From Active Contours to Minimal Geodesic Paths: New Solutions to Active Contours Problems by Eikonal Equations

In this chapter, we give an overview of part of our previous work based on the minimal path framework and the Eikonal partial differential equation (PDE). We show that by designing adequate Riemannian and Randers geodesic metrics the minimal paths can be utilized to search for solutions to almost all of the active contour problems and to the Euler-Mumford elastica problem, which allows to blend the advantages from minimal geodesic paths and those original approaches, i.e. the active contours and elastica curves. The proposed minimal path-based models can be applied to deal with a broad variety of image analysis tasks such as boundary detection, image segmentation and tubular structure extraction. The numerical implementations for the computation of minimal paths are known to be quite efficient thanks to the Eikonal solvers such as the Finsler variant of the fast marching method.