2014/03/19 by Guillaume Bal, Bal, Guillaume, Chenxi Guo +3
Engineering · Mathematics · #Electrical and Bioimpedance Tomography #Numerical methods in inverse problems #Reservoir Engineering and Simulation Methods #math.AP
paper · pdf · doi:10.48550/arxiv.1403.4964
arxiv created 2014/03/19 · arxiv updated 2014/03/21
We consider the imaging of anisotropic conductivity tensors γ=(γij)1≤ i,j≤ 2 from knowledge of several internal current densities J=γ∇ u where u satisfies a second order elliptic equation ∇⋅(γ∇ u)=0 on a bounded domain X⊂ℝ2 with prescribed boundary conditions on ∂ X. We show that γ can be uniquely reconstructed from four \em well-chosen functionals J and that noise in the data is differentiated once during the reconstruction. The inversion procedure is local in the sense that (most of) the tensor γ(x) can be reconstructed from knowledge of the functionals J in the vicinity of x. We obtain the existence of an open set of boundary conditions on ∂ X that guaranty stable reconstructions by using the technique of complex geometric optics (CGO) solutions. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modality called Current Density Imaging or Magnetic Resonance Electrical Impedance Tomography.