N-dimensional skeletonization: A unified mathematical framework

We present a skeletonization algorithm defined by explicit Boolean conditions which are dimension independent. The proposed procedure leads to new thinning algorithms in two dimensions (2D) and three dimensions (3D). We establish the mathematical properties of the resulting skeleton referred to as the MB skeleton. From a topological point of view, we prove that the algorithm preserves connectivity in 2D and 3D. From a metric point of view, we show that the MB skeleton is located on a median hypersurface (MHS) that we define. This MHS does not correspond to the standard notion of median axis/surface in 2D/3D, as it combines the various distances associated with the hypercubic grid. The MHS specificities prove to make the skeleton robust with respect to noise and rotation. Then we present the algorithmic properties of the MB skeleton: First, the algorithm is fully parallel, which means that no spatial subiterations are needed. This property, together with the symmetry of the Boolean n-dimensional patterns, leads to a perfectly isotropic skeleton. Second, we emphasize the extreme conciseness of the Boolean expression, and derive the computational efficiency of the procedure.