Homography and Lines

Homography has interesting applications in several fields.

A homographic transformation generally maps lines to lines. Point-line duality also makes it possible to describe directly how line equations are transformed.

The transformation $\mathbf{H}_{ij}$ that maps points $\mathbf{x}_i$ from image $i$ to points $\mathbf{x}_j$ in image $j$ maps lines according to the dual relation:

$$\begin{array}{rl} \mathbf{x}_j & \sim \mathbf{H}_{ij} \math... ...f{l}_j & \sim \mathbf{H}_{ij}^{-\top} \mathbf{l}_i. \end{array}$$ (1.118)

Indeed, if a line $\mathbf{l}_i$ is represented by the equation

$$\mathbf{l}_i^\top\mathbf{x}_i=0,$$ (1.119)

and $\mathbf{x}_j\sim\mathbf{H}_{ij}\mathbf{x}_i$, then the corresponding line in image $j$ must satisfy

$$\mathbf{l}_j^\top\mathbf{x}_j=0,$$ (1.120)

from which

$$\mathbf{l}_j \sim \mathbf{H}_{ij}^{-\top}\mathbf{l}_i.$$ (1.121)

is obtained.

By examining points at infinity, it can be seen that an improper point has coordinates

$$(x,y,0)^{\top}.$$

Thus, there is a special line

$$\mathbf{l}_{\infty}=(0,0,1)^{\top}$$

that contains all these points. Line $\mathbf{l}_{\infty}$ is called the line at infinity of the projective plane and, in the representation using homogeneous coordinates, separates proper points from improper points.

The principle of duality therefore explains how, given a transformation $\mathbf{M}$ (projective or homographic), the transformation that maps a point $\mathbf {x}$ to $\mathbf{x}'$ is written as

$$\mathbf{x}' \sim \mathbf{M}\mathbf{x},$$ (1.122)

whereas the transformation that maps a line $\mathbf{l}$ becomes

$$\mathbf{l}' \sim \mathbf{M}^{-\top}\mathbf{l}.$$ (1.123)