Bad channel repair via interpolation

Spherical spline interpolation (EEG)

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(r_i) at any point r_i on the surface of the sphere can be represented by:

where the C = (c₁, ..., c_N)^T are constants which must be estimated. The function g_m(⋅) of order m is given by:

where P_n(x) are Legendre polynomials of order n.

To estimate the constants C, we must solve the following two equations simultaneously:

where G_ss ∈ R^N×N is a matrix whose entries are G_ss[i, j] = g_m(cos(r_i, r_j)) and X ∈ R^N×1 are the potentials V(r_i) measured at the good channels. T_s = (1, 1, ..., 1)^⊤ 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:

C_i is the same as matrix \begin{bmatrix} T_s^T 0T_s G_ss but with its first column deleted, therefore giving a matrix of dimension (N + 1)×N.

Now, to estimate the potentials X̂ ∈ R^M×1 at the bad channels, we have to do:

where G_ds ∈ R^M×N computes g_m(r_i, r_j) between the bad and good channels. T_d = (1, 1, ..., 1)^⊤ is a column vector of dimension M. Plugging in equation :eq:`estimate_constant` in :eq:`estimate_data`, we get

To interpolate bad channels, one can simply do:

>>> evoked.interpolate_bads(reset_bads=False)  # doctest: +SKIP

and the bad channel will be fixed.