International Journal of Computer Vision

, Volume 60, Issue 3, pp 225–240

Optimal Mass Transport for Registration and Warping

  • Steven Haker
  • Lei Zhu
  • Allen Tannenbaum
  • Sigurd Angenent
Article

Abstract

Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge–Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the L2 Kantorovich–Wasserstein or “Earth Mover's Distance” under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extend this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.

elastic registration image warping optimal transport mass-preservation gradient flows 

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References

  1. Ambrosio, L. 2000. Lecture notes on optimal transport problems. Lectures given at Euro Summer School. Available on http://cvgmt.sns.it/papers/amb00a/.Google Scholar
  2. Angenent, S., Haker, S., and Tannenbaum, A. 2003. Minimizing flows for the Monge-Kantorovich problem. SIAM J. Math. Analysis, 35:61–97.CrossRefGoogle Scholar
  3. Angenent, S., Haker, S., Tannenbaum, A., and Kikinis, R. 1999a. On area preserving maps of minimal distortion. In System Theory: Modeling, Analysis, and Control, T. Djaferis and I. Schick (Eds.), Kluwer: Holland, pp. 275–287.Google Scholar
  4. Angenent, S., Haker, S., Tannenbaum, A., and Kikinis, R. 1999b. Laplace-Beltrami operator and brain surface flattening. IEEE Trans. on Medical Imaging, 18:700–711.CrossRefGoogle Scholar
  5. Aruliah, D.A., Aschery, U.M., Haberz, E., and Oldenburgx, D. 2001. A method for the forward modelling of 3D electromagnetic quasistatic problems. Mathematical Models and Methods in Applied Sciences, 11:1–21.CrossRefGoogle Scholar
  6. Benamou, J.-D. and Brenier,Y. 2000.Acomputational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84:375–393.CrossRefGoogle Scholar
  7. Brenier, Y. 1991. Polar factorization and monotone rearrangement of vector-valued functions. Com. Pure Appl. Math., 64:375–417.Google Scholar
  8. Bro-Nielsen, M. and Gramkow, C. 1996. Fast fluid registration of medical images. In Visualization in Biomedical Imaging, K. Höhne and R. Kikinis (Eds.), Lecture Notes in Computer Science, vol. 1131, Springer-Verlag: New York, pp. 267–276.Google Scholar
  9. Christensen, G.E., Rabbit, R.D., and Miller, M. 1996. Deformable templates using large deformation kinetics. IEEE Trans. on Image Processing, 5:1435–1447.CrossRefGoogle Scholar
  10. Christensen, G.E., Rabbit, R.D., and Miller, M. 1993. A deformable neuroanatomy handbook based on viscous fluid mechanics. In 27th Ann. Conf. on Inf. Sciences and Systems, pp. 211–216.Google Scholar
  11. Christensen, G.E. and Johnson, H.J. 2001. Consistent image registration. IEEE Trans. on Medical Imaging, 20:568–582.CrossRefGoogle Scholar
  12. Cullen, M. and Purser, R. 1984. An extended Lagrangian theory of semigeostrophic frontogenesis. J. Atmos. Sci., 41:1477–1497.CrossRefGoogle Scholar
  13. Dacorogna, B. and Moser, J. 1990. On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal.Non Linéaire, 7:1–26.Google Scholar
  14. Fry, D. 1993. Shape recognition using metrics on the space of shapes.Ph.D. Thesis, Harvard University.Google Scholar
  15. Gangbo, W. 1994. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mechanics Anal., 128:381–399.Google Scholar
  16. Gangbo, W. and McCann, R. 1996. The geometry of optimal transportation. Acta Math., 177:113–161.Google Scholar
  17. Gangbo, W. and McCann, R. 1999. Shape recognition via Wasserstein distance. Technical Report, School of Mathematics, Georgia Institute of Technology.Google Scholar
  18. Haker, S., Angenent, S., Tannenbaum, A., and Kikinis, R. 2000. Nondistorting flattening maps and the 3D visualization of colon CT images. IEEE Trans. of Medical Imaging.Google Scholar
  19. Haralick, R. and Shapiro, L. 1992. Computer and Robot Vision. Addison-Wesley: New York.Google Scholar
  20. Hinterberger, W. and Scherzer, O. 2001. Models for image interpolation based on the optical flow. Computing, 66:231–247.CrossRefGoogle Scholar
  21. Kaijser, T. 1998. Computing the Kantorovich distance for images. Journal of Mathematical Imaging and Vision, 9:173–191.CrossRefGoogle Scholar
  22. Kantorovich, L.V. 1948. On a problem of monge. Uspekhi Mat. Nauk., 3:225–226.Google Scholar
  23. Knott, M. and Smith, C. 1984. On the optimal mapping of distributions. J. Optim. Theory, 43:39–49.Google Scholar
  24. Levina, E. and Bickel, P. 2001. The earth mover's distance is the Mallow's distance: Some insights from statistics. In Proceedings IEEE Int. Conf. on Computer Vision, vol. 2, pp. 251–256.CrossRefGoogle Scholar
  25. McCann, R. 2001. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11:589–608.Google Scholar
  26. McCann, R. 1997. A convexity principle for interacting gases. Adv. Math., 128:153–179.CrossRefGoogle Scholar
  27. Miller,M., Christensen, G., Amit,Y., and Grenander,U. 1992. Mathematical textbook of deformable neuroanatomies. Proc. National Academy of Science, 90:11944–11948.Google Scholar
  28. Moser, J. 1965. On the volume elements on a manifold. Trans. Amer. Math. Soc., 120:286–294.Google Scholar
  29. Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. 1992. Numerical Recipes in C: The Art of Scientific Computing, 2nd edn., Cambridge University Press: Cambridge U.K.Google Scholar
  30. Rachev S. and Rüschendorf, L. 1998. Mass Transportation Problems, vols. I and II, Probability and its Applications. Springer: New York.Google Scholar
  31. Rubner, Y. 1999. Perceptual metrics for image database navigation. Ph.D. Thesis, Stanford University.Google Scholar
  32. Rubner, Y., Tomasi, C., and Guibas, J. 1998. The earth mover's distance as a metric for image retrieval. Technical Report STAN-CS-TN-98-86, Department of Computer Science, Stanford University.Google Scholar
  33. Strang, G. 1986. Introduction to Applied Mathematics. Wellesley-Cambridge Press: Wellesley, MA.Google Scholar
  34. Taylor,M. 1996. Partial Differential Equations III. Springer-Verlag: New York.Google Scholar
  35. Thirion, J.-P. 1995. Fast non-rigid matching of non-rigid images. In Medical Robotics and Computer Aided Surgery (MRCAS' 95), Baltimore, p. 4754.Google Scholar
  36. Toga, A. 1999. Brain Warping. Academic Press: San Diego.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Steven Haker
    • 1
  • Lei Zhu
    • 2
  • Allen Tannenbaum
    • 2
  • Sigurd Angenent
    • 3
  1. 1.Surgical Planning Laboratory Brigham and Women's Hospital and Harvard Medical SchoolBostonUSA
  2. 2.Departments of Electrical, Computer and Biomedical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of MathematicsUniversity of WisconsinMadisonUSA

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