“Spherical parametrization and remeshing” by Praun and Hoppe

  • ©Emil Praun and Hugues Hoppe




    Spherical parametrization and remeshing



    The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scale-distortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semi-regular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-of-detail, morphing, compression, and smooth surface subdivision.


    1. ALEXA, M. 2000. Merging polyhedral shapes with scattered features. The Visual Computer, 16(1), pp. 26–37.Google ScholarDigital Library
    2. ALEXA, M. 2002. Recent advances in mesh morphing. Computer Graphics Forum, 21(2), pp. 173–196.Google ScholarCross Ref
    3. ARVO, J. 1995. Stratified sampling of spherical triangles. ACM SIGGRAPH 95, pp. 437–438. Google Scholar
    4. BUSS, S., AND FILLMORE, J. 2001. Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics, 20(2), pp. 95–126. Google ScholarDigital Library
    5. CIGNONI, P., MONTANI, C., ROCCHINI, C., AND SCOPIGNO, R. 1998. A general method for recovering attribute values on simplified meshes. IEEE Visualization 1998, pp. 59–66. Google ScholarDigital Library
    6. COHEN, J., OLANO, M., AND MANOCHA, D. 1998. Appearance-preserving simplification. ACM SIGGRAPH 98, pp. 115–122. Google Scholar
    7. DAVIS, G. 1996. Wavelet image compression construction kit. http://www.geoffdavis.net/dartmouth/wavelet/wavelet.html.Google Scholar
    8. ECK, M., DEROSE, T., DUCHAMP, T., HOPPE, H., LOUNSBERY, M., AND STUETZLE, W. 1995. Multiresolution analysis of arbitrary meshes. ACM SIGGRAPH 95, pp. 173–182. Google Scholar
    9. ECKSTEIN, I., SURAZHSKY, V., AND GOTSMAN, C. 2001. Texture mapping with hard constraints. Eurographics 2001, pp. 95–104.Google Scholar
    10. FLOATER, M. 1997. Parametrization and smooth approximation of surface triangulations. CAGD 14(3), pp. 231–250. Google ScholarDigital Library
    11. FUNKHOUSER, T., MIN, P., KAZHDAN, M., CHEN, J., HALDERMAN, A., DOBKIN, D., AND JACOBS, D. 2003. A search engine for 3D models. ACM Transactions on Graphics, 22(1), January 2003, pp. 83–105. Google ScholarDigital Library
    12. FURUTI, C. 1997. http://www.progonos.com/furuti/.Google Scholar
    13. GREENE, N. 1986. Environment mapping and other applications. IEEE Computer Graphics and Applications, 6(11). Google ScholarDigital Library
    14. GRIMM, C. 2002. Simple manifolds for surface modeling and parametrization. Shape Modeling International 2002. Google Scholar
    15. GU, X., GORTLER, S., AND HOPPE, H. 2002. Geometry images. ACM SIGGRAPH 2002, pp. 355–361. Google ScholarDigital Library
    16. GOTSMAN, C., GU, X., AND SHEFFER, A. 2003. Fundamentals of spherical parameterization for 3D meshes. ACM SIGGRAPH 2003. Google ScholarDigital Library
    17. GUSKOV, I., VIDIMČE, K., SWELDENS, W., AND SCHRÖDER, P. 2000. Normal meshes. ACM SIGGRAPH 2000, pp. 95–102. Google ScholarDigital Library
    18. HAKER, S., ANGENENT, S., TANNENBAUM, S., KIKINIS, R., SAPIRO, G., AND HALLE, M. 2000. Conformal surface parametrization for texture mapping. IEEE TVCG, 6(2), pp. 181–189. Google Scholar
    19. HOPPE, H. 1996. Progressive meshes. ACM SIGGRAPH 96, pp. 99–108. Google Scholar
    20. HORMANN, K., GREINER, G., AND CAMPAGNA, S. 1999. Hierarchical parametrization of triangulated surfaces. Vision, Modeling, and Visualization 1999, pp. 219–226.Google Scholar
