Skip to main content
Log in

Deformable Image Registration Based on Functions of Bounded Generalized Deformation

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

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, log in via an institution to check access.

Access this article

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

Similar content being viewed by others

Notes

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

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

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

References

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • 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.

  • 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.

  • 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 

  • 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.

  • 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.

  • 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.

  • 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.

  • 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 

  • 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.

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • 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 

  • 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 

  • 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 

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

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  • 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.

  • 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.

  • 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.

  • 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 

  • 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 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • 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

Authors and 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

Communicated by Xavier Pennec.

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)

Rights and permissions

Reprints and permissions

About this article

Check for updates. 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 129, 1341–1358 (2021). https://doi.org/10.1007/s11263-021-01439-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-021-01439-x

Keywords

Navigation