Journal of Mathematical Imaging and Vision

, Volume 39, Issue 3, pp 259–268

# A New Solution to the Relative Orientation Problem Using Only 3 Points and the Vertical Direction

• Amir Hashemi
• Franck Jung
• Jean-Pierre Guedon
Article

## Abstract

This paper presents a new method to solve the relative pose between two images, using three pairs of homologous points and the knowledge of the vertical direction. The vertical direction can be determined in two ways: The first requires direct physical measurements such as the ones provided by an IMU (inertial measurement unit). The other uses the automatic extraction of the vanishing point corresponding to the vertical direction in an image. This knowledge of the vertical direction solves two unknowns among the three parameters of the relative rotation, so that only three homologous couples of points are requested to position a couple of images. Rewriting the coplanarity equations thus leads to a much simpler solution. The remaining unknowns resolution is performed by “hiding a variable” approach. The elements necessary to build a specific algebraic solver are given in this paper, allowing for a real-time implementation. The results on real and synthetic data show the efficiency of this method.

### Keywords

Relative orientation Vertical direction Minimal solution

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