π CKi (θ) = HKi (θ − ) + HKi (θ + ), i = 1, ..., m 2 2
(23) (24)
As shown clearly in Eqs. (21) and (22), DK (θ) and CK (θ) can be expressed in terms of support functions and reduced to addition of DK1 (θ), ..., DKm (θ) and CK1 (θ), ..., CKm (θ), respectively. Thus a complex object can be modeled as a Minkowski addition of multiple sub-objects. 1) Example I: As shown in Fig. 4, such an object is modeled as a Minkowski addition of two smooth objects (i.e., two elliptical objects), each represented by a support function.
30 20 10 0 −10 Sub−object 1 Sub−object 2 MA
−20 −30 −30
−20
−10
0
10
20
30
(a) Object shapes
Fig. 5. Complex object modeling using Minkowski addition, illustrative Example II
30 Sub−object 1 Sub−object 2 MA
25
so that the Minkowski addition of multiple polygons Ki (for i = 1, ..., m) can be calculated easily. As shown in Fig. 5, such a complex object is modeled as a Minkowski addition of two non-smooth objects (i.e., two triangular objects), each represented by an EGI.
20 15 10 5 0 0
1
2
3
4
5
6
7
(b) Support functions Fig. 4. Complex object modeling using Minkowski addition, illustrative Example I
2) Example II: For non-smooth objects, any convex polygon can be represented by its EGI. Suppose that the EGI parameters of polygon K1 and K2 are K1 : {l1 , .., lM , α1 , ..., αM } K2 : {lM+1 , .., lM+N , αM+1 , ..., αM+N }
3) Example III: As mentioned in the previous section, some symmetric non-smooth objects can also be represented by support functions indirectly. As shown in Fig. 6, this object is modeled as a Minkowski addition of an elliptical sub-object and a rectangular sub-object. The support function representation of the rectangular sub-object is easily obtained by utilizing its symmetry property (see Eq. (14)). In some cases, a complex object (e.g., Fig. 6) can also be modeled as a Minkowski addition of smooth and non-smooth subobjects, both of which can be represented by support functions. This largely facilitates modeling and estimation of complex extended objects. 30 20
From Eq. (12)
10
M
1X DK1 (θ) = lj | sin(θ − αj )| 2 j=1
M+N 1 X DK2 (θ) = lj | sin(θ − αj )| 2
(25)
0 −10
(26)
Sub−object 1 Sub−object 2 MA
−20
j=M+1
−30 −30
Then
−20
−10
0
10
20
30
(a) Object shapes
DK1 ⊕K2 (θ) M+N 1 X = DK1 (θ) + DK2 (θ) = lj | sin(θ − αj )| 2 j=1
30
(27)
20
Similarly, we have
15
CK1 ⊕K2 (θ) = CK1 (θ) + CK2 (θ) =
Sub−object 1 Sub−object 2 MA
25
M+N 1 X lj | cos(θ − αj )| 2 j=1
10
(28)
Thus the EGI parameters of K1 ⊕ K2 can be easily obtained as K1 ⊕ K2 : {l1 , .., lM+N , α1 , ..., αM+N } Clearly, K1 ⊕ K2 is obtained by naturally merging their EGIs. In this manner we can obtain a set of vectors by mixing EGIs
5 0 0
1
2
3
4
5
6
7
(b) Support functions Fig. 6. Complex object modeling using Minkowski addition, illustrative Example III
D. System Model As an example of the object K in Fig. 7, it is described as a Minkowski addition of two rectangular sub-objects K1 and K2 , each represented by an EGI. For simplicity here, it is assumed that these two sub-objects have the same minor and major axis lengths (i.e., lk,1 and lk,2 ) but differ in the angle of orientation αk and γk . Thus the EGI parameters of K1 and K2 can be easily obtained as π 3π , αk + π, αk + } 2 2 π 3π } K2 : {lk,1 , lk,2 , lk,1 , lk,2 , γk , γk + , γk + π, γk + 2 2
K1 : {lk,1 , lk,2 , lk,1 , lk,2 , αk , αk +
Clearly, the shape state vector ek = [lk,1 , lk,2 , αk , γk ]′ is ′ ′ included in the state vector xk = [(xm k ) , lk,1 , lk,2 , αk , γk ] to be estimated. By Eqs. (12)–(13) and (27)–(28), we have
C(θk ) = CK1 ⊕K2 (θ) = CK1 (θ) + CK2 (θ) = lk,1 | cos(θk − αk )| + lk,2 | sin(θk − αk )| + lk,1 | cos(θk − γk )| + lk,2 | sin(θk − γk )|
cov[vk ] = Rk = diag[Rkr , Rkβ , RkD , RkC ]
IV.
Object modeling using Minkowski addition
1) State Dynamics: To simplify analysis, we consider the case that the target is moving at a nearly constant velocity (CV) [27] in the two-dimension (2-D) Cartesian coordinate system. Then the discrete-time dynamics is xk+1 = FkCV xk + Γk wk
Since the measurements z k = [rk , βk , Dk , Ck ]′ are obtained from different physical channels, the noise vk is generally assumed to be zero-mean Gaussian with independent elements and covariance
For this problem, the measurement equation is severely nonlinear, so a nonlinear filter should be used to estimate both kinematics and object shape jointly. Here the estimation process can be implemented by using the unscented filter (UF) [28] [29] for the nonlinear part of (30) and leave the remaining part to the Kalman filter.
