Epipolargeometrie

Epipolargeometrie is the study of the geometric relationship between two or more views of a scene captured by cameras and the projective constraints that relate image points to corresponding 3D points. It provides the mathematical framework for stereo vision, where information from multiple views is used to infer depth and reconstruct the scene. The key objects are the epipole, epipolar lines, and the matrices that encode the correspondence constraints between images, notably the fundamental matrix and the essential matrix.

The epipole in an image is the projection of the other camera’s center. The line in one image that corresponds to a point in the other image is called the epipolar line. For any pair of corresponding points x in image 1 and x′ in image 2, the epipolar constraint holds: x′^T F x = 0, where F is the fundamental matrix. This constraint reduces the correspondence problem from a 2D search to a 1D search along epipolar lines.

When the camera intrinsics are known, the essential matrix E describes the relative motion between calibrated cameras and satisfies x′^T E x = 0. The relationship between F and E is F = K′^{-T} E K^{-1}, where K and K′ are the intrinsic matrices of the two cameras. Epipolargeometrie underpins rectification, which aligns epipolar lines to image scanlines to simplify matching, and facilitates triangulation for 3D reconstruction.

Applications include stereo matching, depth estimation, motion estimation, and camera calibration. Limitations arise from non-rigid scenes, occlusions, calibration errors, and noisy feature correspondences.