Don't know much about this kind of thing, but it seems like it would be useful, if possible, to store the actual displacement (dx,dy) vector to the surface instead of just its magnitude, the SDF. I can't evaluate the tradeoff of storing an extra value, but it seems like it would make the recursion in the first section a lot easier: when you want to test if a particular box contains the object, if [v+d, v-d] contains 0, you would know a lot more where to look: you need only continue the iteration on subdivisions that are in the correct direction.