Hessian Eigenfunctions for Triangular Mesh Parameterization

Daniel Mejia, Oscar Ruiz-Salguero, Carlos A. Cadavid

Abstract

Hessian Locally Linear Embedding (HLLE) is an algorithm that computes the nullspace of a Hessian functional H for Dimensionality Reduction (DR) of a sampled manifold M. This article presents a variation of classic HLLE for parameterization of 3D triangular meshes. Contrary to classic HLLE which estimates local Hessian nullspaces, the proposed approach follows intuitive ideas from Differential Geometry where the local Hessian is estimated by quadratic interpolation and a partition of unity is used to join all neighborhoods. In addition, local average triangle normals are used to estimate the tangent plane TxM at x 2 M instead of PCA, resulting in local parameterizations which reflect better the geometry of the surface and perform better when the mesh presents sharp features. A high frequency dataset (Brain) is used to test our algorithm resulting in a higher rate of success (96:63%) compared to classic HLLE (76:4%).

References

  1. Athanasiadis, T., Zioupos, G., and Fudos, I. (2013). Efficient computation of constrained parameterizations on parallel platforms. Computers & Graphics, 37(6):596-607. Shape Modeling International (SMI) Conference 2013.
  2. Belkin, M. and Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373-1396.
  3. Desbrun, M., Meyer, M., and Alliez, P. (2002). Intrinsic parameterizations of surface meshes. Computer Graphics Forum, 21(3):209-218.
  4. Desikan, R. S., Ségonne, F., Fischl, B., Quinn, B. T., B, C. D., Blacker, D., Buckner, R. L., Dale, A. M., Maguire, R. P., Hyman, B. T., Albert, M. S., and Killiany, R. J. (2006). An automated labeling system for subdividing the human cerebral cortex on +mri scans into gyral based regions of interest. NeuroImage, 31(3):968-980.
  5. Donoho, D. L. and Grimes, C. (2003). Hessian eigenmaps: Locally linear embedding techniques for highdimensional data. Proceedings of the National Academy of Sciences, 100(10):5591-5596.
  6. Floater, M. S. (1997). Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design, 14(3):231-250.
  7. Kharevych, L., Springborn, B., and Schröder, P. (2006). Discrete conformal mappings via circle patterns. ACM Transactions on Graphics, 25(2):412-438.
  8. Lafon, S. and Lee, A. (2006). Diffusion maps and coarsegraining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 28(9):1393-1403.
  9. Lee, Y., Kim, H. S., and Lee, S. (2002). Mesh parameterization with a virtual boundary. Computers & Graphics, 26(5):677-686.
  10. Lévy, B., Petitjean, S., Ray, N., and Maillot, J. (2002). Least squares conformal maps for automatic texture atlas generation. In ACM, editor, ACM SIGGRAPH conference proceedings.
  11. Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. J. (2008). A local/global approach to mesh parameterization. In Proceedings of the Symposium on Geometry Processing, SGP 7808, pages 1495-1504, Aire-la-Ville, Switzerland, Switzerland. Eurographics Association.
  12. Roweis, S. T. and Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323-2326.
  13. Ruiz, O. E., Mejia, D., and Cadavid, C. A. (2015). Triangular mesh parameterization with trimmed surfaces. International Journal on Interactive Design and Manufacturing (IJIDeM), pages 1-14.
  14. Sheffer, A. and de Sturler, E. (2001). Parameterization of faceted surfaces for meshing using angle-based flattening. Engineering with Computers, 17(3):326-337.
  15. Smith, J. and Schaefer, S. (2015). Bijective parameterization with free boundaries. ACM Transactions on Graphics, 34(4):70:1-70:9.
  16. Sun, X. and Hancock, E. R. (2008). Quasi-isometric parameterization for texture mapping. Pattern Recognition, 41(5):1732-1743.
  17. Tenenbaum, J. B., De Silva, V., and Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319-2323.
  18. Yoshizawa, S., Belyaev, A., and Seidel, H.-P. (2004). A fast and simple stretch-minimizing mesh parameterization. In Shape Modeling Applications, 2004. Proceedings, pages 200-208.
  19. Zayer, R., Lévy, B., and Seidel, H.-P. (2007). Linear angle based parameterization. In ACM/EG Symposium on Geometry Processing conference proceedings.
  20. Zhang, Z. and Zha, H. (2002). Principal manifolds and nonlinear dimension reduction via local tangent space alignment. SIAM Journal of Scientific Computing , 26:313-338.
Download


Paper Citation


in Harvard Style

Mejia D., Ruiz-Salguero O. and Cadavid C. (2016). Hessian Eigenfunctions for Triangular Mesh Parameterization . In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016) ISBN 978-989-758-175-5, pages 75-82. DOI: 10.5220/0005668200730080


in Bibtex Style

@conference{grapp16,
author={Daniel Mejia and Oscar Ruiz-Salguero and Carlos A. Cadavid},
title={Hessian Eigenfunctions for Triangular Mesh Parameterization},
booktitle={Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016)},
year={2016},
pages={75-82},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005668200730080},
isbn={978-989-758-175-5},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2016)
TI - Hessian Eigenfunctions for Triangular Mesh Parameterization
SN - 978-989-758-175-5
AU - Mejia D.
AU - Ruiz-Salguero O.
AU - Cadavid C.
PY - 2016
SP - 75
EP - 82
DO - 10.5220/0005668200730080