Epipolarity

Epipolarity is a fundamental concept in computer vision and photogrammetry that describes the geometric relationship between two images of the same 3D scene taken from different viewpoints. When a 3D point is observed from two different camera positions, the projection of this point onto the first image lies on a line in the second image, and vice versa. This line is called the epipolar line.

The epipolar constraint arises because the two camera centers and the 3D point all lie in the same plane. This plane intersects the image planes of the two cameras along the epipolar lines. The intersection of this plane with the line connecting the two camera centers is called the epipolar line. The point where this line intersects the image plane is called the epipole.

The relationship between the two images can be mathematically described by the fundamental matrix or the essential matrix. The fundamental matrix is a 3x3 matrix that relates corresponding points in two uncalibrated image views. The essential matrix is similar but requires knowledge of the camera intrinsics and extrinsics, specifically the relative rotation and translation between the two cameras.

Understanding epipolarity is crucial for tasks such as stereo vision, 3D reconstruction, and structure from motion. It allows for efficient matching of points between images and provides constraints that help in recovering the 3D structure of the scene. Without epipolar geometry, finding corresponding points between images would be a much more computationally expensive and ambiguous process.