Comparing Position Estimates

 

We must establish a common denominator for comparison of position estimates given measurements. We have pursued a statistical procedure for doing so. Our objective is to maximize the posterior probability of position given the bearings, $p(x,\; y\; |\; R)$. Position is discretized in intervals of 30 meters, agreeing with the discretization of Digital Elevation Maps. At first, we must specify a prior density for position, $p(x,\; y)$. Currently we use a uniform distributions to signify absence of prior knowledge. Secondly, we must specify the likelihood the bearings (the measurements) given position, $p(R\; |\; x,\; y)$. In practice it is difficult to determine the likelihood of a bearing given position; it is appropriate to decompose this expression in terms of the possible interpretations of the bearings:

$p(R|x,\; y)\; =\; \sum$_I p(R|x, y, I) p(I|x, y).

The final posterior density is proportional to $p(R|x,y)\; p(x,y)$.

Experience with the mountain detection system has led to the conclusion that, for fixed pose and correspondences, the errors in measuring bearings are fairly independent:

$p(R\; |x,\; y,\; I)\; =\; \prod$_i=1^m p(r_i|I,Γ).

We assume that when a landmark $I$_i is visible from $x,\; y$, the errors in the bearing $r$_i follow a Gaussian law. We also measure the height of the mountain associated with the bearing, and assume errors in height follow a Gaussian law. We consider that any error that exceeds 18 degrees is a mistake, that is, if a mountain at 40 degrees is associated to a bearing at 60 degrees, we consider that association a mistake. Mistakes are assumed distributed uniformly on the interval [0, 360] degrees. Every bearing is associated with all mountains within a $\pm 18$ degrees interval; bearings that are not associated to any mountain are mistakes. Since bearings do not need to be associated to all possible mountains, the summation in expression 1 can be calculated quickly.