1  Mixture Reduction on Matrix Lie Groups  Josip  ́ Cesi ́ c,   Member, IEEE,   Ivan Markovi ́ c,   Member, IEEE,   and Ivan Petrovi ́ c,   Member, IEEE ∗  Abstract —Many physical systems evolve on matrix Lie groups and mixture filtering designed for such manifolds represent an inevitable tool for challenging estimation problems. However, mixture filtering faces the issue of a constantly growing number of   components,   hence   require   appropriate   mixture   reduction techniques.   In   this   letter   we   propose   a   mixture   reduction approach   for   distributions   on   matrix   Lie   groups,   called   the concentrated   Gaussian   distributions   (CGDs).   This   entails appropriate reparametrization of CGD parameters to compute the KL divergence, pick and merge the mixture components. Furthermore,   we   also   introduce   a   multitarget   tracking   filter on   Lie   groups   as   a   mixture   filtering   study   example   for   the proposed reduction method. In particular, we implemented the probability hypothesis density filter on matrix Lie groups. We validate   the   filter   performance   using   the   optimal   subpattern assignment   metric   on   a   synthetic   dataset   consisting   of   100  randomly generated multitarget scenarios.  Index Terms —Mixture reduction, estimation on matrix Lie groups, multitarget tracking, probability hypothesis density filter.  I. I NTRODUCTION  M ANY   statistical   and   engineering   problems   require modeling   of   complex   multi-modal   data,   wherein mixture   distributions   became   an   inevitable   tool   [1],   [2], primarily in traditional application domains like radar and sonar tracking [3], and later in different modern fields such as computer vision [4], speech recognition [5] or multimedia processing [6]. Approaches relying on mixture distributions often face the problem of large or an ever increasing number of mixture components, hence the growth of components must be controlled by approximating the original mixture with the mixture of a reduced size [7]–[9]. For example, in the case of multitarget tracking applications, by employing conventional Gaussian mixture based filters [10], [11], during the recursion process the number of components inevitably increases. This appears firstly due to appearance of newly birthed or spawned components,   and   secondly,   due   to   inclusion   of   multiple measurements, which results in geometrical increase in the number of components. Another   important   aspect   of   estimation   is   the   state space geometry, hence many works have been dedicated to appropriate uncertainty modeling and estimation techniques for a wide range of applications [12]–[15], motivated by theoretical and implementation difficulties caused by treating a   constrained   problem   naively   with   Euclidean   tools.   For example,   Lie   groups   are   natural   ambient   (state)   spaces for description of the dynamics of rigid body mechanical  ∗ J.  ́ Cesi ́ c, I. Markovi ́ c and I. Petrovi ́ c are with the University of Zagreb Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia. E-mail:   { name.surname@fer.hr }  This   work   has   been   supported   from   the   Unity   Through   Knowledge Fund (no. 24/15) under the project Cooperative Cloud based Simultaneous Localization and Mapping in Dynamic Environments (cloudSLAM) and the European Union’s Horizon 2020 research and innovation programme under grant agreement No 688117 (SafeLog).  systems. In [16] it has been observed that the distribution of the pose of a differential drive mobile robot is not a Gaussian distribution in Cartesian coordinates, but rather a distribution on the special Euclidean group SE (2) . Similarly, [17]   discussed   the   uncertainty   association   with   3D   pose employing the SE (3)   group. Furthermore, attitude estimation arises naturally on the SO (3)   group [15]. In [18] a feedback particle filter on matrix Lie groups was proposed, while in [19], [20] authors proposed an extended Kalman filter on matrix Lie groups (LG-EKF), building the theory upon the concentrated   Gaussian   distribution   (CGD)   on   matrix   Lie groups [21]. In this letter we address finite mixtures of distributions on   matrix   Lie   groups.   