Signal Space Separation Beamformer

Measurement at time t is a column vector m(t) with dimensions M × 1, where M is the number of MEG channels.

Because the MEG sensors are located in a source-free volume, the magnetic field, B, can be expressed as a gradient of a scalar potential, Ψ, \( {\mathbf{B}} = - \nabla \Uppsi \), which is a solution of Laplace equation, \( \nabla^{2} \Uppsi = 0 \). Such a solution can be represented as a linear combination of basis functions, e.g., spherical harmonics, aswhere Y _nm are spherical harmonic functions, θ and φ denote spherical angles, r = |r| is the distance from the expansion center, and a _nm and b _nm are expansion coefficients. The first term on the right-hand side of Eq. 1 diverges at the origin and it represents sources within the sensor shell; the second term diverges at infinity and corresponds to sources outside the sensor shell. Contributions of the internal and external sources can be separated and the external terms can be discarded to reduce the environmental noise.

The MEG is measured by SQUID sensors which typically consist of several sensing coils. Magnetic fields for a given sensor at all coil positions, as expressed by gradient of Eq. 1, are combined. Then, the sensor array measurement can be expressed in terms of spherical harmonics as \( {\mathbf{m}}\left( t \right) \approx {\mathbf{SX}}\left( t \right) \), where the matrix S = [S _in S _ext] contains the basis vectors and has dimension M × D, D is the number of the basis vectors (D < M), and X(t) is time dependent column vector of the basis vector amplitudes. The S _in contains internal and S _ext external expansion terms (see Eq. 1). The “≈” sign is used because the expansion is truncated at n _int internal terms and n _ext external terms. The time dependent coefficients X(t) can be estimated as \( {\tilde{\mathbf{X}}}\left( t \right) \approx {\mathbf{S}}^{ + } {\mathbf{m}}\left( t \right) \), where S ⁺ is pseudoinverse of S. The MEG measurement with external interference filtered out can be obtained as \( {\tilde{\mathbf{m}}}\left( t \right) = {\mathbf{Vm}}\left( t \right) \), where the matrix \( {\mathbf{V}} = {\mathbf{S}}_{in} {\mathbf{PS}}^{ + } \) and P = [I 0] (Taulu and Kajola 2005).

Only the scalar beamformers will be discussed. Equations for beamformer are well known (e.g., Sekihara et al. 2004) and general forms of the power and power normalized by noise (pseudo-Z ²) are:where r is the vector of source position, and η is 3 × 1 (or 2 × 1) source orientation vector. For conventional beamformers, \( {\varvec{\Upphi}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) = {\mathbf{L}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) \) is M × 1 lead field matrix, \( {\mathbf{C}}_{M} = {\mathbf{C}}_{m} \) is covariance matrix of the measurement with dimension M × M, and \( {\mathbf{C}}_{N} = {\mathbf{C}}_{\nu } \) is the noise covariance matrix computed from instrumental noise time courses, v(t). The lead field matrix can be separated into the position and orientation parts \( {\mathbf{L}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) = {\mathbf{L}}\left( {\mathbf{r}} \right){\varvec{\upeta}}\left( {\mathbf{r}} \right) \) and the source orientation which maximizes either the P or Z ² can be found by procedure described in (Sekihara et al. 2004).

For the SSS beamformer, \( {\mathbf{C}}_{M} = {\mathbf{C}}_{x} \) is the covariance matrix of the time dependent expansion coefficients X(t) with dimension D × D and is related to the covariance matrix of measurement by \( {\mathbf{C}}_{x} = {\mathbf{S}}^{ + } {\mathbf{C}}_{m} {\mathbf{S}}^{ + T} \), \( {\mathbf{C}}_{N} = {\mathbf{C}}_{x\nu } \) is the noise covariance matrix computed from instrumental noise transformed into the SSS basis and is related to the conventional noise covariance matrix by \( {\mathbf{C}}_{x\nu } = {\mathbf{S}}^{ + } {\mathbf{C}}_{\nu } {\mathbf{S}}^{ + T} \), and \( {\varvec{\Upphi}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) = {\varvec{\Upgamma}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) \) is D × 1 SSS lead field matrix. The Γ is related to L by \( {\varvec{\Upgamma}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) = {\mathbf{S}}^{ + } {\mathbf{L}}\left( {{\mathbf{r}},{\varvec{\upeta}}} \right) \). The Γ can also be decomposed into the position and orientation parts, and source orientation which maximizes P or Z ² can be found the same way as for the conventional beamformers. To compute the SSS beamformer, the data and noise are first transformed into the SSS basis and then the covariance matrices C_x and C_xv are computed. Instead of directly computing the lead field matrix in the SSS basis, we have used the relationship Γ = S ⁺ L. The SSS noise covariance matrix C_xv is non-diagonal. External terms were omitted in S ⁺, and simulations were done either with the non-diagonal form of C_xv, or the C_xv was diagonalized by setting the off-diagonal terms to zero.

