Dual signal subspace projection (DSSP): a novel algorithm for removing large interference in biomagnetic measurements

We assume that the biomagnetic measurements are conducted by using an array of M sensors. We denote the output of the mth sensor at time t by ${b}_{m}(t)$ and the column vector containing the outputs of all sensors is ${\boldsymbol{b}}(t)$: ${\boldsymbol{b}}(t)={[{b}_{1}(t),\ldots ,{b}_{M}(t)]}^{{\rm{T}}}$. The ${\boldsymbol{b}}(t)$ is called the data vector. We use a data model in which the measured data consists of the signal magnetic field generated from signal sources of interest, the interference magnetic field generated from interference sources, and the sensor noise. That is, the data vector is expressed as

$$\begin{eqnarray}&&{\boldsymbol{b}}(t)={{\boldsymbol{b}}}_{{\rm{s}}}(t)+{{\boldsymbol{b}}}_{{\rm{I}}}(t)+{\boldsymbol{\varepsilon }},\end{eqnarray} \tag{ 1 }$$

where ${{\boldsymbol{b}}}_{{\rm{s}}}(t)$, the signal vector, represents the signal magnetic field, ${{\boldsymbol{b}}}_{{\rm{I}}}(t)$, the interference vector, represents the interference magnetic field, and ${\boldsymbol{\varepsilon }}$, the noise vector, represents the sensor noise. The spatio-temporal data matrices of ${\boldsymbol{b}}(t)$, ${{\boldsymbol{b}}}_{{\rm{s}}}(t)$, and ${{\boldsymbol{b}}}_{{\rm{I}}}(t)$ are, respectively, denoted by ${\boldsymbol{B}}$, ${{\boldsymbol{B}}}_{{\rm{S}}}$, and ${{\boldsymbol{B}}}_{{\rm{I}}}$, such that

$$\begin{eqnarray}&&{\boldsymbol{B}}=[{\boldsymbol{b}}({t}_{1}),\ldots ,{\boldsymbol{b}}({t}_{K})],\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{\rm{S}}}=[{{\boldsymbol{b}}}_{{\rm{s}}}({t}_{1}),\ldots ,{{\boldsymbol{b}}}_{{\rm{s}}}({t}_{K})],\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{{\rm{I}}}=[{{\boldsymbol{b}}}_{{\rm{I}}}({t}_{1}),\ldots ,{{\boldsymbol{b}}}_{{\rm{I}}}({t}_{K})],\end{eqnarray} \tag{ 4 }$$

where the data acquisition takes place at ${t}_{1},\ldots ,{t}_{K}$ and $M\lt K$ is assumed.

The M × K spatio-temporal data matrix ${\boldsymbol{B}}$ is modeled as

$$\begin{eqnarray}&&{\boldsymbol{B}}={{\boldsymbol{B}}}_{{\rm{S}}}+{{\boldsymbol{B}}}_{{\rm{I}}}+{{\boldsymbol{B}}}_{\varepsilon },\end{eqnarray} \tag{ 5 }$$

where ${{\boldsymbol{B}}}_{\varepsilon }$ is a matrix whose columns consist of the noise vectors ${\boldsymbol{\varepsilon }}$ at ${t}_{1},\ldots ,{t}_{K}$. The spatio-temporal data model in equation (5) is used in our analysis.

Let us assume that a unit-magnitude source exists at ${\boldsymbol{r}}$. When the source is oriented in the x-, y-, and z-directions, the corresponding outputs of the jth sensor are, respectively, denoted ${l}_{j}^{x}({\boldsymbol{r}})$, ${l}_{j}^{y}({\boldsymbol{r}})$, and ${l}_{j}^{z}({\boldsymbol{r}})$. We define an $M\times 3$ matrix ${\boldsymbol{L}}({\boldsymbol{r}})$ in which the jth row is equal to the row vector $[{l}_{j}^{x}({\boldsymbol{r}}),{l}_{j}^{y}({\boldsymbol{r}}),{l}_{j}^{z}({\boldsymbol{r}})]$. This matrix ${\boldsymbol{L}}({\boldsymbol{r}})$, which is called the lead field matrix, represents the sensitivity of the whole sensor array at the location of ${\boldsymbol{r}}$.

Let us assume that, in total, Q sources exist. Their locations are denoted by ${{\boldsymbol{r}}}_{1},\ldots ,{{\boldsymbol{r}}}_{Q}$, their orientations by ${{\boldsymbol{\eta }}}_{1},\ldots ,{{\boldsymbol{\eta }}}_{Q}$ ⁶ , and their intensities by ${s}_{1}(t),\ldots ,{s}_{Q}(t)$. Then, the signal vector ${{\boldsymbol{b}}}_{{\rm{s}}}(t)$ is expressed as

$$\begin{eqnarray}&&{{\boldsymbol{b}}}_{{\rm{s}}}(t)=\displaystyle \sum _{p=1}^{Q}{s}_{p}(t){\boldsymbol{L}}({{\boldsymbol{r}}}_{p}){{\boldsymbol{\eta }}}_{p}=\displaystyle \sum _{p=1}^{Q}{s}_{p}(t){{\boldsymbol{l}}}_{p},\end{eqnarray} \tag{ 6 }$$

where ${{\boldsymbol{l}}}_{p}={\boldsymbol{L}}({{\boldsymbol{r}}}_{p}){{\boldsymbol{\eta }}}_{p}$, and ${{\boldsymbol{l}}}_{p}$ is called the lead field vector of the pth source. The equation above indicates that the signal vector ${{\boldsymbol{b}}}_{{\rm{s}}}(t)$ is expressed as a linear combination of the lead field vectors. In other words, the vector ${{\boldsymbol{b}}}_{{\rm{s}}}(t)$ lies within the span of the lead field vectors, ${{\boldsymbol{l}}}_{1},\ldots ,{{\boldsymbol{l}}}_{Q}$, i.e.

$$\begin{eqnarray}&&{{\boldsymbol{b}}}_{{\rm{s}}}(t)\in \mathrm{span}\{{{\boldsymbol{l}}}_{1},\ldots ,{{\boldsymbol{l}}}_{Q}\}.\end{eqnarray} \tag{ 7 }$$

This subspace, $\mathrm{span}\{{{\boldsymbol{l}}}_{1},\ldots ,{{\boldsymbol{l}}}_{Q}\}$, is referred to as the (spatial-domain) signal subspace [11].

