weighted.mean: Weighted mean rotation

Description

Compute the weighted geometric or projected mean of a sample of rotations.

Usage

# S3 method for SO3
weighted.mean(
  x,
  w = NULL,
  type = "projected",
  epsilon = 1e-05,
  maxIter = 2000,
  ...
)
# S3 method for Q4
weighted.mean(
  x,
  w = NULL,
  type = "projected",
  epsilon = 1e-05,
  maxIter = 2000,
  ...
)

Value

Weighted mean of the sample in the same parametrization.

Arguments

x: $n\times p$ matrix where each row corresponds to a random rotation in matrix form ($p=9$) or quaternion ($p=4$) form.
w: vector of weights the same length as the number of rows in x giving the weights to use for elements of x. Default is NULL, which falls back to the usual mean function.
type: string indicating "projected" or "geometric" type mean estimator.
epsilon: stopping rule for the geometric method.
maxIter: maximum number of iterations allowed before returning most recent estimate.
...: only used for consistency with mean.default.

Details

This function takes a sample of 3D rotations (in matrix or quaternion form) and returns the weighted projected arithmetic mean $\widehat{\bm S}_P$ or geometric mean $\widehat{\bm S}_G$ according to the type option. For a sample of $n$ rotations in matrix form $\bm{R}_i\in SO(3), i=1,2,\dots,n$, the weighted mean is defined as $$\widehat{\bm{S}}=argmin_{\bm{S}\in SO(3)}\sum_{i=1}^nw_id^2(\bm{R}_i,\bm{S})$$ where $d$ is the Riemannian or Euclidean distance. For more on the projected mean see moakher02 and for the geometric mean see manton04.

moakher02

Examples

Run this code

Rs <- ruars(20, rvmises, kappa = 0.01)

# Find the equal-weight projected mean
mean(Rs)

# Use the rotation misorientation angle as weight
wt <- abs(1 / mis.angle(Rs))
weighted.mean(Rs, wt)

rot.dist(mean(Rs))

# Usually much smaller than unweighted mean
rot.dist(weighted.mean(Rs, wt))

# Can do the same thing with quaternions
Qs <- as.Q4(Rs)
mean(Qs)
wt <- abs(1 / mis.angle(Qs))
weighted.mean(Qs, wt)
rot.dist(mean(Qs))
rot.dist(weighted.mean(Qs, wt))