A COMPARISON AND EVALUATION OF MOTION INDEXING ...

zmin = ¯z + ¯x ′ · sin(θmin) − ¯z ′ · cos(θmin), (4.4)
where ¯x =
i wixi, ¯z =
i wizi, ¯x ′ =
i wix ′ i and ¯z ′ =
i wiz ′ i.
The input to the equations are the point clouds P = (pi) i ∈ [1 : K] with pi =
(xi, yi, zi) T and point cloud P ′ = (p ′ i) i ∈ [1 : K] with p ′ i = (x ′ i, y ′ i, z ′ i) T . The weights
wi ∈ ℜ, 1 ≤ i ≤ K are defined such that K
i=1 wi = 1
The 3-D point cloud distance was proposed by Kovar and Gleicher[19] and are in-
variant only under a three-dimensional subgroup of the six-dimensional group of rigid
3-D Euclidean transformations. Rotations around y axis and translations in xz plane
are considered which are not sufficient for all classes of motions. The 3-D point cloud
distance is not invariant to scaling too.
4.2.2 Quaternion-based Distance:
Unit quaternions are a good mathematical tool to represent joint rotations.
Joint configurations in a motion stream can be encoded using quaternions. Each pose
configuration, (tr, Rr, Rb), consists of a translational parameter tr and a rotational
parameter Rr that determines the root coordinate system. The remaining rotational
parameters Rb describe the joint angles within the skeletal model.
Since joint orientations are represented by quaternions, a local distance measure is
defined to compare poses described by quaternions:
C Quat : j × j → [0, 1]
If j = (tr, Rr, Rb) and j ′ = (t ′ r, R ′ r, R ′ b) are two pose configurations and qb and q ′ b
denote the unit quaternions describing the relative joint rotations Rb and R ′ b, respec-
tively, then the local distance between unit quaternions is calculated by using the inner
product between single quaternions: C Quat (j, j′) =
b∈B wb · 2/π · arccos | qb • q ′ b |,
where | qb • q ′ b | is the absolute value of the inner product of qb and q ′ b and gives the
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