Abstract
In this paper we take first steps in addressing the 3D Digital Subplane Recognition Problem. Let us consider a digital plane \(P: 0 \le ax+by-cz+d <c\) (w.l.o.g. \(0 \le a \le b \le c\)) and a finite subplane S of P defined as the points (x, y, z) of P such that \((x,y) \in \left[ x_0,x_1\right] \times \left[ y_0,y_1\right] \). 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 \(\left\{ \frac{ax+by+d}{c} \right\} , (x,y) \in \left[ x_0,x_1\right] \times \left[ y_0,y_1\right] \) of the subplane. This corresponds to a bilinear congruence sequence. We show that one can determine if the sequence contains a value \(\epsilon \) 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 \(m=x_1-x_0 < n = y_1-y_0\), \(O(m\log \left( \min (a,c-a)\right) \) or \(O(n\log \left( \min (b,c-b)\right) \) otherwise.
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Andres, E., Ouattara, D., Largeteau-Skapin, G., Zrour, R. (2015). Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence. In: Barneva, R., Bhattacharya, B., Brimkov, V. (eds) Combinatorial Image Analysis. IWCIA 2015. Lecture Notes in Computer Science(), vol 9448. Springer, Cham. https://doi.org/10.1007/978-3-319-26145-4_12
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