ECCV 2006: Computer Vision – ECCV 2006 pp 135-146

# Direct Solutions for Computing Cylinders from Minimal Sets of 3D Points

• Christian Beder
• Wolfgang Förstner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3951)

## Abstract

Efficient direct solutions for the determination of a cylinder from points are presented. The solutions range from the well known direct solution of a quadric to the minimal solution of a cylinder with five points. In contrast to the approach of G. Roth and M. D. Levine (1990), who used polynomial bases for representing the geometric entities, we use algebraic constraints on the quadric representing the cylinder. The solutions for six to eight points directly determine all the cylinder parameters in one step: (1) The eight-point-solution, similar to the estimation of the fundamental matrix, requires to solve for the roots of a 3rd-order-polynomial. (2) The seven-point-solution, similar to the six-point-solution for the relative orientation by J. Philip (1996), yields a linear equation system. (3) The six-point-solution, similar to the five-point-solution for the relative orientation by D. Nister (2003), yields a ten-by-ten eigenvalue problem. The new minimal five-point-solution first determines the direction and then the position and the radius of the cylinder. The search for the zeros of the resulting 6th order polynomials is efficiently realized using 2D-Bernstein polynomials. Also direct solutions for the special cases with the axes of the cylinder parallel to a coordinate plane or axis are given. The method is used to find cylinders in range data of an industrial site.

## Keywords

Point Cloud Range Data Cylinder Axis Direct Solution Bernstein Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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