I would like to have tips on the following problem (solution methods, problem category, etc.).

The context is about the simultaneous registration of multiple images ($ n $ images, $ f_ {i, j} $ the function that maps coordinates $ (x_i, y_i) $ to $ (x_j, y_j) $,

To let $ f_ {i, j} ^ k (x, y) $ a polynomial function of order $ k $ from $ mathbb {R} ^ 2 $ to $ mathbb {R} ^ 2 $With $ i, j in [1, n]$,

For example when $ k = 0 $Such functions can be expressed as $ f_ {i, j} ^ 0 (x, y) = (a_ {ij} ^ 0, a_ {ij} ^ 1) $,

Or when $ k = 1 $: $ f_ {i, j} ^ 1 (x, y) = (a_ {ij} ^ 0 + b_ {ij} ^ 0 x + c_ {ij} ^ 0 y, a_ {ij} ^ 1 + b_ {ij} ^ 1 y + c_ {ij} ^ 1 y) $,

Accepted $ k $ is fixed, is waived in the following.

This family of functions should take into account as much as possible in terms of property: $ f_ {i, j} = f_ {i, k} (f_ {k, j}) $,

I want to use this property to reduce the number of unknowns and get robust estimates of the coefficients.

In addition, I have a large number of matches that can be expressed as such: $ v = f_ {i, j} (u) $,

My problem is this: How can I calculate the functions robustly? $ f_ {1, i} $ ?

If $ k = 0 $The problem is quite simple: $ f_ {i, j} = f_ {i, k} (f_ {k, j}) = f_ {i, k} + f_ {k, j} $ so I can write an over-constrained linear system and solve it with a Moore-Penrose inverse or an algorithm like RANSAC.

If k = 1 or 2, I do not know exactly how to proceed. I think I could try to design a custom convergence scheme with a given order of equation resolution and some convergence iterations.

As an example, if I solve $ f_ {1,2} $to get then $ f_ {1,3} $ I can use my matches of the form $ v = f_ {1,3} (u) $ but also the games like $ v = f_ {2,3} (u) => f_ {1,2} (v) = f_ {1,3} (u) $

Thanks, Thomas