Homography

Homography is the geometrical relationship between two sets of points-in this case, sets of detected or measured corner points on planes in 3-space (see Figure [A]. Since planar homography is made up of linear relationships, the homographies this project finds can be represented by 3 x 3 matrices representing the transformations between sets of planes. This project finds the homographies between the 3-D plane of each image that was taken and the plane of the model points, which is (x, y, z = 1) and centered on (0, 0, 1), representing them as 3x3 transformation (rotation / translation) matrices of the form

$H = {\it s} * A * [ R_1 R_2 \vert t ]$

, where s is an arbitrary scalar and A is a matrix as specified below. R is the rotation transformation matrix between the two planes and t is the 3x1 translation vector.

Since a homography is made up of rotations and transformations from the model plane to an image plane, knowing the homography (based on relationships between individual pairs of points) tells us about the rotations and translations necessary to get from the origin to each image. Another part of the homography is the matrix containing information on the intrinsic camera parameters: the vertical and horizontal focal lengths of the camera, the amount of skew it introduces, and the point on any image, called the principal point of the camera, corresponding to the center of the camera lens. This matrix is known as the fundamental matrix of the camera, and is individual to the single camera being used.