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December  2021, 3(4): 701-727. doi: 10.3934/fods.2021020

Learning landmark geodesics using the ensemble Kalman filter

Andreas Bock , and  Colin J. Cotter

Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

* Corresponding author: Andreas Bock

Received  March 2021 Revised  July 2021 Published  December 2021 Early access  August 2021

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.

Keywords: Ensemble Kalman filter, large deformation diffeomorphic metric mapping, Bayesian inverse problem, landmark matching, derivative-free optimisation.
Mathematics Subject Classification: Primary: 62F15, 65C05; Secondary: 34A55.
Citation: Andreas Bock, Colin J. Cotter. Learning landmark geodesics using the ensemble Kalman filter. Foundations of Data Science, 2021, 3 (4) : 701-727. doi: 10.3934/fods.2021020
References:
[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar
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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.  Google Scholar

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[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.  Google Scholar

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A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

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A. Trouvé, Diffeomorphisms groups and pattern matching in image analysis, Int. J. Comput. Vis., 28 (1998), 213-221.  doi: 10.1023/A:1008001603737.  Google Scholar

[31]

A. Trouvé, An infinite dimensional group approach for physics based models in pattern recognition, (1995). Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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L. Younes, Shapes and Diffeomorphisms, Applied Mathematical Sciences, 171, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-12055-8.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

U. Grenander and M. I. Miller, Computational anatomy: An emerging discipline, Quart. Appl. Math., 56 (1998), 617-694.  doi: 10.1090/qam/1668732.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer-Verlag, New York, 2005. doi: 10.1007/b138659.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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 Google Scholar

[25] S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559.  doi: 10.1017/S0962492910000061.  Google Scholar

[30]

A. Trouvé, Diffeomorphisms groups and pattern matching in image analysis, Int. J. Comput. Vis., 28 (1998), 213-221.  doi: 10.1023/A:1008001603737.  Google Scholar

[31]

A. Trouvé, An infinite dimensional group approach for physics based models in pattern recognition, (1995). Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[35]

L. Younes, Shapes and Diffeomorphisms, Applied Mathematical Sciences, 171, Springer-Verlag, Berlin, 2010 doi: 10.1007/978-3-642-12055-8.  Google Scholar

Figure 1.  A matching between landmarks where the geodesics are shown
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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
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Figure 3.  Log data misfits for $ M = N_E = 50 $ for different values of $ \xi $ using three different targets
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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
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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
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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
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Figure 7.  Convergence of $ E^k $ where $ M = 10 $
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Figure 8.  Convergence of $ E^k $ where $ M = 50 $
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Figure 9.  Convergence of $ E^k $ where $ M = 150 $
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Figure 7 where $ M = 10 $">Figure 10.  Evolution of the relative error $ \mathcal{R}^k $ corresponding to the misfits in Figure 7 where $ M = 10 $
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Figure 8 where $ M = 150 $">Figure 11.  Evolution of the relative error $ \mathcal{R}^k $ corresponding to the misfits in Figure 8 where $ M = 150 $
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Figure 9 where $ M = 50 $">Figure 12.  Evolution of the relative error $ \mathcal{R}^k $ corresponding to the misfits in Figure 9 where $ M = 50 $
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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
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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
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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
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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
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