  Conventional edge-based active contours often require the normal component of
an edge indicator function on the optimal contours to approximate zero, while
the tangential component can still be significant. In real images, the full
gradients of the edge indicator function along the object boundaries are often
small. Hence, the curve evolution of edge-based active contours can terminate
early before converging to the object boundaries with a careless contour
initialization. We propose a novel Geodesic Snakes (GeoSnakes) active contour
that requires the full gradients of the edge indicator to vanish at the optimal
solution. Besides, the conventional curve evolution approach for minimizing
active contour energy cannot fully solve the Euler-Lagrange (EL) equation of
our GeoSnakes active contour, causing a Pseudo Stationary Phenomenon (PSP). To
address the PSP problem, we propose an auxiliary curve evolution equation,
named the equilibrium flow (EF) equation. Based on the EF and the conventional
curve evolution, we obtain a solution to the full EL equation of GeoSnakes
active contour. Experimental results validate the proposed geometrical
interpretation of the early termination problem, and they also show that the
proposed method overcomes the problem.