In general, the signal subspace is unknown because the source locations and the source orientations are unknown. However, a subspace that approximates the signal subspace can be obtained from an augmented lead field over the source space, which is the region where a source may exist. To derive the augmented lead field, we define voxels over a source space. Denoting the locations of the voxels by ${{\boldsymbol{r}}}_{1},\ldots ,{{\boldsymbol{r}}}_{N}$, the augmented lead field matrix over all voxel locations is defined as

$$\begin{eqnarray}&&{\boldsymbol{F}}=[{\boldsymbol{L}}({{\boldsymbol{r}}}_{1}),\ldots ,{\boldsymbol{L}}({{\boldsymbol{r}}}_{N})].\end{eqnarray} \tag{ 8 }$$

It is clear that (ignoring the voxel discretization error) the column span of ${\boldsymbol{F}}$ includes the signal subspace, that is

$$\begin{eqnarray}&&{\rm{column}}\;{\rm{span}}\;{\rm{of}}\quad {\boldsymbol{F}}\quad \supseteq \quad \mathrm{span}\{{{\boldsymbol{l}}}_{1},\ldots ,{{\boldsymbol{l}}}_{Q}\}.\end{eqnarray} \tag{ 9 }$$

The column span of ${\boldsymbol{F}}$ is called the pseudo signal subspace in this paper, and the projector onto the pseudo signal subspace can be derived by applying eigenvalue decomposition to ${\boldsymbol{F}}{{\boldsymbol{F}}}^{{\rm{T}}}$, such that⁷

$$\begin{eqnarray}{\boldsymbol{F}}{{\boldsymbol{F}}}^{{\rm{T}}}=[{{\boldsymbol{e}}}_{1},\ldots ,{{\boldsymbol{e}}}_{M}]\left[\begin{array}{cccc}{\gamma }_{1} & 0 & \cdots & 0\\ 0 & {\gamma }_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {\gamma }_{M}\end{array}\right]\left[\begin{array}{c}{{\boldsymbol{e}}}_{1}^{{\rm{T}}}\\ \vdots \\ {{\boldsymbol{e}}}_{M}^{{\rm{T}}}\end{array}\right].\end{eqnarray} \tag{ 10 }$$

We assume that the eigenvalues ${\gamma }_{1},\ldots ,{\gamma }_{\zeta }$ are distinctively larger than the other eigenvalues. Then, the set of eigenvectors $\{{{\boldsymbol{e}}}_{1},\ldots ,{{\boldsymbol{e}}}_{\zeta }\}$ is an orthonormal basis of the column span of the augmented lead field. Accordingly, the pseudo signal subspace projector ${\boldsymbol{P}}$ is derived as

$$\begin{eqnarray}&&{\boldsymbol{P}}=[{{\boldsymbol{e}}}_{1},\ldots ,{{\boldsymbol{e}}}_{\zeta }]{[{{\boldsymbol{e}}}_{1},\ldots ,{{\boldsymbol{e}}}_{\zeta }]}^{{\rm{T}}},\end{eqnarray} \tag{ 11 }$$

and the following relationship holds:

$$\begin{eqnarray}&&{\boldsymbol{P}}{{\boldsymbol{b}}}_{{\rm{s}}}(t)={{\boldsymbol{b}}}_{{\rm{s}}}(t).\end{eqnarray} \tag{ 12 }$$

Now we define the time-domain signal subspace, which is different from the spatial domain signal subspace defined in section 2.2. For the arguments on the time-domain signal subspace, we denote row vectors consisting of the signal-source time courses by ${{\boldsymbol{s}}}_{p}$: ${{\boldsymbol{s}}}_{p}=[{s}_{p}({t}_{1}),\ldots ,{s}_{p}({t}_{K})]$ ($p=1,\ldots ,Q$). We assume that Q is smaller than the number of sensors M. This assumption is called the low-rank signal assumption. Also, the vectors ${{\boldsymbol{s}}}_{1},\ldots ,{{\boldsymbol{s}}}_{Q}$ are assumed to be linearly independent. Using equations (3) and (6), we have the following relation

$$\begin{eqnarray}{{\boldsymbol{B}}}_{{\rm{S}}} & = & \left[\displaystyle \sum _{p=1}^{Q}{s}_{p}({t}_{1}){{\boldsymbol{l}}}_{p},\ldots ,\displaystyle \sum _{p=1}^{Q}{s}_{p}({t}_{K}){{\boldsymbol{l}}}_{p}\right]\\ & = & \left[\begin{array}{c}\displaystyle \sum _{p=1}^{Q}[{s}_{p}({t}_{1}),\ldots ,{s}_{p}({t}_{K})]{l}_{1}^{p}\\ \vdots \\ \displaystyle \sum _{p=1}^{Q}[{s}_{p}({t}_{1}),\ldots ,{s}_{p}({t}_{K})]{l}_{M}^{p}\end{array}\right]=\left[\begin{array}{c}\displaystyle \sum _{p=1}^{Q}{{\boldsymbol{s}}}_{p}{l}_{1}^{p}\\ \vdots \\ \displaystyle \sum _{p=1}^{Q}{{\boldsymbol{s}}}_{p}{l}_{M}^{p}\end{array}\right].\end{eqnarray} \tag{ 13 }$$

Here, ${l}_{1}^{p},\ldots ,{l}_{M}^{p}$ are the elements of the lead field vector ${{\boldsymbol{l}}}_{p}$: ${{\boldsymbol{l}}}_{p}=[{l}_{1}^{p},\ldots ,{l}_{M}^{p}]$.

The row vector whose elements form the jth row of the matrix ${{\boldsymbol{B}}}_{{\rm{S}}}$ is denoted ${{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}$. Equation (13) then implies the relation

$$\begin{eqnarray}&&{{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}=\displaystyle \sum _{p=1}^{Q}{l}_{j}^{p}{{\boldsymbol{s}}}_{p}.\end{eqnarray} \tag{ 14 }$$

Namely, ${{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}$ is expressed as a linear combination of the vectors ${{\boldsymbol{s}}}_{1},\ldots ,{{\boldsymbol{s}}}_{Q}$. The row vector ${{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}$ thus lies within the span of these vectors. That is

$$\begin{eqnarray}&&{{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}\in \mathrm{span}\{{{\boldsymbol{s}}}_{1},\ldots ,{{\boldsymbol{s}}}_{Q}\}.\end{eqnarray} \tag{ 15 }$$

This subspace, $\mathrm{span}\{{{\boldsymbol{s}}}_{1},\ldots ,{{\boldsymbol{s}}}_{Q}\}$, is defined as the time-domain signal subspace, and denoted ${{ \mathcal K }}_{{\rm{S}}}$:

$$\begin{eqnarray}&&{{ \mathcal K }}_{{\rm{S}}}=\mathrm{span}\{{{\boldsymbol{s}}}_{1},\ldots ,{{\boldsymbol{s}}}_{Q}\}.\end{eqnarray} \tag{ 16 }$$

It can be shown under the low-rank signal assumption that ${{ \mathcal K }}_{{\rm{S}}}$ is equal to the row span of ${{\boldsymbol{B}}}_{{\rm{S}}}$, i.e.,

$$\begin{eqnarray}&&{{ \mathcal K }}_{{\rm{S}}}=\mathrm{span}\{{{\boldsymbol{\beta }}}_{1}^{{\rm{S}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{S}}}\}.\end{eqnarray} \tag{ 17 }$$

The proof is presented in section A.1 in the appendix.

