Abstract
We present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane, targeted at robust registration between curves that require large deformations. This condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the Hamiltonian equations of motion; this means that each iteration of the nonlinear optimisation algorithm produces a geodesic (up to numerical discretisation). We ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition (conjugate momentum is normal to the curve). Explicitly introducing and solving for a reparameterisation variable allows the use of a point-to-point matching condition. The equations are discretised using the variational particle-mesh method. We provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations, to show that it is a viable, efficient and accurate method for obtaining geodesics between curves.
Similar content being viewed by others
References
Allassonnière, S., Kuhn, E., & Trouvé, A. (2008). Map estimation of statistical deformable template via nonlinear mixed effect models: Deterministic and stochastic approaches. In Proc. of mathematical foundations of computational anatomy.
Allassonnière, S., Kuhn, E., & Trouvé, A. (2010). Construction of Bayesian deformable models via stochastic approximation algorithm: A convergence study. Bernoulli, 16(3), 641–678.
Bauer, M., Harms, P., & Michor, P. W. (2010). Sobolev metrics on shape space of surfaces. Preprint.
Bauer, M., Harms, P., & Michor, P. W. (2012). Curvature weighted metrics on shape space of hypersurfaces in n-space. Differential Geometry and Its Applications, 30(1), 33–41.
Beg, M., Miller, M., Trouvé, A., & Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2), 139–157. doi:10.1023/B:VISI.0000043755.93987.aa.
Besl, P., & McKay, N. (1992). A method for registration of 3D shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 239–256.
Charpiat, G., Faugeras, O., & Keriven, R. (2005). Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations of Computational Mathematics, 1, 1–58.
Chetverikov, D., Svirko, D., Stepanov, D., & Krsek, P. (2002). The trimmed iterative closest point algorithm. In International conference on pattern recognition (pp. 545–548).
Cootes, T., Marsland, S., Twining, C., Smith, K., & Taylor, C. (2004). Groupwise diffeomorphic non-rigid registration for automatic model building. In T. Pajdla & J. Matas (Eds.), Lecture notes in computer science: Vol. 3024. Computer vision—ECCV 2004 (pp. 316–327). Berlin: Springer.
Cotter, C. J. (2008). The variational particle-mesh method for matching curves. Journal of Physics A: Mathematical and Theoretical, 41(34), 344,003 URL http://stacks.iop.org/1751-8121/41/i=34/a=344003.
Cotter, C., & Holm, D. (2010). Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2(1), 51–68.
Cotter, S., Dashti, M., Robinson, J., & Stuart, A. (2009). Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Problems, 25, 115,008.
Davies, R., Twining, C., Cootes, T., Waterton, J., & Taylor, C. (2002). 3D statistical shape models using direct optimisation of description length. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science: Vol. 2352. Computer vision: ECCV 2002 (pp. 1–17). Berlin: Springer.
Durrleman, S., Pennec, X., Trouvé, A., Gerig, G., & Ayache, N. (2009). Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets. In Lecture notes in computer science: Vol. 5761. Medical image computing and computer-assisted intervention MICCAI 2009 (pp. 297–304).
Feldmar, J., & Ayache, N. (1996). Rigid, affine and locally affine registration of free-form surfaces. International Journal of Computer Vision, 18, 99–119.
Fishbaugh, J., Durrleman, S., & Gerig, G. (2011). Estimation of smooth growth trajectories with controlled acceleration from time series shape data. In Proc. of medical image computing and computer assisted intervention (MICCAI’11).
Fletcher, P. T. (2004). Statistical variability in nonlinear spaces: Application to shape analysis and DT-MRI. Ph.D. thesis, Department of Computer Science, University of North Carolina.
Fletcher, P. T., Lu, C., Pizer, M., & Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging, 23, 995–1005.
Fletcher, P. T., Venkatasubramanian, S., & Joshi, S. (2008). Robust statistics on Riemannian manifolds via the geometric median. In Computer vision and pattern recognition (pp. 1–8).
Frank, J., & Reich, S. (2004). The hamiltonian particle-mesh method for the spherical shallow water equations. Atmospheric Science Letters, 5(5), 89–95.
Gambaruto, A. M., Taylor, D. J., & Doorly, D. J. (2008). Modelling nasal airflow using a Fourier descriptor representation of geometry. International Journal for Numerical Methods in Fluids, 2071–2091. doi:10.1002/fld.1866.
Gay-Balmaz, F., & Ratiu, T. S. (2011). Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 3(1), 41–79.
Glaunes, J., Trouve, A., & Younes, L. (2006). Modeling planar shape variation via hamiltonian flows of curves. In H. Krim Jr. & Y. A. (Eds.), Statistics and analysis of shapes. Basel: Birkhäuser.
Glaunès, J., Qiu, A., Miller, M. I., & Younes, L. (2008). Large deformation diffeomorphic metric curve mapping. International Journal of Computer Vision, 80(3), 317–336. doi:10.1007/s11263-008-0141-9.
