Abstract
We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. We evaluate the performance of our model w.r.t. shape-based classification of pathological malformations of the human knee and show that it outperforms the standard Euclidean as well as a recent nonlinear approach especially in presence of sparse training data. To provide insight into the model’s ability of capturing natural biological shape variability, we carry out an analysis of specificity and generalization ability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alexandrino, M.M., Bettiol, R.G.: Lie Groups and Geometric Aspects of Isometric Actions, vol. 8. Springer, Heidelberg (2015)
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)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Log-Euclidean metrics for fast and simple calculus on diffusion tensors. Magn. Reson. Med. 56(2), 411–421 (2006)
Bogo, F., Romero, J., Loper, M., Black, M.J.: FAUST: dataset and evaluation for 3D mesh registration. In: CVPR. IEEE (2014)
Botsch, M., Sumner, R., Pauly, M., Gross, M.: Deformation transfer for detail-preserving surface editing. In: VMV, pp. 357–364 (2006)
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), (2016)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)
Charpiat, G., Faugeras, O., Keriven, R., Maurel, P.: Distance-based shape statistics. In: ICASSP, pp. V925–V928. IEEE (2006)
Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78(1), 1–215 (2005)
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)
Davies, R., Twining, C., Taylor, C.: Statistical Models of Shape: Optimisation and Evaluation. Springer, London (2008). https://doi.org/10.1007/978-1-84800-138-1
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)
Durrleman, S., Prastawa, M., Charon, N., Korenberg, J.R., Joshi, S., Gerig, G., Trouvé, A.: Morphometry of anatomical shape complexes with dense deformations and sparse parameters. NeuroImage 101, 35–49 (2014)
Fletcher, P.T., Lu, C., Joshi, S.: Statistics of shape via principal geodesic analysis on lie groups. In: CVPR, vol. 1, p. I-95. IEEE (2003)
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)
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
Fuchs, M., Jüttler, B., Scherzer, O., Yang, H.: Shape metrics based on elastic deformations. J. Math. Imaging Vis. 35(1), 86–102 (2009)
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)
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)
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)
Hefny, M.S., Okada, T., Hori, M., Sato, Y., Ellis, R.E.: A liver Atlas using the special euclidean group. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9350, pp. 238–245. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24571-3_29
Heimann, T., Meinzer, H.P.: Statistical shape models for 3D medical image segmentation: a review. Med. Image Anal. 13(4), 543–563 (2009)
Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic Atlas construction for computational anatomy. NeuroImage 23, S151–S160 (2004)
Kircher, S., Garland, M.: Free-form motion processing. ACM Trans. Graph. 27(2), 12 (2008)
Rumpf, M., Wirth, B.: An elasticity-based covariance analysis of shapes. Int. J. Comput. Vis. 92(3), 281–295 (2011)
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)
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)
Zhang, C., Heeren, B., Rumpf, M., Smith, W.A.: Shell PCA: statistical shapemodelling in shell space. In: ICCV, pp. 1671–1679. IEEE (2015)
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 datasets OAI (The Osteoarthritis Initiative 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.) and FAUST [4] as well as for the open-source software Deformetrica [13].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
1 Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary material 2 (mp4 29602 KB)
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Ambellan, F., Zachow, S., von Tycowicz, C. (2019). A Surface-Theoretic Approach for Statistical Shape Modeling. In: Shen, D., et al. Medical Image Computing and Computer Assisted Intervention – MICCAI 2019. MICCAI 2019. Lecture Notes in Computer Science(), vol 11767. Springer, Cham. https://doi.org/10.1007/978-3-030-32251-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-32251-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-32250-2
Online ISBN: 978-3-030-32251-9
eBook Packages: Computer ScienceComputer Science (R0)