Let us define the time-domain interference subspace, which plays a key role in the proposed algorithm. We express the interference vector as

$$\begin{eqnarray}&&{{\boldsymbol{b}}}_{{\rm{I}}}(t)=\displaystyle \sum _{j=1}^{P}{\sigma }_{j}(t){{\boldsymbol{\xi }}}_{j},\end{eqnarray} \tag{ 18 }$$

where P is the number of interference sources. Their intensities are denoted by ${\sigma }_{j}(t)$ ($j=1,\ldots ,P$), and their lead field vectors by ${{\boldsymbol{\xi }}}_{j}$ ($j=1,\ldots ,P$). We define row vectors consisting of the interference-source time courses as ${{\boldsymbol{\sigma }}}_{p}$: ${{\boldsymbol{\sigma }}}_{p}=[{\sigma }_{p}({t}_{1}),\ldots ,{\sigma }_{p}({t}_{K})]$ ($p=1,\ldots ,P$). The vectors ${{\boldsymbol{\sigma }}}_{1},\ldots ,{{\boldsymbol{\sigma }}}_{P}$ are assumed to be linearly independent. The span of ${{\boldsymbol{\sigma }}}_{1},\ldots ,{{\boldsymbol{\sigma }}}_{P}$ is defined to be the time domain interference subspace, and is denoted by ${{ \mathcal K }}_{{\rm{I}}}$, i.e. ${{ \mathcal K }}_{{\rm{I}}}=\mathrm{span}\{{{\boldsymbol{\sigma }}}_{1},\ldots ,{{\boldsymbol{\sigma }}}_{P}\}$. The row vector whose elements form the jth row of the matrix ${{\boldsymbol{B}}}_{{\rm{I}}}$ is denoted by ${{\boldsymbol{\beta }}}_{j}^{{\rm{I}}}$. Using the same arguments used to derive equation (14), we can derive the relation

$$\begin{eqnarray}&&{{\boldsymbol{\beta }}}_{j}^{{\rm{I}}}=\displaystyle \sum _{p=1}^{P}{\xi }_{j}^{p}{{\boldsymbol{\sigma }}}_{p},\end{eqnarray} \tag{ 19 }$$

where ${\xi }_{1}^{p},\ldots ,{\xi }_{M}^{p}$ are the elements of the lead field vector ${{\boldsymbol{\xi }}}_{p}$: ${{\boldsymbol{\xi }}}_{p}=[{\xi }_{1}^{p},\ldots ,{\xi }_{M}^{p}]$. It is clear that ${{\boldsymbol{\beta }}}_{j}^{{\rm{I}}}$ is expressed as a linear combination of the vectors ${{\boldsymbol{\sigma }}}_{1},\ldots ,{{\boldsymbol{\sigma }}}_{P}$ and that the vector ${{\boldsymbol{\beta }}}_{j}^{{\rm{I}}}$ lies within the interference subspace (i.e., ${{\boldsymbol{\beta }}}_{j}^{{\rm{I}}}\in {{ \mathcal K }}_{{\rm{I}}}$). Under the low-rank signal assumption $(P+Q\lt M)$, it can be shown that $\mathrm{span}\{{{\boldsymbol{\beta }}}_{1}^{{\rm{I}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{I}}}\}$ is also equal to ${{ \mathcal K }}_{{\rm{I}}}$.

In the proposed DSSP algorithm, the sensor measurements ${\boldsymbol{B}}$ are first projected onto the inside and the outside of the pseudo signal subspace. Let us define ${{\boldsymbol{B}}}_{\mathrm{in}}$ and ${{\boldsymbol{B}}}_{\mathrm{out}}$ as the projections of ${\boldsymbol{B}}$ onto the inside and the outside of the pseudo signal subspace

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{in}}={\boldsymbol{P}}{\boldsymbol{B}},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{out}}=({\boldsymbol{I}}-{\boldsymbol{P}}){\boldsymbol{B}},\end{eqnarray} \tag{ 21 }$$

where ${\boldsymbol{I}}$ is the identity matrix. Using equations (5) and (12), we have

$$\begin{eqnarray}{{\boldsymbol{B}}}_{\mathrm{in}} & = & {\boldsymbol{P}}{\boldsymbol{B}}={\boldsymbol{P}}({{\boldsymbol{B}}}_{{\rm{S}}}+{{\boldsymbol{B}}}_{{\rm{I}}}+{{\boldsymbol{B}}}_{\varepsilon })\\ & = & {{\boldsymbol{B}}}_{{\rm{S}}}+{\boldsymbol{P}}{{\boldsymbol{B}}}_{{\rm{I}}}+{\boldsymbol{P}}{{\boldsymbol{B}}}_{\varepsilon },\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{{\boldsymbol{B}}}_{\mathrm{out}}=({\boldsymbol{I}}-{\boldsymbol{P}}){\boldsymbol{B}}=({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{{\rm{I}}}+({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{\varepsilon }.\end{eqnarray} \tag{ 23 }$$

Note here that, due to the 'dull' cut-off property⁸ of the projector ${\boldsymbol{P}}$, neither ${\boldsymbol{P}}{{\boldsymbol{B}}}_{{\rm{I}}}$ nor $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{{\rm{I}}}$ becomes negligibly small.

