A new alternative elliptic PDE in EIT Imaging
Bull. Korean Math. Soc. 2012 Vol. 49, No. 6, 1291-1302
Printed November 30, 2012
Sungwhan Kim
Hanbat National University
Abstract : In this paper, we introduce a new elliptic PDE: $$ \left\{ \begin{array}{cc} \nabla \cdot\left( \frac{|\gamma^\omega(\mathbf{r})|^2}{\sigma} \nabla v_\omega (\mathbf{r})\right)=0,& \quad \mathbf{r} \in \Omega, \\ v_\omega (\mathbf{r}) =f(\mathbf{r}),& \quad \mathbf{r} \in \partial \Omega, \end{array} \right. $$ where $\gamma^\omega = \sigma + i \omega \epsilon$ is the admittivity distribution of the conducting material $\Omega$ and it is shown that the introduced elliptic PDE can replace the standard elliptic PDE with conductivity coefficient in EIT imaging. Indeed, letting $v_0$ be the solution to the standard elliptic PDE with conductivity coefficient, the solution $v_\omega$ is quite close to the solution $v_0$ and can show spectroscopic properties of the conducting object $\Omega$ unlike $v_0$. In particular, the potential $v_\omega$ can be used in detecting a thin low-conducting anomaly located in $\Omega$ since the spectroscopic change of the Neumann data of $v_\omega$ is inversely proportional to thickness of the thin anomaly.
Keywords : electrical impedance tomography, alternative elliptic PDE, spectroscopic change, anomaly thickness
MSC numbers : Primary 35J25, 35A35, 35R30
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