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

Eric Andres 1 Dimitri Ouattara 1 Gaëlle Largeteau-Skapin 1 Rita Zrour 1
1 XLIM-SIC - SIC
Université de Poitiers, XLIM - XLIM
Abstract : In 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.
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Seventeenth International Workshop on Combinatorial Image Analysis, Nov 2015, Calcutta, India. Springer Verlag, Lecture Notes in Computer Science, 9448, pp.157-171, 2015, Lecture Notes in Computer science. 〈10.1007/978-3-319-26145-4_12〉
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Eric Andres, Dimitri Ouattara, Gaëlle Largeteau-Skapin, Rita Zrour. Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence. Seventeenth International Workshop on Combinatorial Image Analysis, Nov 2015, Calcutta, India. Springer Verlag, Lecture Notes in Computer Science, 9448, pp.157-171, 2015, Lecture Notes in Computer science. 〈10.1007/978-3-319-26145-4_12〉. 〈hal-01211478〉

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