We   propose   a   novel   approach   to CGD mixture reduction, which required finding solutions for computing Kullback-Leibler divergence of CGD components and CGD component merging. Furthermore, since previous methods require choosing the appropriate tangent space, we also provide an extensive analysis on the choice thereof. As a study example, we use the proposed reduction method in a multitarget tracking scenario. We introduce the probability hypothesis density filter (PHD) on matrix Lie groups with approximation based on a finite mixture of CGDs. II. M ATHEMATICAL   P RELIMINARIES  We now introduce theoretical preliminaries concerning Lie groups; however, for a more rigorous introduction the reader is directed to [22]. A Lie group G is a group which has the structure of a smooth manifold; moreover, a tangent space  T X   ( G )   is associated to   X   ∈   G such that the tangent space placed at the group identity, called Lie algebra   g , is transferred by applying corresponding action to   X . In this paper we are interested in matrix Lie groups which are usually the ones considered in engineering and physical sciences. The Lie algebra   g   ⊂   R n × n   associated to a   p -dimensional matrix   Lie   group   G   ⊂   R n × n   is   a   p -dimensional   vector space. The matrix exponential   exp G   and matrix logarithm   log G  establish a local diffeomorphism between the two  exp G   :   g   →   G   and   log G   :   G   →   g .   (1) Furthermore,   a   natural   relation   exists   between   g   and   the Euclidean space   R p   given through a linear isomorphism  [ · ] ∨  G   :   g   →   R p   and   [ · ] ∧  G   :   R p   →   g .   (2) For   x   ∈   R p   and   X   ∈   G we use the following notation [23]  exp ∧  G   ( x ) = exp G ([ x ] ∧  G   )   and   log ∨  G   ( X ) = [log G ( X )] ∨  G   .   (3) Lie groups are generally non-commutative, i.e.,   XY   6   =   Y X . However, the non-commutativity can be captured by the so- called adjoint representation of G on   g   [24]  X   exp ∧  G   ( y ) = exp ∧  G   (Ad G ( X ) y ) X,   (4)  arXiv:1708.06252v1 [cs.SY] 18 Aug 2017
2  which can be seen as a way of representing the elements of the group as a linear transformation of the group’s algebra. The adjoint representation of   g ,   ad G , is in fact the differential of  Ad G   at the identity. Another important result for working with Lie group elements is the Baker-Campbell-Hausdorff (BCH) formula, which enables representing the product of Lie group members as a sum in the Lie algebra. We will use the following BCH formulae [24], [25]  log ∨  G   (exp ∧  G   ( x ) exp ∧  G   ( y )) =   y   +   φ G ( y ) x   +   O ( || y || 2 ) ,   (5)  log ∨  G   (exp ∧  G   ( x   +   y ) exp ∧  G   ( − x )) = Φ G ( x ) y   +   O ( || y || 2 ) ,   (6) where   φ G ( y ) =   ∑ ∞  n =0  B n   ad G   ( y ) n  n !   ,   B n   are Bernoulli numbers, and   Φ G ( x ) =   φ G ( x ) − 1 . For many common groups used in engineering and physical sciences closed form expressions for  φ G ( · )   and   Φ G ( · )   can be found [17], [24]; otherwise, a truncated series expansion is used.  