Abstract
We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space ℝd. Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in ℝ2 and ℝ3. Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and experiments with 3D curves extracted from brain cortical surfaces are presented.
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Allassonnière, S., Trouvé, A., & Younes, L. (2005). Geodesic shooting and diffeomorphic matching via textured meshes. In EMMCVPR (pp. 365–381).
Avants, B., & Gee, J. C. (2004). Geodesic estimation for large deformation anatomical shape and intensity averaging. NeuroImage, 23, 139–150.
Bakircioglu, M., Grenander, U., Khaneja, N., & Miller, M. I. (1998). Curve matching on brain surfaces using frenet distances. Human Brain Mapping, 6(5–6), 329–333.
Bakircioglu, M., Joshi, S., & Miller, M. (1999). Landmark matching on brain surfaces via large deformation diffeomorphisms on the sphere. In Image processing : Vol. 3661. Proc. SPIE medical imaging 1999 (pp. 710–715). SPIE: Bellingham.
Beg, M. F. (2003). Variational and computational methods for flows of diffeomorphisms in image matching and growth in computational anatomy. Ph.D. dissertation, Johns Hopkins University.
Beg, M. F., Miller, M. I., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157.
Besl, P., & McKay, N. (1992). A method for registration of 3-d shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(2), 239–256.
Camion, V., & Younes, L. (2001). Geodesic interpolating splines. In M. Figueiredo, J. Zerubia, & K. Jain (Eds.), Lecture notes in computer sciences : Vol. 2134. EMMCVPR 2001. Berlin: Springer.
Cao, Y., Miller, M., Winslow, R., & Younes, L. (2005a). Large deformation diffeomorphic metric mapping of vector fields. IEEE Transactions on Medical Imaging, 24, 1216–1230.
Cao, Y., Miller, M. I., Winslow, R. L., & Younes, L. (2005b). Large deformation diffeomorphic metric mapping of fiber orientations. In ICCV (pp. 1379–1386). Los Alamitos: IEEE Comput. Soc.
Cox, M. F., & Cox, M. A. A. (2001). Multidimensional scaling. Boca Raton: Chapman and Hall.
Dupuis, P., Grenander, U., & Miller, M. I. (1998). Variational problems on flows of diffeomorphisms for image matching. Quaterly of Applied Mathematics, 56, 587–600.
Durrleman, S., Pennec, X., Trouve, A., & Ayache, N. (2007). Measuring brain variability via sulcal lines registration: a diffeomorphic approach. In Int. conf. med. image comput. comput. assist. interv. (pp. 675–682).
Feldmar, J., & Ayache, N. (1996). Rigid, affine and locally affine registration of free-form surfaces. International Journal of Computer Vision, 18(2), 99–119.
Fillard, P., Arsigny, V., Pennec, X., Hayashi, K., Thompson, P., & Ayache, N. (2007). Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines. Neuroimage, 34, 639–650.
Gee, J. C., & Bajcsy, R. K. (1999). Elastic matching: Continuum mechanical and probabilistic analysis. In A. W. Toga (Ed.), Brain warping (pp. 183–196). San Diego: Academic Press.
Glaunès, J. (2005). Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes etl l’anatomie numérique. Ph.D. dissertation, Université Paris 13.
Glaunès, J., Trouvé, A., & Younes, L. (2004). Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching. In CVPR (pp. 712–718). Los Alamitos: IEEE Comput. Soc.
Glaunès, J., Trouvé, A., & Younes, L. (2006). Modeling planar shape variation via hamiltonian flows of curves. In H. Krim & A. Yezzi (Eds.), Statistics and analysis of shapes. Boston: Birkhauser.
Grenander, U., & Miller, M. I. (1998). Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, 56(4), 617–694.
Han, X., Xu, C., & Prince, J. L. (2001). A topology preserving deformable model using level set. In CVPR’2001 (Kauai, HI) (Vol. 2, pp. 765–770). Los Alamitos: IEEE Comput. Soc.
Han, X., Xu, C., Braga-Neto, U., & Prince, J. (2002). Topology correction in brain cortex segmentation using a multiscale, graph-based algorithm. IEEE Transactions on Medical Imaging, 21, 109–121.
Helm, P. A., Younes, L., Beg, M. F., Ennis, D. B., Leclercq, C., Faris, O. P., McVeigh, E., Kass, D., Miller, M. I., & Winslow, R. L. (2006). Evidence of structural remodeling in the dyssynchronous failing heart. Circulation Research, 98, 125–132.
Joshi, S. C., & Miller, M. I. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing, 9(8), 1357–1370.
Joshi, M., Cui, J., Doolittle, K., Joshi, S., Van Essen, D., Wang, L., & Miller, M. I. (1999). Brain segmentation and the generation of cortical surfaces. NeuroImage, 9, 461–476.
Joshi, S. C., Davis, B., Jomier, M., & Gerig, G. (2004). Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage, 23, 151–160.
Joshi, A. A., Shattuck, D. W., Thompson, P. M., & Leahy, R. M. (2007). Registration of cortical surfaces using sulcal landmarks for group analysis of meg data. In International congress series: Vol. 1300. New frontiers in biomagnetism. Proceedings of the 15th international conference on biomagnetism (pp. 229–232), Vancouver, BC, Canada, 21–25 August 2006.
