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.