We now analyze the subspace spanned by the row vectors of ${{\boldsymbol{B}}}_{\mathrm{in}}$. We can write

$$\begin{eqnarray}&&{\boldsymbol{P}}{{\boldsymbol{b}}}_{{\rm{I}}}(t)=\displaystyle \sum _{j=1}^{P}{\sigma }_{j}(t){\boldsymbol{P}}{{\boldsymbol{\xi }}}_{j}=\displaystyle \sum _{j=1}^{P}{\sigma }_{j}(t){\tilde{{\boldsymbol{\xi }}}}_{j},\end{eqnarray} \tag{ 24 }$$

where ${\tilde{{\boldsymbol{\xi }}}}_{j}={\boldsymbol{P}}{{\boldsymbol{\xi }}}_{j}$. The equation above indicates that although the projector ${\boldsymbol{P}}$ modifies the lead field vectors, it never changes the time courses of the interference sources. Therefore, defining ${\tilde{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}$ to be the jth row vector of the matrix ${\boldsymbol{P}}{{\boldsymbol{B}}}_{{\rm{I}}}$, we can show

$$\begin{eqnarray}&&{\tilde{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}=\displaystyle \sum _{p=1}^{P}{\tilde{\xi }}_{j}^{p}{{\boldsymbol{\sigma }}}_{p},\end{eqnarray} \tag{ 25 }$$

where ${\tilde{\xi }}_{1}^{p},\ldots ,{\tilde{\xi }}_{M}^{p}$ are the elements of the modified lead field vector ${\tilde{{\boldsymbol{\xi }}}}_{p}$. This equation indicates that the row vector of the matrix ${\boldsymbol{P}}{{\boldsymbol{B}}}_{{\rm{I}}}$ lies within the time-domain interference subspace, i.e., ${\tilde{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}\in {{ \mathcal K }}_{{\rm{I}}}$ ($j=1,\ldots ,M$). We denote a row vector of ${{\boldsymbol{B}}}_{\mathrm{in}}$ by ${{\boldsymbol{\beta }}}_{j}^{\mathrm{in}}$ and the span of ${{\boldsymbol{\beta }}}_{1}^{\mathrm{in}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{\mathrm{in}}$ by ${{ \mathcal K }}_{\mathrm{in}}$. Then, taking the relations ${{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}\in {{ \mathcal K }}_{{\rm{S}}}$ and ${\tilde{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}\in {{ \mathcal K }}_{{\rm{I}}}$ into consideration, equation (22) implies that

$$\begin{eqnarray}&&{{ \mathcal K }}_{\mathrm{in}}={{ \mathcal K }}_{{\rm{S}}}\cup {{ \mathcal K }}_{{\rm{I}}}\cup {{ \mathcal K }}_{\varepsilon },\end{eqnarray} \tag{ 26 }$$

where the span of the row vectors of ${\boldsymbol{P}}{{\boldsymbol{B}}}_{\varepsilon }$ is denoted by ${{ \mathcal K }}_{\varepsilon }$. The proof of (26) is presented in section A.3 in the appendix.

We next analyze the subspace spanned by the row vectors of ${{\boldsymbol{B}}}_{\mathrm{out}}$. We can write

$$\begin{eqnarray}&&({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{b}}}_{{\rm{I}}}(t)=\displaystyle \sum _{j=1}^{P}{\sigma }_{j}(t)({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{\xi }}}_{j}=\displaystyle \sum _{j=1}^{P}{\sigma }_{j}(t){\bar{{\boldsymbol{\xi }}}}_{j},\end{eqnarray} \tag{ 27 }$$

where ${\bar{{\boldsymbol{\xi }}}}_{j}=({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{\xi }}}_{j}$. Using the same arguments employed to derive equation (25) and denoting the jth row vector of the matrix $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{{\rm{I}}}$ by ${\bar{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}$, we can show

$$\begin{eqnarray}&&{\bar{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}=\displaystyle \sum _{p=1}^{P}{{\boldsymbol{\sigma }}}_{p}{\bar{\xi }}_{j}^{p},\end{eqnarray} \tag{ 28 }$$

where ${\bar{\xi }}_{1}^{p},\ldots ,{\bar{\xi }}_{M}^{p}$ are the elements of the modified lead field vector ${\bar{{\boldsymbol{\xi }}}}_{p}$. This equation indicates that the relation ${\bar{{\boldsymbol{\beta }}}}_{j}^{{\rm{I}}}\in {{ \mathcal K }}_{{\rm{I}}}$ holds, and the row span of $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{{\rm{I}}}$ is equal to ${{ \mathcal K }}_{{\rm{I}}}$.

Thus, denoting the jth row vector of ${{\boldsymbol{B}}}_{\mathrm{out}}$ by ${{\boldsymbol{\beta }}}_{j}^{\mathrm{out}}$ and the span of ${{\boldsymbol{\beta }}}_{1}^{\mathrm{out}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{\mathrm{out}}$ by ${{ \mathcal K }}_{\mathrm{out}}$, equation (23) implies that ${{ \mathcal K }}_{\mathrm{out}}$ is given by

$$\begin{eqnarray}&&{{ \mathcal K }}_{\mathrm{out}}={{ \mathcal K }}_{{\rm{I}}}\cup {\bar{{ \mathcal K }}}_{\varepsilon },\end{eqnarray} \tag{ 29 }$$

where ${\bar{{ \mathcal K }}}_{\varepsilon }$ is the subspace spanned by the row vectors of $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{\varepsilon }$. Equations (26) and (29) imply that the intersection of the two subspaces ${{ \mathcal K }}_{\mathrm{in}}$ and ${{ \mathcal K }}_{\mathrm{out}}$ is the time-domain interference subspace ${{ \mathcal K }}_{{\rm{I}}}$:

$$\begin{eqnarray}&&{{ \mathcal K }}_{{\rm{I}}}={{ \mathcal K }}_{\mathrm{in}}\cap {{ \mathcal K }}_{\mathrm{out}},\end{eqnarray} \tag{ 30 }$$

if ${{ \mathcal K }}_{\varepsilon }\cap {\bar{{ \mathcal K }}}_{\varepsilon }=\varnothing$ holds⁹ , where $\varnothing$ indicates the empty set. It is actually straightforward to show ${{ \mathcal K }}_{\varepsilon }\cap {\bar{{ \mathcal K }}}_{\varepsilon }=\varnothing$, because the relationship

$$\begin{eqnarray*}&&{\boldsymbol{P}}{{\boldsymbol{B}}}_{\varepsilon }{(({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{\varepsilon })}^{{\rm{T}}}={\boldsymbol{P}}{{\boldsymbol{B}}}_{\varepsilon }{{\boldsymbol{B}}}_{\varepsilon }^{{\rm{T}}}({\boldsymbol{I}}-{\boldsymbol{P}})={\rho }^{2}{\boldsymbol{P}}({\boldsymbol{I}}-{\boldsymbol{P}})=0,\end{eqnarray*}$$

holds. This means that the rows of ${\boldsymbol{P}}{{\boldsymbol{B}}}_{\varepsilon }$ are orthogonal to the rows of $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{\varepsilon }$. In the equation above, we assume that ${{\boldsymbol{B}}}_{\varepsilon }{{\boldsymbol{B}}}_{\varepsilon }^{{\rm{T}}}={\rho }^{2}{\boldsymbol{I}}$, where ${\rho }^{2}$ is the power of the sensor noise.

