From Triangulation to Simplex Mesh: a Simple and Efficient Transformation

In the field of 3D images, relevant information can be difficult to interpret without further computer-aided processing. Generally, and this is particularly true in medical imaging, a segmentation process is run and coupled with a visualization of the delineated structures to help understanding the underlying information. To achieve the extraction of the boundary structures, deformable models are frequently used tools. Amongst all techniques, Simplex Meshes are valuable models thanks to their good propensity to handle a large variety of shape alterations altogether with a fine resolution and stability. However, despite all these great characteristics, Simplex Meshes are lacking to cope satisfyingly with other related tasks, as rendering, mechanical simulation or reconstruction from iso-surfaces. As a consequence, Triangulation Meshes are often preferred. In order to face this problem, we propose an accurate method to shift from a model to another, and conversely. For this, we are taking advantage of the fact that triangulation and simplex meshes are topologically duals, turning it into a natural swap between these two models. A difficulty arise as they are not geometrically equivalent, leading to loss of information and to geometry deterioration whenever a transformation between these dual meshes takes place. Therefore, some effort as to be carried out to minimizing the shape degradation using an appropriate interpolation to find the position of the vertices in the dual mesh. An accurate and effective transformation technique is described in this paper, where we present a direct method to perform an interpolation of a simplex mesh to obtain its dual, and/or vice-versa. This original method is based on the distance minimization between the tangent planes of the mesh and vertices of each face. Finally, probing resulting mesh shiftings in both directions are commented.