Dynamic contour models have become popular after Terzopoulos, Kass and others introduced the snake model [21,38,40]. In this approach, an energy-minimizing contour, called a ``snake'', is controlled by a combination of the following three forces or energies:
where s is the parameterization of the contour,
is a point on the contour,
and 
and 
are the first and second derivatives of the
contour, respectively.
The internal energy of the contour, 
,
characterizes the regularization properties (stretchness and smoothness) of the snake.
The image energy, 
, represents the potential
due to image forces, and
represents the potential created by external constraint
forces, if there are any. The potentials are defined so that they decrease along the
direction of the forces and the potentials are low near salient image
features. Once an appropriate initialization of the contour is specified, the
snake can quickly converge to the nearby energy minimum, using a
variational approach.
This snake model provides a powerful interactive tool for image segmentation. However, this approach is ``myopic'' because of its use of strictly local information. The implementation of the original snake model is vulnerable to image noise and its initial position. Numerous provisions have been made in the literature to improve the robustness and stability of the snakes [3,4,9,25]. For example, Cohen [8] introduced a ``balloon force'' which can either inflate or deflate the contour. This force helps the snake to trespass spurious isolated weak image edges, and counters its tendency to shrink. The resulting snake is more robust to the initial position and image noise, but human intervention is needed to decide whether an inflationary or deflationary force is needed. Amini et al. [1] suggested using dynamic programming to minimize the energy function. Their method exhaustively searches all the admissible solutions, and each iteration results in a locally optimum contour. Geiger et al. [16] have proposed to solve the problem in a single iteration by allowing the contour to be searched from anywhere in a large area around the initialization position. Neuenschwander et al. [29] proposed to let the user specify the two end points of the desired contour. As the optimization process progresses, the edge information is propagated from the end points towards the center. Fua and Brechbuhler [15] proposed to reach the desired goal by imposing attractor and tangent hard constraints, where the attractor constraint forces the contour to go towards or pass by a particular point in the image, and the tangent constraint forces the contour to have a certain tangent at a particular point. The idea of active contour has been successfully extended to perform tasks such as edge and subjective contour detection, motion tracking, stereo matching and image segmentation [43,26,32,44,48].
Different regularization constraints can be imposed to
obtain deformable templates with dramatically different properties.
A free-form deformable template model using regularization constraints
for dilatational-viscous fluid was proposed by Christensen et al. [6] which accommodates large deformation kinematics.
A partial differential equation is derived based on the fluid
kinematics which governs the evolution of the template.
