Skip to main content

Advertisement

SpringerLink
Log in

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Finding a meaningful 1–1 correspondence between different data, such as images or surface data, has important applications in various fields. It involves the optimization of certain energy functionals over the space of all diffeomorphisms. This type of optimization problems (called the diffeomorphism optimization problems, DOPs) is especially challenging, since the bijectivity of the mapping has to be ensured. Recently, a method, called the Beltrami holomorphic flow (BHF), has been proposed to solve the DOP using quasi-conformal theories (Lui et al. in J Sci Comput 50(3):557–585, 2012). The optimization problem is formulated over the space of Beltrami coefficients (BCs), instead of the space of all diffeomorphisms. BHF iteratively finds a sequence of BCs associated with a sequence of diffeomorphisms, using the gradient descent method, to minimize the energy functional. The use of BCs effectively controls the smoothness and bijectivity of the mapping, and hence makes it easier to handle the constrained optimization problem. However, the algorithm is computationally expensive. In this paper, we propose an efficient splitting algorithm, based on the classical alternating direction method of multiplier (ADMM), to solve the DOP. The basic idea is to split the energy functional into two energy terms: one involves the BC whereas the other involves the quasi-conformal map. Alternating minimization scheme can then be applied to minimize the energy functional. The proposed method significantly speeds up the previous BHF approach. It also extends the previous BHF algorithm to Riemann surfaces of arbitrary topologies, such as multiply-connected shapes. Experiments have been carried out on synthetic together with real surface data, which demonstrate the efficiency and efficacy of the proposed algorithm to solve the DOP.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

Similar content being viewed by others

References

  1. Lui, L.M., Wong, T.W., Zeng, W., Gu, X.F., Thompson, P.M., Chan, T.F., Yau, S.T.: Optimization of surface registrations using Beltrami holomorphic flow. J. Sci. Comput. 50(3), 557–585 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On the Laplace–Beltrami operator and brain surface flattening. IEEE Trans. Med. Imaging 18(8), 700–711 (1999)

    Article  Google Scholar 

  3. Durrleman, S., Pennec, X., Trouve, A., Thompson, P., Ayache, N.: Measuring brain variability via sulcal lines registration: a diffeomorphic approach. In: Medical Image Computing and Computer-Assisted Intervention (MICCAI 2007) Lecture Notes in Computer Science 4791, pp. 675–682 Springer, Berlin, Heidelberg (2007)

  4. Durrleman, S., Pennec, X., Trouve, A., Thompson, P., Ayache, N.: Inferring brain variability from diffeomorphic deformations of currents: an integrative approach. Med. Image Anal. 12, 626–637 (2008)

    Article  Google Scholar 

  5. Fischl, B., Sereno, M.I., Tootell, R.B., Dale, A.M.: High-resolution intersubject averaging and a coordinate system for the cortical surface. Hum. Brain Mapp. 8, 272–284 (1999)

    Article  Google Scholar 

  6. Gardiner, F.: Quasiconformal Teichmüller Theory. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  7. Glaunès, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. J. Math. Imaging Vis. 20, 179–200 (2004)

    Article  Google Scholar 

  8. Gu, X., Wang, Y., Chan, T., Thompson, P., Yau, S.-T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23(8), 949–958 (2004)

    Article  Google Scholar 

  9. Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph. 6(8), 181189 (2000)

    Google Scholar 

  10. Hurdal, M., Stephenson, K.: Cortical cartography using the discrete conformal approach of circle packings. NeuroImage 23, S119S128 (2004)

    Article  Google Scholar 

  11. Hurdal, M.K., Stephenson, K.: Discrete conformal methods for cortical brain flattening. Neuroimage 45, 86–98 (2009)

    Article  Google Scholar 

  12. Joshi, S., Miller, M.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9(8), 13571370 (2000)

    Article  MathSciNet  Google Scholar 

  13. Ju, L., Stern, J., Rehm, K., Schaper, K., Hurdal, M., Rottenberg, D.: Cortical surface flattening using least square conformal mapping with minimal metric distortion. In: IEEE International Symposium on Biomedical, Imaging pp. 77–80 (2004)

  14. Lehto, O., Virtanen, K.: Quasiconformal Conformal Mappings in the Plane. Springer, New York (1973)

    Book  Google Scholar 

  15. Gardiner, F., Lakic, N.: Quasiconformal Teichmuller Theory. American Mathematical Society, Providence (2000)

    Google Scholar 

  16. Leow, A., Yu, C., Lee, S., Huang, S., Protas, H., Nicolson, R., Hayashi, K., Toga, A., Thompson, P.: Brain structural mapping using a novel hybrid implicit/explicit framework based on the level-set method. NeuroImage 24(3), 910–927 (2005)

    Article  Google Scholar 

  17. Lepore, N., Leow, P.T.A.D. : Landmark matching on the sphere using distance functions. In: IEEE International Symposium on Biomedical Imaging (ISBI2006), April 6–9 2006.

