Longitudinal Image Analysis via Path Regression on the Image Manifold

  • Shi-Hui Ying
  • Xiao-Fang Zhang
  • Ya-Xin PengEmail author
  • Ding-Gang Shen


Longitudinal image analysis plays an important role in depicting the development of the brain structure, where image regression and interpolation are two commonly used techniques. In this paper, we develop an efficient model and approach based on a path regression on the image manifold instead of the geodesic regression to avoid the complexity of the geodesic computation. Concretely, first we model the deformation by diffeomorphism; then, a large deformation is represented by a path on the orbit of the diffeomorphism group action. This path is obtained by compositing several small deformations, which can be well approximated by its linearization. Second, we introduce some intermediate images as constraints to the model, which guides to form the best-fitting path. Thirdly, we propose an approximated quadratic model by local linearization method, where a closed form is deduced for the solution. It actually speeds up the algorithm. Finally, we evaluate the proposed model and algorithm on a synthetic data and a real longitudinal MRI data. The results show that our proposed method outperforms several state-of-the-art methods.


Longitudinal image analysis Path regression Diffeomorphism group Image registration Infant brain development 

Mathematics Subject Classification

68T05 97K80 


  1. 1.
    Banerjee, M., Chakraborty, R., Ofori, E., Vaillancourt, D., Vemuri, B.: Nonlinear regression on Riemannian manifolds and its applications to neuro-image analysis. In: Medical Image Computing and Computer-Assisted Intervention, pp. 719–727 (2015)Google Scholar
  2. 2.
    Banerjee, M., Chakraborty, R., Ofori, E., Okun, M., Vaillancourt, D., Vemuri, B.: A nonlinear regression technique for manifold valued data with applications to medical image analysis. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4424–4432 (2016)Google Scholar
  3. 3.
    Erus, G., Doshi, J., An, Y., Verganelakis, D., Resnick, S.: Longitudinally and inter-site consistent multi-atlas based parcellation of brain anatomy using harmonized atlases. NeuroImage 166, 71–78 (2018)CrossRefGoogle Scholar
  4. 4.
    Serag, A., Aljabar, P., Counsell, S., Boardman, J., Hajnal, J., Rueckert, D.: LISA: longitudinal image registration via spatio-temporal atlases. IEEE Trans. Med. Imaging 25(5), 334–337 (2012)Google Scholar
  5. 5.
    Ying, S., Wu, G., Wang, Q., Shen, D.: Groupwise registration via graph shrinkage on the image manifold. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2323–2330 (2013)Google Scholar
  6. 6.
    Ying, S., Wu, G., Wang, Q., Shen, D.: Hierarchical unbiased graph shrinkage (HUGS): a novel groupwise registration for large data set. NeuroImage 84(1), 626–638 (2014)CrossRefGoogle Scholar
  7. 7.
    Zhang, Y., Wei, H., Cronin, M., He, N., Yan, F., Liu, C.: Longitudinal atlas for normative human brain development and aging over the lifespan using quantitative susceptibility mapping. NeuroImage 171, 176–189 (2018)CrossRefGoogle Scholar
  8. 8.
    Bai, Y., Shen, K.: Alternating direction method of multipliers for (l1–l2)-regularized logistic regression model. J. Oper. Res. Soc. China 4(2), 243–253 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhang, Q., Liu, Y., Zhou, W., Wang, Z.: A sequential regression model for big data with attributive explanatory variables. J. Oper. Res. Soc. China 3(4), 475–488 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fletcher, P.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. 105, 171–185 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sun, Z., Lelieveldt, B., Staring, M.: Fast linear geodesic shape regression using coupled logdemons registration. In: International Symposium on Biomedical Imaging, pp. 1276–1279 (2015)Google Scholar
  12. 12.
    Batzies, E., Hüper, K., Machado, L., Leite, F.: Geometric mean and geodesic regression on Grassmannians. Linear Algorithms Appl. 466, 83–101 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fishbaugh, J., Durrleman, S., Prastawa, M., Gerig, G.: Geodesic shape regression with multiple geometries and sparse parameters. Med. Image Anal. 39, 1–17 (2017)CrossRefGoogle Scholar
  14. 14.
    Cheng, M., Wu, H.: Local linear regression on manifolds and its geometric interpretation. JASA 108, 1421–1434 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Aswani, A., Bickel, P., Tomlin, C.: Regression on manifolds: estimation of the exterior derivative. Ann. Stat. 39(1), 48–81 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nilsson, J., Sha, F., Jordan, M.: Regression on manifolds using kernel dimension reduction. In: International Conference on Machine Learning, pp. 697–704 (2007)Google Scholar
  17. 17.
    Bickel, P.J., Li, B.: Local Polynomial Regression on Unknown Manifolds. Lecture Notes-Monograph Series, vol. 54, pp. 177–186 (2007)Google Scholar
  18. 18.
    Shi, X., Styner, M., Lieberman, J., Ibrahim, J., Lin, W., Zhu, H.: Intrinsic regression models for manifold-valued data. In: Medical Image Computing and Computer-Assisted Intervention, pp. 192–199 (2009)Google Scholar
  19. 19.
    Hinkle, J., Fletcher, P., Joshi, S.: Intrinsic polynomials for regression on Riemannian manifolds. J. Math. Image Vis. 50, 32–52 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yang, Y., Dunson, D.: Bayesian manifold regression. Ann. Stat. 44, 876–905 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, P., Sun, R., Huang, T.: A geometric method for computation of geodesic on parametric surfaces. Comput. Aided Geom. Des. 38, 24–37 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Diffeomorphic demons: efficient non-parametric image registration. NeuroImage 45, S61–S72 (2009)CrossRefGoogle Scholar
  23. 23.
    Lorenzi, M., Ayache, N., Frison, G., Pennec, X.: Lcc-demons: a robust and accurate symmetric diffeomorphic registration algorithm. NeuroImage 81, 470–483 (2013)CrossRefGoogle Scholar
  24. 24.
    Beg, M., Miller, M., Trouve, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61, 139–157 (2005)CrossRefGoogle Scholar
  25. 25.
    Kimmel, R., Sethian, J.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. USA 95, 8431–8435 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    ing, S., Li, D., Xiao, B., Peng, Y., Du, S., Xu, M.: Nonlinear image registration with bidirectional metric and reciprocal regularization. PLoS ONE. (2017).
  27. 27.
    Cachier, P., Bardinet, E., Dormont, D., Pennec, X., Ayache, N.: Iconic feature based nonrigid registration: the PASHA algorithm. Comput. Vis. Image Underst. 89, 272–298 (2003)CrossRefGoogle Scholar
  28. 28.
    He, B., Yuan, X.: Alternating direction method of multipliers for linear programming. J. Oper. Res. Soc. China 4(4), 425–436 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    He, B., Xu, M., Yuan, X.: Block-wise ADMM with a relaxation factor for multiple-block convex programming. J. Oper. Res. Soc. China 6(4), 485–505 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.Department of Radiology and BRICUniversity of North Carolina at Chapel HillChapel HillUSA

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