NDLI: A Posteriori Error Estimate and Convergence Analysis for Conductivity Image Reconstruction in MREIT

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Liu, Jijun Woo, EungJe Seo, Jinkeun
Copyright Year	2010
Abstract	Magnetic resonance electrical impedance tomography (MREIT) takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object. When we inject current into the imaging object, there occurs a distribution of internal magnetic flux density $\mathbf{B}=(B_x,B_y,B_z)$. In MREIT we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire $B_z$ data. The harmonic $B_z$ algorithm was invented in 2001 to reconstruct cross-sectional conductivity images from $B_z$ data sets subject to multiple injection currents. Utilizing internal $B_z$ data, it overcomes the inherent ill posedness in electrical impedance tomography. We can set up the inverse problem in MREIT as a coefficient identification problem of finding $\sigma$ appearing in $\nabla\cdot(\sigma\nabla u)=0$ from acquired data of the z component of $\nabla\times(\sigma\nabla u)$. The harmonic $B_z$ algorithm has shown an excellent performance in numerical simulations and phantom experiments. Experimental MREIT studies have now reached the stage of in vivo animal and human imaging experiments. However, there is not much work on rigorous mathematical theories of error estimate and convergence analysis yet. The purpose of this paper is to provide a posteriori error estimate in MREIT conductivity image reconstructions. This enables us to evaluate a difference between a reconstructed conductivity image and the unknown true conductivity image. We also describe a convergence analysis of the harmonic $B_z$ algorithm, which improves the previous result of [J. J. Liu, J. K. Seo, M. Sini, and E. J. Woo, SIAM J. Appl. Math., 67 (2007), pp. 12591282] in the sense that assumptions on the conductivity are much relaxed.
Starting Page	2883
Ending Page	2903
Page Count	21
File Format	PDF
ISSN	00361399
DOI	10.1137/090781292
e-ISSN	1095712X
Issue Number	8
Volume Number	70
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-09-23
Access Restriction	Subscribed
Subject Keyword	error estimate Inverse problems Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation conductivity convergence MREIT Helmholtz decomposition relative contrast image reconstruction Hydro- and aero-acoustics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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