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 $v = f_{2,3}(u) \Rightarrow f_{1, 2}(v) = f_{1,3}(u)$


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