Nothing Special   »   [go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Variational Image Registration Using Inhomogeneous Regularization

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Notes

  1. A super diagonal tensor is the generalization of a diagonal matrix to higher order tensors, where the entries outside the main diagonal are zero.

References

  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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

    Article  Google Scholar 

  8. Christensen, G.E., Johnson, H.J.: Consistent image registration. IEEE Trans. Med. Imaging 20(7), 568–582 (2001)

    Article  Google Scholar 

  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)

  10. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Evans, L.: Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (1998)

    Google Scholar 

  12. Girdziušas, R., Laaksonen, J.: Gaussian process regression with fluid hyperpriors. Neural Information Processing, pp. 567–572. Springer, Berlin (2004)

    Chapter  Google Scholar 

  13. Guo, H., Rangarajan, A., Joshi, S.: Diffeomorphic point matching. Handbook of Mathematical Models in Computer Vision, pp. 205–219. Springer, New York (2006)

    Chapter  Google Scholar 

  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)

  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. Johnson, H., Christensen, G.: Consistent landmark and intensity-based image registration. IEEE Trans. Med. Imaging 21(5), 450–461 (2002)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  18. Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  22. McOwen, R.: Partial differential equations: methods and applications. Tsinghua University Press, Haidian (1996)

    MATH  Google Scholar 

  23. Modersitzki, J.: Numerical Methods for Image Registration. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  24. Opfer, R.: Multiscale kernels. Adv. Comput. Math. 25(4), 357–380 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

  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)

  30. Sotiras, A., Paragios, N., et al.: Deformable image registration: a survey (2012)

  31. Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. Med. Image Anal. 8(3), 325–342 (2004)

    Article  Google Scholar 

  32. Steinke, F., Schölkopf, B.: Kernels, regularization and differential equations. Pattern Recognit. 41(11), 3271–3286 (2008)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  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. 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)

  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)

  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)

  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)

  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)

  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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Basel, 4056 , Basel, Switzerland

    Christoph Jud, Marcel Lüthi, Thomas Albrecht, Sandro Schönborn & Thomas Vetter

Authors
  1. Christoph Jud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel Lüthi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Albrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sandro Schönborn
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Thomas Vetter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Jud.

Appendix

Appendix

1.1 Matrix Products

1.1.1 Khatri-Rao Product

Given a matrix \(A\in \mathrm{I\!R}^{m \times q}\) and a matrix \(B\in \mathrm{I\!R}^{n \times q}\), the Khatri-Rao product of \(A\) and \(B\) is the matching column-wise Kronecker product

$$\begin{aligned} A\odot B = (a_1\bullet b_1, a_2\bullet b_2, \cdots a_q\bullet b_q) \in \mathrm{I\!R}^{mn\times q}. \end{aligned}$$

1.1.2 Hadamard Product

Given a matrix \(A\in \mathrm{I\!R}^{m \times n}\) and a matrix \(B\in \mathrm{I\!R}^{m \times n}\), the Hadamard product of \(A\) and \(B\) is the point-wise matrix product

$$\begin{aligned} A \star B = \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} a_{11}b_{11} &{} a_{12}b_{12} &{} \cdots &{} a_{1n}b_{1n} \\ a_{21}b_{21} &{} a_{22}b_{22} &{} \cdots &{} a_{2n}b_{2n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m1}b_{m1} &{} a_{m2}b_{m2} &{} \cdots &{} a_{mn}b_{mn} \end{array} \right) \in \mathrm{I\!R}^{m \times n}. \end{aligned}$$

1.1.3 Kronecker Product

Given a matrix \(A\in \mathrm{I\!R}^m \times \mathrm{I\!R}^n\) and a matrix \(B\in \mathrm{I\!R}^q \times \mathrm{I\!R}^r\), the Kronecker product of \(A\) and \(B\) is given as

$$\begin{aligned} A \bullet B = \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} a_{11}B &{} a_{12}B &{} \cdots &{} a_{1n}B \\ a_{21}B &{} a_{22}B &{} \cdots &{} a_{2n}B \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m1}B &{} a_{m2}B &{} \cdots &{} a_{mn}B \end{array} \right) \in \mathrm{I\!R}^{mq \times nr}. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jud, C., Lüthi, M., Albrecht, T. et al. Variational Image Registration Using Inhomogeneous Regularization. J Math Imaging Vis 50, 246–260 (2014). https://doi.org/10.1007/s10851-014-0497-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-014-0497-0

Keywords

Search

Navigation