Where is the photograph? Using Grassmann-Cayley algebra

This work proposes a geometric to calculate the position of a camera within a non-calibrated system. The solution is formalized using Grassmann-Cayley algebra.

Computational problem.

Goal: To compute the position of the camera from a single photograph took by the camera itself.

Input: The coordinates of some points of reference in the tridimensional space, and the coordinates of their projections to the image plane.

Output: The coordinates representing the position of the camera in the tridimensional space.

Experimental results.

The method was tested using a Point Gray Bumblebee camera, and a AICON 3D Systems GMbH calibration grid. After an statistical analysis we found that the method has a precision and accuracy of the order of milimeters in computing the position of the camera.

In addition, the method can be used with reference points within a canonical projective, affine or euclidean frame.


[1] Gerardo Altamirano Gómez. “¿Dónde está el fotógrafo? Usando álgebra de Grassmann-Cayley.”. Master thesis. Baja California, México: Centro de Investigación Cientı́fica y de Educación Superior de Ensenada, 2012.

[2] Mariana Chan-Ley et.al. “Método no-calibrado para localización automática de una cámara aplicable en robótica”. In: Congreso Mexicano de Robótica (COMROB), 2018.

[3] Mariana Chan-Ley et.al. "Self-localization of an uncalibrated camera through invariant properties and coded target location," Appl. Opt. 59, D239-D245, 2020.