The problem addressed here is that of determining the first-order ray-transfer characteristics of an unknown optical system from the positions and directions of rays entering and leaving the system. Mathematically, the problem is a rather daunting one of solving a set of linear equations subject to a set of nonlinear constraints for a total of 10 or 14 unknowns. The nonlinearities are removed by rephrasing the problem, expressed initially in terms of the ray-transfer matrix of the system, into one in terms of the point characteristic matrix. The problem is then solved by converting it to a linear equation in a vector of the unknowns.
W. F. Harris,
"Determination of the paraxial ray-transfer characteristics of an optical system from rays passing through it," Optical Engineering 33(9), (1 September 1994). https://doi.org/10.1117/12.175668