Tetrahedral meshes from planar cross-sections

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 finite element analysis. In Ref. [1] (C. Bajaj, E. Coyle and K. Lin, Graphical Models and Image Processing 58 (6) (1996) 524–543), we provided a solution to the construction of a surface triangular mesh from planar -section contours. Here we provide an approach to tetrahedralize the solid region bounded by planar contours and the surface mesh. It is a difficult task because the solid can be of high genus (several through holes) as well as have complicated branching regions. We develop an algorithm to effectively 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 flexible 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 (J. Ruppert, R. Seidel, On the difficulty of tetrahedralizing three-dimensional non-convex polyhedra, in: Proceedings 5th Annual ACM Symposium Comput. Geom., 1989, p. 380–392) 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 classification of two common untetrahedralizable categories which can be better processed if they do occur.

[1]  B. Geiger.  Three-dimensional modeling of human organs and its application to diagnosis and surgical planning , 1993 .

[2]  Roger I. Tanner,et al.  Generation of unstructured tetrahedral meshes by advancing front technique , 1993 .

[3]  O. Zienkiewicz,et al.  Finite element Euler computations in three dimensions , 1988 .

[4]  Paresh Parikh,et al.  Generation of three-dimensional unstructured grids by the advancing-front method , 1988 .

[5]  S. Lo.  A NEW MESH GENERATION SCHEME FOR ARBITRARY PLANAR DOMAINS , 1985 .

[6]  Jean-Daniel Boissonnat,et al.  Shape reconstruction from planar cross sections , 1988, Comput. Vis. Graph. Image Process..

[7]  Chandrajit L. Bajaj,et al.  Convex Decomposition of Polyhedra and Robustness , 1992, SIAM J. Comput..

[8]  N. Weatherill,et al.  Unstructured grid generation using iterative point insertion and local reconnection , 1995 .

[9]  Bernard Chazelle,et al.  Triangulating a non-convex polytype , 1989, SCG '89.

[10]  Edward J. Coyle,et al.  Arbitrary Topology Shape Reconstruction from Planar Cross Sections , 1996, CVGIP Graph. Model. Image Process..

[11]  J. Sullivan,et al.  A normal offsetting technique for automatic mesh generation in three dimensions , 1993 .

[12]  N. Weatherill,et al.  Efficient three‐dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints , 1994 .

[13]  Peter Hansbo,et al.  On advancing front mesh generation in three dimensions , 1995 .

[14]  David A. Field.  The legacy of automatic mesh generation from solid modeling , 1995, Comput. Aided Geom. Des..

[15]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[16]  Kwun-Nan Kevin Lin.  Surface and 3D triangular meshes from planar cross sections , 1997 .

[17]  B. Joe.  Tetrahedral mesh generation in polyhedral regions based on convex polyhedron decompositions , 1994 .

[18]  S. Lo.  Volume discretization into tetrahedra—II. 3D triangulation by advancing front approach , 1991 .

[19]  Raimund Seidel,et al.  On the difficulty of tetrahedralizing 3-dimensional non-convex polyhedra , 1989, SCG '89.

[20]  Klaus-Jürgen Bathe,et al.  On automatic mesh construction and mesh refinement in finite element analysis , 1989 .

[21]  Philippe A. Tanguy,et al.  Three‐dimensional adaptive finite element computations and applications to non‐Newtonian fluids , 1991 .

[22]  William H. Frey,et al.  An apporach to automatic three‐dimensional finite element mesh generation , 1985 .

[23]  Paul-Louis George,et al.  The advancing-front mesh generation method revisited , 1994 .