median.SO3: Median rotation

Description

Compute the sample projected or geometric median.

Usage

median(x, ...)
  ## S3 method for class 'SO3':
median(x, type = "projected",
    epsilon = 1e-05, maxIter = 2000, ...)
  ## S3 method for class 'Q4':
median(x, type = "projected",
    epsilon = 1e-05, maxIter = 2000, ...)

Arguments

$n\times p$ 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.

maxIter

maximum number of iterations allowed before returning most recent estimate.

...

additional arguments.

Value

Estimate of the projected or geometric median in the same parameterization.

Details

The median-type estimators are defined as $$\widetilde{\bm{S}}=argmin_{\bm{S}\in SO(3)}\sum_{i=1}^nd(\bm{R}_i,\bm{S}).$$ If the choice of distance metric $d$ is Riemannian then the estimator is called the geometric median, and if the distance metric in Euclidean then it is called the projected median. The algorithm used in the geometric case is discussed in Hartley et al. (2011) and the projected case is in Stanfill et al. (2013).

References

Hartley R, Aftab K and Trumpf J (2011). "L1 rotation averaging using the Weiszfeld algorithm." In 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3041-3048. IEEE.

Stanfill B, Genschel U and Hofmann H (2013). "Point Estimation of the Central Orientation of Random Rotations." Technometrics, 55(4), pp. 524-535.