# Constrained orthogonal procrustes using the polar decomposition

I want to solve the orthogonal procrustes problem with the additional constraint the rotation matrix is orientation-preserving: $$\min_R \|AR-B\|\quad \textrm{s.t.}\quad R^TR=I,\ \det R = 1$$ where the norm is the Frobenius norm. The standard approach is to compute the SVD of $A^TB$: $$A^TB = UDV^T.$$

If $\det A^TB > 0$, the solution is simply $R=UV^T$. Otherwise, it is $U\Sigma V^T$, where $\Sigma$ is formed by replacing the smallest singular value of $D$ by $-1$, and the rest of them by $1$.

I want to find $R$ without using SVD, by using the polar decomposition $A^TB = UP$ instead. If $\det A^TB>0$, there is no problem: $R$ and $U$ coincide.

If $\det A^TB <0$, is there any way of finding $R$ by using (a modification of) the polar decomposition?

-