A Proof of Convergence of the Horn--Schunck Optical Flow Algorithm in Arbitrary Dimension
Bibliographic record
Abstract
The 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 $n\geq1$, 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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".