Article Contents
Article Contents

# Learning landmark geodesics using the ensemble Kalman filter

• * Corresponding author: Andreas Bock
• We study the problem of diffeomorphometric geodesic landmark matching where the objective is to find a diffeomorphism that, via its group action, maps between two sets of landmarks. It is well-known that the motion of the landmarks, and thereby the diffeomorphism, can be encoded by an initial momentum leading to a formulation where the landmark matching problem can be solved as an optimisation problem over such momenta. The novelty of our work lies in the application of a derivative-free Bayesian inverse method for learning the optimal momentum encoding the diffeomorphic mapping between the template and the target. The method we apply is the ensemble Kalman filter, an extension of the Kalman filter to nonlinear operators. We describe an efficient implementation of the algorithm and show several numerical results for various target shapes.

Mathematics Subject Classification: Primary: 62F15, 65C05; Secondary: 34A55.

 Citation:

• Figure 1.  A matching between landmarks where the geodesics are shown

Figure 2.  Template-target configurations for different values of $M$. Left to right: 10, 50, 150. Linear interpolation has been used between the landmarks to improve the visualisation

Figure 3.  Log data misfits for $M = N_E = 50$ for different values of $\xi$ using three different targets

Figure 4.  Progression of Algorithm 1 for various targets using $M = 10$ and $N_E = 10$. Computation times for 50 iterations: 6s for each configuration

Figure 5.  Progression of Algorithm 1 for various targets using $M = 50$ and $N_E = 50$. Computation times for 50 iterations (top to bottom): 2m8s, 2m9s, 1m29s

Figure 6.  Progression of Algorithm 1 for various targets using $M = 150$ and $N_E = 100$. Computation times for 50 iterations (top to bottom): 5m22s, 5m23s, 5m23s

Figure 7.  Convergence of $E^k$ where $M = 10$

Figure 8.  Convergence of $E^k$ where $M = 50$

Figure 9.  Convergence of $E^k$ where $M = 150$

Figure 10.  Evolution of the relative error $\mathcal{R}^k$ corresponding to the misfits in Figure 7 where $M = 10$

Figure 11.  Evolution of the relative error $\mathcal{R}^k$ corresponding to the misfits in Figure 8 where $M = 150$

Figure 12.  Evolution of the relative error $\mathcal{R}^k$ corresponding to the misfits in Figure 9 where $M = 50$

Table 1.  Global parameters used for Algorithm 1

 Variable Value Description $n$ 50 Kalman iterations $T$ 15 time steps $\tau$ 1 landmark size (cf. (2)) $\epsilon$ 1e-05 absolute error tolerance

Table 2.  Relative error at the last iteration of algorithm 1 for different values of $N_E$ for fixed $M = 10$. The rows correspond to the configurations in Figure 4

Table 3.  Relative error at the last iteration of algorithm 1 for different values of $N_E$ for fixed $M = 50$. The rows correspond to the configurations in Figure 5

Table 4.  Relative error at the last iteration of algorithm 1 for different values of $N_E$ for fixed $M = 150$. The rows correspond to the configurations in Figure 6

