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

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Book cover Multimodal Brain Image Analysis and Mathematical Foundations of Computational Anatomy (MBIA 2019, MFCA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11846))

Abstract

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.

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Notes

  1. 1.

    https://scikit-optimize.github.io.

  2. 2.

    https://doi.org/10.12752/4.ATEZ.1.0.

  3. 3.

    http://graphics.stanford.edu/~niloy/research/shape_space/shape_space_sig_07.html.

References

  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). https://doi.org/10.1007/978-3-030-19385-0_5

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-33718-5_1

    Chapter  Google Scholar 

  11. Gallier, J.: Logarithms and square roots of real matrices existence, uniqueness and applications in medical imaging. arXiv preprint arXiv:0805.0245 (2018)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  15. Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-30232-9_7

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-24571-3_16

    Chapter  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-47157-0_6

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  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. Woods, R.P.: Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18(3), 769–788 (2003)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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.

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Authors and Affiliations

  1. Therapy Planning Group, Zuse Institute Berlin, Berlin, Germany

    Felix Ambellan, Stefan Zachow & Christoph von Tycowicz

Authors
  1. Felix Ambellan
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  2. Stefan Zachow
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  3. Christoph von Tycowicz
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Corresponding author

Correspondence to Felix Ambellan .

Editor information

Editors and Affiliations

  1. The University of Texas at Arlington, Arlington, TX, USA

    Dajiang Zhu

  2. Indiana University – Purdue University Indianapolis, Indianapolis, IN, USA

    Jingwen Yan

  3. University of Pittsburgh, Pittsburgh, PA, USA

    Heng Huang

  4. University of Pennsylvania, Philadelphia, PA, USA

    Li Shen

  5. University of Southern California, Marina Del Rey, CA, USA

    Paul M. Thompson

  6. Harvard Medical School, Boston, MA, USA

    Carl-Fredrik Westin

  7. Inria Sophia-Antipolis, Sophia-Antipolis, France

    Xavier Pennec

  8. University of Utah, Salt Lake City, UT, USA

    Sarang Joshi

  9. University of Copenhagen, Copenhagen, Denmark

    Mads Nielsen

  10. University of Virginia, Charlottesville, VA, USA

    Tom Fletcher

  11. Inria, Paris, France

    Stanley Durrleman

  12. University of Copenhagen, Copenhagen, Denmark

    Stefan Sommer

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Ambellan, F., Zachow, S., von Tycowicz, C. (2019). An As-Invariant-As-Possible \(\text {GL}^+(3){}\)-Based Statistical Shape Model. In: Zhu, D., et al. Multimodal Brain Image Analysis and Mathematical Foundations of Computational Anatomy. MBIA MFCA 2019 2019. Lecture Notes in Computer Science(), vol 11846. Springer, Cham. https://doi.org/10.1007/978-3-030-33226-6_23

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  • DOI: https://doi.org/10.1007/978-3-030-33226-6_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-33225-9

  • Online ISBN: 978-3-030-33226-6

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