3D modeling using the Delaunay triangulation

A major problem in medical imaging is the three dimensional reconstruction of human organs from parallel cross sections. Besides visualization, such 3D models are needed for radiation therapy planning, surgical planning and simulation, rapid prototyping and volumetric measurements. We proposed a solution to the reconstruction problem, which is based on the Delaunay triangulation [1]. The video displays the different steps of this algorithm. As a demonstration object, we use the human pelvis. The pelvis is an ideal structure to demonstrate both the usefulness of shape reconstruction and the algorithmic concept of our met hod. Firstly, it is a nontrivial exercise to imagine the pelvis from its cross-sections. Even with a radiological training, it is hard to see the shape of the oblique pelvic inlet. Secondly, it is important that the cross-sections show a variety of topologies, like branching and holes. The pelvic data is a series of 23 axial MRIs (magnetic resonance images). The bone contours were outlined manually since the image characteristic of MRI is not sufficient to employ automatic segmentation methods. In each cross-section, we get one or several, possibly nested closed polygons. We calculate the 2D Delaunay triangulation of the polygon vertices. There are several situations where Steiner points are added automatically onto the polygons: when segments are not contained (conforming Delaunay triangulation), when there are obtuse angles opposite of polygon segments (to guarantee that the internal Voronoi skeleton stays inside the polygons). We also add Steiner points inside polygons at the orthogonal projection of external Voronoi vertices of adjacent sections (which improves com-