Harmonic Surface Mapping with Laplace-Beltrami Eigenmaps

  • Yonggang Shi
  • Rongjie Lai
  • Kyle Kern
  • Nancy Sicotte
  • Ivo Dinov
  • Arthur W. Toga
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5242)


In this paper we propose a novel approach for the mapping of 3D surfaces. With the Reeb graph of Laplace-Beltrami eigenmaps, our method automatically detects stable landmark features intrinsic to the surface geometry and use them as boundary conditions to compute harmonic maps to the unit sphere. The resulting maps are diffeomorphic, robust to natural pose variations, and establish correspondences for geometric features shared across population. In the experiments, we demonstrate our method on three subcortical structures: the hippocampus, putamen, and caudate nucleus. A group study is also performed to generate a statistically significant map of local volume losses in the hippocampus of patients with secondary progressive multiple sclerosis.


Caudate Nucleus Subcortical Structure Level Contour Reeb Graph Spherical Parameterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Thompson, P.M., Hayashi, K.M., de Zubicaray, G.I., Janke, A.L., Rose, S.E., Semple, J., Hong, M.S., Herman, D.H., Gravano, D., Doddrell, D.M., Toga, A.W.: Mapping hippocampal and ventricular change in Alzheimer disease. NeuroImage 22(4), 1754–1766 (2004)CrossRefGoogle Scholar
  2. 2.
    Gerig, G., Styner, M., Jones, D., Weinberger, D., Lieberman, J.: Shape analysis of brain ventricles using SPHARM. In: Proc. Workshop on Mathematical Methods in Biomedical Image Analysis, pp. 171–178 (2001)Google Scholar
  3. 3.
    Davies, R.H., Twining, C.J., Allen, P.D., Cootes, T.F., Taylor, C.J.: Shape discrimination in the hippocampus using an MDL model. In: Proc. IPMI, pp. 38–50 (2003)Google Scholar
  4. 4.
    Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imag. 23(8), 949–958 (2004)CrossRefGoogle Scholar
  5. 5.
    Shi, Y., Thompson, P.M., de Zubicaray, G., Rose, S.E., Tu, Z., Dinov, I., Toga, A.W.: Direct mapping of hippocampal surfaces with intrinsic shape context. NeuroImage 37(3), 792–807 (2007)CrossRefGoogle Scholar
  6. 6.
    Christensen, G.E., Rabbitt, R.D., Miller, M.I.: Deformable templates using large deformation kinematics. IEEE Trans. Imag. Process. 5(10), 1435–1447 (1996)CrossRefGoogle Scholar
  7. 7.
    Joshi, S., Miller, M.I.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Imag. Process. 9(8), 1357–1370 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, L., Miller, J.P., Gado, M.H., McKeel, D.W., Rothermich, M., Miller, M.I., Morris, J.C., Csernansky, J.G.: Abnormalities of hippocampal surface structure in very mild dementia of the alzheimer type. NeuroImage 30(1), 52–60 (2006)CrossRefGoogle Scholar
  9. 9.
    Pizer, S.M., Fritsch, D.S., Yushkevich, P.A., Johnson, V.E., Chaney, E.L.: Segmentation, registration, and measurement of shape variation via image object shape. IEEE Trans. Med. Imag. 18(10), 851–865 (1999)CrossRefGoogle Scholar
  10. 10.
    Styner, M., Gerig, G., Lieberman, J., Jones, D., Weinberger, D.: Statistical shape analysis of neuroanatomical structures based on medial models. Med. Image. Anal. 7(3), 207–220 (2003)CrossRefGoogle Scholar
  11. 11.
    Yushkevich, P.A., Zhang, H., Gee, J.C.: Continuous medial representation for anatomical structures. IEEE Trans. Med. Imag. 25(12), 1547–1564 (2006)CrossRefGoogle Scholar
  12. 12.
    Reuter, M., Wolter, F., Peinecke, N.: Laplace-Beltrami spectra as Shape-DNA of surfaces and solids. Computer-Aided Design 38, 342–366 (2006)CrossRefGoogle Scholar
  13. 13.
    Qiu, A., Bitouk, D., Miller, M.I.: Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator. IEEE Trans. Med. Imag. 25(10), 1296–1306 (2006)CrossRefGoogle Scholar
  14. 14.
    Uhlenbeck, K.: Generic properties of eigenfunctions. Amer. J. of Math. 98(4), 1059–1078 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reeb, G.: Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction nemérique. Comptes Rendus Acad. Sciences 222, 847–849 (1946)zbMATHGoogle Scholar
  16. 16.
    Shewchuk, J.R.: Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. Theory & Applications 22(1–3), 21–74 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Ann. J. Math. 86, 109–160 (1964)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Shi, Y., Thompson, P.M., Dinov, I., Osher, S., Toga, A.W.: Direct cortical mapping via solving partial differential equations on implicit surfaces. Med. Image. Anal. 11(3), 207–223 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yonggang Shi
    • 1
  • Rongjie Lai
    • 2
  • Kyle Kern
    • 3
  • Nancy Sicotte
    • 3
  • Ivo Dinov
    • 1
  • Arthur W. Toga
    • 1
  1. 1.Lab of Neuro ImagingUCLA School of MedicineLos AngelesUSA
  2. 2.Department of MathematicsUCLALos AngelesUSA
  3. 3.Department of NeurologyUCLA School of MedicineLos AngelesUSA

Personalised recommendations