An SL(2) Invariant Shape Median

  • Benjamin BerkelsEmail author
  • Gina Linkmann
  • Martin Rumpf


Median averaging is a powerful averaging concept on sets of vector data in finite dimensions. A generalization of the median for shapes in the plane is introduced. The underlying distance measure for shapes takes into account the area of the symmetric difference of shapes, where shapes are considered to be invariant with respect to different classes of affine transformations. To obtain a well-posed problem the perimeter is introduced as a geometric prior. Based on this model, an existence result can be established in the class of sets of finite perimeter. As alternative invariance classes other classical transformation groups such as pure translation, rotation, scaling, and shear are investigated. The numerical approximation of median shapes uses a level set approach to describe the shape contour. The level set function and the parameter sets of the group action on every given shape are incorporated in a joint variational functional, which is minimized based on step size controlled, regularized gradient descent. Various applications show in detail the qualitative properties of the median.


Variational methods Shape space TV-model Level set method 


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  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, New York (2000) zbMATHGoogle Scholar
  2. 2.
    Avants, B., Gee, J.C.: Geodesic estimation for large deformation anatomical shape averaging and interpolation. NeuroImage 23, 139–150 (2004). Supplement 1 CrossRefGoogle Scholar
  3. 3.
    Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005) CrossRefGoogle Scholar
  4. 4.
    Berkels, B., Linkmann, G., Rumpf, M.: A shape median based on symmetric area differences. In: Deussen, O., Keim, D., Saupe, D. (eds.) Vision, Modeling, and Visualization Proceedings, pp. 399–407 (2008) Google Scholar
  5. 5.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) zbMATHGoogle Scholar
  6. 6.
    Bhatia, K.K., Hajnal, J.V., Puri, B.K., Edwards, A.D., Rueckert, D.: Consistent groupwise non–rigid registration for atlas construction. In: IEEE International Symposium on Biomedical Imaging: Nano to Macro, vol. 1, pp. 908–911 (2004) Google Scholar
  7. 7.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997) zbMATHCrossRefGoogle Scholar
  8. 8.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001) zbMATHCrossRefGoogle Scholar
  9. 9.
    Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Found. Comput. Math. 5(1), 1–58 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Charpiat, G., Faugeras, O., Keriven, R., Maurel, P.: Distance-based shape statistics. In: Acoustics, Speech and Signal Processing, 2006 (ICASSP 2006), vol. 5 (2006) Google Scholar
  11. 11.
    Chen, S.E., Parent, R.E.: Shape averaging and its applications to industrial design. IEEE Comput. Graph. Appl. 9(1), 47–54 (1989) CrossRefGoogle Scholar
  12. 12.
    Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models—their training and application. Comput. Vis. Image Underst. 61(1), 38–59 (1995) CrossRefGoogle Scholar
  13. 13.
    Dupuis, D., Grenander, U., Miller, M.: Variational problems on flows of diffeomorphisms for image matching. Q. Appl. Math. 56, 587–600 (1998) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Felzenszwalb, P.F.: Representation and detection of deformable shapes. IEEE Trans. Pattern Anal. Mach. Intell. 27(2), 208–220 (2005) CrossRefGoogle Scholar
  15. 15.
    Fletcher, P.T., Lu, C., Joshi, S.: Statistics of shape via principal geodesic analysis on Lie groups. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition CVPR, vol. 1, pp. 95–101 (2003) Google Scholar
  16. 16.
    Fletcher, T., Venkatasubramanian, S., Joshi, S.: Robust statistics on Riemannian manifolds via the geometric median. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2008) Google Scholar
  17. 17.
    Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948) Google Scholar
  18. 18.
    Fuchs, M., Jüttler, B., Scherzer, O., Yang, H.: Shape metrics based on elastic deformations. J. Math. Imaging Vis. 35(1), 86–102 (2009) CrossRefGoogle Scholar
  19. 19.
    Jiang, X., Schiffmann, L., Bunke, H.: Computation of median shapes. In: Proceedings of the Fourth IEEE Asian Conference on Computer Vision (ACCV’00), pp. 300–305 (2000) Google Scholar
  20. 20.
    Jiang, X., Abegglen, K., Bunke, H., Csirik, J.: Dynamic computation of generalised median strings. Pattern Anal. Appl. 6(3), 185–193 (2003) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23, 151–160 (2004). Supplement 1 CrossRefGoogle Scholar
  22. 22.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16, 81–121 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: a general framework. Int. J. Comput. Vis. 4(1–2), 61–84 (2001) CrossRefGoogle Scholar
  25. 25.
    Miller, M., Trouvé, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Eng. 4, 375–405 (2002) CrossRefGoogle Scholar
  26. 26.
    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Rueckert, D., Frangi, A.F., Schnabel, J.A.: Automatic construction of 3D statistical deformation models using nonrigid registration. In: Niessen, W., Viergever, M. (eds.) Medical Image Computing and Computer–Assisted Intervention, MICCAI. Lecture Notes in Computer Science, vol. 2208, pp. 77–84. Springer, Berlin (2001) Google Scholar
  28. 28.
    Rumpf, M., Wirth, B.: An elasticity approach to principal modes of shape variation. In: Proceedings of the Second International Conference on Scale Space Methods and Variational Methods in Computer Vision (SSVM 2009). Lecture Notes in Computer Science, vol. 5567, pp. 709–720. Springer, Berlin (2009) CrossRefGoogle Scholar
  29. 29.
    Rumpf, M., Wirth, B.: A nonlinear elastic shape averaging approach. SIAM J. Imaging Sci. 2(3), 800–833 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Sokolowski, J., Zochowski, A.: Optimality conditions for simultaneous topology and shape optimization. SIAM J. Control Optim. 42, 1198–1221 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sokołowski, J., Zolésio, J.-P.: Introduction to shape optimization. Springer, Berlin (1992). Shape sensitivity analysis zbMATHGoogle Scholar
  32. 32.
    Sundaramoorthi, G., Jackson, J.D., Yezzi, A., Mennucci, A.C.: Tracking with Sobolev active contours. In: CVPR ’06: Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 674–680. IEEE Computer Society, Washington (2006) Google Scholar
  33. 33.
    Wirth, B., Bar, L., Rumpf, M., Sapiro, G.: Geodesics in shape space via variational time discretization. In: EMMCVPR’09, to appear (2009) Google Scholar
  34. 34.
    Yezzi, A., Soatto, S.: Deformotion: Deforming motion, shape average and the joint registration and approximation of structures in images. Int. J. Comput. Vis. 53(2), 153–167 (2003) CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für Numerische SimulationUniversität BonnBonnGermany

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