A. Concentrated Gaussian distribution  Herein we introduce the concept of the concentrated Gaussian distribution which is used to define random variables on matrix Lie group. A random variable   X   ∈   G has a CGD with the mean   μ   and covariance   Σ , i.e.,   X   ∼ G ( X ;   μ,   Σ) , if  X   = exp ∧  G   ( ξ ) μ ,   (7) where   μ   ∈   G,   and   ξ   ∼   N   ( ξ ;   0 p × 1 ,   Σ)   is   a   zero-mean ‘classical’ Gaussian random variable with the covariance   Σ   ⊂  R p × p   [17], [20]. Note that in this way, we are directly defining the CGD covariance in the pertaining Lie algebra   g , while the mean is defined on the group G. Given that, the previous definition (7) then induces a pdf of  X   over G as follows [17], [20]  1 =  ∫  R p  1  √ (2 π ) p | Σ |   exp ∧  G  (  −   1 2   || ξ || 2 Σ  )  d ξ  =  ∫  G  β   exp ∧  G  (  −   1 2   ||   log ∨  G   ( Xμ − 1 ) || 2 Σ  )  d X   (8) where   || x || 2 Σ   =   x T Σ − 1 x . Therein the change of coordinates  ξ   = log ∨  G   ( Xμ − 1 ) , with the pertaining differentials   d X   =  | Φ( ξ ) | d ξ , resulted with the CGD normalizing constant  β   = 1 /  √  (2 π ) p | Φ(log ∨  G   ( Xμ − 1 ))ΣΦ(log ∨  G   ( Xμ − 1 )) T | .   (9) Note that this change of variables is valid if all eigenvalues of   Σ   are small, i.e., almost all the mass of the distribution is concentrated in a small neighborhood around the mean value [20]. The pdf over   X   is now fully determined by (8) and (9). III. CGD   MIXTURE REDUCTION  With the theoretical preliminaries setup, we continue with mixture reduction on matrix Lie groups. A finite mixture of our present interest is given as the weighted sum of CGDs  N ∑  i =1  w i G ( X ;   μ i ,   Σ i )   ,   (10) where   w i   are component weights and   N   is the total number of mixture components. An illustration of (10) is given in Fig.   1.   Component   reduction   procedures   typically   require three building blocks: (i) component distance measure, (ii) component picking algorithm, and (iii) component merging. While various solutions exist for ‘classical’ Gaussian mixtures [7]–[9], questions remain on how to approach the component number reduction for CGD mixtures on matrix Lie groups. Therefore, first we focus on the the fundamental question of how to measure the distance between two CGD components.  A. Component distance measure  Our aim is to use a standard information-theoretic measure between two CGD components and we propose to use the Kullback-Leibler (KL) divergence [26]. Let   G i   =   G ( X ;   μ i ,   Σ i )  and   G j   =   G ( X ;   μ j   ,   Σ j   )   be two mixture components with  p i ( X )   and   p j   ( X )   as their respective pdfs. Since there is nothing   intrinsic   in   the   definition   of   KL   divergence   that requires the underlying space to be Euclidean, by definition  D KL ( G i ,   G j   ) =  ∫  G  p i ( X ) log  (   p i ( X )  p j   ( X )  )  d X .   (11) In order to evaluate the integral (11), we need to employ the change of coordinates as in (8), but this time from the direction of the group G, i.e., from   X   ∈   G to   ξ   ∈   R p . Note that in (8) the change evolved around the distribution mean   μ ; however, since in (11) generally   μ i   6   =   μ j   , we cannot apply the same approach. Hence, before evaluating (11), we first discuss how to change the coordinates on the level of a single distribution. Let   G ( X ;   μ,   Σ)   be a CGD, and if we change the coordinates using   X   = exp ∧  G   ( ξ ) μ t ,   μ t   ∈   G, where   μ t   6   =   μ , we get  1 =  ∫  G  β   exp ∧  G  (  −   1 2   ||   log ∨  G   ( Xμ − 1 ) || 2 Σ  )  d X   (12)  CoC  ≈  ∫  R p  η   exp ∧  G  (  −   1 2   ||   log ∨  G   (exp ∧  G   ( ξ ) μ t μ − 1 ) || 2 Σ  )  d ξ  (6)  ≈  ∫  R p  η   exp ∧  G  (  −   1 2   || Φ G ( r t )( ξ   −   r t ) || 2 Σ  )  d ξ  =  ∫  R p  η   exp ∧  G  (  −   1 2   || ξ   −   r t || 2  φ G   ( r t )Σ φ T G   ( r t )  )  d ξ ,  where   r t   = log ∧  G   ( μμ − 1  t   ) ,   η   approximately evaluates to  η   =   β | Φ( ξ ) |   =   | Φ( ξ ) |  √ (2 π ) p | Σ | · | Φ(log ∨  G   (exp ∧  G   ( ξ ) μ t μ − 1 )) | ≈   1  √ (2 π ) p | φ G ( r t )Σ φ G ( r t ) T |   ,   (13) and we obtain   ξ   ∼ N   ( ξ ;   r t , φ G ( r t )Σ φ T G ( r t )) .  Remark 1.   Covariance of a CGD represents the uncertainty relevant only to the tangent space of its own mean. In [24] authors studied how the covariance changes if looked at from the perspective of a value which is different than the distribution mean. They dubbed this procedure ‘distribution unfolding’. For example, if we unfold   G ( X ;   μ,   Σ)   around an arbitrary   μ t   ∈   G, using   (5)   and following [24] we get  ξ t   = log ∨  G  (   exp ∧  G   ( ξ ) μμ − 1  t  )  ≈   log ∨  G  ( μμ − 1  t  )   +   φ G  (   log ∨  G   ( μμ − 1  t   ) ) ξ .   (14)
