Topology-Preserving General Operators in Arbitrary Binary Pictures K´ alm´anPal´agyi Department of Image Processing and Computer Graphics, University of Szeged, Hungary palagyi@inf.u-szeged.hu Abstract. A general operator may transform a binary picture by chang- ing 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 pro- duce the same result for each input picture. A transition function is said to be equivalent if it specifies a pair of equivalent parallel and sequen- tial operators. This paper establishes a necessary and sufficient condi- tion 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. 1 Introduction A binary picture on a digital space is a mapping that assigns a color of black or white to each point [12]. A reduction (or reductive) operator transforms a binary picture only by changing some black points to white ones; an operator that never turns a black point into white is called an addition (or an augmenta- tive operator); a general (or reductive-augmentative) operator may change both black and white points [5]). Parallel operator s can alter all points that satisfy their transition function s simultaneously, while sequential operator s traverse the points of a picture, and may alter just the actually visited point. The author introduced the notions of equivalent reductions [14] that can be extended for general operators. Two general operators are said to be equivalent if they produce the same result for each input picture. Sequential operators with the same transition function may produce different results for different visiting orders (raster scans) of points. An order-independent transition function [15] produces the same result for arbitrary visiting orders. A transition function is called equivalent if it specifies a pair of equivalent parallel and sequential operators. Various algorithms (e.g., thinning [4], shrinking [5], generation of skeleton by influence zones (SKIZ) [16], warping of binary images [2], or narrow band algorithm in level set methods [6]) are required to preserve topology . Topology E. Bayro-Corrochano and E. Hancock (Eds.): CIARP 2014, LNCS 8827, pp. 22–29, 2014. c  Springer International Publishing Switzerland 2014