IWCIA 2015: Combinatorial Image Analysis pp 157-171

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

• Eric Andres
• Dimitri Ouattara
• Gaelle Largeteau-Skapin
• Rita Zrour
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9448)

## 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 (xyz) 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.

## Keywords

Digital planes Digital subplane recognition problem  Congruence sequence

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Eric Andres
• 1
Email author
• Dimitri Ouattara
• 1
• Gaelle Largeteau-Skapin
• 1
• Rita Zrour
• 1
1. 1.Laboratoire XLIM, SIC, UMR CNRS 7252Université de PoitiersFuturoscope ChasseneuilFrance