3  G  w a ,   G ( μ a ,   Σ a )  μ a  w b ,   G ( μ b ,   Σ b )  μ b  w c ,   G ( μ c ,   Σ c )  μ c  G  g   w j   ,   N   ( r,   Σ φ j   )  w i ,   N   (0 ,   Σ i )  μ i  G  g  w ∗ ,   N   ( r ∗ ,   Σ ∗ )  G  w ∗ ,   G ( μ ∗ ,   Σ Φ ∗  )  Fig. 1: Illustration of a finite mixture of CGDs (left) and the component merging procedure (right).  By computing the expectation and covariance of   (14) , we obtain a reparametrized distribution,   ξ t   ∼   N   ( ξ t ;   r t ,   Σ φ ) , where  r t   = log ∨  G  ( μμ − 1  t  )   (15)  Σ φ   =   φ G ( r t )Σ φ T  G   ( r t ) .   (16)  This pdf is equal to the one obtained through the change of   coordinates   in   (12) .   However,   obtaining   this   result   by using the procedure of coordinates change through a pdf is important from the perspective of KL divergence evaluation. An illustration of unfolding a component   j   around   μ i , using  (15)   and   (16) , is given in Fig. 1.  The KL divergence between two CGDs   G i   =   G ( μ i ,   Σ i )   and  G j   =   G ( μ j   ,   Σ j   )   can now be evaluated as  D KL ( G i ,   G j   )   ≈  ∫  R p  p i ( ξ ) log  (   p i ( ξ )  p j   ( ξ )  )  d ξ   =   D KL ( N i ,   N j   )   , p i ( ξ )   ∼ N i   =   N   ( ξ ;   r i ,   Σ φ i   )   , r i   = log ∧  G   ( μ i μ − 1  t   )   ,   (17)  p j   ( ξ )   ∼ N j   =   N   ( ξ ;   r j   ,   Σ φ j   )   , r j   = log ∧  G   ( μ j   μ − 1  t   )   ,  and   Σ φ   =   φ G ( r )Σ φ T G ( r ) .   By   employing   the   change   of coordinates, we can evaluate the KL divergence of two CGDs similarly as in the case of ‘classical’ Gaussian distributions, but   with   reparametrized   means   and   covariances.   The   KL divergence is then equal to  D KL ( N i ,   N j   ) = 1 2  (  tr ( Σ φ j  − 1 Σ φ i  )   −   K   + log R  | Σ φ j   | | Σ φ i   |   (18)  + ( r j   −   r i ) T (Σ φ j   ) − 1 ( r j   −   r i )  )  ,  where   tr(   .   )   and   |   .   |   designate matrix trace and determinant, respectively, while   K   is the mean vector dimension. Finally, for mixture components it is necessary to use the scaled symmetrized KL divergence [27], which also takes component weights into account  D s KL ( w i N i , w j   N j   ) = 1 2  (  ( w i   −   w j   ) log R  w i  w j  +   (19)  w i D KL ( N i ,   N j   ) +   w j   D KL ( N j   ,   N i )  )  .  B. Component picking algorithm  Now   that   we   know   how   to   compute   a   distance   measure between two CGD mixture components, we need to choose an appropriate component picking algorithm which will tell us how to screen the whole mixture and which components to pick for merging. However, with CGD mixtures there is also another momentum. If we have   N   components in the mixture   with   different   weights,   how   should   we   approach the problem of measuring distance, i.e., choosing   μ t   for the change of coordinates? Should all the distances be calculated with respect to the mean of the component with the highest weight or the lowest weight? Or should we ‘reparametrize’ each   component   on   a   pairwise   basis?   In   this   letter   we study the following five scenarios: (i/ii) all components are reparametrized about the mean of the component with the highest/lowest weight, (iii) the reparametrization about the identity element, and (iv/v) components are reparametrized on a pairwise basis by choosing the mean of the component pair with the higher/lower weight. For analyzing the five scenarios we use two common component picking strategies; (i) Exhaustive pairwise [28], and (ii) West’s [29] algorithms. The   Exhaustive   pairwise   algorithm   determines   distances between all components and merges the closest pair, while West’s algorithm sorts the components according to their respective weights, then finds and merges the component most similar to the first one.  