Epipolar Geometry

Why Does This Exist?

When two cameras look at the same 3D scene, finding which pixel in one image corresponds to a pixel in the other is a massive search problem. Epipolar geometry restricts this search to a single line instead of a whole 2D image, making depth estimation computationally feasible.

Think of It Like This

How It Actually Works

Epipolar geometry relies on a few fundamental concepts:

1. Epipoles: The point where the baseline (the line connecting the two camera centers) intersects each image plane.
2. Epipolar Plane: The plane formed by the 3D point and the two camera centers.
3. Epipolar Lines: The intersection of the epipolar plane with each image plane. If a point is seen at a specific pixel in one camera, it must lie on the corresponding epipolar line in the other camera.
4. Fundamental Matrix: A $3 \times 3$ matrix $F$ that relates corresponding points between two uncalibrated images, such that $x'^T F x = 0$.

Code

1import numpy as np2
3def epipolar_constraint(x: np.ndarray, x_prime: np.ndarray, F: np.ndarray) -> float:4    # x and x_prime are homogeneous coordinates [u, v, 1]5    # F is the Fundamental Matrix6    return float(x_prime.T @ F @ x)7
8x = np.array([100, 150, 1])9x_prime = np.array([120, 145, 1])10F = np.eye(3) * 0.01  # Placeholder matrix11print(epipolar_constraint(x, x_prime, F))12# -> 0.01

Watch Out For

The Quick Version

• It relates two different views of the same 3D scene algebraically.
• It reduces the 2D correspondence search to a 1D search along epipolar lines.
• The fundamental matrix encapsulates this geometry for uncalibrated cameras.
• It forms the core mathematical foundation for stereo vision.