Triangulation for points on lines

Triangulation consists in finding a 3D point reprojecting the best as possible onto corresponding image points. It is classical to minimize the reprojection error, which, in the pinhole camera model case, is nonlinear in the 3D point coordinates. We study the triangulation of points lying on a 3D line, which is a typical problem for Structure-From-Motion in man-made environments. We show that the reprojection error can be minimized by finding the real roots of a polynomial in a single variable, which degree depends on the number of images. We use a set of transformations in 3D and in the images to make the degree of this polynomial as low as possible, and derive a practical reconstruction algorithm. Experimental comparisons with an algebraic approximation algorithm and minimization of the reprojection error using Gauss-Newton are reported for simulated and real data. Our algorithm finds the optimal solution with high accuracy in all cases, showing that the polynomial equation is very stable. It only computes the roots corresponding to feasible points, and can thus deal with a very large number of views - triangulation from hundreds of views is performed in a few seconds. Reconstruction accuracy is shown to be greatly improved compared to standard triangulation methods that do not take the line constraint into account.