Epipolar Geometry

The geometric relationship that forces the matching point of a scene point, seen by two cameras, to lie on a specific line.

Epipolar geometry describes the geometric constraint between two viewpoints. A 3D point X together with the two cameras’ optical centers forms a plane (the epipolar plane); its intersection with each image plane is an epipolar line, and the projection of one camera’s optical center onto the other image is an epipole. The core result: the match for a point in the left image must lie on a specific epipolar line in the right image, so a search for correspondences only needs to scan along that line — reducing a 2D search to 1D. Mathematically this is expressed by the 3×3 fundamental matrix F, with corresponding points x and x′ satisfying x′ᵀFx = 0; when both cameras’ intrinsics K and K′ are known, this can be written as the essential matrix E = K′ᵀFK, which encodes the rotation and the direction-only (unscaled) translation between the two cameras, introduced to computer vision by Longuet-Higgins in 1981. F or E is usually estimated from matched points with an algorithm like the eight-point algorithm combined with RANSAC, and relative pose is then decomposed from it. It underlies stereo rectification, stereo matching, triangulation, structure from motion, and visual SLAM initialization.

ExampleStereo rectification for a stereo camera pair is exactly the process of warping both images so all epipolar lines become horizontal and aligned row by row, letting stereo matching search left-right along a single row to solve for disparity.

Also called

Essential Matrix, Fundamental Matrix, Epipolar Constraint

Related

Triangulation · Stereo Matching · Homography · Camera Calibration · Structure from Motion · Random Sample Consensus

Sources

Wikipedia: Epipolar geometry
Wikipedia: Essential matrix