pyriemann.utils.tangentspace.exp_map_logeuclid

pyriemann.utils.tangentspace.exp_map_logeuclid(X, Cref)

Project matrices back to manifold by log-Euclidean exponential map.

The projection of a matrix \(\mathbf{X}\) from tangent space to SPD/HPD manifold with log-Euclidean exponential map according to a reference SPD/HPD matrix \(\mathbf{C}_\text{ref}\) as described in Eq.(3.4) of [1]:

\[\mathbf{X}_\text{original} = \exp \left( \log(\mathbf{C}_\text{ref}) + [D_{\mathbf{C}_\text{ref}} \log] \left(\mathbf{X}\right) \right)\]

where \([D_{\mathbf{C}_\text{ref}} \log] \left( \mathbf{X}\right)\) indicates the differential of the matrix logarithm at point \(\mathbf{C}_\text{ref}\) applied to \(\mathbf{X}\). Calculation is performed according to Eq. (5) in [2].

Parameters:

Xndarray, shape (…, n, n)

Matrices in tangent space.

Crefndarray, shape (n, n)

Reference SPD/HPD matrix.

Returns:

X_originalndarray, shape (…, n, n)

Matrices in SPD/HPD manifold.

Notes

Added in version 0.4.

References

[1]

Geometric Means in a Novel Vector Space Structure on Symmetric Positive‐Definite Matrices V. Arsigny, P. Fillard, X. Pennec, N. Ayache. SIMAX, 2006, 29(1), pp. 328-347.

[2]

A New Canonical Log-Euclidean Kernel for Symmetric Positive Definite Matrices for EEG Analysis G. Wagner vom Berg, V. Röhr, D. Platt, B. Blankertz. IEEE TBME, 2024.