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. INTRODUCTION
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. MATHEMATICAL PRELIMINARIES
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
TX(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 ⊂Rn×n associated to a p-dimensional
matrix Lie group G ⊂Rn×n is a p-dimensional vector
space. The matrix exponential expG and matrix logarithm logG
establish a local diffeomorphism between the two
expG : g →G and
logG : G →g.
(1)
Furthermore, a natural relation exists between g and the
Euclidean space Rp given through a linear isomorphism
[·]∨
G : g →Rp and [·]∧
G : Rp →g.
(2)
For x ∈Rp and X ∈G we use the following notation [23]
exp∧
G(x) = expG([x]∧
G) and
log∨
G(X) = [logG(X)]∨
G .
(3)
Lie groups are generally non-commutative, i.e., XY ̸= 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(AdG(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, adG, is in fact the differential of
AdG 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 + ϕG(y)x + O(||y||2),
(5)
log∨
G(exp∧
G(x + y) exp∧
G(−x)) = ΦG(x)y + O(||y||2),
(6)
where ϕG(y) = P∞
n=0
Bn adG(y)n
n!
, Bn are Bernoulli numbers,
and ΦG(x) = ϕG(x)−1. For many common groups used in
engineering and physical sciences closed form expressions for
ϕG(·) and ΦG(·) 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; 0p×1, Σ) is a zero-mean
‘classical’ Gaussian random variable with the covariance Σ ⊂
Rp×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 =
Z
Rp
1
p
(2π)p|Σ|
exp∧
G

−1
2||ξ||2
Σ

dξ
=
Z
G
β exp∧
G

−1
2|| log∨
G(Xµ−1)||2
Σ

dX
(8)
where ||x||2
Σ = xTΣ−1x. Therein the change of coordinates
ξ = log∨
G(Xµ−1), with the pertaining differentials dX =
|Φ(ξ)|dξ, resulted with the CGD normalizing constant
β = 1/
q
(2π)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
X
i=1
wiG(X; µi, Σi) ,
(10)
where wi 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 Gi = G(X; µi, Σi)
and Gj = G(X; µj, Σj) be two mixture components with
pi(X) and pj(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
DKL(Gi, Gj) =
Z
G
pi(X) log
 pi(X)
pj(X)

dX .
(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 ∈Rp. Note that in (8)
the change evolved around the distribution mean µ; however,
since in (11) generally µi ̸= µj, 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)µt, µt ∈G, where µt ̸= µ, we get
1 =
Z
G
β exp∧
G

−1
2|| log∨
G(Xµ−1)||2
Σ

dX
(12)
CoC
≈
Z
Rp η exp∧
G

−1
2|| log∨
G(exp∧
G(ξ)µtµ−1)||2
Σ

dξ
(6)≈
Z
Rp η exp∧
G

−1
2||ΦG(rt)(ξ −rt)||2
Σ

dξ
=
Z
Rp η exp∧
G

−1
2||ξ −rt||2
ϕG(rt)ΣϕT
G(rt)

dξ ,
where rt = log∧
G(µµ−1
t ), η approximately evaluates to
η = β|Φ(ξ)| =
|Φ(ξ)|
p
(2π)p|Σ| · |Φ(log∨
G(exp∧
G(ξ)µtµ−1))|
≈
1
p
(2π)p|ϕG(rt)ΣϕG(rt)T|
,
(13)
and we obtain ξ ∼N(ξ; rt, ϕG(rt)ΣϕT
G(rt)).
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 µt ∈G, using (5) and following [24] we get
ξt = log∨
G
�exp∧
G(ξ)µµ−1
t


≈log∨
G
�µµ−1
t


+ ϕG
�log∨
G(µµ−1
t )


ξ .
(14)
3
G
wa, G(µa, Σa)
µa
wb, G(µb, Σb)
µb
wc, G(µc, Σc)
µc
G
g
wj, N(r, Σϕ
j )
wi, N(0, Σi)
µi
G
g
w∗, N(r∗, Σ∗)
G
w∗, G(µ∗, ΣΦ∗)
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, ξt ∼N(ξt; rt, Σϕ),
where
rt = log∨
G
�µµ−1
t


(15)
Σϕ = ϕG(rt)ΣϕT
G(rt).
(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 µi, using
(15) and (16), is given in Fig. 1.
The KL divergence between two CGDs Gi = G(µi, Σi) and
Gj = G(µj, Σj) can now be evaluated as
DKL(Gi, Gj) ≈
Z
Rp pi(ξ) log
 pi(ξ)
pj(ξ)

dξ = DKL(Ni, Nj) ,
pi(ξ) ∼Ni = N(ξ; ri, Σϕ
i ) , ri = log∧
G(µiµ−1
t ) ,
(17)
pj(ξ) ∼Nj = N(ξ; rj, Σϕ
j ) , rj = log∧
G(µjµ−1
t ) ,
and Σϕ
=
ϕG(r)ΣϕT
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
DKL(Ni, Nj) =1
2

tr
�Σϕ
j
−1Σϕ
i


−K + logR
|Σϕ
j |
|Σϕ
i |
(18)
+ (rj −ri)T(Σϕ
j )−1(rj −ri)

