The Finite Element Method and Balancing Principle for Magnetic Resonance Imaging
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This work considers a finite element method in combination with balancing principle for a posteriori choice of the regularization parameter for image reconstruction problem appearing in magnetic resonance imaging (MRI). The fixed point iterative algorithm is formulated and it’s performance is demonstrated on the image reconstruction from experimental MR data.
KeywordsMRI Fredholm integral equation of the first kind Finite element method Regularization Balancing principle
The research of LB was supported by the Swedish Research Council grant VR 2018-03661 and by the French Institute in Sweden, FRÖ program. The research of KN was supported by the French Institute in Sweden, TOR program.
- 1.Basistov, Y.A., Goncharsky, A.V., Lekht, E.E., Cherepashchuk, A.M., Yagola, A.G.: Application of the regularization method for increasing of the radiotelescope resolution power. Astron. zh. 56(2), 443–449 (1979) (in Russian)Google Scholar
- 2.Bakushinsky A.B., Kokurin, M.Y., Smirnova, A.: Iterative Methods for Ill-Posed Problems. Walter de Gruyter GmbH&Co. (2011)Google Scholar
- 4.Brown, R.W., Cheng, Y.-C.N., Haacke, E.M., Thompson, M.R., Venkatesan, R.: Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd edn. Wiley, Inc. (1999)Google Scholar
- 6.Ito, K., Jin, B.: Inverse Problems: Tikhonov Theory and Algorithms. Series on Applied Mathematics, vol. 22. World Scientific (2015)Google Scholar
- 7.Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Books on Mathematics (2009)Google Scholar
- 9.Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springers Applied Mathematical Sciences, vol. 160. New York (2005)Google Scholar
- 11.Lazarov, R.D., Lu, S., Pereverzev, S.V.: On the balancing principle for some problems of numerical analysis. Numer. Math. 106(4), 659–689Google Scholar
- 13.Mueller, J.L., Siltanen, S.: Linear and Nonlinear Inverse Problems with Practical Applications. SIAM Computational Science & Engineering (2012)Google Scholar
- 15.Tikhonov, A.N., Leonov, A.S., Yagola, A.G.: Nonlinear Ill-Posed Problems. Chapman & Hall (1998)Google Scholar
- 17.Tikhonov, A.N., Goncharskiy, A.V., Stepanov, V.V., Kochikov, I.V.: Ill-Posed problem in image processing. DAN USSR, Moscow 294(4), 832–837 (1987) (in Russian)Google Scholar