Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active contours

Recently, Caselles et al. have shown the equivalence between a classical snake problem of Kass et al. and a geodesic active contour model. The PDE derived from the geodesic problem gives an evolution equation for active contours which is very powerfull for image segmentation since changes of topology are allowed using the level set implementation. However in Caselles' paper the equivalence with classical snake is only shown for 2D images and 1D curves, by using concepts of Hamiltonian theory which have no meanings for active surfaces. This paper propose to examine the notion of equivalence and to revisite Caselles et al. arguments. Then a notion equivalence is introduced and shown for classical snakes and geodesic active contours in the 2D (active contour) and 3D (active surface) case.