Region-based approximation of probability distributions (for visibility between imprecise points among obstacles)

Let $p$ and $q$ be two imprecise points, given as probability density functions on ${\mathbb{R}}^2$, and let $\mathcal{R}$ be a set of $n$ line segments (obstacles) in ${\mathbb{R}}^2$. We study the problem of approximating the probability that $p$ and $q$ can see each other; that is, that the segment connecting $p$ and $q$ does not cross any segment of $\mathcal{R}$. To solve this problem, we approximate each density function by a weighted set of polygons; a novel approach to dealing with probability density functions in computational geometry.