Epipolar Geometry

Definition

• Epipolar plane: spanned by a 3D point and the two camera centers.
• Baseline: the line between the two camera centers; part of every epipolar plane.
• Epipole: intersection of the baseline with an image plane (can be outside the image or at infinity).
• Epipolar line: intersection of the epipolar plane with an image plane = projection of the other camera’s viewing ray. All epipolar lines go through the epipole.

Epipolar constraint

$$x_0^{T}E{x}_1=0(\text{calibrated})u_0^{T}F{u}_1=0(\text{pixels})$$

with $E=[t{]}_{\times }R$ and $F=K_0^{-T}E{K}_1^{-1}$.

Intuition

The two viewing rays and the baseline are coplanar. The constraint is the statement ”$x_0$ is orthogonal to the normal $[t{]}_{\times }R{x}_1$ of the epipolar plane”.

Use for matching: the corresponding point of $u_1$ must lie on the epipolar line $l_0=F{u}_1$, so the search is 1D instead of 2D.

Appears in

• Lecture 3, Epipolar geometry: definitions and visualizations
• Lecture 3, The Epipolar Constraint: derivation
• Lecture 1, course overview
• Lecture 4, Recap and rectification: epipolar lines become scanlines

Related concepts

• Essential and Fundamental Matrix
• Image Rectification
• Triangulation
• Structure from Motion