Skip to main content
SpringerLink
Log in

Inconsistent Surface Registration via Optimization of Mapping Distortions

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

Abstract

We address the problem of registering two surfaces, of which a natural bijection between them does not exist. More precisely, only a partial subset of the source surface is assumed to be in correspondence with a subset of the target surface. We call such a problem an inconsistent surface registration (ISR) problem. This problem is challenging as the corresponding regions on each surface and a meaningful bijection between them have to be simultaneously determined. In this paper, we propose a variational model to solve the ISR problem by minimizing mapping distortions. Mapping distortions are described by the Beltrami coefficient as well as the differential of the mapping. Registration is then guided by feature landmarks and/or intensities, such as curvatures, defined on each surface. The key idea of the approach is to control angle and scale distortions via quasiconformal theory as well as minimizing landmark and/or intensity mismatch. A splitting method is proposed to iteratively search for the optimal corresponding regions as well as the optimal bijection between them. Bijectivity of the mapping is easily enforced by a thresholding of the Beltrami coefficient. We test the proposed method on both synthetic and real examples. Experimental results demonstrate the efficacy of our proposed model.

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

Similar content being viewed by others

Notes

  1. https://github.com/sylqiu/incon_reg.

References

  1. Fidentis database. https://www.facebase.org/facial_norms/notes/6262. Accessed 1 Oct 2018

  2. Amberg, B., Romdhani, S., Vetter, T.: Optimal step nonrigid ICP algorithms for surface registration. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE (2007)

  3. Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48). Princeton University Press, Princeton (2008)

    Google Scholar 

  4. Audenaert, E.A., Van Houcke, J., Almeida, D.F., Paelinck, L., Peiffer, M., Steenackers, G., Vandermeulen, D.: Cascaded statistical shape model based segmentation of the full lower limb in CT. Comput. Methods Biomech. Biomed. Eng. 22(6), 644–657 (2019)

    Article  Google Scholar 

  5. Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. Natl. Acad. Sci. 103(5), 1168–1172 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choi, G., Leung-Liu, Y., Gu, X., Lui, L.M.: Parallelizable global conformal parameterization of simply-connected surfaces via partial welding. SIAM J. Imaging Sci. (2020) (in press)

  7. Choi, G.P.T., Ho, K.T., Lui, L.M.: Spherical conformal parameterization of genus-0 point clouds for meshing. SIAM J. Imaging Sci. 9(4), 1582–1618 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, G.P.T., Lui, L.M.: A linear formulation for disk conformal parameterization of simply-connected open surfaces. Adv. Comput. Math. 44(1), 87–114 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choi, P.T., Lam, K.C., Lui, L.M.: Flash: fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces. SIAM J. Imaging Sci. 8(1), 67–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. In: Computer Graphics Forum, vol. 21, Issue 3, pp. 209–218. Wiley Online Library (2002)

  11. Gu, X., Yau, S.T.: Global conformal surface parameterization. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 127–137. Eurographics Association (2003)

  12. Ho, K.T., Lui, L.M.: QCMC: quasi-conformal parameterizations for multiply-connected domains. Adv. Comput. Math. 42(2), 279–312 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Igarashi, T., Moscovich, T., Hughes, J.F.: As-rigid-as-possible shape manipulation. ACM Trans. Graph. (TOG) 24, 1134–1141 (2005)

    Article  Google Scholar 

  14. Kovalsky, S.Z., Aigerman, N., Basri, R., Lipman, Y.: Large-scale bounded distortion mappings. ACM Trans. Graph. (TOG) 34(6), 191 (2015)

    Article  Google Scholar 

  15. Lai, R., Wen, Z., Yin, W., Gu, X., Lui, L.M.: Folding-free global conformal mapping for genus-0 surfaces by harmonic energy minimization. J. Sci. Comput. 58(3), 705–725 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lam, K.C., Gu, X., Lui, L.M.: Landmark constrained genus-one surface teichmüller map applied to surface registration in medical imaging. Med. Image Anal. 25(1), 45–55 (2015)

