It's a trivial question, but one I'm not 100% clear about.
Given two matrices
$$P_{\{1,2\}} = \left[ \begin{array}{cc}R & t \\ \textbf{0} & 1 \end{array}\right]$$
where $R$ is a 3x3 orthonormal rotation matrix and t is a 3x1 translation, I want to find the homography $H$, such that $H = \left[ \begin{array}{cc}sR & t \\ \textbf{0} & 1 \end{array}\right]$ that gives
$$P_1 H P_2 = I$$
For my purpose I am just minimizing over the Euler angles, scale factor and the translation norms (I'm actually looking for a minimization over a number of these pairs).
In an effort to reduce the parameters in the optimization, I am looking at quaternions. However, I don't quite understand if the parallel holds up. For the same problem can I just do the following?
$$q_1 q_H q_2$$
If so, what should the results look like? Should it be $[1,0,0,0]$? Also, how can I find that scale factor $s$ using quaternions? The translation component should still stay the same (just minimize the Euclidean distance).