Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence

Abstract

International audienceIn this paper we take first steps in addressing the 3D Digital Subplane Recognition Problem. Let us consider a digital plane P : 0 ≤ ax + by − cz + d < c (w.l.o.g. 0 ≤ a ≤ b ≤ c) and a finite subplane S of P dened as the points (x, y, z) of P such that (x, y) ∈ [x0, x1] × [y0, y1]. The Digital Subplane Recognition Problem consists in determining the characteristics of the subplane S in less than linear (in the number of voxels) complexity. We discuss approaches based on remainder values ax+by+d c , (x, y) ∈ [x0, x1] × [y0, y1] of the subplane. This corresponds to a bilinear congruence sequence. We show that one can determine if the sequence contains a value in logarithmic time. An algorithm to determine the minimum and maximum of such a bilinear congruence sequence is also proposed. This is linked to leaning points of the subplane with remainder order conservation properties. The proposed algorithm has a complexity in, if m = x1 −x0 < n = y1 −y0, O(m log (min(a, c − a)) or O(n log (min(b, c − b)) otherwise