Abstract
We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Thompson, D.W.: On Growth and Forms. Cambridge University Press, Cambridge (1917)
Bookstein, F.L.: Morphometric tools for landmark data; geometry and biology. Cambridge University Press, Cambridge (1991)
Davatzikos, C.: Spatial transformation and registration of brain images using elastically deformable models. Comp. Vision and Image Understanding 66(2), 207–222 (1997)
Joshi, S.C., Miller, M.I.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Processing 9(8), 1357–1370 (2000)
Camion, V., Younes, L.: Geodesic interpolating splines. In: Figueiredo, M., Zerubia, J., Jain, A.K. (eds.) EMMCVPR 2001. LNCS, vol. 2134, pp. 513–527. Springer, Heidelberg (2001)
Glaunès, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. Journal of Mathematical Imaging and Vision, MIA 2002 special (20) (2004)
Chui, H., Rangarajan, A.: A new point matching algorithm for non-rigid registration. Computer Vision and Image Understanding 89, 114–141 (2003)
Wang, Y., Peterson, B.S., Staib, L.H.: 3d brain surface matching based on geodesics and local geometry. Computer Vision and Image Understanding 89, 252–271 (2003)
Davies, R.H., Cootes, T.F., Taylor, C.J.: 3D statistical shape models using direct optimisation of description length. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 3–20. Springer, Heidelberg (2002)
Glaunès, J., Trouvé, A., Younes, L.: Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching. In: CVPR, pp. 712–718. IEEE Computer Society Press, Los Alamitos (2004)
deRham, G.: Variétés différentiables, formes, courants, formes harmoniques. Act. Sci. Indust. 1222 (1955)
Morgan, F.: Geometric measure theory, 2nd edn. Acad. Press, INC., New York (1995)
do Carmo, M.P.: Differential Forms and Applications. Springer, Heidelberg (1994)
Wahba, G.: Spline Models for Observational Data. In: CBMS-NSF Regional conference series. SIAM, Philadelphia (1990)
Trouvé, A.: An infinite dimensional group approach for physics based models. Technical report, electronically (1995), available at http://www.cis.jhu.edu
Dupuis, P., Grenander, U., Miller, M.I.: Variational problems on flows of diffeomorphisms for image matching. Quaterly of Applied Math. 56, 587–600 (1998)
Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting in computational anatomy. Technical report, Center for Imaging Science, Johns Hopkins University (2003)
Vaillant, M., Miller, M.I., Younes, L., Trouvé, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23, 161–169 (2004)
Miller, M.I., Trouvé, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering 4, 375–405 (2002)
USF HumanID 3D faces database, courtesy of Professor Sudeep Sarkar, University of South Florida, Tampa FL. http://marthon.csee.usf.edu/HumanID/
Chupin, M., Hasboun, D., Baillet, S., Kinkingnéhun, S., Dubois, B., Garnero, L.: Competitive segmentation of the hippocampus and the volumetry in alzheimer’s disease. In: 10th Annual Meeting of the Organization for Human Brain Mapping, June 13-17 (2004)
Hirani, A.N.: Discrete exterior calculus. PhD thesis, California Institute of Technology (2003)
Cohen-Steiner, D., Morvan, J.-M.: Restricted delaunay triangulations and normal cycle. In: SCG 2003: Proceedings of the nineteenth annual symposium on Computational geometry, San Diego, California, USA, pp. 312–321. ACM Press, New York (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Vaillant, M., Glaunès, J. (2005). Surface Matching via Currents. In: Christensen, G.E., Sonka, M. (eds) Information Processing in Medical Imaging. IPMI 2005. Lecture Notes in Computer Science, vol 3565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505730_32
Download citation
DOI: https://doi.org/10.1007/11505730_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26545-0
Online ISBN: 978-3-540-31676-3
eBook Packages: Computer ScienceComputer Science (R0)