Epipolar Geometry

Introduction

Epipolar geometry is the geometry of stereo vision. When a 3D point $P$ is viewed by two cameras at different positions, it creates corresponding points $p_1$ and $p_2$ in the two image planes. The projection $p_2$ is restricted to the epipolar line, $L$.

Key Concepts

• Epipolar Points ($e_1$ and $e_2$): The points where the baseline intersects the image planes
• Epipolar Lines: For a point in one image, its corresponding epipolar line in the other image is where its matching point must lie
• Epipolar Plane: The plane containing the 3D point and both camera centers
• Fundamental Matrix ($F$): The algebraic representation of epipolar geometry
• Essential Matrix ($E$): The specialized case of $F$ for calibrated cameras Given a point

Mathematical Framework

The fundamental matrix $F$ relates corresponding points in stereo images. For corresponding points $x$ and $x'$, we have:

The epipolar constraint equation:

$$x'^T F x = 0$$

The essential matrix $E$ is related to the fundamental matrix by:

$$E = K'^T F K$$

where $K$ and $K'$ are the camera calibration matrices.

For a point $x$ in the first image, its epipolar line $l'$ in the second image is given by:

$$l' = Fx$$

The fundamental matrix has the following properties:

• Rank 2: $det(F) = 0$
• 7 degrees of freedom
• For any epipolar line $l'$: $l' = Fx$

Applications

The epipolar geometry has several important applications in computer vision:

• Stereo Vision: Computing depth from stereo images using the relation $Z = \frac{fB}{d}$, where:
  • $Z$ is the depth
  • $f$ is the focal length
  • $B$ is the baseline distance
  • $d$ is the disparity
• 3D Reconstruction: Using triangulation to recover 3D points
• Camera Calibration: Estimating camera parameters
• Motion Estimation: Computing relative camera motion