C. Merging the components  A component merging algorithm for Gaussian components in  R p   was proposed in [28]:  r ∗ =   1  w ∗  ∑  i  w i r i ,   Σ ∗ =   1  w ∗  ∑  i  (  w i  ( Σ i   +   r i r T  i  ))  −   r ∗ ( r ∗ ) T  where   w ∗   =   ∑  i   w i ,   w i N   ( r i ,   Σ i )   represents   the   i -th component, and   w ∗ N   ( r ∗ ,   Σ ∗ )   is the resulting component. Although   merging   works   for   an   arbitrary   number   of components, in our case we will always merge two. However, the previous expressions are defined for Gaussians in   R p   and the question arises how to apply the same approach for CGD mixtures? We propose to use the same principle as for computing the KL divergence described in Section III-A, i.e., the components to be merged need to be first reparametrized about the tangent space of the same mean, since covariances are only relevant with respect to their own mean. Once we compute the resulting component,   w ∗ N   ( r ∗ ,   Σ ∗ ) , we need to map it back to the group G. Given a lemma from [20] and following convention (7), the procedure evaluates to  μ ∗ = exp ∧  G   ( r ∗ ) μ t ,   Σ Φ ∗  = Ad r ∗  G   Φ G ( r ∗ )Σ ∗ ( Ad r ∗  G   Φ G ( r ∗ ) ) T ,   (20) where   Ad r ∗  G   = Ad G (exp ∧  G   ( r ∗ )) . We can notice that covariance reparametrization was necessary to make it relevant from the perspective of the tangent space of the newly computed  μ ∗ . An illustration of merging and reparametrization (20) of component   j   with respect to   μ i   is given in Fig. 1. IV. S TUDY EXAMPLE   - PHD   FILTER ON   L IE GROUPS  MTT is a complex problem consisting of many challenges and PHD filter presents itself as one of the solutions to MTT.
4  The reason why PHD filter is interesting for the present letter   is   because   one   of   its   implementations   is   based   on Gaussian mixtures (GM-PHD) [10]. Besides Gaussians, other distributions can be used and in our previous work [30] we proposed a mixture approximation of the PHD filter based on the von Mises distribution on the unit circle. In this letter, as a study example, we implement a PHD filter tailored for Lie groups (LG-PHD), based on the mixture of CGDs and the reduction schemes presented in the previous section. The LG-PHD can be potentially applied in MTT scenarios where the target state is modelled as a pose in SE (2)   or SE (3)  The PHD filter propagates the   intensity function   D k − 1 , and operates by evaluating two steps—prediction and update. By assuming   D k − 1   and birth intensity being Gaussian mixtures [10], the GM-PHD prediction results with another Gaussian mixture (Prop.   1   in [10]). Similarly, if   D k − 1   and birth intensity are given with CGD mixtures, the LG-PHD prediction results with another CGD mixture, relying on the LG-EKF prediction applied to each mixture component [23]. The   product   of   two   Gaussians   evaluates   to   a   scaled Gaussian, hence the update step of GM-PHD can be calculated analytically (Prop.   2   in [10]). In contrary, the product of two CGDs, occurring in LG-PHD update, cannot be evaluated directly. Hence, we apply approximations following the same train of thought as in LG-EKF prediction [23] where given posterior   p ( X k − 1 | Z 1: k − 1 )   and motion model   p ( X k | X k − 1 ) , it approximates the joint distribution   p ( X k , X k − 1 | Z 1: k − 1 ) , and then marginalizes obtaining   p ( X k | Z 1: k − 1 ) . Similarly, given  p ( X k | Z 1: k − 1 )   and likelihood   p ( Z k | X k ) , we approximate the joint   distribution   p ( X k , Z k | Z 1: k − 1 ) ,   and   then   marginalize obtaining   p ( X k | Z k ) .   Final   LG-PHD   formulae   are   nearly identical to GM-PHD, except for Jacobian matrices.  A. Experiments  In order to validate the performance of the proposed LG-PHD filter, and compare different reduction approaches that are applied after update steps, we devised appropriate Monte Carlo   simulation   scenarios.   