Essential and Fundamental Matrix

Definitions

$$E=[t{]}_{\times }RF=K_0^{-T}E{K}_1^{-1}{x}_0^{T}E{x}_1=0,u_0^{T}F{u}_1=0$$

$R=R_{1\to 0}$, $t$ = camera 1’s center in camera 0’s frame, $x_i=K_i^{-1}{u}_i$.

	$E$	$F$
coordinates	calibrated rays	pixels
needs intrinsics	yes	no
rank	2	2
degrees of freedom	5 (3 + 3 − 1 scale)	7 (9 − 1 scale − 1 for $\det F=0$)
epipoles	$E{\tilde{e}}_1=0={\tilde{e}}_0^{T}E$	$F{e}_1=0=e_0^{T}F$
epipolar lines	${\tilde{l}}_0=E{x}_1$, ${\tilde{l}}_1=E^T{x}_0$	$l_0=F{u}_1$, $l_1=F^T{u}_0$

Relative pose from E

$$E=U\text{diag}(1,1,0)V^T\Rightarrow t=\pm u_3,R=UW{V}^T\text{ or }U{W}^T{V}^T,W=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$

4 candidates; the correct one has the triangulated point in front of both cameras. $t$ is only known up to scale.

Estimation: $F$ with the (normalized) 8-point algorithm inside RANSAC, then $E=K_0^{T}F{K}_1$, projected to singular values $(1,1,0)$. $E$ alone needs only 5 points (5-point algorithm).

Appears in

• Lecture 3, Essential Matrix
• Lecture 3, Fundamental Matrix
• Lecture 3, Recovering the relative pose
• Lecture 3, From F to E

Related concepts

• Epipolar Geometry
• Eight-Point Algorithm
• Intrinsic Parameters
• Extrinsic Parameters