In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image rectification, image registration, or camera motion—rotation and translation—between two images. Once camera resectioning has been done from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene (see Augmented reality).
3D plane to plane equation We have two cameras a and b, looking at points P i {\displaystyle P_{i}} in a plane. Passing from the projection
b p i = (
b u i ;
b v i ; 1 ) {\displaystyle {}^{b}p_{i}=\left({}^{b}u_{i};{}^{b}v_{i};1\right)} of P i {\displaystyle P_{i}} in b to the projection
a p i = (
a u i ;
a v i ; 1 ) {\displaystyle {}^{a}p_{i}=\left({}^{a}u_{i};{}^{a}v_{i};1\right)} of P i {\displaystyle P_{i}} in a:
a p i =
b z i
a z i K a ⋅ H a b ⋅ K b − 1 ⋅
b p i {\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
a z i {\displaystyle {}^{a}z_{i}} and
b z i {\displaystyle {}^{b}z_{i}} are the z coordinates of P in each camera frame and where the homography matrix H a b {\displaystyle H_{ab}} is given by
H a b = R − t n T d {\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}} .
R {\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. Ka and Kb are the cameras' intrinsic parameter matrices.
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