Deformable Image Registration Based on Functions of Bounded Generalized Deformation

Abstract

Functions of bounded deformation (BD) are widely used in the theory of elastoplasticity to describe the possibly discontinuous displacement fields inside elastoplastic bodies. BD functions have been proved suitable for deformable image registration, the goal of which is to find the displacement field between a moving image and a fixed image. Recently BD functions have been generalized to symmetric tensor fields of bounded generalized variation. In this paper, we focus on the first-order symmetric tensor fields, i.e., vector-valued functions, of bounded generalized variation. We specify these functions as functions of bounded generalized deformation (BGD) since BGD functions are natural generalizations of BD functions. We propose a BGD model for deformable image registration problems by regarding concerned displacement fields as BGD functions. BGD model employs not only the first-order but also higher-order coupling information of components of the displacement field. It turns out that BGD model allows for jump discontinuities of displacements while, in contrast to BD model, at the same time is able to employ higher-order derivatives of displacements in smooth regions. As a result, BGD model tends to capture possible discontinuities of displacements appeared around edges of the target objects while keep the smoothness of displacements inside the target objects as well. This characteristic enables BGD model to obtain better registration results than BD model and other variational models. To our knowledge, it is the first time in literature to use BGD functions for image registration. A first-order adaptive primal–dual algorithm is adopted to solve the proposed BGD model. Numerical experiments on 2D and 3D images show both effectiveness and advantages of BGD model.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    https://www.dir-lab.com.

  2. 2.

    https://www.dir-lab.com/ReferenceData.html.

  3. 3.

    https://www.dir-lab.com/Results.html.

References

  1. Aganj, I., Yeo, B. T. T., Sabuncu, M. R., & Fischl, B. (2013). On removing interpolation and resampling artifacts in rigid image registration. IEEE Transactions on Image Processing, 22(2), 816–827.

    MathSciNet  MATH  Article  Google Scholar 

  2. Alahyane, M., Hakim, A., Laghrib, A., & Raghay, S. (2018). Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 12(5), 1055–1081.

    MathSciNet  MATH  Article  Google Scholar 

  3. Alam, F., Rahman, S. U., Ullah, S., & Gulati, K. (2018). Medical image registration in image guided surgery: Issues, challenges and research opportunities. Biocybernetics and Biomedical Engineering, 38(1), 71–89.

    Article  Google Scholar 

  4. Ambrosio, L., Coscia, A., & Dal Maso, G. (1997). Fine properties of functions with bounded deformation. Archive for Rational Mechanics and Analysis, 139(3), 201–238.

    MathSciNet  MATH  Article  Google Scholar 

  5. Balle, F., Beck, T., Eifler, D., Fitschen, J. H., Schuff, S., & Steidl, G. (2019). Strain analysis by a total generalized variation regularized optical flow model. Inverse Problems in Science and Engineering, 27(4), 540–564.

    MathSciNet  Article  Google Scholar 

  6. Barroso, A. C., Fonseca, I., & Toader, R. (2000). A relaxation theorem in the space of functions of bounded deformation. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 29(1), 19–49.

    MathSciNet  MATH  Google Scholar 

  7. Bonettini, S., & Ruggiero, V. (2012). On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration. Journal of Mathematical Imaging and Vision, 44(3), 236–253.

    MathSciNet  MATH  Article  Google Scholar 

  8. Bouaziz, S., Tagliasacchi, A., & Pauly, M. (2013). Sparse iterative closest point. In Proceedings of the eleventh Eurographics/ACMSIGGRAPH symposium on geometry processing (pp. 113–123). Genova, Italy: Eurographics Association.

  9. Bredies, K. (2013). Symmetric tensor fields of bounded deformation. Annali di Matematica Pura ed Applicata, 192(5), 815–851.

    MathSciNet  MATH  Article  Google Scholar 

  10. Bredies, K., & Holler, M. (2014). Regularization of linear inverse problems with total generalized variation. Journal of Inverse and Ill-posed Problems, 22(6), 871–913.

    MathSciNet  MATH  Article  Google Scholar 

  11. Bredies, K., Kunisch, K., & Pock, T. (2010). Total generalized variation. SIAM Journal on Imaging Sciences, 3(3), 492–526.

    MathSciNet  MATH  Article  Google Scholar 

  12. Burger, M., Modersitzki, J., & Ruthotto, L. (2013). A hyperelastic regularization energy for image registration. SIAM Journal on Scientific Computing, 35(1), B132–B148.

    MathSciNet  MATH  Article  Google Scholar 

  13. Castillo, R., Castillo, E., Guerra, R., Johnson, V. E., McPhail, T., Garg, A. K., et al. (2009). A framework for evaluation of deformable image registration spatial accuracy using large landmark point sets. Physics in Medicine & Biology, 54(7), 1849–1870.

