This paper concerns the reconstruction of an anisotropic conductivity tensor γ from internal current densities of the form J = γ∇u, where u solves a second-order elliptic equation ∇ · (γ∇u) = 0 on a bounded domain X with prescribed boundary conditions. A minimum number of such functionals equal to n + 2, where n is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that γ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of J. In the special case where γ is scalar, it can be reconstructed with no loss of derivatives. We provide a precise statement of what components may be reconstructed with a loss of zero or one derivatives.
We consider the imaging of anisotropic conductivity tensors γ = (γ ij ) 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 ⊂ R 2 with prescribed boundary conditions on ∂X. We show that γ can be uniquely reconstructed from four 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.
This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupledphysics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic tomography.We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are found to hold for a lesser number of measurements than those required in recently established explicit reconstruction procedures for the nonlinear problem.2010 Mathematics Subject Classification. Primary: 35R30, 35S05; Secondary: 35J47.
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