Impedance imaging has received a lot of attention in the past two decades, as a means for non-destructively imaging the interior of a conductive object. One injects a known electrical current pattern into an object at the exterior boundary, then measures the induced potential (voltage) on some portion of the boundary. The goal is to recover information about the interior conductivity of the object, which (we hope) influences the voltages we measure. Of course one can also use multiple input currents and measured voltages. A variation on this problem is that of "boundary identification," in which some portion of the object's boundary is unknown (and inaccessible), and must be determined from this type of current in-voltage out data. This might be due to the boundary having been corroded or otherwise damaged.
Chiew, Esther and Selhorst-Jones, Vincent, "Determining the Shape of a Resistor Grid" (2008). Mathematical Sciences Technical Reports (MSTR). 33.