The Horn and Schunck (HS) method, which amounts to the Jacobi iterative scheme in the interior of the image, was one of the first optical flow algorithms. In this article, we prove the convergence of the HS method, whenever the problem is well-posed. Our result is shown in the framework of a generalization of the HS method in dimension n ≥ 1, with a broad definition of the discrete Laplacian. In this context, the condition for the convergence is that the intensity gradients are not all contained in a same hyperplane. Two other articles ([17] and [13]) claimed to solve this problem in the case n = 2, but it appears that both of these proofs are erroneous. Moreover, we explain why some standard results on the convergence of the Jacobi method do not apply for the HS problem, unless n = 1. It is also shown that the convergence of the HS scheme implies the convergence of the Gauss-Seidel and SOR schemes for the HS problem.

中文翻译：

---

任意维上的角和萧克光学流算法的收敛性证明。

Horn and Schunck（HS）方法相当于图像内部的Jacobi迭代方案，是最早的光流算法之一。在本文中，只要问题适当提出，我们就证明了HS方法的收敛性。我们的结果在n≥1维的HS方法的泛化框架中得到展示，其中对离散拉普拉斯算子进行了广义定义。在这种情况下，收敛的条件是强度梯度并非全部包含在同一超平面中。其他两篇文章（[17]和[13]）声称在n = 2的情况下解决了这个问题，但似乎这两个证明都是错误的。此外，我们解释了为什么关于Jacobi方法收敛的一些标准结果不适用于HS问题，除非n = 1。