    Article  Google Scholar 

  14. Castillo, R., Castillo, E. M., Fuentes, D. T., Ahmad, M., Wood, A. M., Ludwig, M. S., et al. (2013). A reference dataset for deformable image registration spatial accuracy evaluation using the copdgene study archive. Physics in Medicine & Biology, 58(9), 2861–2877.

    Article  Google Scholar 

  15. Chambolle, A., & Pock, T. (2011). A first-order primal–dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1), 120–145.

    MathSciNet  MATH  Article  Google Scholar 

  16. Chumchob, N. (2013). Vectorial total variation-based regularization for variational image registration. IEEE Transactions on Image Processing, 22(11), 4551–4559.

    MathSciNet  MATH  Article  Google Scholar 

  17. Chumchob, N., Chen, K., & Brito-Loeza, C. (2011). A fourth-order variational image registration model and its fast multigrid algorithm. Multiscale Modeling & Simulation, 9(1), 89–128.

    MathSciNet  MATH  Article  Google Scholar 

  18. Dal Maso, G. (2013). Generalised functions of bounded deformation. Journal of the European Mathematical Society, 15(5), 1943–1997.

    MathSciNet  MATH  Article  Google Scholar 

  19. Du, K., Bayouth, J. E., Cao, K., Christensen, G. E., Ding, K., & Reinhardt, J. M. (2012). Reproducibility of registration-based measures of lung tissue expansion. Medical Physics, 39(3), 1595–1608.

    Article  Google Scholar 

  20. Evans, L. C. (2010). Partial differential equations (2nd ed.). Providence, RI: AMS.

    Google Scholar 

  21. Gao, Y., Liu, F., & Yang, X. (2018). Total generalized variation restoration with non-quadratic fidelity. Multidimensional Systems and Signal Processing, 29(4), 1459–1484.

    MATH  Article  Google Scholar 

  22. Goldstein, T., Li, M., & Yuan, X. (2015). Adaptive primal–dual splitting methods for statistical learning and image processing. In Advances in neural information processing systems (pp. 2089–2097). Montreal, CA: Curran Associates, Inc.

  23. Hajinezhad, D., Hong, M., Zhao, T., & Wang, Z. (2016). Nestt: A nonconvex primal-dual splitting method for distributed and stochastic optimization. In Advances in neural information processing systems (pp. 3215–3223). Barcelona, Spain: Curran Associates, Inc.

  24. Haker, S., Zhu, L., Tannenbaum, A., & Angenent, S. (2004). Optimal mass transport for registration and warping. International Journal of Computer Vision, 60(3), 225–240.

    Article  Google Scholar 

  25. Heinrich, M. P., Handels, H., & Simpson, I. J. (2015). Estimating large lung motion in copd patients by symmetric regularised correspondence fields. In International conference on medical image computing and computer-assisted intervention (pp. 338–345). Springer.

  26. Hermann, S. (2014). Evaluation of scan-line optimization for 3d medical image registration. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 3073–3080). Columbus, OH: IEEE.

  27. Hermann, S., & Werner, R. (2013). High accuracy optical flow for 3d medical image registration using the census cost function. In Pacific-rim symposium on image and video technology (pp. 23–35). Berlin, Heidelberg: Springer.

  28. Hömke, L., Frohn-Schauf, C., Henn, S., & Witsch, K. (2007). Total variation based image registration. In Image processing based on partial differential equations (pp. 343–361). Berlin, Heidelberg: Springer.

  29. Hossny, M., Nahavandi, S., & Creighton, D. (2008). Comments on ‘Information measure for performance of image fusion’. Electronics Letters, 44(18), 1066–1067.

    Article  Google Scholar 

  30. König, L., & Rühaak, J. (2014). A fast and accurate parallel algorithm for non-linear image registration using normalized gradient fields. In 2014 IEEE 11th international symposium on biomedical imaging (ISBI) (pp. 580–583). Beijing: IEEE.

  31. Lax, P. D. (2002). Functional analysis. Hoboken: Wiley.

    Google Scholar 

  32. Lin, F. H., & Yang, X. (2002). Geometric measure theory: An introduction. Beijing: Science Press.

    Google Scholar 

  33. Lombaert, H., Grady, L., Pennec, X., Ayache, N., & Cheriet, F. (2014). Spectral log-demons: Diffeomorphic image registration with very large deformations. International Journal of Computer Vision, 107(3), 254–271.

    Article  Google Scholar 

  34. Mainardi, L., Passera, K. M., Lucesoli, A., Vergnaghi, D., Trecate, G., Setti, E., et al. (2008). A nonrigid registration of MR breast images using complex-valued wavelet transform. Journal of Digital Imaging, 21(1), 27–36.

