Triangulation (computer vision)

In computer vision triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function from 3D to 2D for the cameras involved, in the simplest case represented by the camera matrices.

The triangulation problem is in theory trivial, each point in an image corresponds to a line in 3D space such that all points on the line are projected to that first point in the image. If a pair of corresponding points in two, or more images, can be found it must be the case that they are all the projection of a common 3D point P. The set of lines generated by the image points must intersect at P and the algebraic formulation of the coordinates of P can be computed in a variety of of ways, as is presented below.

In practice, however, the problem is that the image coordinates of corresponding points cannot be measured with arbitrary accuracy. Instead, various types of noise, such as geometric noise from lens distortion or interest point detection error, lead to inaccuracies in the measured image coordinates. As a consequence, the lines generated by the corresponding image points do not always intersect in 3D space. The problem, then, is to find a 3D point which optimally fits the measured image points. It is in this respect that the various triangulation methods differ, they are based on different types of optimization criteria.