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.