D(θk ) = DK1 ⊕K2 (θk ) = DK1 (θk ) + DK2 (θk ) = lk,1 | sin(θk − αk )| + lk,2 | cos(θk − αk )| + lk,1 | sin(θk − γk )| + lk,2 | cos(θk − γk )|
Fig. 7.
2) Measurement Equation: It is assumed that a high resolution sensor is fixed at (Xo , Yo ) and provides measurements of down-range extent D and cross-range extent C along LOS, as well as the range r and bearing β measurements of the object centroid, given by p (xk − Xo )2 + (yk − Yo )2 zk = arctan((yk − Yo )/(xk − Xo )) + vk (30) D(θk ) C(θk )
(29)
with FkCV = diag(F, F, I4 ), Γk = diag(Γ, Γ, I4 ) 1 0 0 0 2 1 T T /2 0 1 0 0 I4 = , F = , Γ= 0 0 1 0 0 1 T 0 0 0 1
where I4 describes the approximately unchanged object extension over time, and the uncertainty of the target state is embodied in the process noise wk . Given more information, we can design Fk specifically for different scenarios, which leads to different centroid state and object extension transition.
S IMULATION R ESULTS AND P ERFORMANCE E VALUATION
To illustrate the effectiveness of the proposed approach, simulation examples are provided in this section. In the simulation, we focus on the case of a stationary sensor platform and a moving extended object. Consider a scenario in which an extended object (as an example in Fig. 7) moves at a nearly CV in the 2-D Cartesian coordinate system with the ′ initial kinematic state xm 0 = [1000m, 60m/s, 2000m, 30m/s] . The true object is modeled as a Minkowski addition of two rectangular objects with different angles of orientation, both of which have the same lengths of minor and major axes. For this object at the initial time, the shape state e0 = [5m, 25m, π3 , π6 ]′ . The sensor is fixed at the origin (0, 0) and it provides measurements of range, bearing, and the target extent along the LOS every T = 2s. Each measurement is corrupted by zero-mean white Gaussian noise with standard deviations σr = 10m, σβ = 0.02rad, σD = 15m, σC = 15m. Simulation results for this scenario are shown in Figs. 8 and 9, which give the trajectory of the object and tracking performance. Evaluation of estimation performance has always been centered on the root-mean-square error (RMSE): RMSE(ˆ x) = (
M 1 X ke xi k2 )1/2 M i=1
(31)
where x ei is the estimation error on the ith of the M MonteCarlo runs. However, use of the average Euclidean error (AEE) is better than RMSE, as analyzed convincingly in [30]. The AEE has a more direct, natural interpretation and is less dominant by large errors. It is defined as AEE(ˆ x) =
M 1 X ke xi k M i=1
(32)
8000 2100 Y
7000 2000
6000 1900 900
1000 1100 1200 1300 X Y
Y
5000 4000 3000
7960 7940 7920 7900 7880 7860
2000 1000 0 Fig. 8.
2000
4000
6000
X
8000
1.275 1.28 1.285 1.29 4 X x 10 True Object Estimated Object 10000
12000
14000
Trajectory of the extended object
In this simulation, AEE is chosen as the measure and comparison results are presented in terms of AEE for the kinematic state (i.e., position and velocity) over 100 Monte Carlo runs. From Figs. 9(a) and 9(b), it can be seen that the proposed approach can estimate the kinematic state well. To objectively evaluate the quality of shape estimation, the Hausdorff distance is adopted to measure the degree of resemblance between the estimated shape and the true one. Since a convex body K is completely determined by its support function HK (·), the Hausdorff distance can be defined via the support functions as [22] dH (A, B) = kHA − HB k∞ = sup |HA (θk ) − HB (θk )|
(33)
θk ∈[0,2π]
where k·k∞ denotes the infinity norm of a function and “sup” denotes the “supremum” or “least upper bound”. For Eq. (33), the Hausdorff distance calculates the distance between the bodies A and B. For convex bodies, this is equivalent to the alternative definition dH (A, B) = max{max d(x, A), max d(x, B)} x∈B
x∈A
(34)
which applies to arbitrary compact sets, where d(x, A) denotes the distance from the point x to the set A. The Hausdorff distance is widely used for object matching in computer vision, object recognition, and shape analysis [23]. The Hausdorff distance of object shape estimation is calculated by Eq. (33), which is hard to use directly because [0, 2π) as the range of θ is a continuous set. For this reason, [0, 2π) is 2π replaced by the discrete set {i · N , i = 0, 1, ...Ns − 1} based s on uniform angle sampling, where Ns is the number of sample points (here Ns = 1000). Then Eq. (33) becomes ˆ = sup |HK (θ) − H ˆ (θ)| dH (K, K) K 2π θ∈{i· N ,i=0,1,...Ns −1} s
(35)
ˆ and the true shape K. Clearly, between the estimated shape K the shorter the Hausdorff distance is, the better the shape estimation is. As shown in Fig. 9(c), the Hausdorff distance demonstrates that the object shape can be estimated accurately. Overall, simulation results illustrated the effectiveness of what we proposed. V.
