Approximate mathematical morphology

Mathematical morphology is a well-established discipline whose aim is to provide speciic tools, based originally on Minkowski operators in aane spaces, for feature selection in complex objects, primarily patterns and images. Complexity of morphological operations makes it desirable to propose a theoretical scheme of an approximate morphological calculus within a general paradigm of soft computing. We propose a scheme based on ideas of rough set theory. In this scheme, the underlying space of points (e.g. pixels) is partitioned into disjoint cells (classes) by means of some primitive attributes (features) and morphological operations are performed on classes, which allows for compression of data. In our chapter, we discuss topological foundations of morphology, in particular spaces of rough fractals as well as morphological approximate operations and we also point to plausible applications by proving the approximate collage theorem.