The uncertainty in the recovered image flow
values results from sensor uncertainties
and noise and from the image processing techniques used to extract
and track features.
We use a static camera calibration
technique to model the uncertainty in 3-D to 2-D feature locations.
The strategy used to find the 2-D uncertainty in the features 2-D
representation is to utilize the recovered camera parameters and the 3-D
world coordinates of a known set of points
and compute the corresponding pixel coordinates, for points distributed
throughout the image plane a number of times, find the actual feature
pixel coordinates and construct 2-D histograms for the displacements from
the recovered coordinates for the experiments performed. The number of
the experiments giving a certain displacement error would be the
axis of this histogram, while the
and
axis are the displacement
error. The three dimensional histogram functions are then normalized such
that the volume under the histogram is equal to
unit volume and the
resulting normalized function is used as the distribution of pixel displacement
error.
The spatial uncertainty in the image processing technique can be modeled
by using synthesized images and corrupting them, then applying the
feature extraction mechanism to both images and computing the resulting
spatial histogram for the error in finding features. The probability density
function for the error in finding the flow vectors can thus be computed
as a spatial convolution of the sensor and strategy uncertainties.
We then eliminate the unrealistic motion estimates by using the physical
(geometric and mechanical) limitations of the manipulating hand.
Assuming that feature points lie on a planar surface on the hand, then
we can develop bounds on the coefficients of the motion equations,
which are second degree functions in and
in three dimensions,
and
.
The 2-D uncertainties are then used to recover the 3-D uncertainties in
the motion and structure parameters. The system is linearized by either
dividing the parameter space into three subspaces for the translational,
rotational and structure parameters and solving iteratively or using other
linearization techniques and/or assumptions to solve a linear system of
random variables [4,5,6,31,32,34]. As an example, the recovered
3-D translational velocity cumulative density functions for an actual world motion, ,
and
, is shown in figure 12. It should be noted that the recovered distributions
represents a fairly accurate estimation of the actual 3-D motion.
