Kabsch Algorithm

Given corresponding point sets and in , the Kabsch algorithm finds the orientation-preserving rigid motion that minimizes the sum of squared distances

(1)

It is therefore a solution of the determinant-constrained orthogonal Procrustes problem.

Let the centroids be

			(2)
			(3)

and let and be the matrices whose th columns are and , respectively. Compute the singular value decomposition

(4)

with the singular values in nonincreasing order, and define the diagonal matrix

(5)

The minimizing rotation matrix and translation are then

			(6)
			(7)

When , the last entry of prevents a reflection (Kabsch 1976, Kabsch 1978, Lawrence et al. 2019).