Mass Preserving Mappings and Image Registration (original) (raw)

Abstract

Image registration is the process of establishing a common geometric reference frame between two or more data sets from the same or different imaging modalities possibly taken at different times. In the context of medical imaging and in particular image guided therapy, the registration problem consists of finding automated methods that align multiple data sets with each other and with the patient. In this paper we propose a method of mass preserving elastic registration based on the Monge-Kantorovich problem of optimal mass transport.

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References

  1. S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, “On area preserving maps of minimal distortion,” in System Theory: Modeling, Analysis, and Control, edited by T. Djaferis and I. Schick, Kluwer, Holland, 1999, pp. 275–287.
    Google Scholar
  2. S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, “Laplace-Beltrami operator and brain surface flattening,” IEEE Trans. on Medical Imaging 18 (1999), pp. 700–711.
    Article Google Scholar
  3. J.-D. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,” Numerische Mathematik 84 (2000), pp. 375–393.
    Article MATH MathSciNet Google Scholar
  4. Y. Brenier, “Polar factorization and monotone rearrangement of vector-valued functions,” Com. Pure Appl. Math. 64 (1991), pp. 375–417.
    Article MathSciNet Google Scholar
  5. G. E. Christensen, R. D. Rabbit, and M. I. Miller, “Deformable templates using large deformation kinematics,” IEEE Trans. of Medical Imag. 5 (1996) pp. 1435–1447.
    Article Google Scholar
  6. M. Cullen and R. Purser, “An extended Lagrangian theory of semigeostrophic frontogenesis,” J. Atmos. Sci. 41 (1984), pp. 1477–1497.
    Article MathSciNet Google Scholar
  7. C. Davatzikos, “Spatial transformation and registration of brain images using elastically deformable models,” Comp. Vis. and Image Understanding 66 (1997), pp. 207–222.
    Article Google Scholar
  8. B. Dacorogna and J. Moser, “On a partial differential equation involving the Jacobian determinant,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 1–26.
    MATH MathSciNet Google Scholar
  9. W. Gangbo, “An elementary proof of the polar factorization of vector-valued functions,” Arch. Rational Mechanics Anal. 128 (1994), pp. 381–399.
    Article MATH MathSciNet Google Scholar
  10. W. Gangbo and R. McCann, “The geometry of optimal transportation,” Acta Math. 177 (1996), pp. 113–161.
    Article MATH MathSciNet Google Scholar
  11. A. F. Goldszal, C. Davatzikos, D. L. Pham, M. X. H. Yan, R. N. Bryan, and S. M. Resnick, “An image processing protocol for qualitative and quantitative volumetric analysis of brain images”, J. Comp. Assist. Tomogr., 22 (1998) pp. 827–837.
    Article Google Scholar
  12. S. Haker, S. Angenent, A. Tannenbaum, and R. Kikinis, “Nondistorting flattening maps and the 3D visualization of colon CT images,” IEEE Trans. of Medical Imag., July 2000.
    Google Scholar
  13. S. Haker and A. Tannenbaum, “Optimal transport and image registration,” submitted to IEEE Trans. Image Processing, January 2001.
    Google Scholar
  14. N. Hata, A. Nabavi, S. Warfield, W. Wells, R. Kikinis and F. Jolesz, “A volumetric optical flow method for measurement of brain deformation from intraoperative magnetic resonance images,” Proc. Second International Conference on Medical Image Computing and Computer-assisted Interventions (1999), pp. 928–935.
    Google Scholar
  15. L. V. Kantorovich, “On a problem of Monge,” Uspekhi Mat. Nauk. 3 (1948), pp. 225–226.
    Google Scholar
  16. H. Lester, S. R. Arridge, K. M. Jansons, L. Lemieux, J. V. Hajnal and A. Oatridge, “Non-linear registration with the variable viscosity fluid algorithm,” Information Processing in Medical Imaging (1999), pp. 238–251.
    Google Scholar
  17. J. B. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Image Analysis 2 (1998), pp. 1–36.
    Article Google Scholar
  18. J. Moser, “On the volume elements on a manifold,” Trans. Amer. Math. Soc. 120 (1965), pp. 286–294.
    Article MATH MathSciNet Google Scholar
  19. W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Edition, Cambridge University Press, Cambridge U.K., 1992.
    Google Scholar
  20. S. Rachev and L. Rüschendorf, Mass Transportation Problems, Volumes I and II, Probability and Its Applications, Springer, New York, 1998.
    MATH Google Scholar
  21. A. Toga, Brain Warping, Academic Press, San Diego, 1999.
    Google Scholar

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Authors and Affiliations

  1. Department of Radiology, Surgical Planning Laboratory Brigham and Women’s Hospital, Boston, MA, 02115, USA
    Steven Haker & Ron Kikinis
  2. Departments of Electrical and Computer and Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250, USA
    Allen Tannenbaum

Authors

  1. Steven Haker
  2. Allen Tannenbaum
  3. Ron Kikinis

Editor information

Editors and Affiliations

  1. Image Sciences Institute, University Medical Center Utrecht, Heidelberglaan 100, 3584 CX, Utrecht, The Netherlands
    Wiro J. Niessen & Max A. Viergever &

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Haker, S., Tannenbaum, A., Kikinis, R. (2001). Mass Preserving Mappings and Image Registration. In: Niessen, W.J., Viergever, M.A. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2001. MICCAI 2001. Lecture Notes in Computer Science, vol 2208. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45468-3\_15

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