procrustes solves for two matrices A and B the
`Procrustes Problem' of finding an orthogonal matrix Q such that
A-B*Q has the minimal Frobenius norm. kabsch determines a best rotation of a given vector set into a
second vector set by minimizing the weighted sum of squared deviations.
The order of vectors is assumed fixed.
procrustes(A, B)kabsch(A, B, w = NULL)
procrustes returns a list with components P, which is
B*Q, then Q, the orthogonal matrix, and d, the
Frobenius norm of A-B*Q. kabsch returns a list with U the orthogonal matrix applied,
R the translation vector, and d the least root mean square
between U*A+R and B.
procrustes(A,B) uses the svd decomposition
to find an orthogonal matrix Q such that A-B*Q has a
minimal Frobenius norm, where this norm for a matrix C is defined
as sqrt(Trace(t(C)*C)), or norm(C,'F') in R. Solving it with B=I means finding a nearest orthogonal matrix.
kabsch solves a similar problem and uses the Procrustes procedure
for its purpose. Given two sets of points, represented as columns of the
matrices A and B, it determines an orthogonal matrix
U and a translation vector R such that U*A+R-B
is minimal.
Kabsch, W. (1976). A solution for the best rotation to relate two sets of vectors. Acta Cryst A, Vol. 32, p. 9223.
svd## Procrustes
U <- rortho(5) # random orthogonal matrix
P <- procrustes(U, eye(5))
## Kabsch
P <- matrix(c(0, 1, 0, 0, 1, 1, 0, 1,
0, 0, 1, 0, 1, 0, 1, 1,
0, 0, 0, 1, 0, 1, 1, 1), nrow = 3, ncol = 8, byrow = TRUE)
R <- c(1, 1, 1)
phi <- pi/4
U <- matrix(c(1, 0, 0,
0, cos(phi), -sin(phi),
0, sin(phi), cos(phi)), nrow = 3, ncol = 3, byrow = TRUE)
Q <- U %*% P + R
K <- kabsch(P, Q)
# K$R == R and K$U %*% P + c(K$R) == QRun the code above in your browser using DataLab