The Signal-Space Projection (SSP) method

General concepts

Unlike many other noise-cancellation approaches, SSP does not require additional reference sensors to record the disturbance fields. Instead, SSP relies on the fact that the magnetic field distributions generated by the sources in the brain have spatial distributions sufficiently different from those generated by external noise sources. Furthermore, it is implicitly assumed that the linear space spanned by the significant external noise patterns has a low dimension.

Without loss of generality we can always decompose any n-channel measurement b(t) into its signal and noise components as

Further, if we know that b_n(t) is well characterized by a few field patterns b₁…b_m, we can express the disturbance as

where the columns of U constitute an orthonormal basis for b₁…b_m, c_n(t) is an m-component column vector, and the error term e(t) is small and does not exhibit any consistent spatial distributions over time, i.e., C_e = E{ee^⊤} = I. Subsequently, we will call the column space of U the noise subspace. The basic idea of SSP is that we can actually find a small basis set b₁…b_m such that the conditions described above are satisfied. We can now construct the orthogonal complement operator

and apply it to b(t) in Equation :eq:`additive_model` yielding

since P_⟂b_n(t) = P_⟂(Uc_n(t) + e(t)) ≈ 0 and P_⟂b_s(t) ≈ b_s(t). The projection operator P_⟂ is called the signal-space projection operator and generally provides considerable rejection of noise, suppressing external disturbances by a factor of 10 or more. The effectiveness of SSP depends on two factors:

• The basis set b₁…b_m should be able to characterize the disturbance field patterns completely and
• The angles between the noise subspace space spanned by b₁…b_m and the signal vectors b_s(t) should be as close to π ⁄ 2 as possible.

If the first requirement is not satisfied, some noise will leak through because P_⟂b_n(t) ≠ 0. If the any of the brain signal vectors b_s(t) is close to the noise subspace not only the noise but also the signal will be attenuated by the application of P_⟂ and, consequently, there might by little gain in signal-to-noise ratio.

Since the signal-space projection modifies the signal vectors originating in the brain, it is necessary to apply the projection to the forward solution in the course of inverse computations.

For more information on SSP, please consult the references listed in :footcite:t:`TescheEtAl1995,UusitaloIlmoniemi1997`.

Estimation of the noise subspace

As described above, application of SSP requires the estimation of the signal vectors b₁…b_m constituting the noise subspace. The most common approach, also implemented in :func:`mne.compute_proj_raw` is to compute a covariance matrix of empty room data, compute its eigenvalue decomposition, and employ the eigenvectors corresponding to the highest eigenvalues as basis for the noise subspace. It is also customary to use a separate set of vectors for magnetometers and gradiometers in the Vectorview system.

EEG average electrode reference

The EEG average reference is the mean signal over all the sensors. It is typical in EEG analysis to subtract the average reference from all the sensor signals b¹(t), ..., bⁿ(t). That is:

where the noise term b^j_n(t) is given by

Thus, the projector vector P_⟂ will be given by P_⟂ = (1)/(n)[1, 1, ..., 1]