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The long term goal of our research is to develop
an efficient means of amplifying the user’s intent dur-
ing freehand motion definition. Analogously to 3D
drawing engines (Fiorentino et al., 2003), the envis-
aged method consists in discovering some key-points
of the motion, then interpolating a smooth trajectory
between each consecutive key-points. The goal of the
current research is to address the former issue, i.e., to
discover key-points in motions. In contrast to planar
or spatial curves, a sensor-captured motion results in
a discrete time series of displacements, that formally,
amounts to a time series of elements in the special
Euclidean group of rigid body motion, commonly de-
noted SE(3). Although an universal definition of key-
points is hard to state, a reasonable assumption is to
identify key-points with change-points, sometimes re-
ferred to as “break-points” (Qin et al., 2001), in the
time series. The latter problem, i.e., the interpolation
of a smooth motion between key-points, is not dis-
cussed in this paper. Nevertheless, it is worth noting
that an adaptation of the method introduced by (Hofer
and Pottmann, 2004) should fulfill the requirement.
We formulate the problem as a change-points de-
tection problem in time series of elements in SE(3).
A major difficulty arises from the particular structure
of the group SE(3) that does not satisfy closure under
linear combination. Consequently, such a structure
sets some serious constraints that prevent numerous
of the common time series data processing or mining
techniques from being applicable. For example, most
of the statistical properties, such as the mean, can-
not be properly estimated in a straightforward man-
ner. However, by exploiting the Lie group structure
of SE(3), we show how to adapt a difference of means
method (“which is an adaptation of an image edge de-
tection technique”, (Agarwal et al., 2006)). In par-
ticular, we show that the change-points on the group
SE(3) can be discovered in its associated Lie algebra
se(3) that form a vector space. The method discussed
by (Agarwal et al., 2006) is suitable only for detect-
ing changes in step functions (i.e., piecewise constant
functions). Our adaptation is formulated in a way that
does not assume such a simple model, and should per-
form well with various piecewise-defined functions.
The contribution of the present work lies into two-
fold. First, a novel method for detecting change-
points in SE(3) is presented and evaluated. Second,
an underlying general approach, which can be easily
adapted to be applied to various Lie groups is sug-
gested.
The remainder of this paper is organized as follow.
In the next section, we discuss about the problematic
and related works.
In Section 3, we briefly present the Lie group the-
ory and we describe the structure of the group SE(3).
In Section 4 we introduce our method for detecting
change-points on SE(3). Then, in Section 5, we pro-
pose a set of evaluations using both synthetic and real
data. Finally, in Section 6, we summarize our key
points.
2 DISCUSSIONS AND RELATED
WORKS
The detection of change-points in time series, which
consists in partitioning the time series in homoge-
neous segments (in some sense), is an important issue
in several domains ((Basseville and Nikiforov, 1993),
(Ide and Inoue, 2005), (Agarwal et al., 2006)). Con-
sequently, numerous attempts at solving this prob-
lem exist ((Basseville and Nikiforov, 1993), (Moskv-
ina and Zhigljavsky, 2003), (Ide and Tsuda, 2007),
(Gombay, 2008) ,(Kawahara and Sugiyama, 2009)).
However, most of the existing techniques apply only
to scalar, or, for certain, vector time series. Fur-
thermore, as pointed out in the introduction section,
some of these methods are restricted to be performed
only with time series that follow simple models ((Bas-
seville and Nikiforov, 1993), (Agarwal et al., 2006)).
Therefore, these methods may not provide us with a
suitable solution to deal with time series of elements
in more elaborated structures.
The difference of means method (Agarwal et al.,
2006) is relying only on linear operators, and thus,
should be easily extended to vector spaces, such as
the set of real matrices (R
n×n
), in which the notions of
mean and distance exist. Still, considering a general
metric group structure, the difficulty remains as the
closure under linear operators may not hold. How-
ever, restricting our attention to metric Lie groups ex-
tends the possibilities. One of the great particular-
ity of this class of groups is the local approximation
of their structure by the tangent space, which, at the
identity, is a Lie algebra forming a vector space. Maps
from the group to its algebra, and inversely, exist in a
neighborhood of the identity, and are referred to as the
logarithmic and the exponential maps (respectively).
This particular nature of Lie groups provide a means
of extending certain methods relying on linear opera-
tions to non-linear groups.
For example, (Lee and Shin, 2002) extend the con-
cept of Linear Time-Invariant (LTI) filters to orienta-
tion data (e.g., quaternion group). The same approach
is used in the work of (Courty, 2008) to define a bi-
lateral motion filter. (Tuzel et al., 2005) propose an
adaptation of the mean shift clustering technique to
CHANGE-POINT DETECTION ON THE LIE GROUP SE(3) FOR SEGMENTING GESTURE-DEFINED SPATIAL
RIGID MOTION
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