homography

Homography is a projective transformation that relates two images of the same planar scene or, more generally, a mapping between two planes in projective space. In homogeneous coordinates, a point x = (x, y, 1)^T is mapped to x' = H x, where H is a 3×3 invertible matrix. The transformation is defined only up to scale, so H and λH represent the same homography. A homography preserves straight lines and incidences, but Euclidean distances, angles, and parallelism are not generally preserved.

Planar scenes: If all points lie on a single plane in 3D and the camera undergoes perspective motion, the mapping between the two images is a homography. Conversely, a homography is the most general projective transformation of a plane; for non-planar scenes there is no exact homography, though it can approximate the relation for a single plane.

Estimation: A homography can be estimated from point correspondences. Each pair yields two linear equations in the eight independent entries of H (fixing one entry, such as h9 = 1). At least four correspondences are required. Common methods include direct linear transformation (DLT), normalization of coordinates for numerical stability, and nonlinear refinement to minimize reprojection error. Robust estimation using RANSAC is often employed to reject outliers.

Applications: image alignment and panorama stitching, planar scene rectification, image warping, and augmented reality tasks involving a tracked plane.

Notes: H belongs to the projective general linear group PGL(3); composition corresponds to matrix multiplication, and all descriptions are up to a scale. When the scene is not planar, a single homography does not exactly model the relation between images; multiple homographies or a fundamental matrix may be needed to describe epipolar geometry.