Journal of Mathematical Imaging and Vision

, Volume 50, Issue 3, pp 246–260 | Cite as

Variational Image Registration Using Inhomogeneous Regularization

  • Christoph JudEmail author
  • Marcel Lüthi
  • Thomas Albrecht
  • Sandro Schönborn
  • Thomas Vetter


We present a generalization of the convolution-based variational image registration approach, in which different regularizers can be implemented by conveniently exchanging the convolution kernel, even if it is nonseparable or nonstationary. Nonseparable kernels pose a challenge because they cannot be efficiently implemented by separate 1D convolutions. We propose to use a low-rank tensor decomposition to efficiently approximate nonseparable convolution. Nonstationary kernels pose an even greater challenge because the convolution kernel depends on, and needs to be evaluated for, every point in the image. We propose to pre-compute the local kernels and efficiently store them in memory using the Tucker tensor decomposition model. In our experiments we use the nonseparable exponential kernel and a nonstationary landmark kernel. The exponential kernel replicates desirable properties of elastic image registration, while the landmark kernel incorporates local prior knowledge about corresponding points in the images. We examine the trade-off between the computational resources needed and the approximation accuracy of the tensor decomposition methods. Furthermore, we obtain very smooth displacement fields even in the presence of large landmark displacements.


Variational image registration Separable filter approximation  Nonstationary filtering Gaussian process regression 


