Source and metric estimation in the eikonal equation using optimization on a manifold

Jérôme Fehrenbach; Lisl Weynans

doi:10.3934/ipi.2022050

Article Contents

2023, Volume 17, Issue 2: 419-440. Doi: 10.3934/ipi.2022050

This issue Previous Article Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data Next Article Existence and uniqueness in anisotropic conductivity reconstruction with Faraday's law

Source and metric estimation in the eikonal equation using optimization on a manifold

Jérôme Fehrenbach^1, , and
Lisl Weynans^2,

1.
Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse CNRS, UPS, F-31062 Toulouse Cedex 9, France
2.
Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France

^*Corresponding author: Jérôme Fehrenbach
^*Corresponding author: Jérôme Fehrenbach

Received: December 2021

Revised: September 2022

Early access: October 12, 2022

Published online: October 12, 2022

Abstract / Introduction Full Text(HTML) Figure(8) / Table(3) Related Papers Cited by

Abstract

We address the estimation of the source(s) location in the eikonal equation on a Riemann surface, as well as the determination of the metric when it depends on a few parameters. The available observations are the arrival times or are obtained indirectly from the arrival times by an observation operator, this frame is intended to describe electro-cardiographic imaging. The sensitivity of the arrival times is computed from $ {{{\rm{Log}}}}_x $ the log map wrt to the source $ x $ on the surface. The $ {{{\rm{Log}}}}_x $ map is approximated by solving an elliptic vectorial equation, using the Vector Heat Method. The $ L^2 $-error function between the model predictions and the observations is minimized using Gauss-Newton optimization on the Riemann surface. This allows to obtain fast convergence. We present numerical results, where coefficients describing the metric are also recovered like anisotropy and global orientation.

Keywords:

Mathematics Subject Classification: Primary: 35F21, 49Q12; Secondary: 35R30.

Citation:

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Figure 1. Some triangle $ ABC $ of the mesh and the associated angles

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Figure 2. Top : location of the successive iterates $ x^k $ (red) and the true source point $ x^* $ (blue), the 3rd iterate $ x^3 $ coincides with $ x^* $. Bottom left: evolution of the cost function $ J $ for cases 1a and 1b. Bottom right: evolution of the distance to the solution $ \|x^k-x^*\| $ in both cases, note that the iterates are the same for cases 1a and 1b

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Figure 3. Top: location of the successive iterates $ x^k $ before splitting (red) and after splitting (green) and the true source points (blue). Bottom left: evolution of the cost function $ J $ for cases 2a and 2b. Bottom right: evolution of the distance to the solution in both cases

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Figure 4. Left: evolution of the cost function $ J $ for cases 3a and 3b. Right: evolution of the distance to the solution $ \|x^k-x^*\| $ in both cases

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Figure 5. Left: location of the successive iterates $ x^k $ (red) and the true source point $ x^* $ (blue). Right: evolution of the parameters of the metric (dashed: case 3a, line: case 3b). Note that it is not optimized during the first 4 steps

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Figure 6. Left: configuration of the torso (electrodes in red) and the heart surface (visible by transparency). Right: observations at the electrodes (mimicking an ECG) obtained with noise-level 1%

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Figure 7. Top: location of the successive iterates $ x^k $ (red) and the true source point $ x^* $ (blue). Bottom left: evolution of the cost function $ J $. Bottom right: evolution of the distance to the solution $ \|x^k-x^*\| $ in both cases

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Figure 8. Time necessary to solve the localization problem of Test case 1 on meshes with increasing size

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Table 1. Reference and estimated activation times for test case 1

True activation time $ \tau^\star $	retrieved $ \tau $ for case 1a	retrieved $ \tau $ for case 1b
0.2	0.229	0.217

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Table 2. Reference and estimated activation times for test case 2

True activation times $ \tau^\star_1 $/$ \tau^\star_2 $	retrieved $ \tau $ for case 2a	retrieved $ \tau $ for case 2b
0/0.2	0.031/0.228	0.017/0.226

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Table 3. Reference and estimated activation times for test case 3

True activation times $ \tau^\star $	retrieved $ \tau $ for case 3a	retrieved $ \tau $ for case 3b
0/0.2	0.019/0.214	0.011/0.0215

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