Skip to main content
SpringerLink
Log in

Registration of Anatomical Images Using Paths of Diffeomorphisms Parameterized with Stationary Vector Field Flows

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

Computational Anatomy aims for the study of variability in anatomical structures from images. Variability is encoded by the spatial transformations existing between anatomical images and a template selected as reference. In the absence of a more justified model for inter-subject variability, transformations are considered to belong to a convenient family of diffeomorphisms which provides a suitable mathematical setting for the analysis of anatomical variability. One of the proposed paradigms for diffeomorphic registration is the Large Deformation Diffeomorphic Metric Mapping (LDDMM). In this framework, transformations are characterized as end points of paths parameterized by time-varying flows of vector fields defined on the tangent space of a Riemannian manifold of diffeomorphisms and computed from the solution of the non-stationary transport equation associated to these flows. With this characterization, optimization in LDDMM is performed on the space of non-stationary vector field flows resulting into a time and memory consuming algorithm. Recently, an alternative characterization of paths of diffeomorphisms based on constant-time flows of vector fields has been proposed in the literature. With this parameterization, diffeomorphisms constitute solutions of stationary ODEs. In this article, the stationary parameterization is included for diffeomorphic registration in the LDDMM framework. We formulate the variational problem related to this registration scenario and derive the associated Euler-Lagrange equations. Moreover, the performance of the non-stationary vs the stationary parameterizations in real and simulated 3D-MRI brain datasets is evaluated. Compared to the non-stationary parameterization, our proposal provides similar results in terms of image matching and local differences between the diffeomorphic transformations while drastically reducing memory and time requirements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

  • Arnold, V. I. (1989). Mathematical methods of classical mechanics. Berlin: Springer.

    Google Scholar 

  • Arsigny, V., Commonwick, O., Pennec, X., & Ayache, N. (2006a). Statistics on diffeomorphisms in a Log-Euclidean framework. In Lecture notes in computer science (LNCS) : Vol. 4190. Proc. of the 9th international conference on medical image computing and computer assisted intervention (MICCAI’06) (pp. 924–931). Berlin: Springer.

    Google Scholar 

  • Arsigny, V., Pennec, X., & Ayache, N. (2006b). Bi-invariant means in Lie groups. Application to left-invariant polyaffine transformations (Research Report RR-5885). INRIA Sophia-Antipolis.

  • Avants, B., & Gee, J. C. (2004). Shape averaging with diffeomorphic flows for atlas creation. In Proc. of the 2nd IEEE international symposium on biomedical imaging: from nano to macro (ISBI’04) (pp. 595–598) 2004.

  • Avants, B., Schoenemann, P. T., & Gee, J. (2006). Lagrangian frame diffeomorphic image registration: morphometric comparison of human and chimpanzee cortex. Medical Image Analysis, 10(3), 397–412.

    Article  Google Scholar 

  • Avants, B. B., Epstein, C. L., Grossman, M., & Gee, J. C. (2008). Symmetric dieomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain. Medical Image Analysis, 12, 26–41.

    Article  Google Scholar 

  • Beg, M. F. (2003). Variational and computational methods for flows of diffeomorphisms in image matching and growth in computational anatomy. Ph.D. thesis, John Hopkins University, USA.

  • Beg, M. F., & Khan, A. (2006). Computing an average anatomical atlas using LDDMM and geodesic shooting. In Proc. of the 3rd IEEE international symposium on biomedical imaging: from nano to macro (ISBI’06) (pp. 1116–1119) 2006.

  • Beg, M. F., & Khan, A. (2007). Symmetric data attachment terms for large deformation image registration. IEEE Transactions on Medical Imaging, 26(9), 1179–1189.

    Article  Google Scholar 

  • Beg, M. F., Miller, M. I., Trouve, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal Computer Vision, 61(2), 139–157.

