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In short, data repair using spherical spline interpolation :footcite:`PerrinEtAl1989` consists of the following steps:
- Project the good and bad electrodes onto a unit sphere
- Compute a mapping matrix that maps N good channels to M bad channels
- Use this mapping matrix to compute interpolated data in the bad channels
Spherical splines assume that the potential V(\boldsymbol{r_i}) at any point \boldsymbol{r_i} on the surface of the sphere can be represented by:
V(\boldsymbol{r_i}) = c_0 + \sum_{j=1}^{N}c_{i}g_{m}(cos(\boldsymbol{r_i}, \boldsymbol{r_{j}}))
where the C = (c_{1}, ..., c_{N})^{T} are constants which must be estimated. The function g_{m}(\cdot) of order m is given by:
g_{m}(x) = \frac{1}{4 \pi}\sum_{n=1}^{\infty} \frac{2n + 1}{(n(n + 1))^m}P_{n}(x)
where P_{n}(x) are Legendre polynomials of order n.
To estimate the constants C, we must solve the following two equations simultaneously:
G_{ss}C + T_{s}c_0 = X
{T_s}^{T}C = 0
where G_{ss} \in R^{N \times N} is a matrix whose entries are G_{ss}[i, j] = g_{m}(cos(\boldsymbol{r_i}, \boldsymbol{r_j})) and X \in R^{N \times 1} are the potentials V(\boldsymbol{r_i}) measured at the good channels. T_{s} = (1, 1, ..., 1)^\top is a column vector of dimension N. Equation :eq:`matrix_form` is the matrix formulation of Equation :eq:`model` and equation :eq:`constraint` is like applying an average reference to the data. From equation :eq:`matrix_form` and :eq:`constraint`, we get:
\begin{bmatrix} c_0 \\ C \end{bmatrix} = {\begin{bmatrix} {T_s}^{T} && 0 \\ T_s && G_{ss} \end{bmatrix}}^{-1} \begin{bmatrix} 0 \\ X \end{bmatrix} = C_{i}X
C_{i} is the same as matrix {\begin{bmatrix} {T_s}^{T} && 0 \\ T_s && G_{ss} \end{bmatrix}}^{-1} but with its first column deleted, therefore giving a matrix of dimension (N + 1) \times N.
Now, to estimate the potentials \hat{X} \in R^{M \times 1} at the bad channels, we have to do:
\hat{X} = G_{ds}C + T_{d}c_0
where G_{ds} \in R^{M \times N} computes g_{m}(\boldsymbol{r_i}, \boldsymbol{r_j}) between the bad and good channels. T_{d} = (1, 1, ..., 1)^\top is a column vector of dimension M. Plugging in equation :eq:`estimate_constant` in :eq:`estimate_data`, we get
\hat{X} = \begin{bmatrix} T_d && G_{ds} \end{bmatrix} \begin{bmatrix} c_0 \\ C \end{bmatrix} = \underbrace{\begin{bmatrix} T_d && G_{ds} \end{bmatrix} C_{i}}_\text{mapping matrix}X
To interpolate bad channels, one can simply do:
>>> evoked.interpolate_bads(reset_bads=False) # doctest: +SKIP
and the bad channel will be fixed.
Examples: