Theoretical aspects of morphological filters by reconstruction

This paper investigates morphological connected filters and, in particular, the so-called filters by reconstruction. A brief background is offered on the theory of morphological filtering. Then, the concept of connectivity is introduced within the morphological framework, which makes it possible to establish connected filters as those that do not introduce discontinuities or, in other words, that extend the input image flat zones. An important subset of connected filters is the class of filters by reconstruction, which allows to build connected filters that treat both the peaks and valleys of an input image while possessing a robustness property called the strong-property. The focus of our research is on the combination, by means of the sup- and inf-operations, of alternating filters by reconstruction when their component filters belong to a granulometry and an antigranulometry (by reconstruction). These operators will be investigated by means of the study of their grain and pore properties. Some commutation properties are introduced that facilitate the manipulation of filters by reconstruction. An important theoretical result of this paper is the establishment of a new family of strong morphological filters. Although most theoretical expressions refer to set operators, results are automatically extendable for non-binary (gray-level) functions.