Klassen, E., Srivastava, A., Mio, W., & Joshi, S. H. (2003). Analysis of planar shapes using geodesic paths on shape spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(3), 372–383.
Leow, A., Thompson, P. M., Protas, H., & Huang, S.-C. (2004). Brain warping with implicit representations. In ISBI (pp. 603–606). Los Alamitos: IEEE Comput. Soc.
McLachlan, R. I., & Marsland, S. (2007). N-particle dynamics of the Euler equations for planar diffeomorphisms. Dynamical Systems, 22(3), 269–290.
Michor, P. W., & Mumford, D. (2007). An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Applied Computational Harmonic Analysis, 23(1), 74–113.
Miller, M. I., Massie, A. B., Ratnanather, J. T., Botteron, K. N., & Csernansky, J. G. (2000). Bayesian construction of geometrically based cortical thickness metrics. NeuroImage, 12, 676–687.
Miller, M. I., Trouvé, A., & Younes, L. (2002). On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4, 375–405.
Mio, W., & Srivastava, A. (2004). Elastic-string models for representation and analysis of planar shapes. In CVPR (2) (pp. 10–15).
Qiu, A., Younes, L., Wang, L., Ratnanather, J. T., Gillepsie, S. K., Kaplan, G., Csernansky, J. G., & Miller, M. I. (2007). Combining anatomical manifold information via diffeomorphic metric mappings for studying cortical thinning of the cingulate gyrus in schizophrenia. NeuroImage, 37, 821–833.
Ratnanather, J. T., Barta, P. E., Honeycutt, N. A., Lee, N., Morris, N. G., Dziorny, A. C., Hurdal, M. K., Pearlson, G. D., & Miller, M. I. (2003). Dynamic programming generation of boundaries of local coordinatized submanifolds in the neocortex: application to the planum temporale. NeuroImage, 20(1), 359–377.
Rettmann, M. E., Han, X., Xu, C., & Prince, J. L. (2002). Automated sulcal segmentation using watersheds on the cortical surface. NeuroImage, 15(2), 329–344.
Schmidt, F. R., Clausen, M., & Cremers, D. (2006). Shape matching by variational computation of geodesics on a manifold. In K. Franke, K.-R. Müller, & B. Nickolay (Eds.), Lecture notes in computer science : Vol. 4174. DAGM-symposium (pp. 142–151). Berlin: Springer.
Sharon, E., & Mumford, D. (2006). 2d-shape analysis using conformal mapping. International Journal of Computer Vision, 70(1), 55–75.
Thompson, P., & Toga, A. (1996). A surface-based technique for warping three-dimensional image of the brain. IEEE Transactions on Medical Imaging, 15(4), 402–417.
Thompson, P. M., Schwartz, C., Lin, R. T., Khan, A. A., & Toga, A. W. (1996). Three–dimensional statistical analysis of sulcal variability in the human brain. Journal of Neuroscience, 16(13), 4261–4274.
Thompson, P. M., Hayashi, K. M., Sowell, E. R., Gogtay, N., Giedd, J. N., Rapoport, J. L., de Zubicaray, G. I., Janke, A. L., Rose, S. E., Semple, J., Doddrell, D. M., Wang, Y., van Erp, T. G., Cannon, T. D., & Toga, A. W. (2004). Mapping cortical change in alzheimer’s disease, brain development, and schizophrenia. NeuroImage, 23, S2–S18.
Trouvé, A. (1995). An infinite dimensional group approach for physics based models (Technical report). Electronically available at http://www.cis.jhu.edu.
Twining, C., Marsland, S., & Taylor, C. (2002). Measuring geodesic distances on the space of bounded diffeomorphisms. In Proceedings of the British machine vision conference (BMVC), Cardiff, September 2002 (Vol. 2, pp. 847–856).
Vaillant, M., & Glaunès, J. (2005). Surface matching via currents. In Inform. proc. in med. imaging : Vol. 3565. Lecture notes in comput. sci. (pp. 381–392). Berlin: Springer.
Welker, W. (1990). Why does cerebral cortex fissure and fold? Cerebral Cortex, 83, 3–136.
Yang, C., Duraiswami, R., Gumerov, N., & Davis, L. (2003). Improved fast gauss transform and efficient kernel density estimation. In IEEE international conference on computer vision (pp. 464–471).
Younes, L. (1998). Computable elastic distances between shapes. SIAM Journal on Applied Mathematics, 58, 565–586.
Zhang, Z. (1994). Iterative point matching for registration of free-form curves and surfaces. International Journal of Computer Vision, 13(2), 119–152.
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J. Glaunès and A. Qiu contributed equally to this work.
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Glaunès, J., Qiu, A., Miller, M.I. et al. Large Deformation Diffeomorphic Metric Curve Mapping. Int J Comput Vis 80, 317–336 (2008). https://doi.org/10.1007/s11263-008-0141-9
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DOI: https://doi.org/10.1007/s11263-008-0141-9