    21. KENT, J., CARLSON, W., AND PARENT, R. 1992. Shape transformation for polyhedral objects. ACM SIGGRAPH 92, pp. 47–54. Google ScholarDigital Library
    22. KHODAKOVSKY, A., SCHRÖDER, P., AND SWELDENS, W. 2000. Progressive geometry compression. ACM SIGGRAPH 2000, 271–278. Google ScholarDigital Library
    23. KOBBELT, L., VORSATZ, J., LABSIK, U., AND SEIDEL, H.-P. 1999. A shrink wrapping approach to remeshing polygonal surfaces. Eurographics 1999, pp. 119–130.Google Scholar
    24. LEE, A., SWELDENS, W., SCHRÖDER, P., COWSAR, L., AND DOBKIN, D. 1998. MAPS: Multiresolution adaptive parametrization of surfaces. ACM SIGGRAPH 98, pp. 95–104. Google Scholar
    25. LEE, A., MORETON, H., AND HOPPE, H. 2000. Displaced subdivision surfaces. ACM SIGGRAPH 2000, pp. 85–94. Google ScholarDigital Library
    26. LÉVY, B., PETITJEAN, S., RAY, N., AND MAILLOT, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM SIGGRAPH 2002, pp. 362–371. Google ScholarDigital Library
    27. LOSASSO, F., HOPPE, H., SCHAEFER, S., AND WARREN, J. 2003. Smooth geometry images. Submitted for publication.Google Scholar
    28. MAILLOT, J., YAHIA, H., AND VERROUST, A. 1993. Interactive texture mapping. ACM SIGGRAPH 93, pp. 27–34. Google Scholar
    29. PRAUN, E., SWELDENS, W., AND SCHRÖDER, P. 2001. Consistent mesh parametrizations. ACM SIGGRAPH 2001, pp. 179–184. Google Scholar
    30. QUICKEN, M., BRECHBÜHLER, C., HUG, J., BLATTMANN, H., SZÉKELY, G. 2000. Parametrization of closed surfaces for parametric surface description, CVPR 2000, pp. 354–360.Google Scholar
    31. SANDER, P., SNYDER, J., GORTLER, S., AND HOPPE, H. 2001. Texture mapping progressive meshes. ACM SIGGRAPH 2001, pp. 409–416. Google ScholarDigital Library
    32. SANDER, P., GORTLER, S., SNYDER, J., AND HOPPE, H. 2002. Signal-specialized parametrization. Eurographics Workshop on Rendering 2002, pp. 87–100. Google ScholarDigital Library
    33. SCHRÖDER, P. AND SWELDENS, W. 1995. Spherical wavelets: Efficiently representing functions on the sphere. ACM SIGGRAPH 95, 161–172. Google Scholar
    34. SHAPIRO, A. AND TAL, A. 1998. Polygon realization for shape transformation. The Visual Computer, 14 (8–9), pp. 429–444.Google Scholar
    35. SHEFFER, A., GOTSMAN, C., AND DYN, N. 2003. Robust spherical parameterization of triangular meshes. 4th Israel-Korea Bi-National Conf. on Geometric Modeling and Computer Graphics, pp. 94–99.Google Scholar
    36. SHEFFER, A., AND HART, J. 2002. Seamster: Inconspicuous low-distortion texture seam layout. IEEE Visualization 2002, pp. 291–298. Google ScholarDigital Library
    37. SORKINE, O., COHEN-OR, D., GOLDENTHAL, R., AND LISCHINSKI, D. 2002. Bounded-distortion piecewise mesh parametrization. IEEE Visualization 2002. Google Scholar
    38. STAUNTON, R. 1999. Hexagonal sampling in image processing. In Advances in imaging and electron physics, Vol. 107, Academic Press.Google Scholar
    39. TURK, G. 1990. Generating random points in triangles. Graphics Gems, Academic Press, pp. 649–650. Google Scholar
    40. TODHUNTER, I., AND LEATHEM, J. G. 1949. Spherical Trigonometry, Macmillan and Co, Limited.Google Scholar
    41. WOOD, Z., DESBRUN, M., SCHRÖDER, P., AND BREEN, D. 2000. Semi-regular mesh extraction from volumes. IEEE Visualization 2000, pp. 275–282. Google ScholarDigital Library

ACM Digital Library Publication: