  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.
