Essential Matrix

1. Motivating the problem: Stereo

Given two views of a scene (the two cameras not necessarily having optical axes), what is the relationship between the location of a scene point in one image and its location in the other?

2. Assumption

We have calibrated cameras, so we know the transformation from one origin to the other:

Some translation $\pmb{t}$ and rotation $\pmb{R}$, this means: $$\pmb{x_2 = Rx_1 + t}$$

3. Essential Matrix

Cross both sides by $t$: $$\pmb{t} \times \pmb{x_2 = t \times Rx_1 + t \times t} \\ = \pmb{t \times Rx_1}$$ Dot both sides with $x_2$: $$\pmb{x_2 \cdot} (\pmb{t} \times \pmb{x_2)} = \pmb{x _2 \cdot (t \times Rx_1)} \\ \pmb{0 = x_2 \cdot (t \times Rx_1)}$$ Suppose $\pmb{E=[t_x]R}$, this means: $$\pmb{x_2^T Ex_1 = 0}$$ $\pmb{E}$ is called the "essential matrix", it is a expression relating the two points on the two image planes.