ABSTRACT
This chapter focuses on the relation between mathematical morphology (MM) (Serra, 1983) operations and rough sets (Pawlak, 1981, 1982; Pawlak and Skowron, 2007c,b,a) mainly based on topological spaces considered in the context of image retrival (see, e.g., (Fashandi, Peters, and Ramanna, 2009)) and the basic image correspondence problem (see, e.g., (Peters, 2009, 2010; Meghdadi, Peters, and Ramanna, 2009)). There are some obvious similarities between MM operations and set approximations in rough set theory. There have been several attempts to link MM and rough sets. Two major works have been published in this area (Polkowski, 1993; Bloch, 2000). L. Polkowski defines hit-or-miss topology on rough sets and proposed a scheme to approximate mathematical morphology within the general paradigm of soft computing (Polkowski, 1993),(Polkowski, 1999). Later, I. Bloch tries to demonstrate a direct link between MM and rough sets through relations, a pair of dual operations and neighbourhood systems (Bloch, 2000). I.Bloch’s approach is carried forward by J.G. Stell, who defines a single framework that includes the principal constructions of both mathematical morphology and rough sets (Stell, 2007). To make this chapter fairly self-contained, background information on the basics of topology is presented, first. The chapter then presents the basics of mathematical morphology. Then principles of rough set theory are considered and the links between them are discussed. Finally, a proposed application of the ideas from these two areas is given in terms of image retrieval.
