I’d like some advice about the following problem (resolution methods, problem category, etc.). The context is about the coregistration of several images all-in-once ($ n$ images, $ f_{i,j}$ the function that maps coordinates $ (x_i, y_i)$ to $ (x_j, y_j)$ .

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, if $ k=0$ , such functions can be expressed as $ f_{i,j}^0(x, y) = (a_{ij}^0, a_{ij}^1)$ . Or if $ 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)$ .

Let’s assume $ k$ is fixed, it will be omitted in the following.

This family of functions should respect as much as possible following the property: $ f_{i,j} = f_{i, k}(f_{k, j})$ .

I want to use this property to decrease the number of unknowns and obtain robust estimates of coefficients. Moreover, I have a large set of matches that can be expressed as such: $ v = f_{i,j}(u)$ .

My problem is the following: how can I compute robustly the functions $ 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 overdetermined linear system and solve it with a Moore-Penrose inverse or an algorithm such as RANSAC.

If k = 1 or 2, I don’t really known how to proceed. I think I could try to design a custom convergence scheme with a predetermined equation resolution order and some iterations to converge.

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

Thanks, Thomas