Abstract
In this work we consider the problem of magnetic resonance imaging (MRI). We propose and formulate a finite element method to handle this problem and present an adaptive algorithm for local mesh refinements. Reconstructions from experimental MR data are shown and used to compare different interpolation techniques.
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References
M. Asadzadeh, K. Eriksson, On adaptive finite element methods for Fredholm integral equations of the second kind. SIAM J. Numer. Anal. 31(3), 831–855 (1994)
Yu.A. Basistov, A.V. Goncharsky, E.E. Lekht, A.M. Cherepashchuk, A.G. Yagola, Application of the regularization method for increasing of the radiotelescope resolution power, Astron. zh. 56(2), 443–449 (1979) (in Russian)
A.B. Bakushinsky, M.Y. Kokurin, A. Smirnova, Iterative Methods for Ill-posed Problems (Walter de Gruyter GmbH & Co., 2011)
L. Beilina, M.V. Klibanov, Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm. Inverse Probl. 26, 125009 (2010)
S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edn. (Springer, New York, 2002)
R.W. Brown, Y.-C.N. Cheng, E.M. Haacke, M.R.T., R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd edn. (Wiley, 1999)
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Dover Books on Mathematics, 2009)
N. Koshev, L. Beilina, An adaptive finite element method for Fredholm integral equations of the first kind and its verification on experimental data. Cent. Eur. J. Math. 11(8), 1489–1509 (2013)
V. Skala, Barycentric coordinates computation in homogeneous coordinates. Comput. Graph. 32, 120–127 (2008). Elsevier
A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear ill-posed problems (Chapman & Hall, 1998)
A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer, London, 1995)
A.N. Tikhonov, A.V. Goncharskiy, V.V. Stepanov, I.V. Kochikov, Ill-posed problem in image processing, DAN USSR, 294(4), 832–837 (1987) (in Russian) (Moscow)
Acknowledgements
The research of LB was done during the sabbatical stay at the IR4M UMR8081, CNRS, Université Paris-Sud, Université Paris-Saclay, France, which was supported by the sabbatical program at the Faculty of Science, University of Gothenburg, Sweden. We wish to thank B. Bayle and L. Meylheuc from ICube Strasbourg for providing the Osteopal samples, and L. Jourdain for her help with the NMR probe fabrication.
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Beilina, L., Guillot, G., Niinimäki, K. (2018). On Finite Element Method for Magnetic Resonance Imaging. In: Beilina, L., Smirnov, Y. (eds) Nonlinear and Inverse Problems in Electromagnetics. PIERS PIERS 2017 2017. Springer Proceedings in Mathematics & Statistics, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-319-94060-1_9
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