About the Frequencies of Some Patterns in Digital Planes Application to Area Estimators

Abstract

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: 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 slope of plane. Actually all the results of this paper can be generalized for the regions of hyperplanes for any dimension d ≥ 3. The proofs of some results used in this article are contained in the extended version of this paper [1].

Farey lines defining Farey diagrams and application to some discrete structures

“…There seem to be some similarities between (m, n)-cubes, which we redefine in this article, and threshold functions on a two-dimensional rectangular grid, for which asymptotic values have been derived in [5]. In [10], it was shown that the number of straight lines with some particular conditions (which we call Farey lines) is O(mn(m + n)) In this work, we derive an asymptotic value whose major term is mn(m + n)/ζ (3). The Farey diagram for discrete segments was also studied by McIlroy in [10].…”

“…Definition 9 ((m, n)-cube [3], see Figure 1). The (m, n)-pattern w i,j (α, β, γ) at the position (i, j) of a discrete plane P α,β,γ is the (m, n)-pattern w defined by : …”

“…In [3], one of the strategies for the enumeration of discrete pieces of planes, was to estimate, in the most precise possible way, the number of vertices in a diagram called a Farey diagram. The upper bound for the cardinality of the pieces of discrete planes (or (m, n) − cubes) obtained in [3], is a homogeneous polynomial of degree 8 : m 3 n 3 (m + n) 2 .…”

Patterns in Discretized Parabolas and Length Estimation

“…The proof of this lemma uses Weyl's argument, as in the proof of Theorem 1 of [2] or Appendix A.1 of [5], but extended to the quadratic case following the same ideas as in [6, p6-7]. It is given in Appendix A.1 of [3].…”

“…F L u (ω) is intuitively the frequency of the pattern ω in the discretized straight lines of slope u. (See [2,4] for more details and [5] for the generalization to slopes of planes).…”