# GENERATING QUALITY TETRAHEDRAL MESHES FROM BINARY VOLUMES

### Mads Fogtmann Hansen, Jakob Andreas Bærentzen, Rasmus Larsen

#### Abstract

This paper presents two new quality measures for tetrahedra which are smooth and well-suited for gradient based optimization. Bothmeasures are formulated as a distance fromthe regular tetrahedron and utilize the fact that the covariance of the vertices of a regular tetrahedron is isotropic. We use these measures to generate high quality meshes from signed distance maps. This paper also describes an approach for computing (smooth) signed distance maps from binary volumes as volumetric data in many cases originate from segmentation of objects from imaging techniques such as CT, MRI, etc. The mesh generation is split into two stages; a candidate mesh generation stage and a compression stage, where the surface of the candidate mesh is moved to the zero iso-surface of the signed distance maps, while one of the quality measures ensures that the quality remains high. We apply the mesh generation algorithm on four examples (torus, Stanford dragon, brain mask, and pig back) and report the dihedral angle, aspect ratio and radius-edge ratio. Even though, the algorithm incorporates none of the mentioned quality measures in the compression stage it receives a good score for all these measures. The minimum dihedral angle is in none of the examples smaller than 15º.

#### References

- Brock, K., Sharpe, M., Dawson, L., Kim, S., and Jaffray, D. (2005). Accuracy of finite element model-based multi-organ deformable image registration. Medical Physics, 32:1647.
- Cheng, S., Dey, T., Edelsbrunner, H., Facello, M., and Teng, S. (2000). Silver exudation. Journal of the ACM (JACM), 47(5):883-904.
- Chew, L. (1989). Constrained Delaunay Triangulations. Algorithmica, 4(1):97-108.
- Ciarlet, P. (1988). Mathematical Elasticity, Vol. I. Studies in Mathematics and its Applications, 20.
- Cootes, T., Taylor, C., Cooper, D., Graham, J., et al. (1995). Active Shape Models-Their Training and Application. Computer Vision and Image Understanding, 61(1):38-59.
- De, S., Lim, Y., Manivannan, M., and Srinivasan, M. (2006). Physically Realistic Virtual Surgery Using the Point-Associated Finite Field (PAFF) Approach. PRESENCE: Teleoperators and Virtual Environments, 15(3):294-308.
- De Floriani, L. and Puppo, E. (1992). An on-line algorithm for constrained Delaunay triangulation. CVGIP: Graphical Models and Image Processing, 54(4):290- 300.
- Edelsbrunner, H. and Guoy, D. (2002). An Experimental Study of Sliver Exudation. Engineering with Computers, 18:229-240.
- Fuchs, A. (1997). Automatic Grid Generation with Almost Regular Delaunay Tetrahedra. SFB 404, Geschäftsstelle.
- Grosskopf, S. and Neugebauer, P. (1998). Fitting geometrical deformable models to registered range images. Lecture notes in computer science, pages 266-274.
- Kobbelt, L., Vorsatz, J., Labsik, U., and Seidel, H. (1999). A Shrink Wrapping Approach to Remeshing Polygonal Surfaces. Computer Graphics Forum, 18(3):119-130.
- Kühnapfel, U., C¸akmak, H., and Maaß, H. (2000). Endoscopic surgery training using virtual reality and deformable tissue simulation. Computers & Graphics, 24(5):671-682.
- Madsen, K., Nielsen, H., and Tingleff, O. (2004). A comparison of tetrahedron quality measures. Technical report, Technical university of Denmark.
- McInerney, T. and Terzopoulos, D. (1996). Deformable models in medical image analysis: a survey. Medical Image Analysis, 1(2):91-108.
- Molino, N., Bridson, R., Teran, J., and Fedkiw, R. (2003). A crystalline, red green strategy for meshing highly deformable objects with tetrahedra. In In 12th Int. Meshing Roundtable, pages 103-114.
- Montagnat, J. and Delingette, H. (2005). 4D deformable models with temporal constraints: application to 4D cardiac image segmentation. Medical Image Analysis, 9(1):87-100.
- Neugebauer, P. and Klein, K. (1997). Adaptive triangulation of objects reconstructed from multiple range images. IEEE Visualization97, Late Breaking Hot Topics.
- Parthasarathy, V., Graichen, C., and Hathaway, A. (1994). A comparison of tetrahedron quality measures. Finite Elements in Analysis and Design, 15(3):255-261.
- Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., and Ayache, N. (2005). Riemannian Elasticity: A Statistical Regularization Framework for Non-linear Registration. LECTURE NOTES IN COMPUTER SCIENCE, 3750:943.
- Radovitzky, R. and Ortiz, M. (2000). Tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing-front-Delaunay triangulation. Computer Methods in Applied Mechanics and Engineering, 187(3-4):543-569.
- Shewchuk, J. (1996). Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. Lecture Notes In Computer Science; Vol. 1148, pages 203- 222.
- Shewchuk, J. (1998). Tetrahedral mesh generation by Delaunay refinement. Proceedings of the fourteenth annual symposium on Computational geometry, pages 86-95.
- Shewchuk, J. (2002a). A comparison of tetrahedron quality measures. Technical report, University of California at Berkeley.
- Shewchuk, J. (2002b). Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery. Proceedings of the 11th International Meshing Roundtable, pages 193-204.
- Suzuki, N., Hattori, A., Ezumi, T., Uchiyama, A., Kumano, T., Ikemoto, A., Adachi, Y., and Takatsu, A. (1998). Simulator for virtual surgery using deformable organ models and force feedback system. Stud Health Technol Inform, 50:227-33.
- Weatherhill, N. and Hassan, O. (1994). Efficient threedimensional Delaunay triangulation with automatic point creation and imposd boundary constraints. International journal for numerical methods in engineering, 37(12):2005-2039.
- Wood, Z., Schroder, P., Breen, D., and Desbrun, M. (2000). Semi-regular mesh extraction from volumes. IEEE Visualization: Proceedings of the conference on Visualization'00, 2000:275-282.

#### Paper Citation

#### in Harvard Style

Fogtmann Hansen M., Andreas Bærentzen J. and Larsen R. (2009). **GENERATING QUALITY TETRAHEDRAL MESHES FROM BINARY VOLUMES** . In *Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)* ISBN 978-989-8111-69-2, pages 5-12. DOI: 10.5220/0001654700050012

#### in Bibtex Style

@conference{visapp09,

author={Mads Fogtmann Hansen and Jakob Andreas Bærentzen and Rasmus Larsen},

title={GENERATING QUALITY TETRAHEDRAL MESHES FROM BINARY VOLUMES},

booktitle={Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)},

year={2009},

pages={5-12},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001654700050012},

isbn={978-989-8111-69-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)

TI - GENERATING QUALITY TETRAHEDRAL MESHES FROM BINARY VOLUMES

SN - 978-989-8111-69-2

AU - Fogtmann Hansen M.

AU - Andreas Bærentzen J.

AU - Larsen R.

PY - 2009

SP - 5

EP - 12

DO - 10.5220/0001654700050012