In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol. XXXVIII, Part 3/W4 — Paris, France, 3-4 September, 2009 
disturbances may destabilize the ziplock’s active vertices. As a 
result convergence may not occur or the snake may get trapped 
near the initial position. As a means of overcoming this 
problem, an external force with a large capture range can be 
applied. 
The Gradient Vector Flow (GVF) field (Xu & Prince, 1997), 
which is an example for such an external force, is used in the 
proposed approach. The GVF field was aimed at addressing two 
issues: a poor convergence to concave regions, and problems 
associated with the initialisation. It is computed as a spatial 
diffusion of the gradient of an edge map derived from the 
image. This computation causes diffuse forces to exist far from 
the object, and crisp force vectors to be near the edges. The 
GVF field points toward the object boundary when very near to 
the boundary, but varies smoothly over homogeneous image 
regions, extending to the image border. The main advantage of 
the GVF field is that it can capture a snake from a long range. 
Thus, the problem of far initialization can be alleviated. 
The Evolution of a ziplock snake is illustrated in Fig. 8. The 
snake is fixed at the head and tail, and it consists of two parts, 
the active and the passive vertices. These parts are separated by 
moving force boundaries. The active parts of the snake consist 
of the head and tail segments. 
O Passive Vertex 
• Active Vertex 
Figure 8. Evolution of a ziplock snake. 
The GVF is defined to be the vector field 
G(x,y) = (u(x,y),v(x, y)) that minimizes the energy 
functional: 
£= ¡¡p{« x 2 +u y 2 + v x 2 +v v 2 ) + IY/| 2 |G-Y/f dxdy 
(8) 
where V/ is the vector field computed from f(x,y) having 
vectors pointing toward the edges. f(x,y) is derived from the 
image and it has the property that it is larger near the image 
edges. 
The regularization parameter ju should be set according to the 
amount of noise present in the image; more noise requires a 
higher value of ¡u . Through use of calculus of variations 
(Courant & Hilbert, 1953), the GVF can be found by solving 
the following Euler equations: 
¿iV 2 u-(ti-f)(f 2 +f 2 ) = 0 
(9) 
(U V 2 v-(v-/ v )(//+/ v 2 ) = 0 
where V“ is the Laplacian operator and f x and f are partial 
derivatives of f with respect to x and y. 
Let ^( 5 ) = (jc(s), y(s)) be a parametric active contour in which 
s is the curve length and x and y are the image coordinates of 
the 2D-curve. The internal snake energy is then defined as 
E mx (V(s)) = j [a(s) | V s (s) | 2 +fi(s) | V ss (s) | 2 ] (10) 
where y and F vs are the first and second derivatives of V with 
respect to 5. The functions a(s) and /3(s) control the 
elasticity and the rigidity of the contour, respectively. The 
global energy 
E = E int (V(s)) + E img (V(s)) (11) 
needs to be minimized, with a(s) = a and /?(.?) = /? being 
constants. Minimization of the energy function of Eq. 11 gives 
rise to the following Euler equations: 
dE jm „ (V(s)) .... 
- aV ss (5) + PV SSSS (s) + 8 =0 (12) 
oV(s) 
where V(5) stands for either x(j') or ^(5), and V ss and 
V ssss denote the second and fourth derivatives of V, 
respectively. After approximation of the derivatives with finite 
differences, and conversion to vector notation with 
Vj = (x,-,>' ; ), the Euler equations take the form 
(13) 
-2A[K-, -2V, + V M ]+[V, -2V M + V M ] + G(u,v) = 0 
where G(w,v) is the GVF vector field . Eq. 13 can be written 
in matrix form as 
KV + G(u,v) = 0 (14) 
where AT is a pentadiagonal matrix. 
Finally, the motion of the GVF ziplock snake can be written in 
the form (Kass et al., 1988) 
V [,] =(K + yIY l *(y F [, " ,1 -/rG(u,v)| vM ) (15) 
where y stands for the viscosity factor (step size) determining 
the rate of convergence and / is the iteration index, k alters 
the strength of the external force. 
It is noteworthy that the proposed model still might fail to 
detect the correct boundaries in the following cases: 
• High variation of curvature at the roundabout border 
resulting in an initialization that is too poor in some parts, 
with the consequence that the snakes becomes and remain 
straight. 
• The roundabout central area lacks sufficient contrast with 
the surroundings, causing the curve to converge to nearby 
features.