Computation of S ⁺ requires inversion of S ^T S, which may be ill-conditioned. The inversion can be successfully completed by regularization. But, it was found that the standard regularization procedures (SVD truncation and Tikhonov regularizations) cannot simultaneously maintain low sensor noise and large SSS interference attenuation. To avoid this problem, we have removed from S all basis vectors, one at a time, and each time re-computed the condition number of the S ^T S based on the remaining vectors. We then removed the basis vector which reduced the condition number most. The procedure was repeated until the condition number was less than a specified value. Such procedure maintains specified span of singular values and yet reduces the SSS sensor noise and maintains large SSS interference attenuation. The vectors which were removed from the matrix S either have low amplitudes or are only slightly different from a linear combination of other vectors. Removal of these vectors will not result in a loss of important spatial topographies, at least within the accuracy of the specified span of singular values.

The SSS is known to attenuate the sensor noise, especially if the condition number of matrix S ^T S is small. We have adjusted the condition number limit to 10⁵, because at this value the sensor noise attenuation by the SSS for a reasonable range of spherical harmonic expansion orders is ≈1.

We report results only on pseudo-t and f dual state beamformer constructs (Vrba and Robinson 2001), but statistically normalized (Barnes and Hillebrand 2003) and event related (Robinson 2004; Cheyne et al. 2007) beamformers were also simulated and exhibit similar behaviour.

A realistic, helmet shaped sensor array with 306 triple sensors (102 radial magnetometers and 204 planar gradiometers with 1.7 cm baseline) was simulated with random sensor gain error of 0.1%. The spontaneous brain activity, the “brain noise”, was modeled by 5000 dipole sources, present in all the following simulations, randomly distributed in a shell bounded by concentric spherical surfaces with 5 and 8 cm radii. Dipoles had random orientations and random amplitudes. After the brain noise simulation was completed, the rms brain noise density over all samples and all channels was normalized to 14 fT/√Hz for planar gradiometers, which resulted in 33.8 fT/√Hz noise density for magnetometers.

Either one or two tangential target sources were placed into the model sphere. Single source was positioned at (0, 0, a) and the two sources were positioned at (0, ±d/2, (a ² − d ²/4)^0.5), where a is the source distance from the model sphere center and d is the source separation. Parameters used were a = 3, 5, 7, 9 cm and d = 0.5, 1, 2 cm, and source orientations were (1, 0, 0). The duration of the time series was T = 100 s, sample rate f _s = 150 Hz, there were 156 triggers associated with the simulated source activity, and 0.2-s pre- and 0.2-s post-trigger intervals.

Source magnitudes were adjusted for signal-to-noise ratio (SNR) measured relative to the brain noise of SNR = 0.1, 1, and 10, with SNR defined as weighted averages (M _m SNR _m + M _p SNR _p)/(M _m + M _p), where M _m and M _p are the numbers of magnetometer and planar gradiometer channels, and SNR _m and SNR _p are the magnetometer and planar gradiometer SNRs. The SNRs were defined as \( {\text{SNR }} = \, q^{ 2} \left| {\mathbf{L}} \right|^{ 2} /\left( {\nu_{rms}^{2} \, M} \right), \) where q is source magnitude, L is lead field vector (sensor response to a unit source), ν _rms is nominal brain noise, and M is the number of magnetometer or planar gradiometer channels.

For single source, the signal was 25 ms wide generalized Lorentzian peak with 50 ms latency relative to triggers, and 30% random amplitude variation. When two sources were used, the signal of the first was the same as that of the single source, and the signal of the second was 40 ms wide peak with 90 ms latency and 40% random amplitude variation.

To assess the spatial resolution of the beamformer, its output peak dimensions were measured for single source simulations. The beamformer scan covered a volume which contained the source. Number of voxels, in which the beamformer output was larger than ½ the associated peak amplitude, was counted. The peak volume was obtained by multiplying this count by voxel volume, and the peak dimension was approximated as a cube root of the peak volume. Voxels were kept sufficiently small such that at least 50 voxels were counted within the peak volume.