An efficient algorithm for exact evaluation of stochastic watersheds

Abstract

The stochastic watershed is a method for unsupervised image segmentation proposed by Angulo and Jeulin (2007). The method first computes a probability density function (PDF), assigning to each piece of contour in the image the probability to appear as a segmentation boundary in seeded watershed segmentation with randomly selected seeds. Contours that appear with high probability are assumed to be more important. This PDF is then post-processed to obtain a final segmentation. The main computational hurdle with the stochastic watershed method is the calculation of the PDF. In the original publication by Angulo and Jeulin, the PDF was estimated by Monte Carlo simulation, i.e., repeatedly selecting random markers and performing seeded watershed segmentation. Meyer and Stawiaski (2010) showed that the PDF can be calculated exactly, without performing any Monte Carlo simulations, but do not provide any implementation details. In a naive implementation, the computational cost of their method is too high to make it useful in practice. Here, we extend the work of Meyer and Stawiaski by presenting an efficient (quasi-linear) algorithm for exact computation of the PDF. We demonstrate that in practice, the proposed method is faster than any previously reported method by more than two orders of magnitude. The algorithm is formulated for general undirected graphs, and thus trivially generalizes to images with any number of dimensions.