Topology-Preserving General Operators in Arbitrary Binary Pictures

A general operator may transform a binary picture by changing both black and white points. Sequential operators traverse the points of a picture, and consider a single point for possible alteration, while parallel operators can alter a set of points simultaneously. An order-independent transition function yields the same sequential operator for arbitrary visiting orders. Two operators are called equivalent if they produce the same result for each input picture. A transition function is said to be equivalent if it specifies a pair of equivalent parallel and sequential operators. This paper establishes a necessary and sufficient condition for order-independent transition functions, a sufficient criterion for equivalent transition functions, and a sufficient condition for topology-preserving parallel general operators in arbitrary binary pictures.