An As-Invariant-As-Possible \(\text {GL}^+(3){}\)-Based Statistical Shape Model

  • Felix AmbellanEmail author
  • Stefan Zachow
  • Christoph von Tycowicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11846)


We describe a novel nonlinear statistical shape model based on differential coordinates viewed as elements of \(\text {GL}^+(3){}\). We adopt an as-invariant-as possible framework comprising a bi-invariant Lie group mean and a tangent principal component analysis based on a unique \(\text {GL}^+(3){}\)-left-invariant, \(\text {O}(3){}\)-right-invariant metric. Contrary to earlier work that equips the coordinates with a specifically constructed group structure, our method employs the inherent geometric structure of the group-valued data and therefore features an improved statistical power in identifying shape differences. We demonstrate this in experiments on two anatomical datasets including comparison to the standard Euclidean as well as recent state-of-the-art nonlinear approaches to statistical shape modeling.


Statistical shape analysis Tangent principal component analysis Lie groups Classification Manifold valued statistics 



The authors are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). Furthermore, we are grateful for the open-access OAI dataset of the Osteoarthritis Initiative, that is a public-private partnership comprised of five contracts (N01-AR-2-2258; N01-AR-2-2259; N01-AR-2-2260; N01-AR-2-2261; N01-AR-2-2262) funded by the National Institutes of Health, a branch of the Department of Health and Human Services, and conducted by the OAI Study Investigators. Private funding partners include Merck Research Laboratories; Novartis Pharmaceuticals Corporation, GlaxoSmithKline; and Pfizer, Inc. Private sector funding for the OAI is managed by the Foundation for the National Institutes of Health. This manuscript was prepared using an OAI public use data set and does not necessarily reflect the opinions or views of the OAI investigators, the NIH, or the private funding partners.


