Homography

From Wikipedia, the free encyclopedia

Homography is a concept in the mathematical science of geometry. It is defined as a relation between two figures, such that to any point in one figure corresponds one and only one point in the other, and vice versa.

1 Computer Vision Applications
- 1.1 3d plane to plane equation
2 Mathematical definition
3 External links

[edit] Computer Vision Applications

In the field of Computer Vision, a homography is defined in 2 dimensional space as a mapping between a point on a ground plane as seen from one camera, to the same point on the ground plane as seen from a second camera. This has many practical applications, most notably it provides a method for compositing 2D or 3D objects into an image or video with the correct pose.

[edit] 3d plane to plane equation

Having two cameras.

Passing from a point $p 1$ to a point $p 2$ .

$p_2 = K_2 \cdot H_{12} \cdot K_1^{-1} \cdot p_1$

where $H 12$ is

$H_{12} = (R + (t \cdot \frac{n^T}{d}))$

$R$ is the rotation matrix. $t$ is the translation vector. $n$ and $d$ are the normal vector of the plane and the distance to the plane respectively.

$K 1$ and $K 2$ are the cameras intrisic parameters matrices.

[edit] Mathematical definition

$p_{a} = \begin{bmatrix} x_{a}\\y_{a}\\1\end{bmatrix}, p_{b} = \begin{bmatrix} x_{b}\\y_{b}\\1\end{bmatrix}, \mathbf{H}_{ab} = \begin{bmatrix} h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33} \end{bmatrix}$

$p_{b} = \mathbf{H}_{ab}p_{a}$

And:

$p_{a} = \mathbf{H}_{ba}p_{b}$

Where:

$\mathbf{H}_{ba} = \mathbf{H}_{ab}^{-1}$

[edit] External links

homest is a GPL C/C++ programming language library for robust, non-linear (based on the Levenberg-Marquardt algorithm) homography estimation from matched point pairs.