Least-squares affine transformation

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Given

$P = [p_1, ..., p_n] in RR^{m times n}$

and

$Q = [q_1, ..., q_n] in RR^{m times n}$

The problem is to find an affine transformation

$f : RR^m -> RR^m : f(x) = Ax + b,$

where $A in RR^{m times m}$ , and $b in RR^m$ , such that the least-squares error

$sum_{i = 1}^n |f(p_i) - q_i|^2 = |AP + b 1_{1 times n} - Q|^2$

is minimized, where $|.|$ is the Frobenius norm. Denoting

$p' = sum_{i = 1}^n p_i / n,$
$r' = sum_{i = 1}^n r_i / n,$
$P' = [p_1 - p', ..., p_n - p'], and$
$Q' = [q_1 - q', ..., q_n - q'],$

the solution is given by

$A = (PR^T) (PP^T)^{-1} and$
$b = r' - Ap'.$

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Least-squares affine transformation