  1. 1.
    Ambellan, F., Lamecker, H., von Tycowicz, C., Zachow, S.: Statistical shape models: understanding and mastering variation in anatomy. In: Rea, P.M. (ed.) Biomedical Visualisation. AEMB, vol. 1156, 1st edn, pp. 67–84. Springer, Cham (2019). Scholar
  2. 2.
    Ambellan, F., Tack, A., Ehlke, M., Zachow, S.: Automated segmentation of knee bone and cartilage combining statistical shape knowledge and convolutional neural networks. Med. Image Anal. 52, 109–118 (2019)CrossRefGoogle Scholar
  3. 3.
    Ambellan, F., Zachow, S., von Tycowicz, C.: A surface-theoretic approach for statistical shape modeling. In: Proceedings of Medical Image Computing and Computer Assisted Intervention (MICCAI) (2019, accepted for publication)Google Scholar
  4. 4.
    Brandt, C., von Tycowicz, C., Hildebrandt, K.: Geometric flows of curves in shape space for processing motion of deformable objects. Comput. Graph Forum 35(2), 295–305 (2016)CrossRefGoogle Scholar
  5. 5.
    Bredbenner, T.L., Eliason, T.D., Potter, R.S., Mason, R.L., Havill, L.M., Nicolella, D.P.: Statistical shape modeling describes variation in tibia and femur surface geometry between control and incidence groups from the osteoarthritis initiative database. J. Biomech. 43(9), 1780–1786 (2010)CrossRefGoogle Scholar
  6. 6.
    Conaghan, P.G., Kloppenburg, M., Schett, G., Bijlsma, J.W., et al.: Osteoarthritis research priorities: a report from a eular ad hoc expert committee. Ann. Rheum. Dis. 73(8), 1442–1445 (2014)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. Int. J. Comput. Vis. 90(2), 255–266 (2010)CrossRefGoogle Scholar
  9. 9.
    Fletcher, P., Lu, C., Pizer, S., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE. Trans. Med. Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  10. 10.
    Freifeld, O., Black, M.J.: Lie bodies: a manifold representation of 3D human shape. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, vol. 7572, pp. 1–14. Springer, Heidelberg (2012). Scholar
  11. 11.
    Gallier, J.: Logarithms and square roots of real matrices existence, uniqueness and applications in medical imaging. arXiv preprint arXiv:0805.0245 (2018)
  12. 12.
    Gao, L., Lai, Y.K., Liang, D., Chen, S.Y., Xia, S.: Efficient and flexible deformation representation for data-driven surface modeling. ACM Trans. Graph 35(5), 158 (2016)CrossRefGoogle Scholar
  13. 13.
    Hasler, N., Stoll, C., Sunkel, M., Rosenhahn, B., Seidel, H.P.: A statistical model of human pose and body shape. Comput. Graph Forum 28(2), 337–346 (2009)CrossRefGoogle Scholar
  14. 14.
    Heeren, B., Zhang, C., Rumpf, M., Smith, W.: Principal geodesic analysis in the space of discrete shells. Comput. Graph Forum 37(5), 173–184 (2018)CrossRefGoogle Scholar
  15. 15.
    Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kilian, M., Mitra, N.J., Pottmann, H.: Geometric modeling in shape space. ACM Trans. Graph. (SIGGRAPH) 26(3), #64, 1–8 (2007)Google Scholar
  17. 17.
    Lawrence, R.C., et al.: Estimates of the prevalence of arthritis and other rheumatic conditions in the united states: part II. Arthritis Rheumatol. 58(1), 26–35 (2008)CrossRefGoogle Scholar
  18. 18.
    Martin, R.J., Neff, P.: Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics. J. Geom. Mech. 8(3), 323–357 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Miller, M.I., Trouvé, A., Younes, L.: Hamiltonian systems and optimal control in computational anatomy: 100 years since d’arcy thompson. Annu. Rev. Biomed. Eng. 17, 447–509 (2015)CrossRefGoogle Scholar
  20. 20.
    Neogi, T., et al.: Magnetic resonance imaging-based three-dimensional bone shape of the knee predicts onset of knee osteoarthritis. Arthritis Rheum. 65(8), 2048–2058 (2013)CrossRefGoogle Scholar
  21. 21.
    Pennec, X., Arsigny, V.: Exponential barycenters of the canonical Cartan connection and invariant means on Lie groups. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 123–166. Springer, Heidelberg (2013). Scholar
  22. 22.
    Thomson, J., O’Neill, T., Felson, D., Cootes, T.: Automated shape and texture analysis for detection of osteoarthritis from radiographs of the knee. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9350, pp. 127–134. Springer, Cham (2015). Scholar
  23. 23.
    Thomson, J., O’Neill, T., Felson, D., Cootes, T.: Detecting osteophytes in radiographs of the knee to diagnose osteoarthritis. In: Wang, L., Adeli, E., Wang, Q., Shi, Y., Suk, H.-I. (eds.) MLMI 2016. LNCS, vol. 10019, pp. 45–52. Springer, Cham (2016). Scholar
  24. 24.
    von Tycowicz, C., Ambellan, F., Mukhopadhyay, A., Zachow, S.: An efficient Riemannian statistical shape model using differential coordinates. Med. Image Anal. 43, 1–9 (2018)CrossRefGoogle Scholar
  25. 25.
    von Tycowicz, C., Schulz, C., Seidel, H.P., Hildebrandt, K.: Real-time nonlinear shape interpolation. ACM Trans. Graph 34(3), 34:1–34:10 (2015)Google Scholar
  26. 26.
    Woods, R.P.: Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18(3), 769–788 (2003)CrossRefGoogle Scholar
  27. 27.
    Zacur, E., Bossa, M., Olmos, S.: Multivariate tensor-based morphometry with a right-invariant riemannian distance on GL+(n). J. Math. Imaging Vis. 50(1–2), 18–31 (2014)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Therapy Planning Group, Zuse Institute BerlinBerlinGermany

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