I recently derived the most general possible version of Pick's Theorem which works for any shape consisting of the union of any number of polygons (these don't have to be simple polygons either) with any number of holes inside of them (the holes also don't have to be simple polygons). The generalization states that:
$A=I+\frac{B}{2}-\frac{P+P_s}{2}+\frac{H+H_s}{2}$
where $I$ is the total number of interior integer coordinate points of the shape, $B$ is the total number of boundary integer coordinate points of the shape, $P$ is the total number of path-connected domains of finite area only containing boundary and interior points of the shape, $P_s$ is the total number of non-intersecting polygons only containing boundary and interior points of the shape, $H$ is the total number of path-connected domains of finite area only containing boundary and exterior points of the shape, and $H_s$ it the total number of non-intersecting polygons only containing boundary and exterior points of the shape.
If the shape in question is itself a single simple polygon then we have $P=1$, $P_s=1$, $H=0$, $H_s=0$, and the generalization reduces to the standard form of Pick's Theorem:
$A=I+\frac{B}{2}-1$
If the shape in question is the union of $P$ non-intersecting polygons with no holes then the generalization reduces to:
$A=I+\frac{B}{2}-P$
If the shape in question is a single polygon with $H$ holes, all of which are non-intersecting polygons, then the generalization reduces to:
$A=I+\frac{B}{2}-1+H$
My question is this: has this generalization been published anywhere? Is there a simpler version of it somewhere to be found?