Path openings and closings

This paper lays the theoretical foundations to path openings and closings. The traditional morphological filter used for the analysis of linear structures in images is the union of openings (or the intersection of closings) by linear segments. However structures in images are rarely strictly straight, and as a result a more flexible approach is needed. An extension to the idea of using straight line segments as structuring elements is to use constrained paths, i.e. discrete, one-pixel thick successions of pixels oriented in a particular direction, but in general forming curved lines rather than perfectly straight lines. However the number of such paths is prohibitive and the resulting algorithm by simple composition is inefficient. In this paper we propose a way to compute openings and closings over large numbers of constrained, oriented paths in an efficient manner, suitable for building filters with applications to the analysis of oriented features, such as for example texture.