`pytransform3d.uncertainty`.frechet_mean#

pytransform3d.uncertainty.frechet_mean(samples, mean0, exp, log, inv, concat_one_to_one, concat_many_to_one, n_iter=20)[source]#

Compute the Fréchet mean of samples on a smooth Riemannian manifold.

The mean is computed with an iterative optimization algorithm [1] [2]:

For a set of samples \(\{x_1, \ldots, x_n\}\), we initialize the estimated mean \(\bar{x}_0\), e.g., as \(x_1\).
For a fixed number of steps \(K\), in each iteration \(k\) we improve the estimation of the mean by
1. Computing the distance of each sample to the current estimate of the mean in tangent space with \(d_{i,k} \leftarrow \log (x_i \cdot \bar{x}_k^{-1})\).
2. Updating the estimate of the mean with \(\bar{x}_{k+1} \leftarrow \exp(\frac{1}{N}\sum_i d_{i,k}) \cdot \bar{x}_k\).
Return \(\bar{x}_K\).

Parameters:

samplesarray-like, shape (n_samples, …): Samples on a smooth Riemannian manifold.
mean0array-like, shape (…): Initial guess for the mean on the manifold.
expcallable: Exponential map from the tangent space to the manifold.
logcallable: Logarithmic map from the manifold to the tangent space.
invcallable: Computes the inverse of an element on the manifold.
concat_one_to_onecallable: Concatenates elements on the manifold.
concat_many_to_onecallable: Concatenates multiple elements on the manifold to one element on the manifold.
n_iterint, optional (default: 20): Number of iterations of the optimization algorithm.

Returns:

meanarray, shape (…): Fréchet mean on the manifold.
mean_diffsarray, shape (n_samples, n_tangent_space_components): Differences between the mean and the samples in the tangent space. These can be used to compute the covariance. They are returned to avoid recomputing them.

pytransform3d.uncertainty.frechet_mean#