Skip to main content
SpringerLink
Log in

Fitting Smooth Paths on Riemannian Manifolds: Endometrial Surface Reconstruction and Preoperative MRI-Based Navigation

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. 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)

    Article  Google Scholar 

  2. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  MATH  Google Scholar 

  3. 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

  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)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  Google Scholar 

  10. 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)

    Google Scholar 

  11. Park, J.: Interpolation and tracking of rigid body orientations. In: ICCAS, pp. 668–673 (2010)

    Google Scholar 

  12. 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)

    Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. Sander, O.: Geodesic finite elements of higher order. Technical report 356 (2013)

    Google Scholar 

  16. Umaria, N., Olliff, J.: Imaging features of pelvic endometriosis. Br. J. Radiol. 74, 556–562 (2001)

    Article  Google Scholar 

  17. 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)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICTEAM Institute, Université Catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Antoine Arnould, Pierre-Yves Gousenbourger & Pierre-Antoine Absil

  2. ISIT UMR 6284 CNRS, Clermont University, Clermont-Ferrand, France

    Chafik Samir & Michel Canis

Authors
  1. Antoine Arnould
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre-Yves Gousenbourger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chafik Samir
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pierre-Antoine Absil
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Michel Canis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre-Yves Gousenbourger .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

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

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-25040-3_53

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25039-7

  • Online ISBN: 978-3-319-25040-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation