An efficient and quasi linear worst-case time algorithm for digital plane recognition

Abstract

This paper introduces a method for the digital naive plane recognition problem. This method is a revision of a previous one. It is the only method which guarantees an O(nlogD) time complexity in the worst-case, where (D - 1) represents the size of a bounding box that encloses the points, and which is very efficient in practice. The presented approach consists in determining if a set of n points in ℤ3 corresponds to a piece of digital naive hyperplane in [4 log9/5 D] + 10 iterations in the worst case. Each iteration performs n dot products. The method determines whether a set of 106 voxels corresponds to a piece of a digital plane in ten iterations in the average which is five times less than the upper bound. In addition, the approach succeeds in reducing the digital naive plane recognition problem in ℤ3 to a feasibility problem on a two-dimensional convex function. This method is especially fitted when the set of points is dense in the bounding box, i.e. when D = O(√n). © 2008 Springer-Verlag Berlin Heidelberg.