I've calculated the $2\times 3$ affine transform between two 2D spaces (images). If I scale both spaces by a factor $s$ in both dimensions, what would be the affine transform between these two new spaces?
By some brute-force experiments, I was able to make some deductions. If the affine transform is given by:
$\textbf{A}_{original} = \left[\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{matrix}\right]$
Then the affine transform relating the scaled spaces is given by:
$\textbf{A}_{transformed} = \left[\begin{matrix}a_{11} & a_{12} & s \times a_{13}\\a_{21} & a_{22} & s \times a_{23}\end{matrix}\right]$
The third column denotes translation - so scaling that makes sense. However, I had assumed that the elements $a_{11}$ and $a_{22}$ would also be scaled - but it seems like that isn't the case.
These transformation matrices are multiplied by a $3 \times 1$ vector where $[\begin{matrix}x & y & 1\end{matrix}]^{T}$.
- Could someone point me in the right direction about why this is the case?
- How would I extend this to a perspective transform (a more general $3\times 3$ transform)?