In this subsection, we present an algorithm to derive an orthonormal basis of ${{ \mathcal K }}_{{\rm{I}}}$. We first extract ${{ \mathcal K }}_{\mathrm{in}}$ and ${{ \mathcal K }}_{\mathrm{out}}$ from ${{\boldsymbol{B}}}_{\mathrm{in}}$ and ${{\boldsymbol{B}}}_{\mathrm{out}}$, respectively by applying singular value decomposition:

$$\begin{eqnarray}{{\boldsymbol{B}}}_{\mathrm{in}}=[{{\boldsymbol{f}}}_{1},\ldots ,{{\boldsymbol{f}}}_{M}]\left[\begin{array}{cccc}{\varphi }_{1} & 0 & \cdots & 0\\ 0 & {\varphi }_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {\varphi }_{M}\end{array}\right]\left[\begin{array}{c}{{\boldsymbol{u}}}_{1}^{{\rm{T}}}\\ \vdots \\ {{\boldsymbol{u}}}_{M}^{{\rm{T}}}\end{array}\right]\end{eqnarray} \tag{ 31 }$$

and

$$\begin{eqnarray}{{\boldsymbol{B}}}_{\mathrm{out}}=[{{\boldsymbol{g}}}_{1},\ldots ,{{\boldsymbol{g}}}_{M}]\left[\begin{array}{cccc}{\phi }_{1} & 0 & \cdots & 0\\ 0 & {\phi }_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {\phi }_{M}\end{array}\right]\left[\begin{array}{c}{{\boldsymbol{v}}}_{1}^{{\rm{T}}}\\ \vdots \\ {{\boldsymbol{v}}}_{M}^{{\rm{T}}}\end{array}\right].\end{eqnarray} \tag{ 32 }$$

For the two sets of singular values, distinctively large singular values are denoted by ${\varphi }_{1},\ldots ,{\varphi }_{\mu }$ and ${\phi }_{1},\ldots ,{\phi }_{\nu }$. We can then derive the subspaces ${{ \mathcal K }}_{\mathrm{in}}$ and ${{ \mathcal K }}_{\mathrm{out}}$ such that

$$\begin{eqnarray}&&{{ \mathcal K }}_{\mathrm{in}}\approx \mathrm{span}\{{{\boldsymbol{u}}}_{1},\ldots ,{{\boldsymbol{u}}}_{\mu }\},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{{ \mathcal K }}_{\mathrm{out}}\approx \mathrm{span}\{{{\boldsymbol{v}}}_{1},\ldots ,{{\boldsymbol{v}}}_{\nu }\}.\end{eqnarray} \tag{ 34 }$$

Additional arguments, presented in section 6, lead to the conclusion that the interference subspace ${{ \mathcal K }}_{{\rm{I}}}$ is equal to the intersection of $\mathrm{span}\{{{\boldsymbol{u}}}_{1},\ldots ,{{\boldsymbol{u}}}_{\mu }\}$ and $\mathrm{span}\{{{\boldsymbol{v}}}_{1},\ldots ,{{\boldsymbol{v}}}_{\nu }\}$. That is ${{ \mathcal K }}_{{\rm{I}}}=\mathrm{span}\{{{\boldsymbol{u}}}_{1},\ldots ,{{\boldsymbol{u}}}_{\mu }\}\cap \mathrm{span}\{{{\boldsymbol{v}}}_{1},\ldots ,{{\boldsymbol{v}}}_{\nu }\}$.

The procedure used to find the intersection is described below. According to [12], an orthonormal basis of the intersection is obtained as a set of the principal vectors whose principal angles are equal to zero. To find those principal vectors, we first define

$$\begin{eqnarray}&&{\boldsymbol{U}}=[{{\boldsymbol{u}}}_{1},\ldots ,{{\boldsymbol{u}}}_{\mu }],\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\boldsymbol{V}}=[{{\boldsymbol{v}}}_{1},\ldots ,{{\boldsymbol{v}}}_{\nu }].\end{eqnarray} \tag{ 36 }$$

Singular-value decomposition of a matrix ${{\boldsymbol{U}}}^{{\rm{T}}}{\boldsymbol{V}}$ is then performed, and the results are expressed as

$$\begin{eqnarray}{{\boldsymbol{U}}}^{{\rm{T}}}{\boldsymbol{V}}={\boldsymbol{Y}}\left[\begin{array}{ccc}\mathrm{cos}({\theta }_{1}) & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0 & \ldots & \mathrm{cos}({\theta }_{\nu })\end{array}\right]{{\boldsymbol{Z}}}^{{\rm{T}}},\end{eqnarray} \tag{ 37 }$$

where ${\boldsymbol{Y}}$ and ${\boldsymbol{Z}}$ are matrices whose columns consist of singular vectors. In the equation above, we use the fact that $\mu \gt \nu$. Equation (37) indicates that the singular values of the matrix ${{\boldsymbol{U}}}^{{\rm{T}}}{\boldsymbol{V}}$ are equal to the cosines of the principal angles between $\mathrm{span}\{{{\boldsymbol{u}}}_{1},\ldots ,{{\boldsymbol{u}}}_{\mu }\}$ and $\mathrm{span}\{{{\boldsymbol{v}}}_{1},\ldots ,{{\boldsymbol{v}}}_{\nu }\}$. The intersection ${{ \mathcal K }}_{{\rm{I}}}$ has the property that the principal angles are equal to zero. Thus, by observing the relation

$$\begin{eqnarray*}&&\mathrm{cos}({\theta }_{1})=\mathrm{cos}({\theta }_{2})\quad =\cdots =\quad \mathrm{cos}({\theta }_{r})\approx 1,\end{eqnarray*}$$

the dimension of ${{ \mathcal K }}_{{\rm{I}}}$ is determined to be r.

