Anatomical structures and tissues are often hard to be segmented in medical images due to their poorly defined boundaries, i.e., low contrast in relation to other nearby false boundaries. The specification of the boundary polarity can help to alleviate part of this problem. Although this information is naturally exploited by boundary-based methods like live wire, region-based methods are usually conceived for undirected graphs. This motivated researchers to investigate extensions to better handle oriented transitions. Recently, an Oriented Image Foresting Transform has been proposed. In this work, we further generalize it by different non-smooth connectivity functions that allow a better handling of ties in its energy formulation. We give direct proof for the optimality of the proposed extensions in terms of a global maxima of an energy function, and show the obtained gains in accuracy over a more challenging 3D dataset of MR images of the foot. We also discuss the incorporation of the boundary polarity for the interactive segmentation of colored images. © 2013 IEEE.