Abstract
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using \(C^1\) piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.
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Abrao, M.S., da C. Goncalves, M.O., Dias, J.A., Podgaec, J.S., Chamie, L.P., Blasbalg, R.: Comparison between clinical examination, transvaginal sonography and magnetic resonance imaging for the diagnosis of deep endometriosis. Hum. Reprod. 22, 3092–3097 (2007)
Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)
Bagaria, S.J., Rasalkar, D.D., Paunipagar, B.K.: Imaging tools for ndometriosis: Role of Ultrasound, MRI and Other Imaging Modalities in Diagnosis and Planning Intervention. In: Chaudhury, K. (ed.) Endometriosis - Basic Concepts and Current Research Trends, InTech (2012). doi:10.5772/29063, ISBN: 978-953-51-0524-4
Bajaj, C.L., Xu, G.L., Zhang, Q.: Bio-molecule surfaces construction via a higher-order level-set method. J. Comput. Sci. Technol. 23(6), 1026–1036 (2008)
Gousenbourger, P.Y., Samir, C., Absil, P.A.: Piecewise-Bézier \({C}^1\) interpolation on Riemannian manifolds with application to 2D shape morphing. In: IEEE ICPR (2014)
Hüper, K., Silva Leite, F.: On the geometry of rolling and interpolation curves on \(S^n, {\rm SO}_n\), and Grassmann manifolds. J. Dyn. Control Syst. 13(4), 467–502 (2007)
Joshi, S., Jermyn, I., Klassen, E., Srivastava, A.: An efficient representation for computing geodesics between n-dimensional elastic shapes. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2007)
Machado, L., Leite, F.S., Krakowski, K.: Higher-order smoothing splines versus least squares problems on Riemannian manifolds. J. Dyn. Control Syst. 16(1), 121–148 (2010)
Massein, A., Petit, E., Darchen, M., Loriau, J., Oberlin, O., Marty, O., Sauvanet, E., Afriat, R., Girard, F., Molini, V., Duchatelle, V., Zins, M.: Imaging of intestinal involvement in endometriosis. Diagn. Interv. Imaging 94(3), 281–291 (2013)
Nira, D.: Linear and nonlinear subdivision schemes in geometric modeling. In: Foundations of computational mathematics, Hong Kong 2008. London Math. Soc. Lecture Note Ser., vol. 363, pp. 68–92. Cambridge Univ. Press, Cambridge (2009)
Park, J.: Interpolation and tracking of rigid body orientations. In: ICCAS, pp. 668–673 (2010)
Rentmeesters, Q.: A gradient method for geodesic data fitting on some symmetric Riemannian manifolds. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 7141–7146 (2011)
Samir, C., Absil, P.A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comput. Math. 12, 49–73 (2012)
Samir, C., Kurtek, S., Srivastava, A., Canis, M.: Elastic shape analysis of cylindrical surfaces for 3D/2D registration in endometrial tissue characterization. IEEE Trans. MI 33, 1035–1043 (2014)
Sander, O.: Geodesic finite elements of higher order. Technical report 356 (2013)
Umaria, N., Olliff, J.: Imaging features of pelvic endometriosis. Br. J. Radiol. 74, 556–562 (2001)
Zhao, H.K., Osher, S., Fedkiw, R.: Fast surface reconstruction using the level set method. In: Proceedings of the IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 194–201 (2001)
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Arnould, A., Gousenbourger, PY., Samir, C., Absil, PA., Canis, M. (2015). Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_53
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DOI: https://doi.org/10.1007/978-3-319-25040-3_53
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