Deep Eikonal Solvers

Moshe Lichtenstein; Gautam Pai; Ron Kimmel

doi:10.1007/978-3-030-22368-7_4

Deep Eikonal Solvers

Moshe Lichtenstein
, Gautam Pai
, Ron Kimmel

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

13 Citations (Scopus)

Abstract

A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces. The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.

Original language	English
Title of host publication	Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings
Editors	Jan Lellmann, Jan Modersitzki, Martin Burger
Pages	38-50
Number of pages	13
DOIs	https://doi.org/10.1007/978-3-030-22368-7_4
Publication status	Published - 2019
Externally published	Yes
Event	7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019 - Hofgeismar, Germany Duration: 30 Jun 2019 → 4 Jul 2019 Conference number: 7

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11603

Conference

Conference	7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019
Abbreviated title	SSVM 2019
Country/Territory	Germany
City	Hofgeismar
Period	30/06/19 → 4/07/19

Keywords

n/a OA procedure
Geodesic distance
The Eikonal equation
Deep learning for PDE

Access to Document

10.1007/978-3-030-22368-7_4

Cite this

Lichtenstein, M., Pai, G., & Kimmel, R. (2019). Deep Eikonal Solvers. In J. Lellmann, J. Modersitzki, & M. Burger (Eds.), Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings (pp. 38-50). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11603). https://doi.org/10.1007/978-3-030-22368-7_4

Lichtenstein, Moshe ; Pai, Gautam ; Kimmel, Ron. / Deep Eikonal Solvers. Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings. editor / Jan Lellmann ; Jan Modersitzki ; Martin Burger. 2019. pp. 38-50 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Deep Eikonal Solvers",

abstract = "A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces. The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.",

keywords = "n/a OA procedure, Geodesic distance, The Eikonal equation, Deep learning for PDE",

author = "Moshe Lichtenstein and Gautam Pai and Ron Kimmel",

note = "Publisher Copyright: {\textcopyright} 2019, Springer Nature Switzerland AG.; 7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019 ; Conference date: 30-06-2019 Through 04-07-2019; 7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019, SSVM 2019 ; Conference date: 30-06-2019 Through 04-07-2019",

year = "2019",

doi = "10.1007/978-3-030-22368-7\_4",

language = "English",

isbn = "9783030223670",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

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Lichtenstein, M, Pai, G & Kimmel, R 2019, Deep Eikonal Solvers. in J Lellmann, J Modersitzki & M Burger (eds), Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11603, pp. 38-50, 7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019, Hofgeismar, Germany, 30/06/19. https://doi.org/10.1007/978-3-030-22368-7_4

Deep Eikonal Solvers. / Lichtenstein, Moshe; Pai, Gautam; Kimmel, Ron.
Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings. ed. / Jan Lellmann; Jan Modersitzki; Martin Burger. 2019. p. 38-50 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11603).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

TY - GEN

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AU - Lichtenstein, Moshe

AU - Pai, Gautam

AU - Kimmel, Ron

N1 - Conference code: 7

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N2 - A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces. The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.

AB - A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces. The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.

KW - n/a OA procedure

KW - Geodesic distance

KW - The Eikonal equation

KW - Deep learning for PDE

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DO - 10.1007/978-3-030-22368-7_4

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SN - 9783030223670

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 38

EP - 50

BT - Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings

A2 - Lellmann, Jan

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Y2 - 30 June 2019 through 4 July 2019

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Lichtenstein M, Pai G, Kimmel R. Deep Eikonal Solvers. In Lellmann J, Modersitzki J, Burger M, editors, Scale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings. 2019. p. 38-50. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-22368-7_4

Deep Eikonal Solvers

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