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Signed Distance Computation Using the Angle Weighted Pseudonormal
May/June 2005 (vol. 11 no. 3)
pp. 243-253
The normals of closed, smooth surfaces have long been used to determine whether a point is inside or outside such a surface. It is tempting to also use this method for polyhedra represented as triangle meshes. Unfortunately, this is not possible since, at the vertices and edges of a triangle mesh, the surface is not C^1 continuous, hence, the normal is undefined at these loci. In this paper, we undertake to show that the angle weighted pseudonormal (originally proposed by Thürmer and Wüthrich and independently by Séquin) has the important property that it allows us to discriminate between points that are inside and points that are outside a mesh, regardless of whether a mesh vertex, edge, or face is the closest feature. This inside-outside information is usually represented as the sign in the signed distance to the mesh. In effect, our result shows that this sign can be computed as an integral part of the distance computation. Moreover, it provides an additional argument in favor of the angle weighted pseudonormals being the natural extension of the face normals. Apart from the theoretical results, we also propose a simple and efficient algorithm for computing the signed distance to a closed C^0 mesh. Experiments indicate that the sign computation overhead when running this algorithm is almost negligible.

[1] G. Thürmer and C. Wüthrich, “Computing Vertex Normals from Polygonal Facets,” J. Graphics Tools, vol. 3, no. 1, pp. 43-46, 1998.
[2] C.H. Séquin, “Procedural Spline Interpolation in Unicubix,” Proc. Third USENIX Computer Graphics Workshop, pp. 63-83, 1986.
[3] B. Payne and A. Toga, “Distance Field Manipulation of Surface Models,” IEEE Computer Graphics and Applications, vol. 12, no. 1, pp. 65-71, 1992.
[4] A. Guéziec, “Meshsweeper: Dynamic Point-to-Polygonal Mesh Distance and Applications,” IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 1, pp. 47-60, Jan.-Mar. 2001.
[5] J. Sethian, Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge Univ. Press, 1999.
[6] S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, first ed. Springer Verlag, Nov. 2002.
[7] Geometric Level Set Methods in Imaging, Vision, and Graphics, S. Osher and N. Paragios, eds. Springer, 2003.
[8] M. Lin and S. Gottschalk, “Collision Detection between Geometric Models: A Survey,” Proc. IMA Conf. Math. of Surfaces, 1998.
[9] M.G. Coutinho, Dynamic Simulations of Multibody Systems. Springer, 2001.
[10] E. Guendelman, R. Bridson, and R.P. Fedkiw, “Nonconvex Rigid Bodies with Stacking,” ACM Trans. Graphics, vol. 22, no. 3, pp. 871-878, 2003.
[11] J. Linhart, “A Quick Point-in-Polyhedron Test,” Computers & Graphics, vol. 14, no. 3, pp. 445-448, 1990.
[12] F.R. Feito and J.C. Torres, “Inclusion Test for General Polyhedra,” Computers & Graphics, vol. 21, no. 1, pp. 23-30, 1997.
[13] P.C.P. Carvalho and P.R. Cavalcanti, “Point in Polyhedron Testing Using Spherical Polygons,” Graphics Gems V, A.W. Paeth, ed., pp. 42-49, AP Professional, 1995.
[14] M. Jones, “The Production of Volume Data from Triangular Meshes Using Voxelisation,” Computer Graphics Forum, vol. 15, no. 5, pp. 311-318, 1996.
[15] F. Dachille and A. Kaufman, “Incremental Triangle Voxelization,” Proc. Graphics Interface 2000, pp. 205-212, 2000.
[16] S. Mauch, “Efficient Algorithms for Solving Static Hamilton-Jacobi Equations,” PhD dissertation, Caltech, 2003.
[17] C. Sigg, R. Peikert, and M. Gross, “Signed Distance Transform Using Graphics Hardware,” Proc. IEEE Visualization '03, pp. 83-90, 2003.
[18] A. Sud, M.A. Otaduy, and D. Manocha, “Difi: Fast 3D Distance Field Computation Using Graphics Hardware,” Proc. Eurographics, vol. 23, no. 3, 2004.
[19] J. Foley, A. van Dam, S. Feiner, and J. Hughes, Computer Graphics: Principles and Practice in C, second ed. Addison-Wesley, 1995.
[20] S.F.F. Gibson, R.N. Perry, A.P. Rockwood, and T.R. Jones, “Adaptively Sampled Distance Fields: A General Representation of Shape for Computer Graphics,” Proc. SIGGRAPH 2000, pp. 249-254, 2000.
[21] J. Huang, Y. Li, R. Crawfis, S.-C. Lu, and S.-Y. Liou, “A Complete Distance Field Representation,” Proc. Visualization 2001, pp. 247-254, 2001.
[22] D.E. Johnson and E. Cohen, “Bound Coherence for Minimum Distance Computations,” Proc. 1999 IEEE Int'l Conf. Robotics and Automation, pp. 1843-1848, 1999.
[23] E. Larsen, S. Gottschalk, M.C. Lin, and D. Manocha, “Fast Proximity Queries with Swept Sphere Volumes,” technical report, Dept. of Computer Science, Univ. of North Carolina Chapel Hill, 1999.
[24] J. Wu and L. Kobbelt, “Piecewise Linear Approximation of Signed Distance Fields,” Proc. Vision, Modeling, and Visualization 2003, 2003.
[25] H. Fuchs, Z. Kedem, and B. Naylor, “On Visible Surface Generation by A Priori Tree Structures,” Proc. Seventh Ann. Conf. Computer Graphics and Interactive Techniques, pp. 124-133, 1980.
[26] H. Gouraud, “Continuous Shading of Curved Surfaces,” IEEE Trans. Computers, vol. 20, no. 6, pp. 623-629, June 1971.
[27] A.S. Glassner, Computing Surface Normals for 3D Models, pp. 562-566. Academic Press, 1990.
[28] N. Max, “Weights for Computing Vertex Normals from Facet Normals,” J. Graphics Tools, vol. 4, no. 2, pp. 1-6, 1999.
[29] H. Aanaes and J.A. Baerentzen, “Pseudo-Normals for Signed Distance Computation,” Proc. Vision, Modeling, and Visualization 2003, 2003.
[30] P. Sander, X. Gu, S. Gortler, H. Hoppe, and J. Snyder, “Silhouette Clipping,” Proc. ACM SIGGRAPH Conf. Computer Graphics, pp. 327-334, 2000.
[31] G. vandenBergen, Collision Detection in Interactive 3D Environments. Morgan Kaufmann, 2004.
[32] S. Gottschalk, “Collision Queries Using Oriented Bounding Boxes,” PhD dissertation, Univ. of North Carolina at Chapel Hill, 2000.
[33] M. Botsch and L. Kobbelt, “A Robust Procedure to Eliminate Degenerate Faces from Triangle Meshes,” Vision, Modeling, Visualization 2001 Proc., 2001.
[34] K. Erleben and H. Dohlmann, personal communications.
[35] J.R. Shewchuk, “Robust Adaptive Floating-Point Geometric Predicates,” Proc. 12th Ann. Symp. Computational Geometry, pp. 141-150, May 1996.

Index Terms:
Mesh, signed distance field, normal, pseudonormal, polyhedron.
Citation:
J. Andreas B?rentzen, Henrik Aan?, "Signed Distance Computation Using the Angle Weighted Pseudonormal," IEEE Transactions on Visualization and Computer Graphics, vol. 11, no. 3, pp. 243-253, May-June 2005, doi:10.1109/TVCG.2005.49
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