Homography and Lines

There are interesting applications of homography in various fields.

A homographic transformation generally transforms lines into lines. In special cases, however, it can transform lines into points, such as in the perspective projection of elements on the horizon: homogeneous coordinates indeed represent points and vectors differently, and when a line reduces to a point, its homogeneous coordinate becomes 0.

The homographic transformation applied to a line (effect of the point-line duality) is exactly the inverse transformation of the one that transforms the corresponding points between the spaces: the transformation $\mathbf{H}_{ij}$ that transforms points $\mathbf{x}_i$ from image $i$ to points $\mathbf{x}_j$ of image $j$ transforms the equations of the lines from image $j$ to image $i$:

$$\begin{array}{rl} \mathbf{x}_j & = \mathbf{H}_{ij} \mathbf{... ...mathbf{l}_i & = \mathbf{H}^{\top}_{ij} \mathbf{l}_j \end{array}$$ (1.82)

Examining points and lines at infinity (for example, at the horizon), one can see how a point at infinity has coordinates $(x,y,0)^{\top}$. Therefore, there exists a special line $\mathbf{l}_{\infty}=(0,0,1)^{\top}$ that connects all these points.

The principle of duality allows for explaining how, given a transformation $\mathbf{M}$ (projective or homographic), the transformation that transforms a point $\mathbf {x}$ into $\mathbf{x}'$ can be expressed as

$$\mathbf{x}' = \mathbf{M}\mathbf{x}$$ (1.83)

while the transformation that transforms a line $\mathbf{l}$ instead becomes

$$\mathbf{l}' = \mathbf{M}^{-\top}\mathbf{l}$$ (1.84)