title: A PROOF OF CONVERGENCE OF THE HORN AND SCHUNCK OPTICAL FLOW ALGORITHM IN ARBITRARY DIMENSION.
abstract: 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 dimensionn≥ 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 casen= 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, unlessn= 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.
authors.affiliation: RUBIC (Research Unit of Biomechanics and Imaging in Cardiology), University of Montreal Hospital Research Center (CRCHUM), Montréal, Québec, H2L-2W5 Canada.
authors.forename: Louis
authors.initials: L
authors.lastname: LE Tarnec
authors.affiliation: Laboratory of Biorheology and Medical Ultrasonics, University of Montreal Hospital Research Center (CRCHUM), Montréal, Québec, H2L-2W5 Canada.
authors.forename: François
authors.initials: F
authors.lastname: Destrempes
authors.affiliation: Laboratory of Biorheology and Medical Ultrasonics, University of Montreal Hospital Research Center (CRCHUM), Montréal, Québec, H2L-2W5 Canada, Department of Radiology, Radio-Oncology and Nuclear Medicine, University of Montreal, Montréal, Québec, H3T-1J4 Canada, Institute of Biomedical Engineering, University of Montreal, Montréal, Québec, H3T-1J4 Canada.
authors.forename: Guy
authors.initials: G
authors.lastname: Cloutier
authors.affiliation: RUBIC (Research Unit of Biomechanics and Imaging in Cardiology), University of Montreal Hospital Research Center (CRCHUM), Montréal, Québec, H2L-2W5 Canada, Department of Radiology, Radio-Oncology and Nuclear Medicine, University of Montreal, Montréal, Québec, H3T-1J4 Canada, Institute of Biomedical Engineering, University of Montreal, Montréal, Québec, H3T-1J4 Canada.
authors.forename: Damien
authors.initials: D
authors.lastname: Garcia
keywords: Horn and Schunck algorithm; Jacobi iterations; Optical flow