Granger, S., & Pennec, X. (2006). Multi-scale em-icp: A fast and robust approach for surface registration. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science: Vol. 2353. Computer vision ECCV 2002 (pp. 69–73). Berlin: Springer.
Greengard, L., & Strain, J. (1991). The fast Gauss transform. SIAM Journal of Scientific Statistical Computing, 12, 79–94.
Grenander, U., & Miller, M. (1998). Computational anatomy: an emerging discipline. Quarterly of Applied Mathematics, LVI(4), 617–694.
Khesin, B., & Wendt, R. (2008). Ergebnisse der Mathematik und Grenzgebiete 3. Folge: Vol. 51. Geometry of infinite-dimensional groups. Berlin: Springer. Chap. 1
Kilian, M., Mitra, N. J., & Pottmann, H. (2007). Geometric modeling in shape space. ACM Transactions on Graphics, 26.
Kurtek, S., Klassen, E., Ding, Z., & Srivastava, A. (2010). A novel riemannian framework for shape analysis of 3d objects. In IEEE computer vision and pattern recognition (pp. 1625–1632).
Kurtek, S., Klassen, E., Ding, Z., Avison, M. J., & Srivastava, A. (2011a). Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces. In Lecture notes in computer science: Vol. 6801. Information processing in medical imaging (pp. 147–158). Berlin: Springer.
Kurtek, S., Klassen, E., Ding, Z., Jacobson, S., Jacobson, J., Avison, M., & Srivastava, A. (2011b). Parameterization-invariant shape comparisons of anatomical surfaces. IEEE Transactions on Medical Imaging, 30(3), 849–858.
McLachlan, R., & Marsland, S. (2007). Discrete mechanics and optimal control for image registration. ANZIAM Journal, 48(C), 1–16.
Michor, P. W., & Mumford, D. (2006). Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society, 8(1), 1–48.
Michor, P. W., & Mumford, D. (2007). An overview of the riemannian metrics on spaces of curves using the hamiltonian approach. Applied and Computational Harmonic Analysis, 23(1), 74–113.
Miller, M., & Younes, L. (2001). Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision, 41, 61–84.
Nocedal, J., & Wright, S. (2006). Practical methods of optimization (2nd edn.). New York: Wiley.
Oliphant, T. (2007). Python for scientific computing. Computing in Science & Engineering, 9(3), 10–20.
Powell, M. (1995). A thin plate spline method for mapping curves into curves in two dimensions. In Computational techniques and applications (pp. 43–57).
Sharon, E., & Mumford, D. (2006). 2d-shape analysis using conformal mapping. International Journal of Computer Vision, 70(1), 55–75. doi:10.1007/s11263-006-6121-z.
Srivastava, A., Klassen, E., Joshi, S. H., & Jermyn, I. H. (2011). Shape analysis of elastic curves in Euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33, 1415–1428.
Sundaramoorthi, G., Mennucci, A., Soatto, S., & Yezzil, A. (2011). A new geometric metric in the space of curves with applications to tracking deforming objects by filtering and prediction. SIAM Journal on Imaging Sciences, 4, 109–145.
Taylor, D., Doorly, D., & Shroter, R. (2009). Airflow in the human nasal cavity: an inter-subject comparison. In ASME summer bioengineering conference (pp. 1071–1072). Amer. Soc. Mechanical Engineers.
Trouvé, A., & Vialard, F. X. (2010, to appear). Shape splines and stochastic shape evolution: a second order point of view. Quarterly of Applied Mathematics.
Trouve, A., & Younes, L. (2005). Local geometry of deformable templates. SIAM Journal on Mathematical Analysis, 37(1), 17–59. doi:10.1137/S0036141002404838.
Vaillant, M., & Glaunes, J. (2005). Surface matching via currents. In Lecture notes in computer science:Vol. 3565. IPMI (pp. 381–392). Berlin: Springer.
Vialard, F. X. (2009). Hamiltonian approach to shape spaces in a diffeomorphic framework: From the discontinuous image matching problem to a stochastic growth model. Ph.D. thesis, ENS Cachan.
Vialard, F. X., Risser, L., Rueckert, D., & Cotter, C. (2012). Diffeomorphic 3d image registration via geodesic shooting using an efficient adjoint calculation. International Journal of Computer Vision. doi:10.1007/s11263-011-0481-8.
Younes, L. (2007). Jacobi fields in groups of diffeomorphisms and applications. Quarterly of Applied Mathematics, 65, 113–134.
Younes, L., Michor, P., Shah, J., & Mumford, D. (2008). A metric on shape spaces with explicit geodesics. Rendiconti Lincei - Math. e Appl., 19(1), 25–57.
Zhang, Z. (1994). Iterative point matching for registration of free-form curves and surfaces. International Journal of Computer Vision, 13, 119–152.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cotter, C.J., Clark, A. & Peiró, J. A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching. Int J Comput Vis 99, 103–121 (2012). https://doi.org/10.1007/s11263-012-0520-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-012-0520-0