Directional adaptive deformable models for segmentation with application to 2D and 3D medical images

In this paper, we address the problem of adapting the functions controlling the material properties of 2D snakes, and show how introducing oriented smoothness constraints results in a novel class of active contour models for segmentation which extends standard isotropic inhomogeneous membrane/thin-plate stabilizers. These constraints, expressed as adaptive L² matrix norms, are defined by two 2^nd-order symmetric and positive definite tensors which are invariant with respect to rigid motions in the image plane. These tensors, equivalent to directional adaptive stretching and bending densities, are quadratic with respect to 1^st- and 2^nd-order derivatives of the image intensity, respectively. A representation theorem specifying their canonical form is established and a geometrical interpretation of their effects if developed. Within this framework, it is shown that, by achieving a directional control of regularization, such non- isotropic constraints consistently relate the differential properties (metric and curvature) of the deformable model with those of the underlying intensity surface, yielding a satisfying preservation of image contour characteristics.