    Article  Google Scholar 

  • Christensen, G. E. (1999). Consistent linear-elastic transformations for image matching. In Lecture notes in computer science (LNCS). Proc. of international conference on information processing and medical imaging (IPMI’99) (pp. 224–237). Berlin: Springer.

    Chapter  Google Scholar 

  • Christensen, G. E., Rabbitt, R. D., & Miller, M. I. (1996). Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing, 5(10), 1435–1447.

    Article  Google Scholar 

  • Cotter, C. J., & Holm, D. D. (2006). Singular solutions, momentum maps and computational anatomy. In Proc. of the 1st international workshop on mathematical foundations of computational anatomy (MFCA’06) (pp. 18–28) 2006.

  • Csernansky, J. G., Wang, L., Joshi, S. C., Ratnanather, J. T., & Miller, M. I. (2004). Computational anatomy and neuropsychiatric disease: probabilistic assessment of variation and statistical inference of group difference, hemispheric asymmetry, and time-dependent change. Neuroimage, 23(1), 56–68.

    Article  Google Scholar 

  • Davis, B., Fletcher, P. T., Bullit, E., & Joshi, S. (2007). Population shape regression from random design data. In Proc. of the 11th IEEE international conference on computer vision (ICCV’07) 2007.

  • DoCarmo, M. P. (1992). Riemannian geometry. Boston: Birkhauser.

    Google Scholar 

  • Dodgas, B., Sattuck, D. W., & Leahy, R. M. (2005). Segmentation of skull and scalp in 3D human MRI using mathematical morphology. Human Brain Mapping, 26(4), 273–285.

    Article  Google Scholar 

  • Dupuis, P., Grenander, U., & Miller, M. (1998). Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, 3, 587–600.

    MathSciNet  Google Scholar 

  • Ebin, D., & Marsden, J. (1970). Groups of diffeomorphisms and the motion of an incompressible fluid. Annals of Mathematics, 92, 102–103.

    Article  MathSciNet  Google Scholar 

  • Garcin, L., & Younes, L. (2006). Geodesic matching with free extremities. Journal of Mathematical Imaging Vision, 25, 329–340.

    Article  MathSciNet  Google Scholar 

  • Gerig, G., Davis, B., Lorenzen, P., Xu, S., Jomier, M., Piven, J., & Joshi, S. (2006). Computational anatomy to assess longitudinal trajectory of brain growth. In Proc. of the 3rd international symposium on 3D data processing, visualization, and transmission (pp. 1041–1047) 2006.

  • Grabowski, J. (1988). Free subgroups of diffeomorphism groups. Fundamenta Mathematicae, 131, 103–121.

    MATH  MathSciNet  Google Scholar 

  • Grenander, U. (1994). General pattern theory. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Hernandez, M. (2008). Variational techniques with applications to segmentation and registration of medical images. Ph.D. Thesis, University of Zaragoza, Spain.

  • Holm, D. D., Ratnanather, J. T., Trouve, A., & Younes, L. (2004). Soliton dynamics in computational anatomy. Neuroimage, 23, 170–178.

    Article  Google Scholar 

  • Joshi, S., Davis, B., Jomier, M., & Gerig, G. (2004). Unbiased diffeomorphic atlas construction for computational anatomy. Neuroimage, 23, 151–160.

    Article  Google Scholar 

  • Leow, A., Klunder, A. D., Jack, C. R., & Toga, A. W. (2006). Longitudinal stability of MRI for mapping brain change using tensor-based morphometry. Neuroimage, 31, 627–640.

    Article  Google Scholar 

  • Lepore, N., Brun, C. A., Chiang, M. C., Chou, Y. Y., Dutton, R. A., Hayashi, K. M., Lopez, O. L., Aizenstein, H. J., Toga, A. W., Becker, J. T., & Thompson, P. M. (2006). Multivariate statistics of the Jacobian matrices in tensor based morphometry and their application to HIV/AIDS. In Lecture notes in computer science (LNCS) : Vol. 4190. Proc. of the 9th international conference on medical image computing and computer assisted intervention (MICCAI’06) (pp. 191–198). Berlin: Springer.

