Direct Reconstructions of Conductivities from Boundary Measurements

Abstract

Abstract. The problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging, geophysics, and nondestructive testing. A. Nachman [Ann. of Math. (2), 143 (1996), pp. 71-96.] proved global uniqueness for the two-dimensional inverse conductivity problem using a constructive method of proof. Based on this proof, Siltanen, Mueller, and Isaacson [Inverse Problems, 16 (2000), pp. 681-699] presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration. The method was verified with nonnoisy rotationally symmetric examples. In this paper the method is extended by introducing a new regularization scheme, which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise.

Numerical solution of the $\mathbb R$-linear Beltrami equation

“…If the matrix A # is invertible, then this is equivalent to In preconditioning large scale problems, the matrix equation see [8,4] for convergence results in an electrical conductivity problem. Then (4.8) is not very accurately solved.…”

“…These assumptions are realistic in several applications; see [8,6,2] and the references therein. For the Beltrami equation and its applications, see [1,5].…”

Electrical impedance tomography: 3D reconstructions using scattering transforms

“…The specific choice of ζ F will certainly affect the solution. For the 2D problem a similar approximation was used in [27,28].…”

“…We emphasize that the matrices S and I +R+R * +S involved in (28) are Hermitian, and that h is a sparse vector. This is utilized in the numerical implementation.…”

“…The 2D Calderón problem was solved for smooth conductivities in [8], and the algorithm was successfully implemented [9][10][11][12][13][14]. Several ideas from these implementations were used recently in a crude simplification of the reconstruction method for 3D [15,16].…”