## Minimal Path Approach Model

Minimal path approach (MPA) is the recent version of the active contour model proposed by Cohen et al.  in 1997. Compared with its predecessors, the main improvements of this model are in the following two areas:

(i) Optimal curve searching with global energy minimization.

(ii) Simplified initialization.

Similar to other active contour models, the contour evolving process of MPA is also based on the minimization of an energy function. A potential field P is created based on the edge map of the original image, and the energy function is expressed as the integration over P along the curve. The goal of the optimization process is to find a curve whose energy function is minimum.

The definition of energy function is given as following: assume a pair of control points are p0 and p1, the energy of the curve C connecting this pair of points is

where C (0) = p0 and C (L) = p1; w is a constant; s is the arc-length parameter; L is the length of the curve C; Q = [0, L ]. The regularization term multiplied by the constant w measures the curve length and is used for smoothness control. By minimizing the energy function E(C), the "shortest" path in the potential field will be picked up as the optimal solution among all the curves linking end points p0 and p1 . Similarly, we can find the curves between every two adjacent end points along the object boundary, so that the closed object contour can be obtained.

Nonetheless, the MPA method does not solve those problems appearing in the traditional active contour model completely. For example, MPA can find the optimal solutionfor curve searching between every two givenpoints. This indeed can overcome the local minima problem. However, this also means a higher requirement to the location accuracy of initial points because the technique for searching these initial points is still open. Therefore, human interaction is often needed to locate the end points precisely.

Another drawback with MPA is that this approach lacks the topography handling ability. For some applications within our study, such as carotid artery lumen contour tracking in MRI sequences, the topology of blood vessel lumen in each cross-section images may change due to bifurcation, and it is impossible to apply MPA directly even though the initial points can be provided precisely. Therefore, a mechanism is needed to track the topology changes for automatic image processing.