Inversion of an Inductive Loss Convolution Integral for Conductivity Imaging
Electrical conductivity imaging in the human body is usually pursued by either electrical impedance tomography or magnetic induction tomography (MIT). In the latter case, multiple coils are almost always used, so that nonlinear reconstruction is preferred. Recent work has shown that single-coil, scanning MIT is feasible through an analytical 3D convolution integral that relates measured coil loss to an arbitrary conductivity distribution. Because this relationship is linear, image reconstruction may proceed by any number of linear methods. Here, a direct method is developed that combines several strategies that are particularly well suited for inverting the convolution integral. These include use of a diagonal regularization matrix that leverages kernel behavior; transformation of the minimization problem to standard form, avoiding the need for generalized singular value decomposition (SVD); centering the quadratic penalty norm on the uniform solution that best explains loss data; use of KKT multipliers to enforce non-negativity and manage the rather small active set; and, assignment of the global regularization parameter via the discrepancy principle. The entire process is efficient, requiring only one SVD, and provides ample controls to promote proper localization of structural features. Two virtual phantoms were created to test the algorithm on systems comprised of ~11,000 degrees of freedom.