The principal vectors are then obtained either as the first r columns of the matrix ${\boldsymbol{U}}{\boldsymbol{Y}}$ or the first r columns of the matrix ${\boldsymbol{V}}{\boldsymbol{Z}}$. Defining the first r columns of ${\boldsymbol{U}}{\boldsymbol{Y}}$ as ${{\boldsymbol{\psi }}}_{1},\ldots ,{{\boldsymbol{\psi }}}_{r}$, the relation

$$\begin{eqnarray}&&{{ \mathcal K }}_{{\rm{I}}}=\mathrm{span}\{{{\boldsymbol{\psi }}}_{1},\ldots ,{{\boldsymbol{\psi }}}_{r}\}\end{eqnarray} \tag{ 38 }$$

holds and the vectors ${{\boldsymbol{\psi }}}_{1},\ldots ,{{\boldsymbol{\psi }}}_{r}$ form an orthonormal basis for ${{ \mathcal K }}_{{\rm{I}}}$. The interference removal is thus carried out by projecting the measured data ${\boldsymbol{B}}$ onto the subspace orthogonal to the interference subspace. Defining a matrix ${\boldsymbol{G}}$ by

$$\begin{eqnarray}&&{\boldsymbol{G}}=[{{\boldsymbol{\psi }}}_{1},\ldots ,{{\boldsymbol{\psi }}}_{r}],\end{eqnarray} \tag{ 39 }$$

the interference removal can be accomplished by right multiplying ${\boldsymbol{B}}$ by $({\boldsymbol{I}}-{\boldsymbol{G}}{{\boldsymbol{G}}}^{{\rm{T}}})$. That is, the interference-removed measurements ${\widehat{{\boldsymbol{B}}}}_{{\rm{S}}}$ are obtained as

$$\begin{eqnarray}&&{\widehat{{\boldsymbol{B}}}}_{{\rm{S}}}={\boldsymbol{B}}({\boldsymbol{I}}-{\boldsymbol{G}}{{\boldsymbol{G}}}^{{\rm{T}}}).\end{eqnarray} \tag{ 40 }$$

Computer simulations were carried out to investigate the validity of the proposed algorithm. We first simulated SCEF measurements distorted by large stimulus-induced artifacts. For data generation, we assumed the sensor array of a 120-channel biomagnetometer [5, 7], which has been specifically developed for SCEF measurements. The biomagnetometer is equipped with 40 vector sensors¹⁰ , which are arranged at 8 × 5 measurement locations covering a 14 cm × 9 cm area.

In this computer simulation, we used a source model consisting of four equal-intensity current vectors; such a source model has been found to be physiologically plausible in our previous studies [8]. The model is shown in figure 1(a). In this model, the two sources aligned along the y-axis are intercellular sources propagating in the nerve axon, representing two anti-directional current dipoles [14], called the leading and the trailing dipoles [15, 16]. The other two sources aligned along the x-axis represent the volume current. The volume current flows from the extracellular milieu to the site between the two dipole sources in the nerve axon. In this computer simulation, the distance between the two dipole sources, as well as that between the two volume current sources, was set to 3 cm.

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The SCEF sources move with a speed of 40–80 cm s⁻¹. In this computer simulation, we assumed that four sources traveled along the y-direction (from $-y$ to $+y$) with a speed of 60 cm s⁻¹. The source-sensor geometry used in the computer simulation is shown in figure 1(b). The magnetic field generated from the four moving sources was computed, and a plot of the simulated measurements of all 120 sensors is shown in figure 2(a). Here, we used no conductor models, and the forward solution was computed using the well-known Biot–Savart law, which is derived based on the quasi-static approximation of Maxwell's equations. The time was set at zero when the y coordinate of the leading dipole was equal to −30 cm. The field contour map at a latency of 5 ms is shown in figure 3(a).

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The artifact data was obtained from artifact-only measurements, in which we applied exactly the same stimulus as used in measuring the SCEF, with a stimulus electrode positioned a few centimeters away from the median nerve of a subject [17]. Since the location of the electrode was not on the nerve, stimuli did not induce the nerve activity but elicited only the artifacts. The artifact-only data is shown in figure 2(b).

The results of adding the artifact data onto the computer-generated signal are shown in figure 2(c), and the field contour map of the artifact-contaminated data at a latency of 5 ms is shown in figure 3(b). Here, the interference-to-signal ratio (ISR) was set at 12. The ISR is defined as the ratio of ${\parallel {{\boldsymbol{B}}}_{{\rm{I}}}\parallel }_{{\rm{F}}}/{\parallel {{\boldsymbol{B}}}_{{\rm{S}}}\parallel }_{{\rm{F}}}$, where ${\parallel {\boldsymbol{X}}\parallel }_{{\rm{F}}}$ indicates the Frobenius norm of a matrix ${\boldsymbol{X}}$. Both the sensor time courses and the contour map show that a significant amount of distortion arises due to contamination by the artifacts.

We then applied the DSSP algorithm to this artifact-contaminated data. Here, a two-dimensional region ($-8\leqslant x\leqslant 8\;{\rm{cm}}$, $-6\leqslant y\leqslant 6$ cm), which is located 7 cm below the sensor plane, was defined as the source space. The augmented lead field was computed with a 0.5 cm voxel grid over this source space. A plot of the singular values $\mathrm{cos}({\theta }_{j})$ in equation (37) is shown in figure 4. The plot shows that the first six singular values are very close to 1. Thus, the dimension of the intersection, r, was determined to be six by thresholding the singular values at 0.99. The artifact-removed sensor time courses are shown in figure 2(d). A field contour map of the artifact-removed data at a latency of 5 ms is shown in figure 3(c). Comparing between figures 2 and 3, we can see that the distortion from the artifacts is significantly reduced.

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We next assessed the effectiveness of the DSSP algorithm on the basis of source reconstruction results. Time-point by time-point source reconstruction is essential for imaging spinal cord activity because of the rapid movements of the sources. We therefore used the recursively applied null-steering (RENS) beamformer algorithm [9, 18] because of its applicability to single time-point data. A snapshot of the source image at 5 ms latency is shown in figure 5. In this figure, the results in (a) are obtained by using the artifact-free sensor data in figure 3(a). The results from the artifact-contaminated sensor data in figure 3(b) are shown in figure 5(b). Due to the presence of the interference, the reconstruction results are significantly distorted. The results from the artifact-removed sensor data in figure 3(c) are shown in figure 5(c). In these results, the four sources are clearly reconstructed and the results are very close to the artifact-free case in figure 5(a), demonstrating the effectiveness of the proposed method.

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We next present a computer simulation on MEG measurements in which the data was overlapped with a large interference caused near the source space. A sensor alignment of the 275-channel whole-head sensor array from the Omega^TM (VMS Medtech, Coquitlam, Canada) neuromagnetometer was used. The coordinate system and source-sensor configuration used in the computer simulation are depicted in figure 6(a). A vertical plane (x = 0 cm) was assumed at the middle of the sensor array, and three sources were assumed to exist on this plane. The coordinates of the sources are $(0,-2,10.3)$, $(0,2.5,10.3)$, and $(0,1,7.3)$ cm. The time courses assigned to the three sources are shown in the top three panels in figure 6(b).