C ONCLUSION
To describe a complex extended object accurately and effectively, we have proposed an approach to model it by using the Minkowski addition of multiple simple sub-objects without losing shape information. These sub-objects are represented by support functions/EGIs in a concise form. Their Minkowski addition can be transformed to a simple addition of support functions or a set of vectors by mixing EGIs. The Minkowski addition fits well with our problem: a) Due to its concise mathematical forms and favorable properties, the complex object modeling is largely simplified and more detailed shape information is described. b) The approaches based on support functions and EGI have been extended and further developed. c) The introduction of the Minkowski addition paves the way for tracking of complex objects, that is, the proposed model can be easily implemented for EOT to estimate the kinematic state and object extension jointly. The effectiveness of the proposed approach has been verified through simulation results. With suitable adoptions of object modeling using Minkowskid addition, a larger range of complex extended object tracking can be handled. R EFERENCES [1] Y. Bar-Shalom and X. R. Li, Estimation and Tracking: Principles, Techniques, and Software. Boston, MA: Artech House, 1993. (Reprinted by YBS Publishing, 1998). [2] X. R. Li and J. Dezert, “Layered multiple-model algorithm with application to tracking maneuvering and bending extended target in clutter,” in Proceedings of the 1st International Conference on Information Fusion (Fusion 1998), Las Vegas, NV, USA, 1998.
70
AEE of Position
AEE of Position
60
[10]
50 40
[11] 30 20
[12]
10 0 0
20
40 60 Time step k
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[13]
(a) AEE of Velocity
AEE of Velocity
4 3
[15]
2
[16]
1
[17]
0 0
20
40 60 Time step k
80
100
[18]
(b) 300
d
Hausdorff distance dH
H
200
[20]
150 100
[21]
50 20
40 60 Time step k
80
100
(c) Fig. 9. performance comparison. (a) AEE of Position. (b) AEE of Velocity. (c) Hausdorff distance.
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Jian Lan
X. Rong Li
Center for Information Engineering Science Research (CIESR) School of Electronics and Information Engineering Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P. R. China Email: [email protected], [email protected]
Abstract—This paper considers modeling for tracking of extended objects using measurements of down-range and crossrange extent. Modeling smooth/non-smooth objects of complex geometric shapes using support functions and extended Gaussian images, although efficient, may not be easily obtained. In view of this, the approaches based on support functions and extended Gaussian images are extended and further developed. In this paper, we attempt to model a complex extended object as a Minkowski addition of multiple simple sub-objects, each represented by a support function or extended Gaussian image. Due to favorable properties of the Minkowski addition, the new approach can make full use of common and individual properties of sub-objects and capture more detailed shape information. This facilitates not only object modeling, but also derivation of tracking algorithms. Thus the proposed approach can be extended to the case with a variety of complex objects. Simulation results demonstrated the benefits of the proposed approach. Keywords—Extended object, down-range and cross-range extent, Minkowski addition, support function, extended Gaussian image.
I.
I NTRODUCTION
Target tracking techniques play an important role in many practical applications. They have been developed extensively with fruitful results in the past several decades [1]. Due to limited sensor resolution capabilities, it is assumed that only a point kinematic measurement of a target (if detectable) is available at any time. Although classical target tracking approaches have been extensively used and well developed, they are no longer suitable for many current tracking scenarios. Unlike the conventional point target tracking, extended object tracking (EOT) uses not only the kinematic measurements (position, velocity, acceleration, etc.) of the object centroid but also high resolution sensor measurements, which can provide extra information to improve target identification and classification. For example, modern high resolution sensors are able to resolve multiple point features on an extended object. In recent years, modeling and tracking of extended objects have received more and more attention. Several models and approaches have been proposed for EOT (see, e.g., [2] [3] Research supported in part by grant for State Key Program for Basic Research of China (973) (2013CB329405), the National Natural Science Foundation of China (61203120), NASA/LEQSF(2013-15)-Phase3-06 through grant NNX13AD29A, Fund for the Doctoral Program of Higher Education of China (20110201120007), and the Fundamental Research Funds for the Central Universities of China.
