4 research outputs found

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

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    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

    Walking in the Farey fan to Compute the Characteristics of a Discrete Straight Line Subsegment

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    International audienceGiven a Digital Straight Line (DSL) of known characteristics (a,b,\mu), we address the problem of computing the characteristics of any of its subsegments. We propose a new algorithm as a smart walk in the so called Farey Fan. We take profit of the fact that the Farey Fan of order n represents in a certain way all the digital segments of length n. The computation of the characteristics of a DSL subsegment is then equivalent to the localization of a point in the Farey Fan. Using fine arithmetical properties of the fan, we design a fast algorithm of theoretical complexity O(log(n)) where n is the length of the subsegment. Experiments show that our algorithm is faster than the one previously proposed by Said and Lachaud in [14,15]

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum
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