Expected size of "Minkowski reflection" of a random subset of $w \times w$ grid about another, given their sizes?

For $U$,$V$ sets of points on Cartesian plane, define the "Minkowski reflection" $U \star V$ of $U$ about $V$ as the set of positions occupied by reflections of every point in $U$ about every point in $V$.

Let $S$ be the Cartesian product of the set of first $w$ natural numbers with itself, and $S_a$, $S_b$ its two random nonempty subsets. Let $a=\frac{|S_a|}{w^2}$, $b=\frac{|S_b|}{w^2}$ be their respective shares in $S$.

What is $F(a,b,w)$, the expected value of $\frac{|S_a \star S_b|}{(3w-2)^2}$ for given $w$?