Abstract
Important engineering applications use unstructured hexahedral meshes for numerical simulations. Hexahedral cells, when compared to tetrahedral ones, tend to be more numerically stable and to require less mesh refinement. However, volume visualization of unstructured hexahedral meshes is challenging due to the trilinear variation of scalar fields inside the cells. The conventional solution consists in subdividing each hexahedral cell into five or six tetrahedra, approximating a trilinear variation by a nonadaptive piecewise linear function. This results in inaccurate images and increases the memory consumption. In this paper, we present an accurate ray-casting volume rendering algorithm for unstructured hexahedral meshes. In order to capture the trilinear variation along the ray, we propose the use of quadrature integration. A set of computational experiments demonstrates that our proposal produces accurate results, with reduced memory footprint. The entire algorithm is implemented on graphics cards, ensuring competitive performance. We also propose a faster approach that, as the tetrahedron subdivision scheme, also approximates the trilinear variation by a piecewise linear function, but in an adaptive and more accurate way, considering the points of minimum and maximum of the scalar function along the ray.
Similar content being viewed by others
References
Bernadon, F.F., Pagot, C.A., Comba, J.L.D., Silva, C.T.: GPU-based tiled ray casting using depth peeling. J. Graph. GPU Game Tools 11(4), 1–16 (2006)
Carr, H., Moller, T., Snoeyink, J.: Artifacts caused by simplicial subdivision. IEEE Trans. Vis. Comput. Graph. 12, 231–242 (2006). doi:10.1109/TVCG.2006.22
Espinha, R., Celes, W.: High-quality hardware-based ray-casting volume rendering using partial pre-integration. In: Proceedings of the XVIII Brazilian Symposium on Computer Graphics and Image Processing, p. 273. IEEE Computer Society, Washington (2005). doi:10.1109/SIBGRAPI.2005.29. http://portal.acm.org/citation.cfm?id=1114697.1115365
Garrity, M.P.: Raytracing irregular volume data. In: Proceedings of the 1990 Workshop on Volume visualization, VVS’90, pp. 35–40. ACM, New York (1990). doi:10.1145/99307.99316
Hajjar, J.E., Marchesin, S., Dischler, J., Mongenet, C.: Second order pre-integrated volume rendering. In: IEEE Pacific Visualization Symposium (2008)
Marchesin, S., de Verdiere, G.: High-quality, semi-analytical volume rendering for amr data. IEEE Trans. Vis. Comput. Graph. 15(6), 1611–1618 (2009). doi:10.1109/TVCG.2009.149
Marmitt, G., Slusallek, P.: Fast ray traversal of tetrahedral and hexahedral meshes for direct volume rendering. In: Proceedings of Eurographics/IEEE-VGTC Symposium on Visualization (EuroVIS), Lisbon, Portugal (2006)
Max, N.L., Williams, P.L., Silva, C.T.: Cell projection of meshes with non-planar faces. In: Data Visualization: The State of the Art, pp. 157–168 (2003)
Miranda, F.M., Celes, W.: Accurate volume rendering of unstructured hexahedral meshes. In: Lewiner, T., Torres, R. (eds.) Sibgrapi 2011 (24th Conference on Graphics, Patterns and Images), pp. 93–100. IEEE, Maceió (2011). doi:10.1109/SIBGRAPI.2011.3. http://www.im.ufal.br/evento/sibgrapi2011/
Moreland, K., Angel, E.: A fast high accuracy volume renderer for unstructured data. In: Proceedings of the 2004 IEEE Symposium on Volume Visualization and Graphics, VV’04, pp. 9–16. IEEE Computer Society, Washington (2004). doi:10.1109/VV.2004.2
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, New York (1992)
Röttger, S., Ertl, T.: A two-step approach for interactive pre-integrated volume rendering of unstructured grids. In: Proceedings of the 2002 IEEE Symposium on Volume Visualization and Graphics, VVS’02, pp. 23–28. IEEE Press, Piscataway (2002) http://portal.acm.org/citation.cfm?id=584110.584114
Röttger, S., Kraus, M., Ertl, T.: Hardware-accelerated volume and isosurface rendering based on cell-projection. In: Proceedings of the Conference on Visualization’00, VIS’00, pp. 109–116. IEEE Computer Society Press, Los Alamitos (2000). http://portal.acm.org/citation.cfm?id=375213.375226
Shirley, P., Tuchman, A.: A polygonal approximation to direct scalar volume rendering. In: Proceedings of the 1990 Workshop on Volume Visualization, VVS’90, pp. 63–70. ACM, New York (1990). doi:10.1145/99307.99322
Weiler, M., Kraus, M., Merz, M., Ertl, T.: Hardware-based ray casting for tetrahedral meshes. In: Proceedings of the 14th IEEE Visualization 2003, VIS’03, p. 44. IEEE Computer Society, Washington (2003). doi:10.1109/VISUAL.2003.1250390
Weiler, M., Mallon, P.N., Kraus, M., Ertl, T.: Texture-encoded tetrahedral strips. In: Proceedings of the 2004 IEEE Symposium on Volume Visualization and Graphics, VV’04, pp. 71–78. IEEE Computer Society, Washington (2004). doi:10.1109/VV.2004.13
Williams, P.L., Max, N.: A volume density optical model. In: Proceedings of the 1992 Workshop on Volume Visualization, VVS’92, pp. 61–68. ACM, New York (1992). doi:10.1145/147130.147151
Williams, P.L., Max, N.L., Stein, C.M.: A high accuracy volume renderer for unstructured data. IEEE Trans. Vis. Comput. Graph. 4, 37–54 (1998)
Acknowledgements
We thank CAPES (Brazilian National Research and Development Council) and CNPq (Brazilian National Council for Scientific and Technological Development) for the financial support to conduct this research. This work was done at the Tecgraf laboratory at PUC-Rio, which is mainly funded by the Brazilian oil company, Petrobras.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miranda, F.M., Celes, W. Volume rendering of unstructured hexahedral meshes. Vis Comput 28, 1005–1014 (2012). https://doi.org/10.1007/s00371-012-0742-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-012-0742-8