A POSTERIORI ERROR ESTIMATE AND CONVERGENCE ANALYSIS FOR CONDUCTIVITY IMAGE RECONSTRUCTION IN MREIT

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 magnet...

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Vydáno v:SIAM journal on applied mathematics Ročník 70; číslo 7/8; s. 2883 - 2903
Hlavní autoři: LIU, JIJUN, SEO, JINKEUN, WOO, EUNGJE
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia Society for Industrial and Applied Mathematics 01.01.2010
Témata:
A posteriori knowledge
Conductivity
Electrical impedance
Electrodes
Error analysis
Estimates
Helmholtz equations
Image reconstruction
Imaging
Magnetic resonance
Magnetic resonance imaging
Mathematical analysis
Mathematics
NMR
Nonlinear equations
Nuclear magnetic resonance
Resistivity
Studies
Tomography
ISSN:0036-1399, 1095-712X
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Shrnutí: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 = (B x , B y , B z ). In MREIT we utilize a magnetic resonance imaging scanner with its main magnetic field in the æ 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 σ appearing in $\nabla .(\sigma \nabla u) = 0$ 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. 1259-1282] in the sense that assumptions on the conductivity are much relaxed.
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ISSN:0036-1399
1095-712X
DOI:10.1137/090781292