One important application of collinearity equations is in the spatial intersection problem, were we estimate approximate intersection of two 3D lines to find model or ground points for a group of corresponding image points. Here, we assume that the following inputs are given
1- Internal orientation parameters of the camera
2- External orientation parameters of the stereo pair
3- Pixel coordinates of a group of corresponding points
The goal is to calculate the model coordinates of all image points. For this purpose, we rewrite the equations of the collinearity as
As we can see, these equations are linear based on unknowns; Therefore, they can be easily solved. In the case where we have a two-point intersection problem, there is one degree of freedom in the problem, but in general, the number of degrees of freedom will be: df=2.n-3 where n is equal to the number of corresponding image points. Therefore, the more image points we have, the more robust geometry we will use to estimate the model point.
Space intersection despite seemingly write as a simple approach, contains delicate specifications that can help us filter out outliers, or detect weak geometries.