I am trying to understand equation 26 given equation 25.
I know that generally, if we have an overdetermined system of linear equations of the form
$Ax = b$
the least squares solution is
$\hat{x} = (A^TA)^{-1}A^Tb$
Applying it to the above equation of
$D [R|t] = C$
where $[R|t]$ is the unknown matrix, we get
$\hat{[R|t]} = (D^TD)^{-1}D^TC$
but that's not exactly what's shown in equation 26. How did (26) come from (25) if they are "solving linearly"?
K is a 3×3 “camera matrix”, R is a 3×3 rotation matrix, and t is a 3×1 translation vector
Thanks!
Best Answer
In fact, equation 25 is $ [R|t] D = C$ and not $D [R|t] = C$ , and indeed $ [R|t] DD^T = CD^T$ , yielding $\hat{[R|t]}= CD^T*(DD^T)^{-1}$