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Undistort points convergence #27993 I have looked into the `undistortPoints()` problem of issue #27916 and have found a solution. The problem is, as @Linhuihang has correctly pointed out, that the fixed-point iterations do not converge. Here are the functions which are optimized for the undistortion problem: $$ \begin{aligned} r^2 &= x'^2 + y'^2 \\ f_1(x') &= \frac{1 + k_4 r^2 + k_5 r^4 + k_6 r^6}{1 + k_1 r^2 + k_2 r^4 + k_3 r^6} (x'' - 2p_1 x' y' - p_2(r^2 + 2 x'^2) - s_1 r^2 + s_2 r^4) = x' \\ f_2(y') &= \frac{1 + k_4 r^2 + k_5 r^4 + k_6 r^6}{1 + k_1 r^2 + k_2 r^4 + k_3 r^6} (y'' - p_1 (r^2 + 2 y'^2) - 2 p_2 x' y' - s_3 r^2 - s_4 r^4) = y' \end{aligned} $$ where $x', y'$ are the undistorted points we want to compute and and $x'', y''$ are the given distorted points. This problem is solved using fixed-point iterations like $$ x'_{k+1} = f_1(x'_k),\quad y'_{k+1} = f_2(y'_k) $$ I guess the issue here is that the distortion function does not necessarily satisfy the [Banach fixed-point theorem](https://en.wikipedia.org/wiki/Banach_fixed-point_theorem), i.e. the slope of the function can be too large. This can be seen in @Linhuihang's comment https://github.com/opencv/opencv/issues/27916#issuecomment-3417883642 - the point series jumps around and doesn't converge. A common solution is to instead do damped fixed-point iterations, so that the updates are "more smooth". $$ x'_{k+1} = (1 - \alpha) x'_k + \alpha f_1(x'_k),\quad y'_{k+1} = (1 - \alpha) y'_k + \alpha f_2(y'_k) $$ I have implemented a simple logic which starts with $\alpha = 1$ (so just like it is now) and reduces $\alpha$ whenever the optimization error would increase. This seems reasonable to me: the initial logic is to do normal fixed-point iterations and to gradually become "more damped" when we notice that we don't converge. Perhaps there is a better way to ensure convergence, but this is the most straightforward modification to the current code that I have found. This problem is not due to the $\tau_x, \tau_y$ parameters; it also occurs when they are zero. In fact, the fixed-point iterations are done when the tilt correction of $\tau_x, \tau_y$ has already been applied. I have added a test to reproduce the problem. This PR fixes #27916. ### Pull Request Readiness Checklist See details at https://github.com/opencv/opencv/wiki/How_to_contribute#making-a-good-pull-request - [x] I agree to contribute to the project under Apache 2 License. - [x] To the best of my knowledge, the proposed patch is not based on a code under GPL or another license that is incompatible with OpenCV - [x] The PR is proposed to the proper branch - [x] There is a reference to the original bug report and related work - [x] There is accuracy test, performance test and test data in opencv_extra repository, if applicable Patch to opencv_extra has the same branch name. - [x] The feature is well documented and sample code can be built with the project CMake
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