Department of Electrical Engineering University of New Orleans New Orleans, LA 70148, U.S.A Email: [email protected]
[4] [5]). In [6] [7], the true shape of an extended object was modeled as a so-called spatial probability distribution. In [8], a random hypersurface model (RHM) was used to describe an extended object. RHM was extended by approximating the object shape with a star-convex one [9]. Apart from this, a new approach of using a random matrix was proposed to estimate the kinematic state and physical extension of extended objects in a Bayesian framework [10]. The random-matrix-based EOT approach was furthered in [11] [12] [13], and in [14] an new random-matrix approach to tracking non-ellipsoidal extended objects was proposed. In addition, modern surveillance sensors are able to provide one or more dimensions of the range extent measurements of an extended object along the line of sight (LOS) [15]. For instance, a high range resolution (HRR) radar can generate an HRR profile of the target and provides a useful measure of down-range extent [16], and an imaging infrared sensor can provide a measure of cross-range extent. In this context, several existing approaches differ mainly in extended object modeling for approximating the true object shape [15] [17] [18] [19]. However, they can only describe basic geometric shapes such as ellipses or rectangles under the assumption that the major axis of object is parallel to its velocity vector. This is not necessarily the case in many practical tracking scenarios. In view of this, two approaches based on support functions and extended Gaussian images (EGI) respectively were proposed in [20]. They can not only model a wide spectrum of smooth and non-smooth extended objects, but also remove the assumption that the orientation of the object is parallel to its velocity vector. These two approaches have been applied to EOT using measurements of down-range and cross-range extent, and a large range of extended object problems can be handled. It is, however, not feasible, effective, or convenient to describe a complex object by using the above approaches directly. So we consider modeling a complex object as a Minkowski addition of multiple simple (smooth/non-smooth) sub-objects without losing important and useful shape information. As illustrated in Fig. 1, such a complex object is modeled as a Minkowski addition of two sub-objects (i.e., an elliptical sub-object A and a rectangular sub-object B). The Minkowski addition “⊕” is necessarily convex. It plays a fundamental role in shape description [21] and fits well with our problem. Smooth sub-objects are represented by support functions. The Minkowski addition of these sub-objects can be
using measurements of down-range and cross-range extent. A. System Model Consider the following system model: xk = f (xk−1 , wk−1 ) zk = h(xk , vk )
Fig. 1.
Illustration of Minkowski addition
directly transformed into a simple addition of their respective support functions. Non-smooth sub-objects are represented by EGI directly. Their Minkowski addition can also be easily obtained by merging their EGI (to be demonstrated later). Some symmetric non-smooth objects can also be modeled in the framework of support functions. For example, we can obtain the support function representations of rectangular objects by utilizing the connection between the EGI and the support function indirectly [20]. In this case, a complex object (e.g., Fig. 1) can also be modeled as a Minkowski addition of smooth and non-smooth sub-objects represented by support functions. This largely facilitates object modeling. Moreover, the down-range and cross-range extent measurements have natural ties with support functions [20]. Tracking algorithms based on the proposed models can be easily developed to estimate the kinematic state and object extension jointly. The proposed modeling for tracking of complex extended objects using Minkowski addition has the following advantages. i) It can make full use of common and individual properties of the sub-objects and capture more detailed shape information of complex extended objects, which facilitates object modeling considerably. ii) Due to the concise mathematical forms and favorable properties of the Minkowski addition, the state vector dimension needed for modeling and estimation can be greatly reduced. Thus the complex object modeling is largely simplified. iii) The approaches based on support functions and EGI are extended and further developed, and the proposed models can be easily implemented for EOT by using Minkowski addition. This paper is organized as follows. Section II reviews the existing work on EOT based on support functions and extended Gaussian images using measurements of down-range and cross-range extent. Section III proposes a model describing a complex extended object as a Minkowski addition of multiple simple sub-objects, each represented by a support function or EGI. Also, we develop a tracking algorithm based on the proposed model. In Section IV, simulation results are presented to demonstrate the effectiveness of the proposed modeling and estimation. The last section concludes the paper. II.
EOT BASED
S UPPORT F UNCTIONS G AUSSIAN IMAGES
ON
It describes the target dynamics and sensor measurements, where xk is the state vector with transition function f , zk is the measurement vector with the measurement function h, k is the time index, and wk ∼ N (0, Qk ) and vk ∼ N (0, Rk ) are independent Gaussian noise. Here it is assumed that a high resolution sensor provides measurements of down-range extent D and cross-range extent C along LOS, as well as the range r and bearing β measurements of the object centroid, given by zk = [rk , βk , Dk , Ck ]′ . Unlike a point target, the state vector ′ ′ ′ of an extended object is given by xk = [(xm k ) , (ek ) ] , which and the vector consists of the centroid kinematic state xm k ek characterizing the shape. Consider xm = [x , x ˙ , y ˙ k ]′ , k k k, y k where (x, y) and (˙x, y˙ ) are the position and velocity in the Cartesian plane, respectively. B. Modeling Based on Support Function In mathematics, the support function H of a non-empty closed convex set K in Euclidean space RN describes the (signed) distance from a fixed reference point to the supporting hyperplane of K [21]. In general, it is practical to assume that the reference point lies in the interior of K (usually taken to be the centroid for simplicity), which makes H always positive for every unit vector [20]. Support functions are frequently used to describe smooth convex figures, since such figures are uniquely determined by H. If K is a convex body in the plane R2 and v = [cos θ, sin θ]′ is a unit vector denoting the viewing direction, the support function H(θ) is H(θ) = sup x′ v
(2)
x∈K
Then it is the distance from the origin O to the supporting line (see Fig. 2) Ls (θ) = {x ∈ R2 |x′ v = H(θ), θ ∈ [0, 2π]} of K perpendicular to v. The support function not only can describe object shapes, but more importantly has natural ties with the down-range and cross-range extent measurements of extended objects. In [20],
AND EXTENDED
In this section, we review briefly the existing work on EOT based on support functions and extended Gaussian images
(1)
Fig. 2.