    Google Scholar 

  • Lorenzen, P., Prastawa, M., Davis, B., Gerig, G., Bullitt, E., & Joshi, S. (2006). Multi-modal image set registration and atlas formation. Medical Image Analysis, 10, 440–451.

    Article  Google Scholar 

  • Michor, P. W., & Mumford, D. (2006). An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Applied and Computational Harmonic Analysis, 23(1), 74–113.

    Article  MathSciNet  Google Scholar 

  • Miller, M. I. (2004). Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms. Neuroimage, 23, 19–33.

    Article  Google Scholar 

  • Miller, M. I., Trouve, A., & Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24, 209–228.

    Article  MathSciNet  Google Scholar 

  • Modersitzki, J. (2004). Numerical methods for image registration. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Nocedal, J., & Wright, S. J. (1999). Numerical optimization. New-York: Springer.

    Book  MATH  Google Scholar 

  • Noether, E. (1918). Invariante variationsprobleme. Nachr. v. d. Ges. d. Wiss. zu Gottingen, 235–257.

  • Qiu, A., Younes, L., Miller, M. I., & Csernansky, J. G. (2007). Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the Alzheimer’s type. Neuroimage, 40, 68–76.

    Article  Google Scholar 

  • Schmid, R. (2004). Infinite dimensional Lie groups with applications to mathematical physics. Journal of Geometry Symmetry in Physics, 1, 54–120.

    MATH  Google Scholar 

  • Staniforth, A., & Cote, J. (1991). Semi-Lagrangian integration schemes for atmospheric models-a review. Mon. Weather Rev., 119, 2206–2223.

    Article  Google Scholar 

  • Thompson, P. M., Giedd, J. N., Woods, R. P., MacDonald, D., Evans, A. C., & Toga, A. W. (2000). Growth patterns in the developing brain detected by using continuum mechanical tensor maps. Nature, 404, 190–193.

    Article  Google Scholar 

  • Thompson, P. M., Mega, M. S., Woods, R. P., Zoumalan, C. I., Lindshield, C. J., Blanton, R. E., Moussai, J., Holmes, C. J., Cummings, J. L., & Toga, A. W. (2001). Cortical change in Alzheimer’s disease detected with a disease-specific population-based brain atlas. Cerebral Cortex, 11(1), 1–16.

    Article  Google Scholar 

  • Wang, L., Swank, J. S., Glick, I. E., Gado, M. H., Miller, M. I., Morris, J. C., & Csernansky, J. G. (2003). Changes in hippocampal volume and shape across time distinguish dementia of the Alzheimer type from healthy aging. Neuroimage, 20, 667–682.

    Article  Google Scholar 

  • Wang, L., Beg, F., Ratnanather, T., Ceritoglu, C., Younes, L., Morris, J. C., Csernansky, J. G., & Miller, M. I. (2007). Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE Transactions on Medical Imaging, 26(4), 462–470.

    Article  Google Scholar 

  • Younes, L. (2007). Jacobi fields in groups of diffeomorphisms and applications. Quarterly of Applied Mathematics, 65, 113–134.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and System Department, Communication Technology Group (GTC), Aragon Institute of Engineering Research (I3A), University of Zaragoza, 50009, Zaragoza, Spain

    Monica Hernandez

  2. Communication Technology Group (GTC), Aragon Institute of Engineering Research (I3A), University of Zaragoza, 50009, Zaragoza, Spain

    Matias N. Bossa & Salvador Olmos

Authors
  1. Monica Hernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matias N. Bossa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Salvador Olmos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Monica Hernandez.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hernandez, M., Bossa, M.N. & Olmos, S. Registration of Anatomical Images Using Paths of Diffeomorphisms Parameterized with Stationary Vector Field Flows. Int J Comput Vis 85, 291–306 (2009). https://doi.org/10.1007/s11263-009-0219-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-009-0219-z

Keywords

Search

Navigation