The Absolute Conic and the Camera Parameters

Consider two camera projections P and P' corresponding to the same camera (internal parameters) but different poses. We saw in exercise 4.3 that the image of the absolute conic (IAC) is independent of the camera pose. In fact, the IAC is directly related to the internal parameter matrix K of the camera defined in equation (1.2).

Given that the absolute conic is in any case invariant under rotations and translations, we can choose coordinates so that the first projection matrix reduces to $P=K(I_{3\times 3}\vert)=(K\vert)$ (c.f. equation 1.1). Hence, for a point $X = (\vec{x}, 0)^{t}$ on the plane at infinity we have projection $u=K\vec{x}$ or $\vec{x}=K^{-1}u$. $\;\;u$ is on the image of the absolute conic exactly when X is on the conic itself, i.e. when

$$u^t K^{-t} K^{-1} u = \vec{x}^t\vec{x} = 0$$

So the image of the absolute conic is given by the matrix K^-tK^-1. If this matrix can be found, by Choleski factorization we can extract K^-1 and thence find the internal parameter matrix K. In fact, as we will see, it is easier to work from the outset with the inverse K^tK of the IAC matrix, called the dual image of the absolute conic (DIAC).