About the frequencies of some patterns in digital planes. Application to area estimators (extended version)
Résumé
In this paper we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are precised. It allows to prove some combinatorial properties of a class of patterns called (m,n)-cubes. This study has also some consequences on local estimators of area: they never converge to the exact area when the resolution tends to zero for almost all region of plane. Actually we can prove with the same technics that this result is true for the regions of hyperplanes for any dimension d>=3.
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