Exact evaluation of targeted stochastic watershed cuts

Seeded segmentation with minimum spanning forests, also known as segmentation by watershed cuts, is a powerful method for supervised image segmentation. Given that correct segmentation labels are provided for a small set of image elements, called seeds, the watershed cut method completes the labeling for all image elements so that the boundaries between different labels are optimally aligned with salient edges in the image. Here, a randomized version of watershed segmentation, the targeted stochastic watershed, is proposed for performing multi-label targeted image segmentation with stochastic seed input. The input to the algorithm is a set of probability density functions (PDFs), one for each segmentation label, defined over the pixels of the image. For each pixel, we calculate the probability that the pixel is assigned a given segmentation label in seeded watershed segmentation with seeds drawn from the input PDFs. We propose an efficient algorithm (quasi-linear with respect to the number of image elements) for calculating the desired probabilities exactly.