Extending anisotropic operators to recover smooth shapes

Anisotropic differential operators are widely used in image enhancement processes. Recently, their property of smoothly extending functions to the whole image domain has begun to be exploited. Strong ellipticity of differential operators is a requirement that ensures existence of a unique solution. This condition is too restrictive for operators designed to extend image level sets: their own functionality implies that they should restrict to some vector field. The diffusion tensor that defines the diffusion operator links anisotropic processes with Riemmanian manifolds. In this context, degeneracy implies restricting diffusion to the varieties generated by the vector fields of positive eigenvalues, provided that an integrability condition is satisfied. We will use that any smooth vector field fulfills this integrability requirement to design line connection algorithms for contour completion. As application we present a segmenting strategy that assures convergent snakes whatever the geometry of the object to be modelled is.