We   applied   two   component picking   strategies,   namely   the   West’s   algorithm   and   the pairwise component picking algorithm. For each we applied the reparametrization approaches as discussed in Section III-B, including the mapping to tangent space of (i) pairwise larger component   T L ,   (ii)   pairwise   smaller   component   T S ,   (iii) identity element   T Id , (iv) largest component   T Max , (v) smallest component   T Min   (West’s algorithm always merges the smallest component hence (ii) and (v) are the same). We generated   100  examples of an MTT scenario and compared the performance of the approaches. The initial number of targets in the scene was a random integer   N 0 | 0   ∈   [5 ,   7] , while the probability of survival was   p S   = 0 . 975   and birth rate was   λ b   = 0 . 25 . All measurements were corrupted with white noise variance  σ 2  xy   = 0 . 5 2   m 2   in distance and   σ φ   = 0 . 1   rad/s in orientation, while clutter was governed by the Poisson distribution with intensity   λ Z   = 5 . The state   X   = ( X pos , X vel )   ∈   SE (2)   ×   R 3  contains position and velocity components. Here we apply the constant velocity motion model [31] given as  f   ( X k − 1 ) =   X k − 1   exp ∧  G  [ T X vel  k − 1  0  ]  .   (21) − 150   − 100   − 50   0   50   100   150  − 100  − 50  0  50  100  x [m]  y [m]  Fig. 2: An example of a multitarget tracking scenario, involving   10  objects, out of which   5   appeared at the beginning, and   5   more were born during the   100   steps long sequence (gray arrows–measurements including false alarms, black arrows–estimated states, black circles– true object birth place, black square–true object death place). TABLE I: Average OSPA over   100   multitarget scenarios.  Exhaustive pairwise   West  T L   T S   T Id   T Max   T Min   T L   T S   T Id   T Max  D t   2.445   2.515 2.764 2.912 3.082   1.910   1.924 2.060 2.125  D d   2.100   2.163 2.419 2.558 2.695   1.415   1.420 1.537 1.605  D c   0.594   0.613 0.627 0.653 0.745   0.737   0.746 0.797 0.792  We derive the pertaining Jacobian  F k − 1 = −   d d s  (  log ∨  G  (  f   ( μ k − 1 ) f   (exp ∧  G   ( s ) μ k − 1 ) − 1 ) )∣ ∣ ∣ ∣ s =0  (22)  =      I   T   Φ SE (2)  (  T   Ad SE (2)  ( μ pos  k − 1  ) μ vel  k − 1  )  Ad SE (2)  ( μ pos  k − 1  )  0   I    ,  where   μ k − 1   = ( μ pos  k − 1 , μ vel  k − 1 )   ∈   SE (2)   ×   R 3   is the mean value, and   T   is discretization time. The probability of measurement detection was   p D   = 0 . 975   and the measurements were arising as SE (2) , hence   h ( X k ) =   X pos  k   and the measurement Jacobian was   H k   =   [ I   0 ] . For illustration purposes, an example of a multitarget scenario with tracking in total   10   targets on SE (2)  is given in Fig. 2 together with LG-PHD results. As a performance metric we used the optimal subpattern asignement (OSPA) metric [32]. In Table I we present the results where for each of the   100   multitarget trajectories the   cumulative   OSPA   D t ,   and   its   localization   component  D d   and cardinality component   D c   were calculated. For both Exhaustive pairwise and West’s picking strategies, relying on mapping to the tangent space of pairwise larger components  T L   generally outperformed the other approaches. V. C ONCLUSION  In   this   letter   we   have   studied   the   problem   of   mixture reduction on matrix Lie groups. We have particularly dealt with the manipulation of CGD components to compute the KL   divergence,   pick   and   merge   the   mixture   components. As a study example, we implemented the LG-PHD filter, a mixture approximation of the PHD filter tailored for MTT with states evolving on matrix Lie groups. Using the OSPA metric we analyzed the performance of the LG-PHD filter with respect to mixture component number reduction.
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