The support function
the down-range extent D(θ) and cross-range extent C(θ) can be directly expressed in terms of support functions:
and (4), respectively. Thus, the object shape state ek with dimension 2N + 1 is given by
D(θ) = H(θ) + H(θ + π) (3) π π C(θ) = H(θ + ) + H(θ − ) (4) 2 2 and an approach based on support functions was proposed to model smooth objects. As an example of an elliptical object, it is approximated by using a 2 × 2 symmetric positive semidefinite matrix Ek as a suitable parametric representation. Let Ek represent an ellipse centered at Xkc , comprising the set of points given by
ek = [ak , ak , bk , ..., ak , bk ]′
{Xk ∈ R2 |(Xk − Xkc )′ Ek−1 (Xk − Xkc ) = 1} where Xk = (xk , yk ) is a point on the boundary of the ellipse. The support function H(θk ) of an ellipse at viewing angle θk is given by [24] H(θk ) = (vk′ Ek vk )1/2 = ([cos θk , sin θk ]Ek [cos θk , sin θk ]′ )1/2
(5)
′
where vk = [cos θk , sin θk ] . Geometric properties of the ellipse are reflected in the algebraic properties of the corresponding matrix Ek , which carries useful and important information about the ellipse extension (i.e., size, shape and orientation). Thus entries of the matrix Ek can be considered to form shape parameters. However, the estimated shape Ek may lose positive semi-definiteness, leading to unpredictable results. So, the Cholesky decomposition Ek = Lk L′k is utilized to handle this problem, where " # (1) Lk 0 Lk = (6) (2) (3) Lk Lk
(0)
(1)
(1)
(N )
(N )
(11)
This method may be extended to other smooth object shapes by using suitable parametric representations. It should be noted that the approach based on support functions can not be applied directly to an extended object modeled as a non-smooth shape such as a rectangle. In [20], another approach based on extended Gaussian images was proposed to deal with this problem. C. Modeling Based on EGI The extended Gaussian image (EGI) is particularly convenient to represent a convex body K concisely [25]. In particular, if we have the EGI parameters of a convex polygon K, the shape of K is uniquely determined. Moreover, the EGI has two other useful properties: translation invariance and orientation equivariance: it is not affected by translation of the object, and rotation of the object leads to an equal rotation of its EGI.
is a lower triangular matrix with positive diagonal entries. Then Eq. (5) can be rewritten as H(θk ) = ([cos θk , sin θk ]Lk L′k [cos θk , sin θk ]′ )1/2
(7)
Fig. 3.
EGI representation of a convex polygon
According to Eqs. (3) and (4), we have D(θk ) = 2([cos θk , sin θk ]Lk L′k [cos θk , sin θk ]′ )1/2
(8)
C(θk ) = 2([− sin θk , cos θk ]Lk L′k [− sin θk , cos θk ]′ )1/2 (9) The shape state ek is changed to consist of the non-zero entries (1) (2) (3) Lk , Lk and Lk and is included in the state vector xk = (1) (2) (3) [xk , x˙ k , yk , y˙ k , Lk , Lk , Lk ]′ . Note that by selecting a suitable parametric representation for the object shape, it may be extended to a variety of object shapes, that is, different parametric representations will lead to different shapes. In this context, the Fourier series expansion is introduced to object modeling based on support functions. As such, more detailed shape information can be described and utilized. As a real-valued integrable periodic function on [0, 2π] (i.e., H(θ) = H(θ + 2π)), the support function H(θk ) of an extended object can be expanded in a Fourier series of a degree N in θk as X (0) (j) (j) H(θk ) = ak + (ak cos jθk + bk cos jθk ) (10) j=1,2,...N
Correspondingly, the down-range extent D(θk ) and crossrange extent C(θk ) can be easily obtained by using Eqs. (3)
Due to these properties, the EGI representation is widely used for object recognition and reconstruction in computer vision, graphics, and image processing. If K is an N -sided polygon whose jth edge has length lj and outer unit normal vector uj = [cos αj , sin αj ]′ , its EGI can be represented by N vectors lj uj for j = 1, 2, ..., N counterclockwise (see Fig. 3), and its EGI parameters are easily obtained as {l1 , ..., lN , α1 , ..., αN } [26]. If the viewing diection is v = [cos θ, sin θ]′ , the downrange and cross-range extent of the object K can be written as [20] N
D(θ) = H(θ) + H(θ + π) =
1X lj | sin(θ − αj )| 2 j=1
(12)
N
π π 1X C(θ) = H(θ − ) + H(θ + ) = lj | cos(θ − αj )| (13) 2 2 2 j=1 For a rectangular object, its EGI parameters can be easily obtained as {lk,1 , lk,2 , lk,1 , lk,2 , αk , αk + π2 , αk + π, αk + 3π 2 }. Thus lk,1 , lk,2 (i.e., the lengths of minor and major axes) and the angle of orientation αk can be considered to form shape parameters and included in the state vector xk =
′ ′ [(xm k ) , lk,1 , lk,2 , αk ] to be estimated. The EGI-based modeling approach works for other non-smooth shapes besides rectangles. It should be noted that the down-range extent D(θ) and cross-range extent C(θ) can be calculated by using the EGI parameters in Eqs. (12) and (13), respectively. This provides a connection between the EGI and support function. For example, the support function representation of a rectangular object is easily obtained by utilizing its symmetric property, that is,
1 H(θk ) = H(θk + π) = D(θk ) 2 1 = (lk,1 | sin(θk − αk )| + lk,2 | cos(θk − αk )|) (14) 2 As such, some non-smooth objects having symmetry can be represented by support functions indirectly. For more details on EGI-based modeling we refer the reader to [20]. III.