    Article  Google Scholar 

  35. McClelland, J. R., Hawkes, D. J., Schaeffter, T., & King, A. P. (2013). Respiratory motion models: A review. Medical Image Analysis, 17(1), 19–42.

    Article  Google Scholar 

  36. Modersitzki, J. (2009). FAIR: Flexible algorithms for image registration (Vol. 6). Philadelphia: SIAM.

    Google Scholar 

  37. Nie, Z., & Yang, X. (2019). Deformable image registration using functions of bounded deformation. IEEE Transactions on Medical Imaging, 38, 1488–1500.

    Article  Google Scholar 

  38. Polzin, T., Niethammer, M., Heinrich, M. P., Handels, H., & Modersitzki, J. (2016). Memory efficient lddmm for lung ct. In International conference on medical image computing and computer-assisted intervention (pp. 28–36). Springer.

  39. Polzin, T., Rühaak, J., Werner, R., Strehlow, J., Heldmann, S., Handels, H., et al. (2013). Combining automatic landmark detection and variational methods for lung ct registration. In Fifth international workshop on pulmonary image analysis (pp. 85–96). Nagoya, Japan: Springer.

  40. Ranftl, R., Bredies, K., & Pock, T. (2014). Non-local total generalized variation for optical flow estimation. In European conference on computer vision (pp. 439–454). Cham: Springer.

  41. Rühaak, J., Polzin, T., Heldmann, S., Simpson, I. J., Handels, H., Modersitzki, J., et al. (2017). Estimation of large motion in lung ct by integrating regularized keypoint correspondences into dense deformable registration. IEEE Transactions on Medical Imaging, 36(8), 1746–1757.

    Article  Google Scholar 

  42. Sotiras, A., Davatzikos, C., & Paragios, N. (2013). Deformable medical image registration: A survey. IEEE Transactions on Medical Imaging, 32(7), 1153–1190.

    Article  Google Scholar 

  43. Suetens, P. (2009). Fundamentals of medical imaging. Cambridge: Cambridge University Press.

    Google Scholar 

  44. Temam, R. (1983). Problèmes mathématiques en plasticité. Montrouge: Gauthier-Villars.

    Google Scholar 

  45. Thévenaz, P., Blu, T., & Unser, M. (2000). Image interpolation and resampling (pp. 393–420). New York: Academic Press.

    Google Scholar 

  46. Thirion, J. P. (1998). Image matching as a diffusion process: An analogy with Maxwell’s demons. Medical Image Analysis, 2(3), 243–260.

    Article  Google Scholar 

  47. Vishnevskiy, V., Gass, T., Szkely, G., & Goksel, O. (2016). Total variation regularization of displacements in parametric image registration. IEEE Transactions on Medical Imaging, 36(2), 385–395.

    Article  Google Scholar 

  48. Vishnevskiy, V., Gass, T., Szekely, G., Tanner, C., & Goksel, O. (2017). Isotropic total variation regularization of displacements in parametric image registration. IEEE Transactions on Medical Imaging, 36(2), 385–395.

    Article  Google Scholar 

  49. Wang, Z., Bovik, A. C., Sheikh, H. R., Simoncelli, E. P., et al. (2004). Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 13(4), 600–612.

    Article  Google Scholar 

  50. Washizu, K. (1975). Variational methods in elasticity and plasticity (2nd ed.). New York: Pergamon Press.

    Google Scholar 

  51. Yoo, J. C., & Han, T. H. (2009). Fast normalized cross-correlation. Circuits, Systems and Signal Processing, 28(6), 819.

    MATH  Article  Google Scholar 

  52. Zhang, J., Ackland, D., & Fernandez, J. (2018). Point-cloud registration using adaptive radial basis functions. Computer Methods in Biomechanics and Biomedical Engineering, 21(7), 498–502.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the multidisciplinary team of liver, billiary and pancreatic tumors in Nanjing Drum Tower Hospital, China for providing CT liver images used in the second 2D numerical experiments. The first author would like to thank Dong WANG for his help with formatting the Latex code of this manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiaoping Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This work is supported by National Natural Science Foundation of China (Grant Nos. 11971229, 12090023) and China’s Ministry of Science and Technology (Grant No. SQ2020YFA070208)

Communicated by Xavier Pennec.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nie, Z., Li, C., Liu, H. et al. Deformable Image Registration Based on Functions of Bounded Generalized Deformation. Int J Comput Vis (2021). https://doi.org/10.1007/s11263-021-01439-x

Download citation

Keywords

  • Functions of bounded generalized deformation
  • Deformable image registration
  • Functions of bounded deformation
  • Primal–dual algorithm