homographies

A homography is a projective transformation of the projective plane, represented by an invertible 3×3 matrix H acting on homogeneous coordinates. If a point is written as x = [X, Y, W]^T, then the transformed point is x' = Hx, defined up to a nonzero scale. In ordinary coordinates, the mapping takes (X/W, Y/W) to (X'/W', Y'/W') via the matrix entries of H. A homography describes how every point and line in one plane corresponds to a point and line in another plane.

Key properties include that a homography maps lines to lines and preserves the cross-ratio of any four collinear points, as well as incidence relations between points and lines. The transformation has eight degrees of freedom (a 3×3 matrix defined up to an overall scale). A special case is when H has the form of an affine transformation (the last row is [0 0 1]); such mappings preserve parallelism. When H does not have this form, the mapping is a full projective transformation, capable of perspective distortions that can move points at infinity to finite positions.

Estimation and applications: In computer vision, a homography often relates two views of a planar scene. It can be estimated from point correspondences x_i ↔ x_i' using the direct linear transformation (DLT) method, typically with at least four correspondences, followed by normalization and optional non-linear refinement to minimize reprojection error. Applications include image stitching and panorama creation, planar scene rectification, and perspective correction in photographs. In general 3D scenes, a single homography is not sufficient to describe all correspondences; the fundamental matrix is used to capture epipolar geometry when the scene is not constrained to a plane.