Electrical Impedance Tomography (EIT)
The caption from figure 1 of the paper you link to


  Figure 1. Left: we insert current from one pair of electrodes
  and measure voltage from other pairs.


This means that the current is injected between an adjacent pair of electrodes. Voltage is sensed between all the other pairs of electrodes. 

I would expect this to be iterated, so that each pair gets connected in turn through the multiplexer shown in figure 7 to the current source, while all the other pairs are sensed for voltage. 

It doesn't matter if one of the current driving pair of electrodes is grounded or not, all that matters is that a current flows between them, and that the differential voltage is sensed between adjacent pairs of voltage sensing electrodes.

I've no more than skimmed the paper for figures 1 and 7, so don't know whether the authors go into detail about how to reconstruct the finger position from the readings. That is the tricky bit. The obvious naive route would be some form of pattern matching, whereby a measurement set is taken with no finger present, and then one with a finger touching, and the difference, suitably scaled for the drive level, is stored. The same thing would be done with fingers in different positions. Then you'd examine the differences and see what systematic alteration was happening. I am sure there are more sophisticated mathematical techniques available.