# Can Kabsch's algorithm also provide the covariance of its solution?

Say you have a collection of data points $$p_i \in \Bbb R^3$$, and a collection of corresponding reference points $$r_i \in \Bbb R^3$$. Kabsch's algorithm, which relies on SVD decomposition, provides an efficient way to determine the rigid transformation that best (in a least squares of errors sense) transforms the $$r_i$$ to $$p_i$$.

This is equivalent to saying that if each $$p_i$$ is known to have been chosen as a 3-D Gaussian variate with unit variance and mean at the transformed value of $$r_i$$, then the solution found is the transformation with the highest likelihood.

Having determined that transformation, which is described by a translation vector $$\vec t$$ and a rotation $$R$$ characterized as the product of three axial rotations $$R = R_z(\psi)R_y(\theta)R_x(\phi)$$ I would like to calculate the covariance matrix among $$\{t_x, t_y, t_z, \psi, \theta, \phi\}$$.

Is there a "standard" or particularly elegant/efficient way to find that covariance, given that we already have the intermediate information used by Kabsch's algorithm?