,
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
DsKL(wiNi, wjNj) =1
2

(wi −wj) logR
wi
wj
+
(19)
wiDKL(Ni, Nj) + wjDKL(Nj, Ni)

.
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 µt 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
Rp was proposed in [28]:
r∗= 1
w∗
X
i
wiri, Σ∗= 1
w∗
X
i

wi
�Σi + rirT
i


−r∗(r∗)T
where w∗
=
P
i wi, wiN(ri, Σi) 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 Rp 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∗)µt, ΣΦ∗= Adr∗
G ΦG(r∗)Σ∗�Adr∗
G ΦG(r∗)

T, (20)
where Adr∗
G = AdG(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 µi is given in Fig. 1.
IV. STUDY EXAMPLE - PHD FILTER ON LIE 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 Dk−1, and
operates by evaluating two steps—prediction and update. By
assuming Dk−1 and birth intensity being Gaussian mixtures
[10], the GM-PHD prediction results with another Gaussian
mixture (Prop. 1 in [10]). Similarly, if Dk−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(Xk−1|Z1:k−1) and motion model p(Xk|Xk−1), it
approximates the joint distribution p(Xk, Xk−1|Z1:k−1), and
then marginalizes obtaining p(Xk|Z1:k−1). Similarly, given
p(Xk|Z1:k−1) and likelihood p(Zk|Xk), we approximate the
joint distribution p(Xk, Zk|Z1:k−1), and then marginalize
obtaining p(Xk|Zk). 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 TL, (ii) pairwise smaller component TS, (iii)
identity element TId, (iv) largest component TMax, (v) smallest
component TMin (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 N0|0 ∈[5, 7], while the probability
of survival was pS = 0.975 and birth rate was λb = 0.25.
All measurements were corrupted with white noise variance
σ2
xy = 0.52 m2 in distance and σφ = 0.1 rad/s in orientation,
while clutter was governed by the Poisson distribution with
intensity λZ = 5. The state X = (Xpos, Xvel) ∈SE(2) × R3
contains position and velocity components. Here we apply
the constant velocity motion model [31] given as
f(Xk−1) = Xk−1 exp∧
G
TXvel
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
TL
TS
TId
TMax
TMin
TL
TS
TId
TMax
Dt
2.445 2.515 2.764 2.912 3.082
1.910 1.924 2.060 2.125
Dd
2.100 2.163 2.419 2.558 2.695
1.415 1.420 1.537 1.605
Dc
0.594 0.613 0.627 0.653 0.745
0.737 0.746 0.797 0.792
We derive the pertaining Jacobian
Fk−1=−d
ds

log∨
G

f(µk−1)f(exp∧
G(s)µk−1)−1
s=0
(22)
=

I
TΦSE(2)

