Learning landmark geodesics using the ensemble Kalman filter

Andreas Bock; Colin J. Cotter

doi:10.3934/fods.2021020

Article Contents

2021, Volume 3, Issue 4: 701-727. Doi: 10.3934/fods.2021020

This issue Previous Article Homotopy continuation for the spectra of persistent Laplacians Next Article Generalized penalty for circular coordinate representation

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
^* Corresponding author: Andreas Bock

Received: February 28, 2021

Revised: June 30, 2021

Early access: August 2021

Published: December 2021

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Abstract

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:

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

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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 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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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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