On mathematical foundations of local deformations analysis

In physics, a large class of problems are concerned with the motion of a muldimensional boundary separating time-dependent domains in which the interface itself satisfies an evolution equation. In mathematics, similar questions have motivated theoretical developments within the general framework of differentiate manifolds. Within the field of image analysis, they have been introduced via the concept of physically-based active contour or surface models or snakes. In snake modelling, a deformable boundary endowed with mechanical properties of the elastic type interacts with an external potential field derived from image data. In the most general case, the interfacial motion is governed by a set of partial differential equations expressing a non-linear coupling between interface intrinsics and external actions. Classically, first-order processes correspond to purely dissipative dynamics whereas second-order processes model dissipative-inertial evolutions. In this paper, we present a general study of the kinematics of deformable non-singular manifolds with codimension 1 evolving according to a first-order dynamics within a d-dimensional space, in terms of their intrinsic geometric properties. We formulate the local equations which describe instantaneous variations of their main differential and integral characteristics. In particular, a physical interpretation of curvature evolution in terms of reaction-diffusion-propagation processes is developed. Delocalizing these equations within the time domain leads to describing local evolution along the stream lines of the deformation field. Within this framework, local ergodicity property of curvature processes is underlined. Integrating further within the space domain leads to global evolution theorems. These results are then applied to the kinematical study of 2D and 3D active models of the inhomogeneous membrane/thin-plate under pressure type (g-snakes) when their optimization is performed via a purely dissipative Lagrangian deformation process. They yield a complete mathematical characterization of the instantaneaous behavior of snake-like models. Moreover, by interpreting elasticity densities as control parameters for the deformation process, it is shown in what extent a dynamical analysis results in a theoretical understanding of the nature of their steady-states.