Conformal Minimal Convex Analysis of Geodesic Active Contours

Geodesic Active Contour Model (GACM) is a great improvement of the traditional Snake Model. It had converted the problem of finding extremity of an energy functional to the problem of finding a geodesic curve in an appropriate Riemannian space. But (geodesic) active contour models are nonconvex in general, so the solutions of the Euler-Lagrange equation of its energy functional are only critical points, but were often taken as minima of the energy functional as granted. This paper focused on the convex analysis of Geodesic Active Contour Models and had proposed a condition to ensure the convexity of the energy functional of GACM. We proved that if the Gaussian curvature K of the image surface I is less than 0 everywhere along a normal geodesic curve

C

(

t

), then its energy functional

E

(

C

) will be convex near C. Further more, it can be induced from our theorem that if ∆ln

h

 > 0 along

C

(

t

), then the energy functional

E

(

C

) will also be convex near C if the image space has its metric tensor taking the form

g

ij

=

h

(|∇

I

(

x

)|)

2

δ

ij

.