Homographies

A homography $\mathrm{H}$ maps a set of points $\mathbf{x}=(x\quad y\quad 1)^\top$ to another set of points $\mathbf{x}^\prime = (x^\prime\quad y^\prime\quad 1)^\top$.

\begin{align*} \mathbf{x}^\prime &\sim \mathrm{H}\mathbf{x} \\ \lambda\begin{pmatrix}x^\prime\\y^\prime\\ 1 \end{pmatrix} &= \begin{pmatrix}h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33}\\ \end{pmatrix}\begin{pmatrix}x\\y\\ 1 \end{pmatrix}\\ \lambda x^\prime &= h_{11}x+h_{12}y+h_{13} \\ \lambda y^\prime &= h_{21}x+h_{22}y+h_{23} \\ \lambda &= h_{31}x+h_{32}y+h_{33} \\ \text{So,} &\\ x^\prime &= \frac{h_{11}x+h_{12}y+h_{13}}{h_{31}x+h_{32}y+h_{33}} \\ y^\prime &= \frac{h_{21}x+h_{22}y+h_{23}}{h_{31}x+h_{32}y+h_{33}} \\ \end{align*}

Rewriting $(x,y,x^\prime,y^\prime) \rightarrow (x_1,x_2,x^\prime_1,x^\prime_2)$, the equation in the linear form,

\begin{align*} -h_{11}x_1-h_{12}x_2-h_{13}+h_{31}x_1x_1^\prime+h_{32}x_2x_1^\prime+h_{33}x_3x_1^\prime &= 0 \\ -h_{21}x_1-h_{22}x_2-h_{23}+h_{31}x_1x_2^\prime+h_{32}x_2x_2^\prime+h_{33}x_3x_2^\prime &= 0 \\ \end{align*}

Now the equations become,

$$\begin{pmatrix}a_x\\a_y\end{pmatrix}h = 0$$

where,

\begin{align*} a_x &= (-x_1 -x_2 -1 \quad 0 \quad 0 \quad 0 \quad x_1x_1^\prime \quad x_2x_1^\prime \quad x_1^\prime)\\ a_y &= ( \quad 0 \quad 0 \quad 0 \quad -x_1 -x_2 -1\quad x_1x_2^\prime \quad x_2x_2^\prime \quad x_2^\prime)\\ h &= (h_{11}\quad h_{12}\quad h_{13}\quad h_{21}\quad h_{22}\quad h_{23}\quad h_{31}\quad h_{32}\quad h_{33})^\top \\ \end{align*}

The matrix has 8 dof (since $h_{33}=1$), each point gives 2 sets of equations, 4 points are requiered to solve the matrix $h$ uniquely with SVD (Singular Value Decomposition).

Homographies

References