T AdSE(2)
�µpos
k−1


µvel
k−1

AdSE(2)
�µpos
k−1


0
I

,
where µk−1 = (µpos
k−1, µvel
k−1) ∈SE(2)×R3 is the mean value,
and T is discretization time. The probability of measurement
detection was pD = 0.975 and the measurements were arising
as SE(2), hence h(Xk) = Xpos
k
and the measurement Jacobian
was Hk =
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 Dt, and its localization component
Dd and cardinality component Dc were calculated. For both
Exhaustive pairwise and West’s picking strategies, relying on
mapping to the tangent space of pairwise larger components
TL generally outperformed the other approaches.
V. CONCLUSION
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.
5
REFERENCES
[1] D. Alspach and H. Sorenson, “Nonlinear bayesian estimation using
gaussian
sum
approximations,”
IEEE
Transactions
on
Automatic
Control, vol. 17, no. 4, pp. 439–448, 1972.
[2] R. Chen and J. S. Liu, “Mixture Kalman filters,” Journal of the Royal
Statistical Society: Series B (Statistical Methodology), vol. 62, no. 3,
pp. 493–508, 2000.
[3] S. S. Blackman, Multiple-target tracking with radar applications,
Gilmour, Ed., 1986.
[4] C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using
real-time tracking,” IEEE Transactions on Pattern Analysis and Machine
Intelligence, vol. 22, no. 8, pp. 747–757, Aug 2000.
[5] J. Goldberger and H. Aronowitz, “A distance measure between GMMs
based on the unscented transform and its application to speaker
recognition,” in Proceedings of Interspeech, 2005, pp. 1985–1989.
[6] A. Nikseresht and M. Gelgon, “Gossip-based computation of a gaussian
mixture model for distributed multimedia indexing,” IEEE Transactions
on Multimedia, vol. 10, no. 3, pp. 385–392, 2008.
[7] D. Salmond, “Mixture reduction algorithms for target tracking,” in IEEE
Colloquium on State Estimation in Aerospace and State Estimation,
1989, pp. 1–4.
[8] A.
Runnalls,
“Kullback-Leibler
approach
to
Gaussian
mixture
reduction,” IEEE Transactions on Aerospace and Electronic Systems,
vol. 43, no. 3, pp. 989–999, 2007.
[9] M. Bukal, I. Markovi´c, and I. Petrovi´c, “Composite distance based
approach to von Mises mixture reduction,” Information Fusion, vol. 20,
no. 1, pp. 136–145, 2014.
[10] B.-N. Vo and W.-K. Ma, “The Gaussian mixture probability hypothesis
density filter,” IEEE Transactions on Signal Processing, vol. 54, no. 11,
pp. 4091–4104, 2006.
[11] B.-T. Vo, B.-N. Vo, and A. Cantoni, “Analytic implementations of the
cardinalized probability hypothesis density filter,” IEEE Transactions on
Signal Processing, vol. 55, no. 7, pp. 3553–3567, 2007.
[12] Y. M. Lui, “Advances in matrix manifolds for computer vision,” Image
and Vision Computing, vol. 30, no. 6-7, pp. 380–388, 2012.
[13] G. Loianno, M. Watterson, and V. Kumar, “Visual inertial odometry for
quadrotors on SE(3),” in IEEE International Conference on Robotics
and Automation (ICRA), 2016, pp. 1544–1551.
[14] J. ´Cesi´c, V. Joukov, I. Petrovi´c, and D. Kuli´c, “Full body human motion
estimation on Lie groups using 3D marker position measurements,” in
IEEE-RAS International Conference on Humanoid Robots (Humanoids),
2016, pp. 826–833.
[15] A. Barrau and S. Bonnabel, “Intrinsic filtering on Lie groups with
applications to attitude estimation,” IEEE Transactions on Automatic
Control, vol. 60, no. 2, pp. 436–449, 2015.
[16] A. W. Long, K. C. Wolfe, M. J. Mashner, and G. S. Chirikjian, “The
banana distribution is Gaussian: A localization study with exponential
coordinates,” in Proceedings of Robotics: Science and Systems (RSS),
Sydney, Australia, 2012.
[17] T. D. Barfoot and P. T. Furgale, “Associating uncertainty with three-
dimensional poses for use in estimation problems,” IEEE Transactions
on Robotics, vol. 30, no. 3, pp. 679–693, Jun. 2014.
[18] C. Zhang, A. Taghvaei, and P. G. Mehta, “Feedback particle filter on
matrix Lie groups,” in Proceedings of the American Control Conference,
2016, pp. 2723–2728.
[19] G. Bourmaud, R. M´egret, A. Giremus, and Y. Berthoumieu, “Discrete
extended Kalman filter on Lie groups,” in European Signal Processing
Conference (EUSIPCO), 2013, pp. 1–5.
[20] G. Bourmaud, R. M´egret, M. Arnaudon, and A. Giremus, “Continuous-
discrete extended Kalman filter on matrix Lie groups using concentrated
Gaussian distributions,” Journal of Mathematical Imaging and Vision,
vol. 51, no. 1, pp. 209–228, 2015.
[21] K. C. Wolfe, M. Mashner, and G. S. Chirikjian, “Bayesian fusion on
Lie groups,” Journal of Algebraic Statistics, vol. 2, no. 1, pp. 75–97,
2011.
[22] G. S. Chirikjian, Stochastic Models, Information Theory, and Lie
Groups.
Birkh¨auser, 2012, vol. 2.
[23] G. Bourmaud, R. M´egret, A. Giremus, and Y. Berthoumieu, “From
intrinsic optimization to iterated extended Kalman filtering on Lie
groups,” Journal of Mathematical Imaging and Vision, vol. 55, no. 3,
pp. 284–303, 2016.
[24] G. Bourmaud, “Estimation de param`etres ´evoluant sur des groupes de
Lie : Application `a la cartographie et `a la localisation d’une cam´era
monoculaire,” Ph.D. dissertation, University of Bordeaux, 2015.
[25] W. Miller, Symmetry Groups and their Applications, 1972.
[26] S. Kullback, Information Theory and Statistics.
New York: Dover
Publications, 1997.
[27] S. Amari, “Alpha-divergence is unique, belonging to both f-divergence
and Bregman divergence classes,” IEEE Transactions on Information
Theory, vol. 55, no. 11, pp. 4925–4931, 2009.
[28] D. J. Salmond, “Mixture reduction algorithms for point and extended
object tracking in clutter,” IEEE Transactions on Aerospace and
Electronic Systems, vol. 45, no. 2, pp. 667–686, 2009.
[29] M. West, “Approximating posterior distributions by mixtures,” Journal
of Royal Statistical Society, Series B, vol. 55, no. 2, pp. 409–442, 1993.
[30] I. Markovi´c, J. ´Cesi´c, and I. Petrovi´c, “Von Mises Mixture PHD Filter,”
IEEE Signal Processing Letters, vol. 20, no. 12, pp. 2229–2233, 2015.
[31] J.
´Cesi´c, I. Markovi´c, and I. Petrovi´c, “Moving object tracking
employing rigid body motion on matrix Lie groups,” in International
Conference on Information Fusion (FUSION), 2016, p. 7.
[32] D. Schuhmacher, B. T. Vo, and B. N. Vo, “A consistent metric for
performance evaluation of multi-object filters,” IEEE Transactions on
Signal Processing, vol. 56, pp. 3447–3457, 2008.