CMRT09: Object Extraction for 3D City Models, Road Databases and Traffic Monitoring - Concepts, Algorithms, and Evaluation 
38 
other than as parallelogram (Fig.4, bottom row), but e.g. 
trapezoid, common quadrilaterals, etc, due to unstable sampling 
characteristics of LiDAR or clutter objects in urban areas. It is 
difficult to decide whether it is actually a moving vehicle part 
or a point set of stationary vehicle with missing parts. Generally, 
these vehicle point sets confuse the shape analysis and deliverer 
only ambiguous geometric features that cannot be adopted for 
robust classification. Therefore, this category of vehicle point 
sets have to be identified and then excluded from candidates 
delivered to movement classification, which means that they 
could be only attributed to uncertain motion status at the 
moment. Those point sets are also undergone the same shape 
analysis process and can be found when the parallelogram 
fitting fails. 
3.2 Movement classification 
G at the identity e, T, , is called the Lie algebra g. The 
exponential map exp is a mapping from Lie algebra elements to 
Lie group elements. The inverse of the exponential map is 
called logarithmic map log. The Lie algebra element of T is 
obtained by performing component-wise log operation on each 
of the M i : 
flog W) 
log(7’) = 
(1) 
l 0 • log (K)J 
(\ 0^ (0 -l'] 
where log(A/ ( ) = a t I 1 + ^.1 I. Equation (1) expresses 
the Lie algebra element of an individual spoke in terms of the 
generator matrices for scaling and 2-d rotation factors. 
As indicated in section 3.1, the point sets of extracted vehicle 
can generally be denoted by spoke model with two parameters, 
whose configuration is formulated as 
'U ' 
x = 
, u,= 
i a i \ 
Vs, 
K E > J 
where k denotes the number of spokes in the model. As inspired 
by the works of Fletcher et al., (2003) and Yarlagadda et ah, 
(2008), the 3D vehicle shape variability is nonlinear and 
represented as a transformation space. Thus the similarity 
between vehicle instances can be measured by group distance 
metric. It has been also confirmed that Lie group PCA can 
better describe the variability of data that is inherently nonlinear 
and statistics on linear models may benefit from the addition of 
nonlinear information. Since our task is intended to classify the 
vehicle motion based on the shape features of vehicle point sets, 
the classification framework for distinguishing generic vehicle 
category can be easily adapted to motion analysis. 
Consequently, a new vehicle configuration Y can be obtained 
by a transformation of X written in matrix form: Y=T(X) where 
f M, . 0 ^ 
r = 
0 . M 
, M' = 
Rf o 
0 e ai 
, R, denotes the 2-d 
k / 
rotation acting on the angle of shear 0 SA . e“‘ denotes the scale 
acting on the extent E. By varying T, different vehicle shape 
(motion status) can be represented as transformations of X. 
based on elaborations in Rossmann (2002), M i is a Cartesian 
product of the scale and angle value groups 'R x SO(2), which 
are the Lie group of 1-d real value and the Lie group of 2-d 
rotation, respectively. Since the Cartesian product of Lie group 
elements is a Lie group and T is the Cartesian product of 
transformation matrices M acting on the individual spokes, T 
forms a Lie group. The group T is a transformation group that 
acts on shape parameters M. However, any vehicle shape X may 
be represented in T as the transformation of a fixed identity 
atom. 
A group is defined as a set of elements together with a binary 
operation (multiplication) satisfying the closure, associative, 
identity and the inverse axioms. A Lie group G is a group 
defined on differentiable manifold. The tangent space of group 
The intrinsic mean p of a set of transformation matrices 7j , 
T 2 , T n of vehicle spoke models is defined as 
p = argmin]Tt/(7j,r 2 ) 2 (2) 
k=\ 
where denotes Riemannian distance on G, and 
d(T t ,T 2 ) = ||log(7]' l 7’ 2 )|| where ||| is the Frobenius norm of the 
resulting algebra elements. The 1-parameter Lie algebra 
element of the spoke model of vehicle point sets is given by 
(AS>) • o ^ 
A M = 
(3) 
v 0 ■ A« 
where A r (/) = / log(Afy), denoting that the Lie algebra element 
is defined at a fixed (a,.,#,) for each spoke, which represents 
the tangent to a geodesic curve parameterized by /. The 
parameter t in (3) sweeps out a 1-parameter sub-group, H v (t) of 
the Lie group G of spoke transformations. For any g e G , the 
distance between g and //,,(/) is defined as 
d{g,H y ) = mind(g,exp[A v (/)]), (4) 
Analogous to the principle components of a vector space, there 
exist 1-parameter subgroups called the principle geodesic 
curves (Fletcher et al., 2003) which describe the essential 
variability of the data points lying on the manifold. The first 
principle geodesic curve for elements of a Lie group G is 
defined as the 1-parameter subgroup H tU (/), where 
n 
v (,) =argmin ^d 2 (fi~ l g„H v ) (5) 
i=i 
Let p n be the projection of p'g, on //,„ , and 
define g <n = pjl/j-'gf. The higher A-th principle geodesic curve 
can be determined recursively based on (5). 
The motion analysis can then be formulated as a binary 
classification problem using Lie distance metrics. The input to 
the Lie distance classifier comprises a set of labeled samples 
Tj from two categories of vehicle status C i - moving vehicles 
and stationary ones. n Y denotes the number of training samples 
for each category. The intrinsic mean p } and the principal 
geodesics H Ul) are computed for each vehicle class C j using