Linear Equations, Matrices and PseudoInverse

objective:

solve polynomial coefficients to fit chromatic aberration data

polynomial model:

`C0 + C1*x + C2*y + C3*x*x + C4*y*y + C5*x*x*x + C6*y*y*y + C7*x*y + C8*x*x*y + C9*x*y*y,
where x and y are pixel coordinates,
and the result approximates red or blue pixel misregistration relative to green at those coordinates.
This model may be too simple (underspecified) or overfit.
  however, gnuplots of model vs data seem reasonable...
4 solutions are wanted:  red and blue in x and y directions

in theory

solving for 10 unknown coefficients wants 10 equations
with 10 sets of independent variables and a set of 10 results

practically

data is noisy

in matrix algebra terms,

we have 10 unknowns C0-9,
an array of measured offsets and a matrix of linearly independent sample variables.
If there were 10 offset measurements and corresponding sample variables,
the matrix algebra expression is MC = O, where:

• M is a 10x10 array of linearly independent sample variables,
• C are the unknown coefficients,
• O are measured offsets.

Chromatic aberration characterized by only 10 measurements is arguably suspect;
doing so would yield a solution for C = MiO, where Mi is the inverse of M.

in reality, a calibration slide has hundreds of sample targets

those measurements comprise an overdetermined system
a approximate solution is appropriate. My favorite approach is Moore-Penrose pseudoinverse.