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)


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.


Digital planes Digital subplane recognition problem  Congruence sequence 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

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