### Abstract

In this paper, we prove the convergence property of the Horn-Schunck optical-flow computation scheme. Horn and Schunck derived a Jacobi-method-based scheme for the computation of optical-flow vectors of each point of an image from a pair of successive digitised images. The basic idea of the Horn-Schunck scheme is to separate the numerical operation into two steps: the computation of the average flow vector in the neighborhood of each point and the refinement of the optical flow vector by the residual of the average flow vectors in the neighborhood. Mitiche and Mansouri proved the convergence property of the Gauss-Seidel- and Jacobi-method-based schemes for the Horn-Schunck-type minimization using algebraic properties of the matrix expression of the scheme and some mathematical assumptions on the system matrix of the problem. In this paper, we derive an alternative proof for the original Horn-Schunck scheme. To prove the convergence property, we develop a method of expressing shift-invariant local operations for digital planar images in the matrix forms. These matrix expressions introduce the norm of the neighborhood operations. The norms of the neighborhood operations allow us to prove the convergence properties of iterative image processing procedures.

Original language | English |
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Title of host publication | Combinatorial Image Analysis - 12th International Workshop, IWCIA 2008, Proceedings |

Pages | 262-273 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 1 Dec 2008 |

Event | 12th International Workshop on Combinatorial Image Analysis, IWCIA 2008 - Buffalo, NY, United States Duration: 7 Apr 2008 → 9 Apr 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4958 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 12th International Workshop on Combinatorial Image Analysis, IWCIA 2008 |
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Country | United States |

City | Buffalo, NY |

Period | 7/04/08 → 9/04/08 |

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### Cite this

*Combinatorial Image Analysis - 12th International Workshop, IWCIA 2008, Proceedings*(pp. 262-273). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4958 LNCS). https://doi.org/10.1007/978-3-540-78275-9-23