Sparse Finite Element Level-Sets for Anisotropic Boundary Detection in 3D Images

Level-Set methods have been successfully applied to 2D and 3D boundary detection problems. The geodesic active contour model has been particularly successful. Several algorithms for the discretisation have been proposed and the banded approach has been used to improve efficiency, which is crucial in 3D boundary detection. In this paper we propose a new scheme to numerically represent and evolve surfaces in 3D. With the new scheme, efficiency and accuracy are further improved. For the representation, space is partitioned into tetrahedra and finite elements are used to define the level-set function. Extreme sparsity is obtained by maintaining data only for tetrahedra that contain the zero level-set. We formulate the evolution PDE in weak form and incorporate a normalisation term. We obtain a stable scheme with consistent sub-grid accuracy without having to rely on any re-initialisation procedure. Boundary detection is performed using an anisotropic extension of the isotropic geodesic model. With the sparse representation, the anisotropic model is computationally feasible. We present experimental results on volumetric data sets including images with a significant amount of noise.