23 June 1993 Mathematical foundations of local deformation analysis
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Abstract
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 instantaneous behavior of snake-like models.
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Nicolas F. Rougon, Nicolas F. Rougon, "Mathematical foundations of local deformation analysis", Proc. SPIE 2035, Mathematical Methods in Medical Imaging II, (23 June 1993); doi: 10.1117/12.146592; https://doi.org/10.1117/12.146592
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