Active Contours on Graphs: Multiscale Morphology and Graphcuts

In this paper we propose two novel methods for formulating and implementing the methodology of geodesic active contours on arbitrary graphs, as applied to multiscale morphology and segmentation. Firstly, we propose approximations to the calculation of the gradient and the divergence of vector functions defined on graphs and use these approximations to apply the technique of geodesic active contours for object detection on graphs. To this end, we extend existing work on graph morphology to multiscale dilation and erosion and implement them recursively usinglevel sets of functions defined on the graph. Second, we propose a graphcut based solution to the geodesic active contour problem on graphs. Appropriate weights are calculated for each edge for which the Riemannian length of a contour can be approximated by the weighted sum of intersections of the contour with the edges of the graph. Finding the minimum Riemannian length contour then becomes equivalent to solving a max flow problem for which efficient solutions have been proposed in the literature.