Prof Laurent D. Cohen, Ceremade, PSL, University Paris Dauphine, France

Active contours were introduced about 30 years ago as an interactive tool for object segmentation through energy minimization. Since active contours have the drawback of being trapped into local minima, we introduced about 20 years ago the minimal path method in order to find the global minimum of the active contour energy. Minimal paths have been used for long as an interactive tool to segment tubular structures as cost minimizing curves. The user usually provides start and end points on the image and gets the minimal path as output. These minimal paths correspond to minimal geodesics according to some adapted metric. They are a way to find a (set of) curve(s) globally minimizing the geodesic active contours energy. Finding a geodesic distance can be solved by the Eikonal equation using the fast and efficient Fast Marching method. Introduced first as a way to find the global minimum of a simplified active contour energy, we have recently extended these methods to cover all kinds of active contour energy terms. Through finding geodesics for new kinds of metrics, we were able to revisit active contour methods and propose the minimization of various active contours terms: Curvature penalization, Region term, Alignment term, as well as front propagation. We will present the mathematical background as well as concrete applications to biomedical and natural images.