bound

Let [Laplace]' [is no subset or equal to] [Laplace] be a subset in which each object has the same size bounding box.

j] [element of] [Laplace]' be two objects whose bounding boxes intersect, where [P.

Let [Laplace]' [is not subset or equal to] [Laplace] be a subset where each object's bounding box has size either [[Alpha].

Each object in the large class has bounding box size [[Alpha].

If we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively, denote the number of bounding box intersection pairs among large and small objects, then by Lemma 3.

Now, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([Laplace]') denote the number of bounding box intersection pairs where one objects is in large class and the other in small class.

Without loss of generality, assume that the largest bounding box in the scene has size [[Alpha].

Each of the objects is essentially a square core, with two wires extending from the opposite corners to the corresponding corners of the bounding box.

On the other hand, the number of bounding box intersections is at least [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Thus, we get a lower bound on the ratio [Rho]