Mathematical morphological operations of boundary-represented geometric objects

The resemblance between the integer number system with multiplication and division and the system of convex objects with Minkowski addition and decomposition is really striking. The resemblance also indicates a computational technique which unifies the two Minkowski operations as a single operation. To view multiplication and division as a single operation, it became necessary to extend the integer number system to the rational number system. The unification of the two Minkowski operations also requires that the ordinary convex object domain must be appended by a notion of inverse objects or negative objects. More interestingly, the concept of negative objects permits further unification. A nonconvex object may be viewed as a mixture of ordinary convex object and negative object, and thereby, makes it possible to adopt exactly the same computational technique for convex as well as nonconvex objects. The unified technique, we show, can be easily understood and implemented if the input polygons and polyhedra are represented by their slope diagram representations.