# GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface Reconstruction from Point Slice Samples

### Oscar Ruiz, Eliana Vasquez, Sebastian Peña, Miguel Granados

#### 2008

#### Abstract

In surface reconstruction from slice samples (typical in medical imaging, coordinate measurement machines, stereolithography, etc.) the available methods attack the geometrical and topological properties of the surface. Topological methods classify the transitions occurred in the 2-manifold between two consecutive slices i and i + 1. Geometrical methods synthesize the surface based on local proximity of the contours in consecutive slices. Superimposed 2D Voronoi Diagrams V Di and V Di+1 for slices i and i + 1, respectively, present topological problems if, for example, a site of V Di lies on an site or an edge of V Di+1 . The usual treatment of this problem in literature is to apply a geometrical disturbance to either V Di or V Di+1 , thus eliminating the degeneracy. In contrast, this article presents the implementation of a method which identifies the degenerate situation, constructs un-instantiated topological constructs, choses a geometrical instantiation based on a virtual disturbance introduced to the actual configuration. The algorithm was successfully applied to remove non-manifold topologies produced by well known algorithms in surface reconstruction.

#### References

- Boissonnat, J. D. (1988). Shape reconstruction from planar cross-sections. Computer Vision, Graphics and Image Processing, pages 1-29.
- Burnikel, C., Fleischer, R., Mehlhorn, K., and Schirra, S. (1999). Efficient exact geometric computation made easy. In Proceedings of the 15th Annual ACM Symposium on Computational Geometry.
- Edelsbrunner, H. and Mücke, E. P. (1990). Simulation of Simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9(1):66-104.
- Funke, S., Klein, C., Mehlhorn, K., , and Schmitt, S. (2005). Controlled perturbation for delaunay triangulations. In Proceedings of S.O.D.A.
- Geiger, B. (1993). Three dimensional modeling of human organs and its application to diagnosis and surgical planning. Research Report 2105, INRIA, SophiaAntipolis, Valbonne, France.
- Ruiz, O., Cadavid, C., and Granados, M. (2002). Evaluation of 2d shape likeness for surface reconstruction. Journal Anales de Ingenieria Grafica, (15):16-24.
- Ruiz, O., Cadavid, C., Granados, M., Pe n˜a, S., and Vasquez, E. (2005). 2d shape similarity as a complement for voronoi-delone methods in shape reconstruction. Elsevier Computer and Graphics, 29(1):81-94.

#### Paper Citation

#### in Harvard Style

Ruiz O., Vasquez E., Peña S. and Granados M. (2008). **GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface Reconstruction from Point Slice Samples** . In *Proceedings of the Third International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2008)* ISBN 978-989-8111-20-3, pages 113-118. DOI: 10.5220/0001098101130118

#### in Bibtex Style

@conference{grapp08,

author={Oscar Ruiz and Eliana Vasquez and Sebastian Peña and Miguel Granados},

title={GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface Reconstruction from Point Slice Samples},

booktitle={Proceedings of the Third International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2008)},

year={2008},

pages={113-118},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001098101130118},

isbn={978-989-8111-20-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Third International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2008)

TI - GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface Reconstruction from Point Slice Samples

SN - 978-989-8111-20-3

AU - Ruiz O.

AU - Vasquez E.

AU - Peña S.

AU - Granados M.

PY - 2008

SP - 113

EP - 118

DO - 10.5220/0001098101130118