Farey Sequences and the Planar Euclidean Medial Axis Test Mask

Authors:
Jérôme Hulin;Édouard Thiel
Affiliations:
Laboratoire d'Informatique Fondamentale de Marseille (LIF, UMR 6166), Aix-Marseille Université, Marseille cedex 9, France 13288;Laboratoire d'Informatique Fondamentale de Marseille (LIF, UMR 6166), Aix-Marseille Université, Marseille cedex 9, France 13288
Venue:
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Year:
2009

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Abstract

The Euclidean test mask $\mathcal{T}$(r) is the minimum neighbourhood sufficient to detect the Euclidean Medial Axis of any discrete shape whose inner radius does not exceed r . We establish a link between $\mathcal{T}$(r) and the well-known Farey sequences, which allows us to propose two new algorithms. The first one computes $\mathcal{T}$(r) in time $\mathcal{O}(r^4)$ and space $\mathcal{O}(r^2)$. The second one computes for any vector $\overrightarrow{v}$ the smallest r for which $\overrightarrow{v} \in\mathcal{T}$(r), in time $\mathcal{O}(r^3)$ and constant space.