M ODELING FOR T RACKING OF C OMPLEX E XTENDED O BJECT U SING M INKOWSKI A DDITION
In this section, we propose an approach to modeling a complex extended object by using a Minkowski addition of multiple simple sub-objects, each represented by a support function or EGI. With this approach, common and individual properties of the sub-objects can be sufficiently utilized to obtain a suitable parametric representation of a complex extended object. As such, more detailed shape information can be captured. This will largely facilitate target tracking based on the proposed models. A. Minkowski Addition If A and B are two arbitrary sets of points in real Euclidean n-dimensional space Rn , their Minkowski addition A ⊕ B is defined as [21] A ⊕ B = {a + b, a ∈ A, b ∈ B}
HλA (θ) = λHA (θ)
(18)
If λ1 , ..., λm are positive real numbers and K1 , ..., Km are convex bodies, then the support function of their linear combination λ1 K1 ⊕ · · · ⊕ λm Km is given by Hλ1 K1 ⊕···⊕λm Km (θ) = λ1 HK1 (θ) + · · · + λm HKm (θ) (19) Obviously from the above equation, the Minkowski addition of convex bodies K1 , ...Km becomes addition of support functions. C. Down-range and Cross-range Extent Here a complex object K is described as a Minkowski addition of multiple simple sub-objects Ki ∀ i = 1, ..., m (i.e., K = K1 ⊕ · · · ⊕ Km ), represented by addition of support functions: HK (θ) = HK1 ⊕···⊕Km (θ) = HK1 (θ) + · · · + HKm (θ) (20) As shown in Eqs. (3) and (4), the down-range extent D and cross-range extent C of an extended object can be expressed in terms of support functions. Thus down-range extent and cross-range extent of K can be easily obtained as follows. Since K = K1 ⊕ · · · ⊕ Km , DK (θ) = DK1 ⊕···⊕Km (θ) = HK1 ⊕···⊕Km (θ) + HK1 ⊕···⊕Km (θ + π) = HK1 (θ) + HK1 (θ + π) + · · · + HKm (θ) + HKm (θ + π) = DK1 (θ) + · · · + DKm (θ)
(21)
(15)
where “+” denotes the vector addition of two points. The Minkowski addition “⊕” is widely used in computational geometry, and it plays a fundamental role in shape description and analysis. The inverse of the “⊕” operator is “⊖”, denoting the Minkowski decomposition operator. If Ki is a compact convex set in Rn , and λi ≥ 0 ∀ i = 1, ..., m, then the vector sum λ1 K1 ⊕ · · · ⊕ λm Km = {λ1 k1 + · · · + λm km , ki ∈ Ki } (16) is called a Minkowski linear combination. The addition and scalar multiplication in Eq. (16) are called Minkowski addition and Minkowski scalar multiplication, respectively. Note that this Minkowski linear combination turns out to yield a convex set. B. Minkowski Addition and Support Functions In the general convex case, if both bodies A and B are represented by support functions for all viewing direction vectors v = [cos θ, sin θ]′ in Rn , the Minkowski addition A⊕B is a unique convex body with HA⊕B (θ) = HA (θ) + HB (θ)
where HA (θ) + HB (θ) is also a support function. For λ ≥ 0, the scalar multiplication is defined as
(17)
CK (θ) = CK1 ⊕···⊕Km (θ)
π π = HK1 ⊕···⊕Km (θ − ) + HK1 ⊕···⊕Km (θ + ) 2 2 π π = HK1 (θ − ) + HK1 (θ + ) + · · · 2 2 π π + HKm (θ − ) + HKm (θ + ) 2 2 = CK1 (θ) + · · · + CKm (θ) (22)
where DKi (θ) = HKi (θ) + HKi (θ + π), i = 1, ..., m π π CKi (θ) = HKi (θ − ) + HKi (θ + ), i = 1, ..., m 2 2
(23) (24)
As shown clearly in Eqs. (21) and (22), DK (θ) and CK (θ) can be expressed in terms of support functions and reduced to addition of DK1 (θ), ..., DKm (θ) and CK1 (θ), ..., CKm (θ), respectively. Thus a complex object can be modeled as a Minkowski addition of multiple sub-objects. 1) Example I: As shown in Fig. 4, such an object is modeled as a Minkowski addition of two smooth objects (i.e., two elliptical objects), each represented by a support function.
30 20 10 0 −10 Sub−object 1 Sub−object 2 MA
−20 −30 −30
−20
−10
0
10
20
30
(a) Object shapes
Fig. 5. Complex object modeling using Minkowski addition, illustrative Example II
30 Sub−object 1 Sub−object 2 MA
25
so that the Minkowski addition of multiple polygons Ki (for i = 1, ..., m) can be calculated easily. As shown in Fig. 5, such a complex object is modeled as a Minkowski addition of two non-smooth objects (i.e., two triangular objects), each represented by an EGI.
