Images as functions and sets

Abstract

Some of the recent theoretical advances in grey-level morphology have drawn their success from relating morphological techniques to other methods of image analysis and research fields. Our goal in this paper is to consider morphological feature detection from this viewpoint. We relate the basic operations underlying grey-level morphological transformations, Minkowski addition and Minkowski subtraction through their functional definitions to surfaces in Euclidean space. This connection provides a direct parallel between the use of linear convolution operations and morphological transformations to describe the geometry of features for an image surface. Also, we show how the equivalent set-theoretical definitions of Minkowski operations leads to a definition of the concept of a grey-level junction. We propose that this new insight suggests morphological techniques may be well-suited to finding image junctions. This idea is developed further elsewhere 1.