Globally Optimal Energy Minimization Techniques - Bezier and Splines in Image Processing and Machine Vision - page 214

(a) (b)

Fig. 10.10. Globally optimal geodesic active contours applied to overlapping ob-
jects. The cells (a) are separated despite the weak intensity gradient between them
(b) (from [7]).

(a) (b)

Fig. 10.11. Segmentation of MRI image of a concave contour, the corpus callosum
in a human brain, from [7]. Image (a) is the original and (b) is the segmentation via
GOGAC.

found by writing the equation for the length of a curve, and then minimizing
the length using techniques from calculus of variations. An entirely equivalent
approach is to define the energy of a curve; then minimizing the energy leads
to the same equations for a geodesic. This latter formulation can better be un-
derstood when we consider how an elastic band stretched between two points
will contract in length to minimize its energy—the final shape of the band is
a geodesic. Thus there is an intimate relationship between the mathematical
formalism of geodesics and the concepts underpinning snakes as proposed by
Kass et al.
The globally minimal geodesic between two sets of points in an isotropic
Riemannian space can be calculated with the fast marching method [2]. This
method computes the surface of minimal action, also known as a distance