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We put a single interference source shown as a filled circle with the label 'interference (location A)' in figure 6(a). As shown in this figure, the interference source was fairly close to the signal sources and it was located just outside the head near the head surface. The coordinate of the interference source was $(-1,-1,-6)$ cm. This interference simulated the noise caused from some types of brain stimulator. In section 5, we actually present experiments in which the MEG data was contaminated by the noise from patient's VNS. The time course of the interference source is presented in the bottom panel of figure 6(b). We generated interference magnetic field with the ISR equal to 100 where the ISR is defined as $\parallel {{\boldsymbol{B}}}_{{\rm{I}}}\parallel /\parallel {{\boldsymbol{B}}}_{{\rm{S}}}\parallel$. The interference magnetic field was computed and overlapped onto the signal magnetic field computed from the activities of the three signal sources.

To generate the magnetic fields, source activities were projected to the sensor time courses through the lead field, which is obtained using the homogeneous spherical head model [19] with the center of the sphere set to $(0,0,4)$ cm. The spatio-temporal data with 1200 time points was generated. In figure 7, the time courses of the signal magnetic field (plus sensor noise) are shown in the top panel. The time courses of interference-overlapped magnetic field are shown in the middle panel. Since the interference magnetic field is 100 times stronger than the signal magnetic field, the sensor time courses in the bottom panel are dominated by the interference magnetic field.

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We set the source space to a region of $-4.5\leqslant x\leqslant 4.5\;{\rm{cm}}$, $-5\leqslant y\leqslant 5\;{\rm{cm}}$ and $5\leqslant z\leqslant 13\;{\rm{cm}}$. This region approximately covers the whole brain and includes the locations of the three sources, but it does not include the location of the interference source. The augmented lead field was computed using a 0.5 cm voxel grid over this region. We then applied the DSSP algorithm for removing the interference. The dimension of the interference subspace r was determined to be 1 by thresholding the cosine of the principal angle (equation (37)) by the value of 0.99. The interference-removed sensor time courses are shown in the bottom panel in figure 7, which shows that the interference is nearly completely removed.

We next evaluate the effect of interference removal based on the quality of reconstructed source images. The adaptive beamformer algorithm [11] was applied for source reconstruction from the sensor data shown in figure 7. The results are shown in figure 8. In figure 8(a), the source reconstruction results obtained from the interference-free magnetic field (the top panel of figure 7) are shown. Three sources can clearly be observed. The reconstruction results obtained using the interference-overlapped magnetic field (the middle panel of figure 7) are shown in figure 8(b). Due to the overlap of the large interference, the results contain significant distortion. The reconstruction results obtained using the artifact-removed sensor data (the bottom panel of figure 7) are shown in figure 8(c) in which the distortion is nearly completely removed, demonstrating the effectiveness of the proposed method.

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As shown in the computer simulation above, the DSSP algorithm allows interference sources to be fairly close to the source space. However the prerequisite exists that the interference sources are located outside the source space. We performed computer simulation in which this prerequisite was not fulfilled. We here put a single interference source at (−1, −1, 5), exactly on the border of the source space. This location is indicated by a filled circle with the label 'interference (location B)' in figure 6(a). The results of source reconstruction are shown in figure 9 where the results without interference removal are shown in (a) and the results with the removal in (b). These results clearly indicate that the DSSP algorithm can still remove the influence of interference but the removal is achieved at the sacrifice of significant amount of signal source intensity.

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A 120-channel biomagnetometer [5, 7] was used for measuring a human SCEF¹¹ . The subject was a healthy male volunteer. The experiment was approved by the ethics committee of Tokyo Medical and Dental University.

The experimental setup is schematically shown in figure 10. As depicted here, the cryostat of the biomagnetometer has a cylindrical body with a protrusion, and this protrusion contains sensors directed upward. The subject lies down in the supine position, and the subject's lower neck is positioned on the upper surface of the protrusion of the cryostat. A stimulus current was applied to the subject's median nerve near his left elbow. The stimulus with an intensity of 10 mA and a duration of 0.3 ms was repeated 2000 times at a repetition rate of 4 Hz. The data acquisition was performed with a sampling frequency of 40 kHz. An analog bandpass filter with a bandwidth of 100–5000 Hz was applied. The signal was averaged across all 2000 measured trials.

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An x-ray image covering the subject's neck and the sensors of the biomagnetometer was obtained to identify the location of the spinal cord. The reconstruction region was determined to be a curved plane containing the spinal cord. An x-ray image with the extracted 2D reconstruction region is shown in figure 11.

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The measured SCEF, averaged over 2000 trials, is shown in figure 12(a). The electric stimulus was given at a latency of 0 ms. Large artifacts are observed particularly in the data before 4 ms, although the peaks between 4 and 8 ms, which are caused by the spinal cord nerve activity, can still be observed. We applied the proposed DSSP algorithm to remove these artifacts. The 2D reconstruction region was used as the source space over which the augmented lead field matrix was computed. The results are shown in figure 12(b). Here, the artifacts are significantly reduced and the SCEF signal, consisting of peaks between 4 and 8 ms, is clearly observed.

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The spinal cord source activity was reconstructed by using the RENS beamformer. Schematic illustration in figure 13 shows relative positions of the subject's neck and median nerve with respect to the reconstruction region, which had an area of $16\times 12\;{\rm{cm}}$ with voxel dimensions of 0.5 cm in the x- and y-directions. The reconstructed source images at a latency of 5.8 ms are shown in figure 14. In this figure, the source image from the original artifact-contaminated sensor data is shown in (a). The image from the artifact-removed data is shown in (b). Both images show the leading dipoles but their directions (indicated by the white arrows) are significantly different.

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To determine which results are physiologically more plausible, we used a source image obtained with the stimulus applied at the median nerve near the subject's wrist; the image is shown in figure 14(c). The signal obtained with wrist stimulation is known to be less affected by artifacts. This is because, with the stimulation near the subject's wrist, it takes 4–5 ms more for the nerve activation to reach the neck region, and thus the spinal cord signal is much less contaminated by stimulus-induced artifacts due to their rapid decay. Therefore, the results from the wrist stimulation should serve as the physiological 'ground truth' for the comparison. The leading dipoles have transverse (the negative x) components in figures 14(b) and (c), while the direction is almost upward in (a), i.e., the current vector has almost no x component in (a).

When the subject's left median nerve is stimulated, the nerve activity is known to propagate from the left median nerve into the spinal cord near the fourth vertebra (c4). Thus, it should be more plausible that the current vector has a transverse, negative x component, as depicted in figure 13. Accordingly, considering the fact that the leading dipoles in (b) and (c) have such negative x components, we can draw the conclusion that the results in figure 14(b) are physiologically more plausible than those in figure 14(a).

