2015/02/16 by Adrian Nachman, Nachman, Adrian, Alexandru Tamasan +3
Engineering · Mathematics · #31A25 #35J60 #35R30 #62P10 #Analysis of PDEs (math.AP) #Electrical and Bioimpedance Tomography #FOS: Mathematics #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1502.04731
openalex publication_date 2015/02/16 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider the inverse problem of recovering an isotropic electrical\nconductivity from interior knowledge of the magnitude of one current density\nfield generated by applying current on a set of electrodes. The required\ninterior data can be obtained by means of MRI measurements. On the boundary we\nonly require knowledge of the electrodes, their impedances, and the\ncorresponding average input currents. From the mathematical point of view, this\npractical question leads us to consider a new weighted minimum gradient problem\nfor functions satisfying the boundary conditions coming from the Complete\nElectrode Model of Somersalo, Cheney and Isaacson. This variational problem has\nnon-unique solutions. The surprising discovery is that the physical data is\nstill sufficient to determine the geometry of the level sets of the minimizers.\nIn particular, we obtain an interesting phase retrieval result: knowledge of\nthe input current at the boundary allows determination of the full current\nvector field from its magnitude. We characterize the non-uniqueness in the\nvariational problem. We also show that additional measurements of the voltage\npotential along one curve joining the electrodes yield unique determination of\nthe conductivity. A nonlinear algorithm is proposed and implemented to\nillustrate the theoretical results.\n