rotations (version 1.5)

mean.SO3: Mean rotation

Description

Compute the sample geometric or projected mean.

Usage

"mean" (x, type = "projected", epsilon = 1e-05, maxIter = 2000, ...)
  "mean" (x, type = "projected", epsilon = 1e-05, maxIter = 2000, ...)

Arguments

x

$n-by-p$ matrix where each row corresponds to a random rotation in matrix form ($p=9$) or quaternion ($p=4$) form.

type

string indicating "projected" or "geometric" type mean estimator.

epsilon

stopping rule for the geometric-mean.

maxIter

maximum number of iterations allowed for geometric-mean.

...

additional arguments.

Value

Estimate of the projected or geometric mean of the sample in the same parameterization.

Details

This function takes a sample of 3D rotations (in matrix or quaternion form) and returns the projected arithmetic mean denoted $S_P$ or geometric mean $S_G$ according to the type option. For a sample of $n$ rotations in matrix form $Ri in SO(3), i=1,2,\dots,n$, the mean-type estimator is defined as $$\widehat{\bm{S}}=argmin_{\bm{S}\in SO(3)}\sum_{i=1}^nd^2(\bm{R}_i,\bm{S})$$ where $d$ is the Riemannian or Euclidean distance. For more on the projected mean see Moakher (2002) and for the geometric mean see Manton (2004). For the projected mean from a quaternion point of view see Tyler (1981).

References

Moakher M (2002). "Means and averaging in the group of rotations." SIAM Journal on Matrix Analysis and Applications, 24(1), pp. 1-16.

Manton JH (2004). "A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups." In Control, Automation, Robotics and Vision Conference, 2004. ICARCV 2004 8th, volume 3, pp. 2211-2216. IEEE.

Tyler DE (1981). "Asymptotic inference for eigenvectors." The Annals of Statistics, 9(4), pp. 725-736.

See Also

median.SO3, bayes.mean, weighted.mean.SO3

Examples

Rs <- ruars(20, rvmises, kappa = 0.01)
mean(Rs)                               #Projected mean
project.SO3(colMeans(Rs))              #Same as mean(Rs)

mean(Rs, type = "geometric")           #Geometric mean
rot.dist(mean(Rs))                     #Bias of the projected mean
rot.dist(mean(Rs, type = "geometric")) #Bias of the geometric mean

Qs <- as.Q4(Rs)
mean(Qs)                               #Projected mean
mean(Qs, type = "geometric")           #Geometric mean
rot.dist(mean(Qs))                     #Bias of the projected mean
rot.dist(mean(Qs, type = "geometric")) #Bias of the geometric mean