INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT

Said Kerrache, Yasushi Nakauchi

Abstract

In this paper, the recently proposed technique of constrained optimal transport is used to interpolate between images under specified constraints. The intensity values in both images are considered as mass distributions, and a flow of minimum kinetic energy is computed to transport the initial distribution to the final one, while satisfying specified constraints on the intermediate mass as well as the the velocity or the momentum field. As an application, the proposed method is used for interpolating between images under constraint on the volume expansion and contraction. This is achieved by imposing bounds on the divergence of the velocity field of the flow. This constraint is discretized then integrated into the problem Lagrangian using the augmented Lagrangian method. A variation of the solution is also presented, where the constraint is decoupled into two constraints coordinated by an additional Lagrange multiplier. This allows a considerable speedup, though numerical robustness decreases in certain cases. Constrained image interpolation by optimal transport has potential applications in image registration. In particular, the proposed method for controlling the volume change has potential application in registration of images under volume change constraints as it is the case for medical images depicting muscle movements or those with contrast enhancing structures.

References

  1. Beauchemin, S. S. and Barron, J. L. (1995). The computation of optical flow. ACM Comput. Surv., 27:433-466.
  2. Beier, T. and Neely, S. (1992). Feature-based image metamorphosis. SIGGRAPH Comput. Graph., 26:35-42.
  3. Benamou, J.-D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84(3):375-393.
  4. Bro-Nielsen, M. and Gramkow, C. (1996). Fast fluid registration of medical images. In Proceedings of the 4th International Conference on Visualization in Biomedical Computing, pages 267-276, London, UK. Springer-Verlag.
  5. Chorin, A. J. and Marsden, J. E. (1993). A Mathematical Introduction to Fluid Mechanics (Texts in Applied Mathematics) (v. 4). Springer.
  6. Cohen, I. M. and Kundu, P. K. (2004). Fluid Mechanics, Third Edition. Academic Press.
  7. Fortin, M. e. and Glowinski, R. e. (1983). Augmented Lagrangian methods: Applications to the numerical solution of boundary-value problems. Studies in Mathematics and its Applications, 15. Amsterdam-New York-Oxford: North-Holland. XIX, 340 p.
  8. Glowinski, R. and Tallec, P. L. (1989). Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM.
  9. Haber, E., Horesh, R., and Modersitzki, J. (2010). Numerical optimization for constrained image registration. Numerical Linear Algebra with Applications, 17:343- 359.
  10. Haber, E. and Modersitzki, J. (2004a). Numerical methods for volume preserving image registration. Inverse Problems, 20(5):1621.
  11. Haber, E. and Modersitzki, J. (2004b). Volume preserving image registration. In Barillot, C., Haynor, D. R., and Hellier, P., editors, Medical Image Computing and Computer-Assisted Intervention MICCAI 2004, volume 3216 of Lecture Notes in Computer Science, pages 591-598. Springer Berlin / Heidelberg.
  12. Haker, S., Zhu, L., Tannenbaum, A., and Angenent, S. (2004). Optimal mass transport for registration and warping. Int. J. Comput. Vision, 60(3):225-240.
  13. Kerrache, S. and Nakauchi, Y. (2011). Computing constrained energy-minimizing flows. In 3rd International Conference on Computer Research and Development. Accepted.
  14. Manning, R. A. and Dyer, C. R. (1999). Interpolating view and scene motion by dynamic view morphing. In Proc. Computer Vision and Pattern Recognition Conf., volume 1, pages 388-394.
  15. Museyko, O., Stiglmayr, M., Klamroth, K., and Leugering, G. (2009). On the application of the mongekantorovich problem to image registration. SIAM J. Img. Sci., 2(4):1068-1097.
  16. Rohlfing, T., Maurer, C. R., Bluemke, D. A., and Jacobs, M. A. (2003). Volume-preserving nonrigid registration of MR breast images using free-form deformation with an incompressibility constraint. Medical Imaging, IEEE Transactions on, 22(6):730-741.
  17. Schaefer, S., McPhail, T., and Warren, J. (2006). Image deformation using moving least squares. ACM Trans. Graph., 25:533-540.
  18. Seitz, S. M. and Dyer, C. R. (1996). View morphing. In SIGGRAPH96, pages 21-30.
  19. Stich, T., Linz, C., Wallraven, C., Cunningham, D., and Magnor, M. (2008). Perception-motivated interpolation of image sequences. In Proceedings of the 5th symposium on Applied perception in graphics and visualization, APGV 7808, pages 97-106, New York, NY, USA. ACM.
  20. Villani, C. (2009). Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften 338. Berlin: Springer. .
  21. Wolberg, G. (1998). Image morphing: A survey. The Visual Computer, 14:360-372.
  22. Zitova, B. (2003). Image registration methods: a survey. Image and Vision Computing, 21(11):977-1000.
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Paper Citation


in Harvard Style

Kerrache S. and Nakauchi Y. (2011). INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT . In Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011) ISBN 978-989-8425-47-8, pages 75-84. DOI: 10.5220/0003375200750084


in Bibtex Style

@conference{visapp11,
author={Said Kerrache and Yasushi Nakauchi},
title={INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT},
booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)},
year={2011},
pages={75-84},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003375200750084},
isbn={978-989-8425-47-8},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2011)
TI - INTERPOLATION BETWEEN IMAGES BY CONSTRAINED OPTIMAL TRANSPORT
SN - 978-989-8425-47-8
AU - Kerrache S.
AU - Nakauchi Y.
PY - 2011
SP - 75
EP - 84
DO - 10.5220/0003375200750084