A Proof of Convergence of the Horn-Schunck Optical Flow Algorithm in Arbitrary Dimension

International audienceThe Horn–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 paper, 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 not all be contained in the same hyperplane. Two other works ([A. Mitiche and A. Mansouri, IEEE Trans. Image Process., 13 (2004), pp. 848–852] and [Y. Kameda, A. Imiya, and N. Ohnishi, A convergence proof for the Horn-Schunck optical-flow computation scheme using neighborhood decomposition, in Combinatorial Image Analysis, Springer, Berlin, 2008, pp. 262–273]) 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 about 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 successive overrelaxation schemes for the HS problem