A homography is a linear transformation that relates two projections of the same planar surface. Mathematically, it is represented by a $3 \times 3$ matrix $H$.
Although the matrix has 9 elements, it is defined up to a scale factor. We typically set $h_{33} = 1$, leaving 8 independent variables that control translation, rotation, scale, shear, and perspective.
A homography can be decomposed to reveal the physical camera motion and plane geometry:
Where $K$ is intrinsics, $R$ is rotation, $\mathbf{t}$ is translation, $\mathbf{n}$ is the plane normal, and $d$ is distance.
| Transformation | DOF | Invariants |
|---|---|---|
| Translation | 2 | Area, Orientation, Parallelism |
| Affine | 6 | Parallelism, Ratio of Areas |
| Homography | 8 | Collinearity, Cross-ratio |