Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
- 2.9k Downloads
This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I0, I1 are given and connected via the diffeomorphic change of coordinates I0○ϕ−1=I1 where ϕ=Φ1 is the end point at t= 1 of curve Φ t , t∈[0, 1] satisfying .Φ t =vt (Φ t ), t∈ [0,1] with Φ0=id. The variational problem takes the form
where ‖vt‖ V is an appropriate Sobolev norm on the velocity field vt(·), and the second term enforces matching of the images with ‖·‖L2 representing the squared-error norm.
In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt, t∈[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ∫01‖vt‖ V dt on the geodesic shortest paths.
Unable to display preview. Download preview PDF.
- Bajcsy, R. and Broit, C. 1982. Matching of deformed images. In Proc. 6th Int. Joint Conf Patt. Recog., pp. 351-353.Google Scholar
- Bajcsy, R., Lieberson, R., and Reivich, M. 1983. A computerized system for the elastic matching of deformed radiographic images to idealized atlas images. Journal of Computer Assisted Tomogra-phy, 7(4):618-625.Google Scholar
- Broit, C. 1981. Optimal registration of deformed images. PhD thesis,University of Pennsylvania.Google Scholar
- Do Carmo, M.P 1976. Differential geometry of curves and surfaces. Prentice-Hall Engineering/Science/Mathematics.Google Scholar
- Do Carmo, M.P 1993. Riemannian Geometry. Birkhauser.Google Scholar
- Christensen, G. 1994. Deformable shape models for anatomy. PhD Thesis, Dept. of Electrical Engineering, Sever Institute of Tech-nology, Washington Univ., St. Louis, MO.Google Scholar
- Dupuis, P., Grenander, U., and Miller, M.I. 1998. Variational prob-lems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, LVI:587-600.Google Scholar
- Grenander, U. and Miller, M.I. 1998. Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, 56:617-694.Google Scholar
- Morton, K.W. and Mayers, D.E 1996. Numerical Solution of Partial Differential Equations. Cambridge University Press, University of Cambridge.Google Scholar
- Robb, R.A. 1999. Biomedical Imaging, Vizualization and Analysis. John Wiley and Sons, Inc., New York, NY.Google Scholar
- Trouvé, A. 1995. An infinite dimensional group approach for physics based models in patterns recognition. Preprint.Google Scholar