Skip to main content
Log in

Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques

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

Abstract

This paper proposes ℒ2- and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric. Such algorithms pair the same points of two images after the images are swapped. Many commonly-used ℒ2 and IT non-rigid registration algorithms are only approximately symmetric. The asymmetry is due to the objective function as well as due to the numerical techniques used in discretizing and minimizing the objective function. This paper analyzes and provides techniques to eliminate both sources of asymmetry.

This paper has five parts. The first part shows that objective function asymmetry is due to the use of standard differential volume forms on the domain of the images. The second part proposes alternate volume forms that completely eliminate objective function asymmetry. These forms, called graph-based volume forms, are naturally defined on the graph of the registration diffeomorphism f, rather than on the domain of f. When pulled back to the domain of f they involve the Jacobian J f and therefore appear “non-standard”. In the third part of the paper, graph-based volume forms are analyzed in terms of four key objective-function properties: symmetry, positive-definiteness, invariance, and lack of bias. Graph-based volume forms whose associated ℒ2 objective functions have the first three properties are completely classified. There is an infinite-dimensional space of such graph-based forms. But within this space, up to scalar multiple, there is a unique volume form whose associated ℒ2 objective function is unbiased. This volume form, which when pulled back to the domain of f is (1+det(J f )) times the standard volume form on Euclidean space, is exactly the differential-geometrically natural volume form on the graph of f. The fourth part of the paper shows how the same volume form also makes the IT objective functions symmetric, positive semi-definite, invariant, and unbiased. The fifth part of the paper introduces a method for removing asymmetry in numerical computations and presents results of numerical experiments. The new objective functions and numerical method are tested on a coronal slice of a 3-D MRI brain image. Numerical experiments show that, even in the presence of noise, the new volume form and numerical techniques reduces asymmetry practically down to machine precision without compromising registration accuracy.

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

Similar content being viewed by others

References

  1. Hajnal, J.V., Hill, D.L.G., Hawkes, D.J.: Medical Image Registration. CRC Press, Boca Raton (2001)

    Google Scholar 

  2. Toga, A.: Brain Warping. Academic Press, San Diego (1999)

    Google Scholar 

  3. Goshtaby, A.A.: 2-D and 3-D Image Registration for Medical Remote Sensing and Industrial Applications. Wiley, Hoboken (2005)

    Google Scholar 

  4. Maintz, J.B.A., Viergever, M.A.: A survey of image registration. Med. Image Anal. MEDIA 2(1), 1–36 (1998)

    Article  Google Scholar 

  5. Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(11), 977–1000 (2003)

    Article  Google Scholar 

  6. Pluim, J.P.W., Maintz, J.B.A., Viergever, M.A.: Mutual information based registration of medical images: a survey. IEEE Trans. Med. Imag. 22(8), 986–1004 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Christensen, G.E., Johnson, H.J.: Invertibility and transitivity analysis for non-rigid image registration. J. Electron. Imag. 12(1), 106–117 (2003)

    Article  Google Scholar 

  10. Ashburner, J., Andersson, J.L.R., Friston, K.J.: High-dimensional image registration using symmetric prior. NeuroImage 9(6), 619–628 (1999)

    Article  Google Scholar 

  11. Beg, M.F., Khan, A.: Symmetric data attachment terms for large deformation image registration. IEEE Trans. Med. Imag. 26(9), 1179–1189 (2007)

    Article  Google Scholar 

  12. Cachier, P., Rey, D.: Symmetrization of the non-rigid registration problem using inversion-invariant energies: application to multiple sclerosis. In: Medical Image Computing and Computer Aided Intervention (MICCAI), Pittsburgh, PA, October 2000, pp. 472–481

  13. Rogelj, P., Kovacic, S.: Symmetric image registration. Med. Image Anal. 10(3), 484–493 (2006)

    Article  Google Scholar 

  14. Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23(1), S151–S160 (2004). Supplement issue on Mathematics in Brain Imaging

    Article  Google Scholar 

  15. Lorenzen, P., Prastawa, M., Davis, B., Gerig, G., Bullitt, E., Joshi, S.: Multi-modal image set registration and atlas formation. Med. Image Anal. MEDIA 10(3), 440–451 (2006)