  1. 1.
    Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A log-euclidean framework for statistics on diffeomorphisms. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2006. Lecture Notes in Computer Science, pp. 924–931 (2006)Google Scholar
  2. 2.
    Avants, B., Schoenemann, P., Gee, J.: Lagrangian frame diffeomorphic image registration: morphometric comparison of human and chimpanzee cortex. Med. Image Anal. 10(3), 397–412 (2006)CrossRefGoogle Scholar
  3. 3.
    Beg, M., Miller, M., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005)CrossRefGoogle Scholar
  4. 4.
    Beuthien, B., Kamen, A., Fischer, B.: Recursive green’s function registration. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2010. Lecture Notes in Computer Science, pp. 546–553 (2010)Google Scholar
  5. 5.
    Bro-Nielsen, M., Gramkow, C.: Fast fluid registration of medical images. In: Proceedings of the 4th International Conference on Visualization in Biomedical Computing, pp. 267–276. Springer, London (1996)Google Scholar
  6. 6.
    Cahill, N.D., Noble, J.A., Hawkes, D.J.: A demons algorithm for image registration with locally adaptive regularization. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2009. Lecture Notes in Computer Science, pp. 574–581. Springer (2009)Google Scholar
  7. 7.
    Chen, M., Lu, W., Chen, Q., Ruchala, K.J., Olivera, G.H.: A simple fixed-point approach to invert a deformation field. Med. Phys. 35, 81 (2008)CrossRefGoogle Scholar
  8. 8.
    Christensen, G.E., Johnson, H.J.: Consistent image registration. IEEE Trans. Med. Imaging 20(7), 568–582 (2001)CrossRefGoogle Scholar
  9. 9.
    Christensen, G.E., Yin, P., Vannier, M.W., Chao, K., Dempsey, J., Williamson, J.F.: Large-deformation image registration using fluid landmarks. In: Proceedings of the 4th IEEE Southwest Symposium on Image Analysis and Interpretation, pp. 269–273 (2000)Google Scholar
  10. 10.
    De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Evans, L.: Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)Google Scholar
  12. 12.
    Girdziušas, R., Laaksonen, J.: Gaussian process regression with fluid hyperpriors. Neural Information Processing, pp. 567–572. Springer, Berlin (2004)CrossRefGoogle Scholar
  13. 13.
    Guo, H., Rangarajan, A., Joshi, S.: Diffeomorphic point matching. Handbook of Mathematical Models in Computer Vision, pp. 205–219. Springer, New York (2006)CrossRefGoogle Scholar
  14. 14.
    Haber, E., Heldmann, S., Modersitzki, J.: A scale-space approach to landmark constrained image registration. In: Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, pp. 612–623. Springer, Heidelberg (2009)Google Scholar
  15. 15.
    Harshman, R.: Foundations of the parafac procedure: models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics, Los Angeles (1970)Google Scholar
  16. 16.
    Johnson, H., Christensen, G.: Consistent landmark and intensity-based image registration. IEEE Trans. Med. Imaging 21(5), 450–461 (2002)CrossRefGoogle Scholar
  17. 17.
    Klein, S., Staring, M., Murphy, K., Viergever, M.A., Pluim, J.P., et al.: Elastix: a toolbox for intensity-based medical image registration. IEEE Trans. Med. Imaging 29(1), 196–205 (2010)CrossRefGoogle Scholar
  18. 18.
    Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Long, Z., Yao, L., Peng, D.: Fast non-linear elastic registration in 2d medical image. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2004. Lecture Notes in Computer Science, pp. 647–654 (2004)Google Scholar
  20. 20.
    Lu, H., Cattin, P., Reyes, M.: A hybrid multimodal non-rigid registration of MR images based on diffeomorphic demons. In: Annual International Conference of the IEEE Engineering in Medicine and Biology Society-EMBC 2010, pp. 5951–5954 (2010)Google Scholar
  21. 21.
    Lüthi, M., Jud, C., Vetter, T.: Using landmarks as a deformation prior for hybrid image registration. In: Proceedings of the 33rd International Conference on Pattern Recognition, pp. 196–205. Springer, Berlin (2011)Google Scholar
  22. 22.
    McOwen, R.: Partial differential equations: methods and applications. Tsinghua University Press, Haidian (1996)zbMATHGoogle Scholar
  23. 23.
    Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  24. 24.
    Opfer, R.: Multiscale kernels. Adv. Comput. Math. 25(4), 357–380 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Papademetris, X., Jackowski, A., Schultz, R., Staib, L., Duncan, J.: Integrated intensity and point-feature nonrigid registration. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI2004. Lecture Notes in Computer Science, pp. 763–770 (2004)Google Scholar
  26. 26.
    Papenberg, N., Olesch, J., Lange, T., Schlag, P., Fischer, B.: Landmark constrained non-parametric image registration with isotropic tolerances. Bildverarbeitung für die Medizin, pp. 122–126 (2009)Google Scholar
  27. 27.
    Rasmussen, C.: Gaussian processes in machine learning. In: Advanced Lectures on Machine Learning. Lecture Notes in Computer Science: Lecture Notes in Artificial Intelligence, vol. 3176, pp. 63–71. Springer, Germany (2004)Google Scholar
  28. 28.
    Schmidt-Richberg, A., Ehrhardt, J., Werner, R., Handels, H.: Diffeomorphic diffusion registration of lung ct images. In: Medical Image Analysis for the Clinic: A Grand Challenge-MICCAI 2010, pp. 165-174 (2010)Google Scholar
  29. 29.
    Schölkopf, B., Steinke, F., Blanz, V.: Object correspondence as a machine learning problem. In: Proceedings of the 22nd International Conference on Machine Learning, pp. 776–783. ACM press, New York (2005)Google Scholar
  30. 30.
    Sotiras, A., Paragios, N., et al.: Deformable image registration: a survey (2012)Google Scholar
  31. 31.
    Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. Med. Image Anal. 8(3), 325–342 (2004)CrossRefGoogle Scholar
  32. 32.
    Steinke, F., Schölkopf, B.: Kernels, regularization and differential equations. Pattern Recognit. 41(11), 3271–3286 (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Thirion, J.: Image matching as a diffusion process: an analogy with maxwell’s demons. Med. Image Anal. 2(3), 243–260 (1998)CrossRefGoogle Scholar
  34. 34.
    Tucker, L.: Implications of factor analysis of three-way matrices for measurement of change. Problems in Measuring Change, pp. 122–137. University of Wisconsin Press, Madison (1963)Google Scholar
  35. 35.
    Twining, C.J., Marsland, S.: Constructing diffeomorphic representations of non-rigid registrations of medical images. In: Information Processing in Medical Imaging. Lecture Notes in Computer Science, pp. 413–425. Springer, Berlin (2003) Google Scholar
  36. 36.
    Vandemeulebroucke, J., Sarrut, D., Clarysse, P.: The popi-model, a point-validated pixel-based breathing thorax model. In: Proceedings of the 15th International Conference on the Use of Computers in Radiation Therapy-ICCR (2007)Google Scholar
  37. 37.
    Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Non-parametric diffeomorphic image registration with the demons algorithm. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2007. Lecture Notes in Computer Science, pp. 319–326 (2007)Google Scholar
  38. 38.
    Vercauteren, T., Pennec, X., Perchant, A., Ayache, N.: Symmetric log-domain diffeomorphic registration: a demons-based approach. In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008. Lecture Notes in Computer Science, pp. 754–761 (2008)Google Scholar
  39. 39.
    Vercauteren, T., Pennec, X., Perchant, A., Ayache, N., et al.: Diffeomorphic demons using itk’s finite difference solver hierarchy. In: Insight Journal-ISC/NA-MIC Workshop on Open Science at MICCAI (2007)Google Scholar
  40. 40.
    Wang, H., Dong, L., O’Daniel, J., Mohan, R., Garden, A., Ang, K., Kuban, D., Bonnen, M., Chang, J., Cheung, R.: Validation of an accelerated’demons’ algorithm for deformable image registration in radiation therapy. Phys. Med. Biol. 50(12), 2887–2905 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Christoph Jud
    • 1
    Email author
  • Marcel Lüthi
    • 1
  • Thomas Albrecht
    • 1
  • Sandro Schönborn
    • 1
  • Thomas Vetter
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland

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