    Article  Google Scholar 

  17. Lam, K.C., Lui, L.M.: Landmark- and intensity-based registration with large deformations via quasi-conformal maps. SIAM J. Imaging Sci. 7(4), 2364–2392 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lam, K.C., Lui, L.M.: Quasi-conformal hybrid multi-modality image registration and its application to medical image fusion. In: International Symposium on Visual Computing, pp. 809–818. Springer (2015)

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

    Article  Google Scholar 

  20. Li, H., Sumner, R.W., Pauly, M.: Global correspondence optimization for non-rigid registration of depth scans. In: Computer Graphics Forum, vol. 27, pp. 1421–1430. Wiley Online Library (2008)

  21. Lipman, Y.: Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. (TOG) 31(4), 108 (2012)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Lui, L.M., Lam, K.C., Yau, S.T., Gu, X.: Teichmuller mapping (t-map) and its applications to landmark matching registration. SIAM J. Imaging Sci. 7(1), 391–426 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lui, L.M., Ng, T.C.: A splitting method for diffeomorphism optimization problem using Beltrami coefficients. J. Sci. Comput. 63(2), 573–611 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lui, L.M., Wen, C.: Geometric registration of high-genus surfaces. SIAM J. Imaging Sci. 7(1), 337–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ng, T., Gu, X., Lui, L.: Teichmüller extremal map of multiply-connected domains using beltrami holomorphic flow. J. Sci. Comput. 60(2), 249–275 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ovsjanikov, M., Corman, E., Bronstein, M., Rodolà, E., Ben-Chen, M., Guibas, L., Chazal, F., Bronstein, A.: Computing and processing correspondences with functional maps. In: SIGGRAPH ASIA 2016 Courses, p. 9. ACM (2016)

  28. Pennec, X., Cachier, P., Ayache, N.: Understanding the demons algorithm: 3D non-rigid registration by gradient descent. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 597–605. Springer (1999)

  29. Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qiu, D., Lam, K.C., Lui, L.M.: Computing quasi conformal folds. arXiv preprint arXiv:1804.03936 (2018)

  31. Rabinovich, M., Poranne, R., Panozzo, D., Sorkine-Hornung, O.: Scalable locally injective mappings. ACM Trans. Graph. (TOG) 36(2), 16 (2017)

    Article  Google Scholar 

  32. Rodolà, E., Cosmo, L., Bronstein, M.M., Torsello, A., Cremers, D.: Partial functional correspondence. In: Computer Graphics Forum, vol. 36, pp. 222–236. Wiley Online Library (2017)

  33. Rusinkiewicz, S., Levoy, M.: Efficient variants of the ICP algorithm. In: 3DIM, vol. 1, pp. 145–152 (2001)

  34. Sorkine, O.: Laplacian mesh processing. In: EUROGRAPHICS05 STAR—state of the art report. Citeseer (2005)

  35. Sorkine, O., Alexa, M.: As-rigid-as-possible surface modeling. In: Symposium on Geometry processing, vol. 4 (2007)

  36. Thirion, J.P.: Image matching as a diffusion process: an analogy with Maxwell’s demons. Med. Image Anal. 2(3), 243–260 (1998)

    Article  Google Scholar 

  37. Wang, H., Dong, L., O’Daniel, J., Mohan, R., Garden, A.S., Ang, K.K., Kuban, D.A., Bonnen, M., Chang, J.Y., Cheung, R.: Validation of an accelerated demons algorithm for deformable image registration in radiation therapy. Phys. Med. Biol. 50(12), 2887 (2005)

    Article  Google Scholar 

  38. Yung, C.P., Choi, G.P., Chen, K., Lui, L.M.: Efficient feature-based image registration by mapping sparsified surfaces. J. Vis. Commun. Image Represent. 55, 561–571 (2018)

    Article  Google Scholar 

  39. Zeng, W., Ming Lui, L., Gu, X.: Surface registration by optimization in constrained diffeomorphism space. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4169–4176 (2014)

Download references

Acknowledgements

L.M. Lui is supported by HKRGC GRF (Ref. 14304715).

Author information

Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Di Qiu & Lok Ming Lui

Authors
  1. Di Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lok Ming Lui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lok Ming Lui.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, D., Lui, L.M. Inconsistent Surface Registration via Optimization of Mapping Distortions. J Sci Comput 83, 64 (2020). https://doi.org/10.1007/s10915-020-01246-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01246-5

Keywords

Search

Navigation