    Article  Google Scholar 

  16. Avants, B.B., Epstein, C.L., Grossman, M., Gee, J.C.: Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain. Med. Image Anal. 12(1), 26–41 (2008)

    Article  Google Scholar 

  17. Tagare, H.D., O’Shea, D., Groisser, D.: Shape based non-rigid correspondence for plane curves. J. Math. Imag. Vis. 16, 57–68 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tagare, H.D.: Shape-based non-rigid correspondence with applications to heart motion analysis. IEEE Trans. Med. Imag. 18(7), 570–579 (1999)

    Article  Google Scholar 

  19. Lorenzen, P., Davis, B., Joshi, S.: Model based symmetric information theoretic large deformation multi-model image registration. In: Intl. Symp. Biomedical Imaging, pp. 720–723 (2004)

  20. Lorenzen, P., Davis, B., Joshi, S.: Unbiased atlas formation via large deformations metric mapping. In: J.S. Duncan, G. Gerig (eds.) Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 411–418 (2005)

  21. Magnotta, V.A., Bockholt, H.J., Johnson, H.J., Christensen, G.E., Andreasen, N.C.: Subcortical, cerebellar and MR based consistent brain image registration. NeuroImage 19(2), 233–245 (2003)

    Article  Google Scholar 

  22. Christensen, G.E., Johnson, H.J., Vannier, M.W.: Synthesizing average 3D anatomical shapes. NeuroImage 32, 146–158 (2006)

    Article  Google Scholar 

  23. Li, B., Christensen, G.E., McLennan, G., Hoffman, E.A., Reinhardt, J.M.: Establishing a normative atlas of the human lung: Inter-subject warping and registration of volumetric CT. Acad. Radiology 10(3), 255–265 (2003)

    Article  Google Scholar 

  24. Lu, W., Parikh, P., El Naqa, I., Nystrom, M., Hubenschmidt, J., Wahab, S., Mutic, S., Sing, A., Christensen, G.E., Bradley, J.D., Low, D.A.: Quantitation of the four-dimensional computed tomography process for lung cancer patients. Med. Phys. 32(4), 890–901 (2005)

    Article  Google Scholar 

  25. Christensen, G.E., He, J., Dill, J.A., Rubinstein, J.T., Vannier, M.W., Wang, G.: Automatic measurement of the labyrinth using image registration and a deformable inner ear atlas. Acad. Radiology 10(9), 988–999 (2003)

    Article  Google Scholar 

  26. Wells, W.M. III, Viola, P., Atsumi, H., Nakajima, S., Kikinis, R.: Multi-modal volume registration by maximization of mutual information. Med. Image Anal. 1(1), 35–51 (1996)

    Article  Google Scholar 

  27. Studholme, C., Hill, D.L.G., Hawkes, D.J.: An overlap invariant entropy measure of 3D medical image alignment. Pattern Recognit. 32(1), 71–86 (1999)

    Article  Google Scholar 

  28. Viola, P., Wells, W.M. III: Alignment by maximization of mutual information. Int. J. Comput. Vis. 24(2), 137–154 (1997)

    Article  Google Scholar 

  29. D’Agostino, E., Maes, F., Vandermeulen, D., Suetens, P.: A viscous fluid model for multimodal non-rigid image registration using mutual information. Med. Image Anal. 7, 565–575 (2003)

    Article  Google Scholar 

  30. Pluim, J.P.W., Maintz, J.B.A., Viergever, M.A.: F-information measures in medical image registration. IEEE Trans. Med. Imag. 23(12), 1508–1516 (2004)

    Article  Google Scholar 

  31. Yanovsky, I., Thompson, P., Osher, S., Leow, A.: Topology preserving log-unbiased non-linear image registration: theory and implementation. In: IEEE Conf. on Computer Vision and Pattern Recognition, June 2007

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hemant D. Tagare.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tagare, H.D., Groisser, D. & Skrinjar, O. Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques. J Math Imaging Vis 34, 61–88 (2009). https://doi.org/10.1007/s10851-008-0129-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-008-0129-7

Keywords

Navigation