Tetrahedral Meshes From Planar

In biomedicine, many three dimensional (3D) objects are sampled in terms of slices such as computed tomography (CT), magnetic resonance imaging (MRI), and ultrasound imaging. It is often necessary to construct surface meshes from the cross sections for visualization, and thereafter construct tetrahedra for the solid bounded by the surface meshes for the purpose of nite element analysis. In paper [1], we provided a solution to the construction of a surface triangular mesh from planar cross-section contours. Here we provide an approach to tetrahedralize the solid region bounded by planar contours and the surface mesh. It is a di cult task because the solid can be of high genus (several through holes) as well as have complicated branching regions. We develop an algorithm to e ectively reduce the solid into prismatoids, and provide an approach to tetrahedralize the prismatoids. Our tetrahedralization approach is similar to the advancing front technique (AFT) for its exible control of mesh quality. The main criticism of AFT is that the remaining interior may be badly shaped or even untetrahedralizable. The emphasis of our prismatoid tetrahedralization approach is on the characterization and prevention of untetrahedralizable parts. Ruppert and Seidel [25] have shown that the problem of deciding whether a polyhedron is tetrahedralizable without adding Steiner points is NP-complete. We characterize this problem under certain constraints, and design one rule to reduce the chance of generating untetrahedralizable shapes. The characterization also leads to the classi cation of two common untetrahedralizable categories which can be better processed if they do occur. 2

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