Distinguishing between multiple Infra Red Transmitters
It's a symmetry problem, you can solve it breaking the simmetry.
Options:


use three transmitter instead of four, the plane and its center is
still determined.
move one transmitter in the middle of the side of the square.
use different duty cycle for one (or more) transmitter.
use linear polarizing filters on the transmitters and a rotating filter on the camera lense.
...


I guess that also SVD could be helpful to find the pattern's center, when the camera is not centered and perpendicular.