Properties of the affinity matrix for multiple closed contour segmentation

In this paper, we describe an algorithm based on a distribution of contours traced by particle moving in directions undergoing Brownian motion for a segmentation problem. We analyze the properties of the affinity matrix, whose elements are the edge-to-edge transition probabilities, from a mathematical point of view and give formal proofs. This analysis is essential because we can use the properties of the affinity matrix to reduce the amount of computation greatly when solving segmentation problems, especially for large real images. The results obtained can be used for robot visual servo to pick multiple objects in the scene