•  [1] M. F. Beg, M. I. Miller, A. Trouvé and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comput. Vis., 61 (2005), 139-157.  doi: 10.1023/B:VISI.0000043755.93987.aa. [2] N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble Kalman inversion, Inverse Problems, 34 (2018), 31pp. doi: 10.1088/1361-6420/aab6d9. [3] B. Charlier, J. Feydy, J. A. Glaunès, F.-D. Collin and G. Durif, Kernel operations on the GPU, with autodiff, without memory overflows, preprint, arXiv: 2004.11127. [4] C. J. Cotter, S. L. Cotter and F.-X. Vialard, Bayesian data assimilation in shape registration, Inverse Problems, 29 (2013), 21pp. doi: 10.1088/0266-5611/29/4/045011. [5] S. L. Cotter, M. Dashti and A. M. Stuart, Approximation of Bayesian inverse problems for PDEs, SIAM J. Numer. Anal., 48 (2010), 322-345.  doi: 10.1137/090770734. [6] P. Dupuis, U. Grenander and M. I. Miller, Variational problems on flows of diffeomorphisms for image matching, Quart. Appl. Math., 56 (1998), 587-600.  doi: 10.1090/qam/1632326. [7] G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res.-Oceans, 99 (1994), 10143-10162.  doi: 10.1029/94JC00572. [8] U. Grenander and M. I. Miller, Computational anatomy: An emerging discipline, Quart. Appl. Math., 56 (1998), 617-694.  doi: 10.1090/qam/1668732. [9] U. Grenander and M. I. Miller, Representations of knowledge in complex systems, J. Roy. Statist. Soc. Ser. B, 56 (1994), 549-603.  doi: 10.1111/j.2517-6161.1994.tb02000.x. [10] S. J. Greybush, E. Kalnay, T. Miyoshi, K. Ide and B. R. Hunt, Balance and ensemble Kalman filter localization techniques, Monthly Weather Review, 139 (2011), 511-522.  doi: 10.1175/2010MWR3328.1. [11] D. D. Holm, A. Trouvé and L. Younes, The Euler-Poincaré theory of metamorphosis, Quart. Appl. Math., 67 (2009), 661-685.  doi: 10.1090/S0033-569X-09-01134-2. [12] M. A. Iglesias, A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems, Inverse Problems, 32 (2016), 45pp. doi: 10.1088/0266-5611/32/2/025002. [13] M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 20pp. doi: 10.1088/0266-5611/29/4/045001. [14] S. C. Joshi and M. I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Trans. Image Process., 9 (2000), 1357-1370.  doi: 10.1109/83.855431. [15] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. doi: 10.1007/b138659. [16] R. E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), 35-45.  doi: 10.1115/1.3662552. [17] L. Kühnel, S. Sommer and A. Arnaudon, Differential geometry and stochastic dynamics with deep learning numerics, Appl. Math. Comput., 356 (2019), 411-437.  doi: 10.1016/j.amc.2019.03.044. [18] F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 598-631. [19] J. Ma, M. I. Miller, A. Trouvé and L. Younes, Bayesian template estimation in computational anatomy, NeuroImage, 42 (2008), 252-261.  doi: 10.1016/j.neuroimage.2008.03.056. [20] J. Mandel, L. Cobb and J. D. Beezley, On the convergence of the ensemble Kalman filter, Appl. Math., 56 (2011), 533-541.  doi: 10.1007/s10492-011-0031-2. [21] M. I. Miller, A. Trouvé and L. Younes, Geodesic shooting for computational anatomy, J. Math. Imaging Vision, 24 (2006), 209-228.  doi: 10.1007/s10851-005-3624-0. [22] D. Mumford, Pattern theory: The mathematics of perception, in Proceedings of the International Congress of Mathematicians, Vol. I, Higher Ed. Press, Beijing, 2002,401-422. [23] D. S. Oliver,  A. C. Reynolds and  N. Liu,  Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge University Press, 2008.  doi: 10.1017/CBO9780511535642. [24] A. Paszke, S. Gross, F. Massa, A. Lerer and J. Bradbury, et al., PyTorch: An imperative style, high-performance deep learning library, preprint, arXiv: 1912.01703 [25] S. Reich and  C. Cotter,  Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804. [26] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), 856-869.  doi: 10.1137/0907058. [27] C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM J. Numer. Anal., 55 (2017), 1264-1290.  doi: 10.1137/16M105959X. [28] T. Schneider, S. Lan, A. Stuart and J. Teixeira, Earth system modeling 2.0: A blueprint for models that learn from observations and targeted high-resolution simulations, Geophysical Research Letters, 44 (2017), 12396-12417.  doi: 10.1002/2017GL076101. [29] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061. [30] A. Trouvé, Diffeomorphisms groups and pattern matching in image analysis, Int. J. Comput. Vis., 28 (1998), 213-221.  doi: 10.1023/A:1008001603737. [31] A. Trouvé, An infinite dimensional group approach for physics based models in pattern recognition, (1995). [32] A. Trouvé and L. Younes, Metamorphoses through Lie group action, Found. Comput. Math., 5 (2005), 173-198.  doi: 10.1007/s10208-004-0128-z. [33] R. Van Der Merwe, A. Doucet, N. De Freitas and E. A. Wan, The unscented particle filter, in Advances in Neural Information Processing Systems, 2000,584–590. [34] C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570. [35] L. Younes, Shapes and Diffeomorphisms, Applied Mathematical Sciences, 171, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-12055-8.

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