A NEW ALTERNATIVE ELLIPTIC PDE IN EIT IMAGING

Abstract

In this paper, we introduce a new elliptic PDE: <TEX>$$\{{\nabla}{\cdot}\(\frac{|{\gamma}^{\omega}(r)|^2}{\sigma}{\nabla}v_{\omega}(r)\)=0,\;r{\in}{\Omega},\\v_{\omega}(r)=f(r),\;r{\in}{\partial}{\Omega},$$</TEX> where <TEX>${\gamma}^{\omega}={\sigma}+i{\omega}{\epsilon}$</TEX> is the admittivity distribution of the conducting material <TEX>${\Omega}$</TEX> and it is shown that the introduced elliptic PDE can replace the standard elliptic PDE with conductivity coefficient in EIT imaging. Indeed, letting <TEX>$v_0$</TEX> be the solution to the standard elliptic PDE with conductivity coefficient, the solution <TEX>$v_{\omega}$</TEX> is quite close to the solution <TEX>$v_0$</TEX> and can show spectroscopic properties of the conducting object <TEX>${\Omega}$</TEX> unlike <TEX>$v_0$</TEX>. In particular, the potential <TEX>$v_{\omega}$</TEX> can be used in detecting a thin low-conducting anomaly located in <TEX>${\Omega}$</TEX> since the spectroscopic change of the Neumann data of <TEX>$v_{\omega}$</TEX> is inversely proportional to thickness of the thin anomaly.