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. Triangulation is sometimes also referred to as reconstruction or intersection. The triangulation problem is in principle trivial. Since each point in an image corresponds to a line in 3D space, all points on the line in 3D are projected to the 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 the projection of a common 3D point x. The set of lines generated by the image points must intersect at x (3D point) and the algebraic formulation of the coordinates of x (3D point) can be computed in a variety of ways, as is presented below. In practice, however, the coordinates of image 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. In the literature there are multiple proposals for how to define optimality and how to find the optimal 3D point. Since they are based on different optimality criteria, the various methods produce different estimates of the 3D point x when noise is involved.
Introduction
In the following, it is assumed that triangulation is made on corresponding image points from two views generated by pinhole cameras. The top image illustrates the epipolar geometry of a pair of stereo cameras of pinhole model. A point x (3D point) in 3D space is projected onto the respective image plane along a line (green) which goes through the camera's focal point, O 1 {\displaystyle \mathbf {O} _{1}} and O 2 {\displaystyle \mathbf {O} _{2}} , resulting in the two corresponding image points y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} . If y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} are given and the geometry of the two cameras are known, the two projection lines (green lines) can be determined and it must be the case that they intersect at point x (3D point). Using basic linear algebra that intersection point can be determined in a straightforward way. The bottom image shows the real case. The position of the image points y 1 {\displaystyle \mathbf {y} _{1}} and y 2 {\displaystyle \mathbf {y} _{2}} cannot be measured exactly. The reason is a combination of factors such as:
… excerpt ends here. Continue reading the full article.