20 15 10 5 0 0
1
2
3
4
5
6
7
(b) Support functions Fig. 4. Complex object modeling using Minkowski addition, illustrative Example I
2) Example II: For non-smooth objects, any convex polygon can be represented by its EGI. Suppose that the EGI parameters of polygon K1 and K2 are K1 : {l1 , .., lM , α1 , ..., αM } K2 : {lM+1 , .., lM+N , αM+1 , ..., αM+N }
3) Example III: As mentioned in the previous section, some symmetric non-smooth objects can also be represented by support functions indirectly. As shown in Fig. 6, this object is modeled as a Minkowski addition of an elliptical sub-object and a rectangular sub-object. The support function representation of the rectangular sub-object is easily obtained by utilizing its symmetry property (see Eq. (14)). In some cases, a complex object (e.g., Fig. 6) can also be modeled as a Minkowski addition of smooth and non-smooth subobjects, both of which can be represented by support functions. This largely facilitates modeling and estimation of complex extended objects. 30 20
From Eq. (12)
10
M
1X DK1 (θ) = lj | sin(θ − αj )| 2 j=1
M+N 1 X DK2 (θ) = lj | sin(θ − αj )| 2
(25)
0 −10
(26)
Sub−object 1 Sub−object 2 MA
−20
j=M+1
−30 −30
Then
−20
−10
0
10
20
30
(a) Object shapes
DK1 ⊕K2 (θ) M+N 1 X = DK1 (θ) + DK2 (θ) = lj | sin(θ − αj )| 2 j=1
30
(27)
20
Similarly, we have
15
CK1 ⊕K2 (θ) = CK1 (θ) + CK2 (θ) =
Sub−object 1 Sub−object 2 MA
25
M+N 1 X lj | cos(θ − αj )| 2 j=1
10
(28)
Thus the EGI parameters of K1 ⊕ K2 can be easily obtained as K1 ⊕ K2 : {l1 , .., lM+N , α1 , ..., αM+N } Clearly, K1 ⊕ K2 is obtained by naturally merging their EGIs. In this manner we can obtain a set of vectors by mixing EGIs
5 0 0
1
2
3
4
5
6
7
(b) Support functions Fig. 6. Complex object modeling using Minkowski addition, illustrative Example III
D. System Model As an example of the object K in Fig. 7, it is described as a Minkowski addition of two rectangular sub-objects K1 and K2 , each represented by an EGI. For simplicity here, it is assumed that these two sub-objects have the same minor and major axis lengths (i.e., lk,1 and lk,2 ) but differ in the angle of orientation αk and γk . Thus the EGI parameters of K1 and K2 can be easily obtained as π 3π , αk + π, αk + } 2 2 π 3π } K2 : {lk,1 , lk,2 , lk,1 , lk,2 , γk , γk + , γk + π, γk + 2 2
K1 : {lk,1 , lk,2 , lk,1 , lk,2 , αk , αk +
Clearly, the shape state vector ek = [lk,1 , lk,2 , αk , γk ]′ is ′ ′ included in the state vector xk = [(xm k ) , lk,1 , lk,2 , αk , γk ] to be estimated. By Eqs. (12)–(13) and (27)–(28), we have
C(θk ) = CK1 ⊕K2 (θ) = CK1 (θ) + CK2 (θ) = lk,1 | cos(θk − αk )| + lk,2 | sin(θk − αk )| + lk,1 | cos(θk − γk )| + lk,2 | sin(θk − γk )|
cov[vk ] = Rk = diag[Rkr , Rkβ , RkD , RkC ]
IV.
Object modeling using Minkowski addition
1) State Dynamics: To simplify analysis, we consider the case that the target is moving at a nearly constant velocity (CV) [27] in the two-dimension (2-D) Cartesian coordinate system. Then the discrete-time dynamics is xk+1 = FkCV xk + Γk wk
Since the measurements z k = [rk , βk , Dk , Ck ]′ are obtained from different physical channels, the noise vk is generally assumed to be zero-mean Gaussian with independent elements and covariance
For this problem, the measurement equation is severely nonlinear, so a nonlinear filter should be used to estimate both kinematics and object shape jointly. Here the estimation process can be implemented by using the unscented filter (UF) [28] [29] for the nonlinear part of (30) and leave the remaining part to the Kalman filter.
D(θk ) = DK1 ⊕K2 (θk ) = DK1 (θk ) + DK2 (θk ) = lk,1 | sin(θk − αk )| + lk,2 | cos(θk − αk )| + lk,1 | sin(θk − γk )| + lk,2 | cos(θk − γk )|
Fig. 7.
2) Measurement Equation: It is assumed that a high resolution sensor is fixed at (Xo , Yo ) and provides measurements of down-range extent D and cross-range extent C along LOS, as well as the range r and bearing β measurements of the object centroid, given by p (xk − Xo )2 + (yk − Yo )2 zk = arctan((yk − Yo )/(xk − Xo )) + vk (30) D(θk ) C(θk )
(29)
with FkCV = diag(F, F, I4 ), Γk = diag(Γ, Γ, I4 ) 1 0 0 0 2 1 T T /2 0 1 0 0 I4 = , F = , Γ= 0 0 1 0 0 1 T 0 0 0 1
where I4 describes the approximately unchanged object extension over time, and the uncertainty of the target state is embodied in the process noise wk . Given more information, we can design Fk specifically for different scenarios, which leads to different centroid state and object extension transition.
