About the frequencies of some patterns in digital planes. Application to area estimators

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 specified. It allows to prove some combinatorial properties of a class of patterns called (m,n)(m,n)-cubes. This study has also some consequences on local estimators of area: we prove that the local estimators restricted to regions of plane never converge to the exact area when the resolution tends to zero for almost all slopes of plane. All the results are generalized for the regions of hyperplanes in any dimension d⩾3d⩾3.