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Kinematics of interface evolution with application to active contour models

In physics, a large class of problems, such as crystal growth or flame fronts propagation, are concerned with the motion of a deformable boundary separating time-dependent domains in which the interface itself satisfies an equation of motion. Recently, similar questions have been introduced in image processing through the concept of physically based active contour models or snakes. In snake modeling, a deformable boundary endowed with elastic properties interacts with a constant external field derived from image properties. In the most general case, the interfacial motion is governed by a set of partial differential equations that nonlinearily couple interface intrinsics and external fields. In this paper, we present a general study of the kinematics of deformable regular (d-1)-dimensional interfaces evolving according to a first- order dynamic in a d-dimensional (d >= 2) space, in terms of their intrinsic geometric properties. We formulate local equations of motion and derive evolution theorems. These results are then applied to the kinematical study of a specific 2-dimensional active contour model when its optimization is performed via a first-order deformation process. This provides a significant insight in the instantaneous behavior of snake-like models as well as the nature of their steady-states.