Hence, if the characteristic vector field
[xi] of a contact metric manifold belongs to the k-nullity distribution, then we have
For a front f : [M.sup.n] [right arrow] [R.sup.n+1] with a (possibly locally defined) unit normal vector field
We first give the fundamental definitions and propositions which are used to compute the energy of the vector field
The i-th (i = [bar.1,n]) attractive force magnitude [F.sub.i] in each point of the vector field
is a continuous function of distance between the robot and the centre of the line segment connecting previous (i-1) and current i nodal points.
Consequently, the fiber flow is associated with a Killing vector field
of ([bar.M], [bar.g]).
Using the mass-metric, if N denotes the unit vector field
normal to [T.sub.x]S in [T.sub.x][F.sub.n](E), the projection of [[nabla].sup.S.sub.v]u of any vector field
u on [T.sub.x]S is
A curl of a vector field
, i.e., E(r), can measure its circulation or rotation about a point.
The vector field
of the jth element of the mth subsystem can be rewritten as [dx.sub.j]/dt = [x.sub.j][f.sup.m.sub.j](x) with the function [f.sup.m.sub.j].
A vector field
f : [R.sup.n] [right arrrow] [R.sup.n] is said to be homogeneous of degree p > 0 if f([lambda]x) = [[lambda].sup.p]f(x) for all x [member of] [R.sup.n] and all [lambda] > 0.
In order to obtain continuity equation in our setup, one can define an energy-momentum tensor corresponding to the displacement vector field
The process of motion estimation is implemented in the measurement domain, and the background extraction and 3-D Recursive Search (3DRS)  are also used to reduce the computational complexity when computing the motion vector field
of to-be-reconstructed video frame.