Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning

TL;DR

The paper introduces Probabilistic Watershed, a method that considers all spanning forests for seeded segmentation, providing a probabilistic framework that connects to existing algorithms like Random Walker and Power Watershed.

Contribution

It establishes the equivalence of Probabilistic Watershed to Random Walker, offers a computationally feasible approach using Kirchhoff's theorem, and reveals new theoretical insights into effective resistance and Power Watershed.

Findings

Probabilistic Watershed computes node-seed connection probabilities.

It proves the equivalence to Random Walker and Power Watershed.

The method is computationally feasible via Kirchhoff's matrix tree theorem.

Abstract

The seeded Watershed algorithm / minimax semi-supervised learning on a graph computes a minimum spanning forest which connects every pixel / unlabeled node to a seed / labeled node. We propose instead to consider all possible spanning forests and calculate, for every node, the probability of sampling a forest connecting a certain seed with that node. We dub this approach "Probabilistic Watershed". Leo Grady (2006) already noted its equivalence to the Random Walker / Harmonic energy minimization. We here give a simpler proof of this equivalence and establish the computational feasibility of the Probabilistic Watershed with Kirchhoff's matrix tree theorem. Furthermore, we show a new connection between the Random Walker probabilities and the triangle inequality of the effective resistance. Finally, we derive a new and intuitive interpretation of the Power Watershed.