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Digital Homeomorphisms in Deformable Registration

  • Pierre-Louis Bazin
  • Lotta Maria Ellingsen
  • Dzung L. Pham
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4584)

Abstract

A common goal in deformable registration applications is to produce a spatial transformation that is diffeomorphic, thereby preserving the topology of structures being transformed. Because this constraint is typically enforced only on the continuum, however, topological changes can still occur within discretely sampled images. This work discusses the notion of homeomorphisms in digital images, and how it differs from the diffeomorphic/homeomorphic concepts in continuous spaces commonly used in medical imaging. We review the differences and problems brought by considering functions defined on a discrete grid, and propose a practical criterion for enforcing digital homeomorphisms in the context of atlas-based segmentation.

Keywords

Image Registration Euler Characteristic Simple Point Deformable Image Registration Deformable Registration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Grenander, U., Miller, M.I.: Computational anatomy: an emerging discipline. Q. Appl. Math. LVI(4), 617–694 (1998)Google Scholar
  2. 2.
    Thompson, P.M., Toga, A.W.: A framework for computational anatomy. Computing and Visualization in Science 5, 1–12 (2002)CrossRefGoogle Scholar
  3. 3.
    Resnick, S.M., Pham, D.L., Kraut, M.A., Zonderman, A.B., Davatzikos, C.: Longitudinal MRI studies of older adults: A shrinking brain. J. Neuroscience 23(8), 3295–3301 (2003)Google Scholar
  4. 4.
    Thompson, P., Hayashi, K., Sowell, E., Gogtay, N., Giedd, J., Rapoport, J., de Zubicaray, G., Janke, A., Rose, S., Semple, J., Doddrell, D., Wang, Y., van Erp, T., Cannon, T., Toga, A.: Mapping cortical change in Alzheimer’s disease, brain development, and schizophrenia. NeuroImage, Special Issue on Mathematics in Brain Imaging (2004)Google Scholar
  5. 5.
    Gerig, G., Joshi, S., Fletcher, T., Gorczowski, K., Xu, S., Pizer, S., Styner, M.: Statistics of populations of images and its embedded objects: Driving applications in neuroimaging. In: Proc. IEEE Int. Symp. Biomedical Imaging, Arlington 2006 (2006)Google Scholar
  6. 6.
    Christensen, G.E.: Deformable shape models for anatomy. Electrical Engineering D.Sc. Dissertation, Washington University, St. Louis, Missouri (1994)Google Scholar
  7. 7.
    Christensen, G.E., Joshi, S.C., Miller, M.I.: Volumetric transformation of brain anatomy. IEEE Trans. Medical Imaging 16(6), 864–877 (1997)CrossRefGoogle Scholar
  8. 8.
    Christensen, G.E., Johnson, H.J.: Consistent image registration. IEEE Trans. Medical Imaging 20(7), 568–582 (2001)CrossRefGoogle Scholar
  9. 9.
    Rohde, G.K., Aldroubi, A., Dawant, B.M.: The adaptive bases algorithm for intensity-based nonrigid image registration. IEEE Trans. Medical Imaging 22(11), 1470–1479 (2003)CrossRefGoogle Scholar
  10. 10.
    Avants, B.B., Gee, J.C.: Shape averaging with diffeomorphic flows for atlas creation. In: Proc. IEEE Int. Symp. Biomedical Imaging, pp. 595–598. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  11. 11.
    Narayanan, R., Fessler, J.A., Park, H., Meyer, C.H.: Diffeomorphic nonlinear transformations: a local parametric approach for image registration. In: Proc. Int. Conf. Information Processing in Medical Imaging, Glenwood Springs 2005 (2005)Google Scholar
  12. 12.
    Noblet, V., Heinrich, C., Heitz, F., Armspach, J.-P.: 3D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization. IEEE Trans. Image Processing 14(5), 553–566 (2005)CrossRefGoogle Scholar
  13. 13.
    Rueckert, D., Sonoda, L., Hayes, C., Hill, D., Leach, M., Hawkes, D.: Non-rigid registration using free-form deformations: Application to breast MR images. IEEE Trans. Medical Imaging 18, 712–721 (1999)CrossRefGoogle Scholar
  14. 14.
    Shen, D., Davatzikos, C.: Hammer: Hierarchical attribute matching mechanism for elastic registration. IEEE Trans. Medical Imaging 21(11) (2002)Google Scholar
  15. 15.
    Vemuri, B.C., Ye, J., Chen, Y., Leonard, C.M.: Image registration via level-set motion: Applications to atlas-based segmentation. Medical Image Analysis 20(1), 1–20 (2003)CrossRefGoogle Scholar
  16. 16.
    Kybic, J., Unser, M.: Fast parametric elastic image registration. IEEE Trans. Image Processing 12(11), 1427–1442 (2003)CrossRefGoogle Scholar
  17. 17.
    Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing 48(3), 357–393 (1989)CrossRefGoogle Scholar
  18. 18.
    Bertrand, G., Couprie, M.: A model for digital topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 229–241. Springer, Heidelberg (1999)Google Scholar
  19. 19.
    Henle, M. (ed.): A Combinatorial Introduction to topology. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  20. 20.
    Sethian, J.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  21. 21.
    Lorensen, W.E., Cline, H.E.: Marching cubes: A high resolution 3D surface construction algorithm. In: Proc. SIGGRAPH’87 1987, vol. 21, pp. 163–169 (1987)Google Scholar
  22. 22.
    O’Neill, B.: Elementary Differential Geometry. Academic Press, San Diego (1997)zbMATHGoogle Scholar
  23. 23.
    Han, X., Xu, C., Prince, J.L.: A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Analysis and Machine Intelligence 25(6), 755–768 (2003)CrossRefGoogle Scholar
  24. 24.
    Malandain, G., Bertrand, G., Ayache, N.: Topological segmentation of discrete surfaces. Int. J. Computer Vision 10(2), 183–197 (1993)CrossRefGoogle Scholar
  25. 25.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhood in cubic grids. Pattern Recognition Letters 15(10), 1003–1011 (1994)CrossRefGoogle Scholar
  26. 26.
    Couprie, M., Bezerra, F., Bertrand, G.: Topological operators for grayscale image processing. J. Electronic Imaging 10(4), 1003–1015 (2001)CrossRefGoogle Scholar
  27. 27.
    Mangin, J.F., Frouin, V., Bloch, I., Régis, J., López-Krahe, J.: From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. J. Mathematical Imaging and Vision 5, 297–318 (1995)CrossRefGoogle Scholar
  28. 28.
    Worth, A.: Internet brain seg mentation repository (1996), http://www.cma.mgh.harvard.edu/ibsr/
  29. 29.
    Bazin, P.L., Pham, D.: Topology-preserving tissue classification of magnetic resonance brain images. In: IEEE Trans. Medical Imaging, Special Issue on Computational Neuroanatomy (in press) (2007)Google Scholar
  30. 30.
    Ellingsen, L.M., Prince, J.L.: Mjolnir: Deformable image registration using feature diffusion. In: Proc. SPIE Medical Imaging Conf., San Diego (2006)Google Scholar
  31. 31.
    Bai, Y., Prince, J.L.: Octree-based topology-preserving isosurface simplification. In: Proc. IEEE Int. Conf. Computer Vision and Pattern Recognition (CVPR’06), New York, IEEE Computer Society Press, Los Alamitos (2006)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Pierre-Louis Bazin
    • 1
  • Lotta Maria Ellingsen
    • 1
  • Dzung L. Pham
    • 1
  1. 1.Johns Hopkins University, BaltimoreUSA

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