A Computational Model of Multidimensional Shape


We develop a computational model of shape that extends existing Riemannian models of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a formulation that accounts for regional differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals or groups, and modeling and simulation of shape dynamics. We give multiple examples of geodesic interpolations and illustrations of the use of the models in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.


  1. Bhattacharya, R., & Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Annals of Statistics, 31(1), 1–29.

    MATH  Article  MathSciNet  Google Scholar 

  2. do Carmo, M. P. (1994). Riemannian geometry. Basel: Birkhauser.

    Google Scholar 

  3. Dryden, I. L., & Mardia, K. V. (1998). Statistical shape analysis. New York: Wiley.

    MATH  Google Scholar 

  4. Fletcher, P., Lu, C., Pizer, S., & Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23(8), 995–1005.

    Article  Google Scholar 

  5. Fletcher, P. T., Venkatasubramanian, S., & Joshi, S. (2008). Robust statistics on Riemannian manifolds via the geometric median. In IEEE international conference on computer vision and pattern recognition.

  6. Fuchs, M., Jüttler, B., Scherzer, O., & Yang, H. (2009). Shape metrics based on elastic deformations. Journal of Mathematical Imaging and Vision, 35(1), 86–102.

    Article  MathSciNet  Google Scholar 

  7. Joshi, S., Klassen, E., Srivastava, A., & Jermyn, I. (2007). An efficient representation for computing geodesics between n-dimensional elastic shapes. In IEEE conference on computer vision and pattern recognition.

  8. Karcher, H. (1977). Riemann center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30, 509–541.

    MATH  Article  MathSciNet  Google Scholar 

  9. Kendall, D. G. (1984). Shape manifolds, Procrustean metrics and complex projective spaces. Bulletin of London Mathematical Society, 16, 81–121.

    MATH  Article  MathSciNet  Google Scholar 

  10. Kendall, D. G., Barden, D., Carne, T. K., & Le, H. (1999). Shape and shape theory. New York: Wiley.

    MATH  Book  Google Scholar 

  11. Kilian, M., Mitra, N., & Pottmann, H. (2007). Geometric modeling in shape space. ACM Transactions on Graphics, 26(3), 64–3–64–8.

    Article  Google Scholar 

  12. Klassen, E., Srivastava, A., Mio, W., & Joshi, S. (2004). Analysis of planar shapes using geodesic paths on shape manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 372–383.

    Article  Google Scholar 

  13. Le, H. L., & Kendall, D. G. (1993). The Riemannian structure of euclidean shape spaces: a novel environment for statistics. Annals of Statistics, 21(3), 1225–1271.

    MATH  Article  MathSciNet  Google Scholar 

  14. Liu, X., Mio, W., Shi, Y., Dinov, I., Liu, X., Lepore, N., Lepore, F., Fortin, M., Voss, P., Lassonde, M., & Thompson, P. M. (2008). Models of normal variation and local contrasts in hippocampal anatomy. In MICCAI.

  15. Liu, X., Shi, Y., Morra, J., Liu, X., Thompson, P. M., & Mio, W. (2009). Mapping hippocampal atrophy with a multi-scale model of shape. In ISBI.

  16. Michor, P., & Mumford, D. (2007). An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Applied and Computational Harmonic Analysis, 23, 74–113.

    MATH  Article  MathSciNet  Google Scholar 

  17. Mio, W., Srivastava, A., & Klassen, E. (2004). Interpolations with elasticae in Euclidean spaces. Quarterly of Applied Mathematics, 62, 359–378.

    MATH  MathSciNet  Google Scholar 

  18. Mio, W., Srivastava, A., & Joshi, S. (2007). On shape of plane elastic curves. International Journal of Computer Vision, 73(3), 307–324.

    Article  Google Scholar 

  19. Mio, W., Bowers, J. C., & Liu, X. (2009). Shape of elastic strings in Euclidean space. International Journal of Computer Vision, 82, 96–112.

    Article  Google Scholar 

  20. Morra, J., Tu, Z., Apostolova, L. G., Green, A., Avedissian, C., Madsen, S., Parikshak, N., Toga, A., Jack, C., Schuff, N., Weiner, M. W., & Thompson, P. M., & the ADNI (2009). Automated mapping of hippocampal atrophy in 1-year repeat MRI data from 490 subjects with Alzheimer’s disease, mild cognitive impairment, and elderly controls. NeuroImage, 45(1), S3–S15. Supplement 1.

    Article  Google Scholar 

  21. Munkres, J. (1984). Elements of algebraic topology. New York: Perseus Books.

    MATH  Google Scholar 

  22. Schmidt, F. R., Clausen, M., & Cremers, D. (2006). Shape matching by variational computation of geodesics on a manifold. In LNCS : Vol. 4174. Pattern Recognition (Proc. DAGM) (pp. 142–151). Berlin: Springer.

    Chapter  Google Scholar 

  23. Shi, Y., Thompson, P., Dinov, I., Osher, S., & Toga, A. (2007a). Direct cortical mapping via solving partial differential equations on implicit surfaces. Medical Image Analysis, 11(3), 207–223.

    Article  Google Scholar 

  24. Shi, Y., Thompson, P., Zubicaray, G., Ross, S., Tu, Z., Dinov, I., & Toga, A. (2007b). Direct mapping of hippocampal surfaces with intrinsic shape context. NeuroImage, 36(3), 792–807.

    Article  Google Scholar 

  25. Srivastava, A., Joshi, S., Mio, W., & Liu, X. (2005). Statistical shape analysis: Clustering, learning and testing. IEEE Transaction on Pattern Analysis and Machine Intelligence, 27, 590–602.

    Article  Google Scholar 

  26. Sundaramoorthi, G., Yezzi, A., & Mennucci, A. (2007). Sobolev active contours. The International Journal of Computer Vision, 73, 345–366.

    Article  Google Scholar 

  27. Younes, L. (1998). Computable elastic distance between shapes. SIAM Journal of Applied Mathematics, 58, 565–586.

    MATH  Article  MathSciNet  Google Scholar 

  28. Younes, L. (1999). Optimal matching between shapes via elastic deformations. Journal of Image and Vision Computing, 17(5/6), 381–389.

    Article  Google Scholar 

  29. Younes, L., Michor, P., Shah, J., & Mumford, D. (2007). A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, Mat. Appl., 9.

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Correspondence to Washington Mio.

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This research was supported in part by NSF grant DMS-0713012 and NIH Roadmap for Medical Research grant U54 RR021813.

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Liu, X., Shi, Y., Dinov, I. et al. A Computational Model of Multidimensional Shape. Int J Comput Vis 89, 69–83 (2010). https://doi.org/10.1007/s11263-010-0323-0

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  • Multidimensional shape
  • Shape of surfaces
  • Elastic shapes