We describe minimal solutions for the rotation and focal length of a pair of cameras. In computer vision, the normal way of solving for the one-to-one mappings induced by rotating a camera or viewing a planar scene is to use 4 point correspondences to solve for an 8 parameter homography. However, under some cases the number of parameters is much smaller than 8. Inspired by similar work in structure and motion we investigated minimal solutions for the case of a rotating camera. Minimal solvers forgo the linearisations historically made to simplify the solution of two-view geometry problems, and instead solve the underlying polynomial equations directly. The solvers can be more complicated than the linear solutions possible for entities such as the fundamental matrix, but they find the true underlying parameters (such as rotation and translation), and thus are not susceptible to overfitting which causes problems for autocalibration. In this work we consider two special cases:

2-Point Algorithm

In the first case we derive a simple cubic equation in 1 unknown that can be used to solve for the focal length of the camera:

Where a and b are simple functions of the input image coordinates u:

3-Point Algorithm

The second algorithm requires solution of simultaneous polynomial equations, and is closely related to the solution of the intersections of 2 ellipses. The simultaneous solution of 2 ellipses can be written as a single matrix equation where the left hand side matrix (p, q, r) is a function of y:

Since the determinant of this matrix (known as the Sylvester matrix) must be 0, we can write down a polynomial in y whose solution gives us the intersections.