The eyes: the biggest offender
The eyeball is a standing dipole — cornea positive, retina negative — of order 1 mV, orders of magnitude larger than cortical potentials. Two motions matter. A blink sweeps the eyelid across the cornea, producing a large, brief, frontally-maximal deflection; a saccade rotates the dipole, producing a step with a lateral (left–right) gradient across frontal channels. Both project through volume conduction onto every electrode, strongest anteriorly. Ocular artifacts are the first thing any EEG pipeline must address.
Regression: subtract a measured reference
If you record the artifact source directly — EOG for the eyes, ECG for the heart — you can regress it out. Model each EEG channel as brain plus a scaled copy of the reference and estimate the scale by least squares. EOG regression is the classic case:
\mathbf{X}_{\text{clean}} = \mathbf{X} - \mathbf{B}\,\mathbf{R}, \qquad \mathbf{B} = \mathbf{C}_{XR}\,\mathbf{C}_{RR}^{-1}Least-squares regression of reference channels R (EOG/ECG) out of the data X; B holds the propagation coefficients.
If you have a sensor watching the artifact directly (an electrode by the eye or on the chest), you can predict how much of it leaked into each brain channel and simply subtract that prediction off.
- \mathbf{R}
- Reference channels that record the artifact itself — EOG for the eyes, ECG for the heart.
- \mathbf{B}
- The best-fit coefficients telling how strongly the reference leaks into each data channel.
- \mathbf{C}_{XR}
- Covariance between the data and the reference (how they move together).
- \mathbf{C}_{RR}^{-1}
- Inverse of the reference's own covariance, which normalizes the fit.
If a blink of size 1 in the EOG adds about 0.3 to a frontal channel, then \mathbf{B} for that channel is about 0.3.
Simple, causal and cheap, but with two well-known failures. First, bidirectional contamination: EOG electrodes also pick up frontal brain activity, so regressing them out removes real signal — overcorrection. Second, the propagation coefficients drift, so a single fixed \mathbf{B} is only an approximation. Reference-based regression is a strong baseline precisely when you have a faithful, non-neural reference; the further the reference strays from that ideal, the more it costs you.
Signal-space projection: remove a direction, not a channel
When you know an artifact's topography \mathbf{u} (its column in \mathbf{A}) but have no reference channel, project it out. SSP builds an orthonormal basis \mathbf{U} for the artifact subspace and applies the projector:
\mathbf{P} = \mathbf{I} - \mathbf{U}\mathbf{U}^{\top}, \qquad \mathbf{X}_{\text{clean}} = \mathbf{P}\,\mathbf{X}The SSP operator removes the artifact subspace spanned by U; any signal component along those directions is lost with it.
Instead of watching the artifact, learn the direction it points to in sensor space and delete everything along that direction — like erasing one axis. Cheap and reference-free, but any brain signal lying along that axis dies with it.
- \mathbf{U}
- The artifact directions — a small set of spatial patterns to remove.
- \mathbf{U}\mathbf{U}^{\top}
- The projector onto the artifact subspace.
- \mathbf{P}
- What remains: projection onto everything at right angles to the artifact.
- \mathbf{I}
- Identity — keep everything, then subtract the bad part.
This is signal-space projection: remove a whole spatial direction rather than dropping a physical channel.
SSP is fast, linear and stable across a session, which is why MEG pipelines lean on it. The price is bluntness: it deletes whole spatial directions, so real brain activity that happens to align with the artifact topography goes too, and each projector removes a degree of freedom from the data. Signal-space separation (SSS) generalises the idea for MEG by splitting the field into components originating inside vs. outside the sensor helmet.
Muscle, heart and body
EMG is the hardest of the routine artifacts because it is broadband and overlaps the beta/gamma range that motor BCIs care about; there is no clean band to notch, and aggressive high-frequency filtering guts real gamma. Cardiac and pulse artifacts are quasi-periodic near 1 Hz and, given an ECG channel, respond well to regression or template subtraction. Motion artifacts are low-frequency and mechanically coupled; the modern trick is to record an auxiliary signal — an accelerometer or the electrode's own impedance — and regress it out, exactly the reference-regression idea applied to a mechanical source.
Reference choice is an artifact decision
Every scalp recording is differential, so the choice of reference shapes which artifacts appear. A common average reference is convenient but has a dark side: it smears any single bad or artifact-laden channel across all the others, so one popping electrode contaminates the whole montage. The surface Laplacian estimates the second spatial derivative, acting as a reference-free spatial high-pass that sharpens local sources and strongly attenuates the broad, distant fields typical of ocular and other far-field artifacts — at the cost of amplifying uncorrelated sensor noise.