  18. Lord, N.A., Ho, J., Vemuri, B., Eisenschenk, S.: Simultaneous registration and parcellation of bilateral hippocampal surface pairs for local asymmetry quantification. IEEE Trans. Med. Imaging 26(4), 471–478 (2007)

    Article  Google Scholar 

  19. Lui, L., Wang, Y., Chan, T., Thompson, P.: Brain anatomical feature detection by solving partial differential equations on general manifolds. Discrete Contin. Dyn. Syst. B 7(3), 605–618 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lui, L., Wang, Y., Chan, T., Thompson, P.: Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl. Numer. Math. 5, 847–858 (2007)

    Article  MathSciNet  Google Scholar 

  21. Lui, L.M., Thiruvenkadam, S., Wang, Y., Chan, T., Thompson, P.: Optimized conformal parameterization of cortical surfaces using shape based matching of landmark curves. SIAM J. Imaging Sci. 3(1), 52–78 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lui, L.M., Wong, T.W., Gu, X.F., Thompson, P.M., Chan, T.F., Yau, S.T.: Hippocampal Shape Registration using Beltrami Holomorphic flow. In: Medical Image Computing and Computer Assisted Intervention(MICCAI). Part II, LNCS, vol. 6362, pp. 323–330 (2010)

  23. Zeng, W., Lui, L.M., Luo, F., Chan, T.F., Yau, S.T., Gu, X.F.: Computing quasiconformal maps using an auxiliary metric and discrete curvature flow. Numer. Math. 121(4), 671–703 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lui, L.M., Lam, K.C., Wong, T.W., Gu, X.F.: Texture map and video compression using Beltrami representation. SIAM J. Imaging Sci. 6(4), 1880–1902 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lvy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. (TOG) 21(3), 362–371 (2002)

    Google Scholar 

  26. Zeng, W., Gu, X.: Registration for 3D surfaces with large deformations using quasi-conformal curvature flow. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR11), Jun 20–25, Colorado Springs. Colorado, USA (2011)

  27. Lyttelton, O., Bouchera, M., Robbinsa, S., Evans, A.: An unbiased iterative group registration template for cortical surface analysis. NeuroImage 34, 1535–1544 (2007)

    Article  Google Scholar 

  28. Glowinski, R., Le Tallee, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. Studies in Applied and Numerical Mathematics. Society for Industrial and Applied Mathematics (1989)

  29. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2010)

    Article  MATH  Google Scholar 

  30. Glowinski, R., Marrocco, A.: Sur lapproximation par elements Lnis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires. Rev. Francaise dAut. Inf. Rech. Oper. R–2, 41–76 (1975)

    MathSciNet  Google Scholar 

  31. Shi, Y., Thompson, P.M., Dinov, I., Osher, S., Toga, A.W.: Direct cortical mapping via solving partial differential equations on implicit surfaces. Med. Image Anal. 11(3), 207–223 (2007)

    Article  Google Scholar 

  32. Tosun, D., Rettmann, M., Prince, J.: Mapping techniques for aligning sulci across multiple brains. Med. Image Anal. 8, 295309 (2004)

    Article  Google Scholar 

  33. Wang, Y., Chiang, M.-C., Thompson, P.M.: Automated surface matching using mutual information applied to Riemann surface structures. Proc. Med. Image Comput. Comput. Assist. Interv. MICCAI 2005(2), 666–674 (2005)

    Google Scholar 

  34. Jacobson, A., Sorkine, O.: A cotangent Laplacian for images as surfaces. ACM Trans. Graph. 25(3), 646–653 (2012)

    Google Scholar 

  35. Weber, O., Myles, A., Zorin, D.: Computing extremal quasiconformal maps. Comput. Graph. Forum 31(5), 16791689 (2012)

    Article  Google Scholar 

  36. Wong, T.W., Zhao, H.: Computation of quasiconformal surface maps using discrete Beltrami flow. UCLA CAM report 12-85 (2013)

  37. Lui, L.M., Lam, K.C., Yau, S.T., Gu, X.F. : Teichmüller extremal mapping and its applications to landmark matching registration. SIAM J. Imaging Sci. (2013). arXiv:1211.2569 ( arXiv:1210.8025)

  38. Ng, T.C., Gu, X.F., Lui, L.M. : Computing Teichmüller extremal map of multiply-connected domains using Beltrami holomorphic flow. J. Sci. Comput. (2013). doi:10.1007/s10915-013-9791-z

  39. Wang, Y., Lui, L., Gu, X., Hayashi, K., Chan, T., Toga, A., Thompson, P., Yau, S.: Brain surface conformal parameterization using riemann surface structure. IEEE Trans. Med. Imaging 26(6), 853–865 (2007)

    Article  Google Scholar 

  40. Wang, Y., Lui, L.M., Chan, T.F., Thompson, P.M.: Optimization of brain conformal mapping with landmarks. Proc. Med. Image Comput. Comput. Assist. Interv. MICCAI 2005, 675–683 (2005)

    Google Scholar 

  41. Zeng, W., Luo, F., Yau, S.-T., Gu, X. : Surface quasi-conformal mapping by solving Beltrami equations. In: IMA Conference on the Mathematics of Surfaces XIII (2009)

  42. Gu, X.F., Yau, S.T.: Computational Conformal Geometry. International Press, Boston (2007)

    Google Scholar 

Download references

Acknowledgments

Lok Ming Lui’s research was supported in part by HKRGC GRF grant (CUHK project ID: 404612) and CUHK FIS grant 1902036.

Author information

Authors and Affiliations

  1. The Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

    Lok Ming Lui & Tsz Ching Ng

Authors
  1. Lok Ming Lui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsz Ching Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lok Ming Lui.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lui, L.M., Ng, T.C. A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients. J Sci Comput 63, 573–611 (2015). https://doi.org/10.1007/s10915-014-9903-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9903-4

Keywords

Search

Navigation