S IMULATION R ESULTS AND P ERFORMANCE E VALUATION
To illustrate the effectiveness of the proposed approach, simulation examples are provided in this section. In the simulation, we focus on the case of a stationary sensor platform and a moving extended object. Consider a scenario in which an extended object (as an example in Fig. 7) moves at a nearly CV in the 2-D Cartesian coordinate system with the ′ initial kinematic state xm 0 = [1000m, 60m/s, 2000m, 30m/s] . The true object is modeled as a Minkowski addition of two rectangular objects with different angles of orientation, both of which have the same lengths of minor and major axes. For this object at the initial time, the shape state e0 = [5m, 25m, π3 , π6 ]′ . The sensor is fixed at the origin (0, 0) and it provides measurements of range, bearing, and the target extent along the LOS every T = 2s. Each measurement is corrupted by zero-mean white Gaussian noise with standard deviations σr = 10m, σβ = 0.02rad, σD = 15m, σC = 15m. Simulation results for this scenario are shown in Figs. 8 and 9, which give the trajectory of the object and tracking performance. Evaluation of estimation performance has always been centered on the root-mean-square error (RMSE): RMSE(ˆ x) = (
M 1 X ke xi k2 )1/2 M i=1
(31)
where x ei is the estimation error on the ith of the M MonteCarlo runs. However, use of the average Euclidean error (AEE) is better than RMSE, as analyzed convincingly in [30]. The AEE has a more direct, natural interpretation and is less dominant by large errors. It is defined as AEE(ˆ x) =
M 1 X ke xi k M i=1
(32)
8000 2100 Y
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1.275 1.28 1.285 1.29 4 X x 10 True Object Estimated Object 10000
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Trajectory of the extended object
In this simulation, AEE is chosen as the measure and comparison results are presented in terms of AEE for the kinematic state (i.e., position and velocity) over 100 Monte Carlo runs. From Figs. 9(a) and 9(b), it can be seen that the proposed approach can estimate the kinematic state well. To objectively evaluate the quality of shape estimation, the Hausdorff distance is adopted to measure the degree of resemblance between the estimated shape and the true one. Since a convex body K is completely determined by its support function HK (·), the Hausdorff distance can be defined via the support functions as [22] dH (A, B) = kHA − HB k∞ = sup |HA (θk ) − HB (θk )|
(33)
θk ∈[0,2π]
where k·k∞ denotes the infinity norm of a function and “sup” denotes the “supremum” or “least upper bound”. For Eq. (33), the Hausdorff distance calculates the distance between the bodies A and B. For convex bodies, this is equivalent to the alternative definition dH (A, B) = max{max d(x, A), max d(x, B)} x∈B
x∈A
(34)
which applies to arbitrary compact sets, where d(x, A) denotes the distance from the point x to the set A. The Hausdorff distance is widely used for object matching in computer vision, object recognition, and shape analysis [23]. The Hausdorff distance of object shape estimation is calculated by Eq. (33), which is hard to use directly because [0, 2π) as the range of θ is a continuous set. For this reason, [0, 2π) is 2π replaced by the discrete set {i · N , i = 0, 1, ...Ns − 1} based s on uniform angle sampling, where Ns is the number of sample points (here Ns = 1000). Then Eq. (33) becomes ˆ = sup |HK (θ) − H ˆ (θ)| dH (K, K) K 2π θ∈{i· N ,i=0,1,...Ns −1} s
(35)
ˆ and the true shape K. Clearly, between the estimated shape K the shorter the Hausdorff distance is, the better the shape estimation is. As shown in Fig. 9(c), the Hausdorff distance demonstrates that the object shape can be estimated accurately. Overall, simulation results illustrated the effectiveness of what we proposed. V.
C ONCLUSION
To describe a complex extended object accurately and effectively, we have proposed an approach to model it by using the Minkowski addition of multiple simple sub-objects without losing shape information. These sub-objects are represented by support functions/EGIs in a concise form. Their Minkowski addition can be transformed to a simple addition of support functions or a set of vectors by mixing EGIs. The Minkowski addition fits well with our problem: a) Due to its concise mathematical forms and favorable properties, the complex object modeling is largely simplified and more detailed shape information is described. b) The approaches based on support functions and EGI have been extended and further developed. c) The introduction of the Minkowski addition paves the way for tracking of complex objects, that is, the proposed model can be easily implemented for EOT to estimate the kinematic state and object extension jointly. The effectiveness of the proposed approach has been verified through simulation results. With suitable adoptions of object modeling using Minkowskid addition, a larger range of complex extended object tracking can be handled. R EFERENCES [1] Y. Bar-Shalom and X. R. Li, Estimation and Tracking: Principles, Techniques, and Software. Boston, MA: Artech House, 1993. (Reprinted by YBS Publishing, 1998). [2] X. R. Li and J. Dezert, “Layered multiple-model algorithm with application to tracking maneuvering and bending extended target in clutter,” in Proceedings of the 1st International Conference on Information Fusion (Fusion 1998), Las Vegas, NV, USA, 1998.
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(c) Fig. 9. performance comparison. (a) AEE of Position. (b) AEE of Velocity. (c) Hausdorff distance.
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