Abstract
We propose a new approach in the context of diffeomorphic image matching with free boundaries. A region of interest is triangulated over a template, which is considered as a grey level textured mesh. A diffeomorphic transformation is then approximated by the piecewise affine deformation driven by the displacements of the vertices of the triangles. This provides a finite dimensional, landmark-type, reduction for this dense image comparison problem. Based on an optimal control model, we analyze and compare two optimization methods formulated in terms of the initial momentum: direct optimization by gradient descent, or root-finding for the transversality equation, enhanced by a preconditioning of the Jacobian. We finally provide a series of numerical experiments on digit and face matching.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Beg, M.F., Miller, M.I., Ratnanather, J.T., Trouvé, A., Younes, L.: Computing Metrics on Diffeomorphisms for Computational Anatomy. International Journal of Computer Vision (2004)
Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flow of diffeomorphisms. Int. J. Comp. Vis. 61, 139–157 (2005)
Bookstein, F.L.: Morphometric tools for landmark data; geometry and biology. Cambridge University Press, Cambridge (1991)
Broit, R., Bajcsy, C.: Matching of deformed images. In: Proc. 6th Int. Conf. of Pattern Recogition, München, pp. 351–353 (1982)
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)
Dupuis, P., Grenander, U., Miller, M.I.: Variational problems on flows of diffeomorphisms for image matching. Quart. Appl. Math. LVI, 587–600 (1998)
Garcin, L., Rangarajan, A., Younes, L.: Non rigid registration of shapes via diffeomorphic point matching and clustering
Glaunes, J., Trouvé, A., Younes, L.: Modeling planar shape variation via hamiltonian flow of curves (submitted)
Glaunes, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. J. Math. Imaging Vision 20(1-2), 179–200 (2004)
Grenander, U., Miller, M.I.: Computational anatomy: An emerging discipline. Quaterly of Applied Mathematics LVI, 617–694 (1998)
Holm, D.D., Ratnanather, J.T., Trouvé, A., Younes, L.: Solition Dynamics in Computational Anatomy. Neuroimage (2004)
Hull, D.G.: Optimal Control Theory for Applications, vol. XIX. Springer, Heidelberg (2003)
Joshi, S., Miller, M.I.: Landmark matching via Large Deformation diffeomorphisms. IEEE Transaction in Image Processing 9, 1357–1370 (2000)
Marsland, S., Twining, C.: Clamped-plate splines and the optimak flow bounded diffeomorphisms. In: Complex Stochastic Systemp and Engineering, pp. 85–103. Clarendon Press, Oxford (2002)
Miller, M.I., Trouvé, A., Younes, L.: On the metrics and Euler Lagrange Equations of Computational Anatomy. Annual Review of Biomedical Engeneering 4, 375–405 (2002)
Miller, M.I., Younes, L.: Group action, diffeomorphism and matching: a general framework. Int. J. Comp. Vis. 41, 61–84 (2001)
Richard, F., Cohen, L.: A new image registration technique with free boundary constraints: application to mammography. Computer Vision and Image Understanding 89(2-3), 166–196 (2003); Special Issue on Nonrigid Registration
Trouvé, A.: Diffeomorphisms Groups and Pattern Matching in Image Analysis. International Journal of Computer Vision 28(3), 213–221 (1998)
Trouvé, A., Younes, L.: Local geometry of deformable template. SIAM Journal of Numerical Analysis (May 2004)
Vaillant, M., Glaunes, J.: Surface Matching via Currents. In: Proceedings of Information Processing in Medical Imaging (IPMI 2005), pp. 381–392 (2005)
Vaillant, M., Miller, M.I., Trouvé, A., Younes, L.: Statistics on diffeomorphisms via tangent space representaion. NeuroImage 23, S161–S169 (2004)
Wahba, G.: Spine Models for Observational Data. SIAM, Philadelphia (1990)
Younes, L.: Optimal matching between shapes via elastic deformations. Image and Vision Computing (1999)
Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58, 565–586 (1998)
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
Allassonnière, S., Trouvé, A., Younes, L. (2005). Geodesic Shooting and Diffeomorphic Matching Via Textured Meshes. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11585978_24
Download citation
DOI: https://doi.org/10.1007/11585978_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30287-2
Online ISBN: 978-3-540-32098-2
eBook Packages: Computer ScienceComputer Science (R0)