Directed Probabilistic Watershed
Authors: Enrique Fita Sanmartin, Sebastian Damrich, Fred A. Hamprecht

The Probabilistic Watershed is a semi-supervised learning algorithm applied on undirected graphs. Given a set of labeled nodes (seeds), it defines a Gibbs probability distribution over all possible spanning forests disconnecting the seeds. It calculates, for every node, the probability of sampling a forest connecting a certain seed with the considered node. We propose the "Directed Probabilistic Watershed", an extension of the Probabilistic Watershed algorithm to directed graphs. Building on the Probabilistic Watershed, we apply the Matrix Tree Theorem for directed graphs and define a Gibbs probability distribution over all incoming directed forests rooted at the seeds. Similar to the undirected case, this turns out to be equivalent to the Directed Random Walker. Furthermore, we show that in the limit case in which the Gibbs distribution has infinitely low temperature, the labeling of the Directed Probabilistic Watershed is equal to the one induced by the incoming directed forest of minimum cost. Finally, for illustration, we compare the empirical performance of the proposed method with other semi-supervised segmentation methods for directed graphs.