Eight-Point Algorithm

Related Concepts

Eight-Point Algorithm; Epipolar Geometry; Essential Matrix; Fundamental Matrix

Definition

The 8-point algorithm is a linear technique to estimate the essential matrix or the fundamental matrix from eight or more point correspondences.

Background

When dealing with multiple images, it is essential to determine the relative geometry between them, which is known as the epipolar geometry. Between two images of a scene, given a pair of corresponding image points $(\vec{{m}}_{i},\vec{{m}}_{i}\prime)$, the following epipolar constraint must be satisfied:

$$\widetilde{{m}}^{\prime}_{i}{^T}\mathbf{M}\widetilde{{m}}_{ i} = 0\;,$$

(1)

where $\widetilde{\vec{{m}}}_{i} = \left [\begin{array}{*{10}c} \vec{{m}}_{i} \\ 1 \end{array} \right ]$ is point $\vec{{m}}_{i}$ in homogeneous coordinates. Similarly, $\widetilde{\vec{{m}}}_{i}\prime$ is point $\vec{{m}}_{i}\prime$ in homogeneous coordinates. Matrix $\mathbf{M}$is a 3 ×3 matrix. If the images are calibrated with known...