Homography (computer vision)

We have two cameras a and b, looking at points ${\displaystyle P_{i}}$ in a plane. Passing from the projection ${\displaystyle {}^{b}p_{i}=\left({}^{b}u_{i};{}^{b}v_{i};1\right)}$ of ${\displaystyle P_{i}}$ in b to the projection ${\displaystyle {}^{a}p_{i}=\left({}^{a}u_{i};{}^{a}v_{i};1\right)}$ of ${\displaystyle P_{i}}$ in a:

${\displaystyle {}^{a}p_{i}={\frac {{}^{b}z_{i}}{{}^{a}z_{i}}}K_{a}\cdot H_{ab}\cdot K_{b}^{-1}\cdot {}^{b}p_{i}}$

where ${\displaystyle {}^{a}z_{i}}$ and ${\displaystyle {}^{b}z_{i}}$ are the z coordinates of P in each camera frame and where the homography matrix ${\displaystyle H_{ab}}$ is given by

${\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}}$.

${\displaystyle R}$ is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance from origin to the plane respectively. K_a and K_b are the cameras' intrinsic parameter matrices.

The figure shows camera b looking at the plane at distance d. Note: From above figure, assuming ${\displaystyle n^{T}P_{i}+d=0}$ as plane model, ${\displaystyle n^{T}P_{i}}$ is the projection of vector ${\displaystyle P_{i}}$ along ${\displaystyle n}$, and equal to ${\displaystyle -d}$. So ${\displaystyle t=t\cdot 1=t\left(-{\frac {n^{T}P_{i}}{d}}\right)}$. And we have ${\displaystyle H_{ab}P_{i}=RP_{i}+t}$ where ${\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}}$.

This formula is only valid if camera b has no rotation and no translation. In the general case where ${\displaystyle R_{a},R_{b}}$ and ${\displaystyle t_{a},t_{b}}$ are the respective rotations and translations of camera a and b, ${\displaystyle R=R_{a}R_{b}^{T}}$ and the homography matrix ${\displaystyle H_{ab}}$ becomes

${\displaystyle H_{ab}=R_{a}R_{b}^{T}-{\frac {(-R_{a}*R_{b}^{T}*t_{b}+t_{a})n^{T}}{d}}}$

where d is the distance of the camera b to the plane.

When the image region in which the homography is computed is small or the image has been acquired with a large focal length, an affine homography is a more appropriate model of image displacements. An affine homography is a special type of a general homography whose last row is fixed to

${\displaystyle h_{31}=h_{32}=0,\;h_{33}=1.}$

• Direct linear transformation
• Epipolar geometry
• Feature (computer vision)
• Fundamental matrix (computer vision)
• Pose (computer vision)
• Photogrammetry

• homest is a GPL C/C++ library for robust, non-linear (based on the Levenberg–Marquardt algorithm) homography estimation from matched point pairs (Manolis Lourakis).
• OpenCV is a complete (open and free) computer vision software library that has many routines related to homography estimation (cvFindHomography) and re-projection (cvPerspectiveTransform).