We have conducted measurements exactly the same as the ones described above but with right median nerve stimulation. The source images from these measurements are shown in figure 15. Here, the source image from the original artifact-contaminated sensor data is shown in (a). The image from the artifact-removed data is shown in (b). Both results show the leading dipole but the direction of the leading dipole (indicated by the white arrow) is again different between these two images. The source image obtained with the stimulus applied at the right median nerve near subject's wrist is shown in (c). In this figure, the current vector should have a transverse, positive x component since the x component of the source should be opposite to that in the left median nerve stimulation. The images in figures 15(b) and (c) show that the dipoles have the expected positive x components. Therefore, we can again conclude that the results in (b) are physiologically more plausible than those in (a), which show the source vector with an almost upward direction. It should be noted that the DSSP algorithm can be used with any type of source reconstruction methods. In these experiments, the RENS beamformer was used for reconstructing the spinal cord activity, primarily because it gives the spatial resolution better than other existing methods¹² applicable to single time point data.

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Here, we prove that, under the low-rank signal assumption ($Q\lt M$), ${{ \mathcal K }}_{{\rm{S}}}$ is equal to the row span of ${{\boldsymbol{B}}}_{{\rm{S}}}$. That is

$$\begin{eqnarray}&&{{ \mathcal K }}_{{\rm{S}}}=\mathrm{span}\{{{\boldsymbol{\beta }}}_{1}^{{\rm{S}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{S}}}\}.\end{eqnarray} \tag{ 45 }$$

Equation (45) is equivalent to claiming that any row vector ${\boldsymbol{x}}$ (${\boldsymbol{x}}\in {{ \mathcal K }}_{{\rm{S}}}$) can be expressed as a linear combination of the row vectors, ${{\boldsymbol{\beta }}}_{1}^{{\rm{S}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{S}}}$. The claim is proved as follows. Since ${\boldsymbol{x}}\in {{ \mathcal K }}_{{\rm{S}}}$, ${\boldsymbol{x}}$ can be expressed as a linear combination of ${{\boldsymbol{s}}}_{p}$:

$$\begin{eqnarray}&&{\boldsymbol{x}}=\displaystyle \sum _{p=1}^{Q}{\gamma }_{p}{{\boldsymbol{s}}}_{p}.\end{eqnarray} \tag{ 46 }$$

If this vector ${\boldsymbol{x}}$ is also expressed as a linear combination of ${{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}$, we have

$$\begin{eqnarray}&&{\boldsymbol{x}}=\displaystyle \sum _{j=1}^{M}{\alpha }_{j}{{\boldsymbol{\beta }}}_{j}^{{\rm{S}}}.\end{eqnarray} \tag{ 47 }$$

Substituting equation (14) into (47) and using equation (46), we get

$$\begin{eqnarray}&&\displaystyle \sum _{j=1}^{M}{\alpha }_{j}\displaystyle \sum _{p=1}^{Q}{l}_{p}^{j}{{\boldsymbol{s}}}_{p}=\displaystyle \sum _{p=1}^{Q}{\gamma }_{p}{{\boldsymbol{s}}}_{p}.\end{eqnarray} \tag{ 48 }$$

Comparing the coefficients of the vector ${{\boldsymbol{s}}}_{p}$ in the left- and right-hand sides of the equation above, we have a set of Q linear equations:

$$\begin{eqnarray*}{\alpha }_{1}{l}_{1}^{1}\quad +\cdots +\quad {\alpha }_{M}{l}_{1}^{M} & = & {\gamma }_{1}\\ {\alpha }_{1}{l}_{2}^{1}\quad +\cdots +\quad {\alpha }_{M}{l}_{2}^{M} & = & {\gamma }_{2}\\ \vdots & & \vdots \\ {\alpha }_{1}{l}_{Q}^{1}\quad +\cdots +\quad {\alpha }_{M}{l}_{Q}^{M} & = & {\gamma }_{Q}.\end{eqnarray*}$$

Under the low-rank signal assumption $M\gt Q$, for any arbitrary set of ${\gamma }_{1},\ldots ,{\gamma }_{Q}$, we have a set of ${\alpha }_{1},\ldots ,{\alpha }_{M}$ that fulfills the above equations. Therefore, equation (47) always holds for any ${\boldsymbol{x}}$ (${\boldsymbol{x}}\in {{ \mathcal K }}_{{\rm{S}}}$), and thus $\mathrm{span}\{{{\boldsymbol{\beta }}}_{1}^{{\rm{S}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{S}}}\}$ is equal to ${{ \mathcal K }}_{{\rm{S}}}$.

Using the same arguments, it can be shown that the row span of ${{\boldsymbol{B}}}_{{\rm{I}}}$, $\mathrm{span}\{{{\boldsymbol{\beta }}}_{1}^{{\rm{I}}},\ldots ,{{\boldsymbol{\beta }}}_{M}^{{\rm{I}}}\}$, is equal to the interference subspace ${{ \mathcal K }}_{{\rm{I}}}$. Also, both the row span of ${\boldsymbol{P}}{{\boldsymbol{B}}}_{{\rm{I}}}$, $\mathrm{span}\{{\tilde{{\boldsymbol{\beta }}}}_{1}^{{\rm{I}}},\ldots ,{\tilde{{\boldsymbol{\beta }}}}_{M}^{{\rm{I}}}\}$, and the row span of $({\boldsymbol{I}}-{\boldsymbol{P}}){{\boldsymbol{B}}}_{{\rm{I}}}$, $\mathrm{span}\{{\bar{{\boldsymbol{\beta }}}}_{1}^{{\rm{I}}},\ldots ,{\bar{{\boldsymbol{\beta }}}}_{M}^{{\rm{I}}}\}$, are equal to ${{ \mathcal K }}_{{\rm{I}}}$.

Here, we present an example of the cutoff property of the pseudo signal subspace projector. The example was derived assuming the source-sensor geometry used in our computer simulation in section 3.1 where the augmented lead field was computed over the two-dimensional source space defined by $-8\leqslant x\leqslant 8\;{\rm{cm}}$, $-6\leqslant y\leqslant 6\;{\rm{cm}}$, and z = 0 cm. The sensors are arranged on a plane of z = 7 cm. Assuming that a source generating interference signal existed at ${\boldsymbol{r}}$, the ratio $R={\parallel {\boldsymbol{P}}{{\boldsymbol{l}}}_{{\rm{I}}}({\boldsymbol{r}})\parallel }^{2}/{\parallel {{\boldsymbol{l}}}_{{\rm{I}}}({\boldsymbol{r}})\parallel }^{2}$ was computed, where ${{\boldsymbol{l}}}_{{\rm{I}}}({\boldsymbol{r}})$ indicates the lead field vector of the interference source. The ratio R expresses the decrease of the total power of the interference after processing by the projector ${\boldsymbol{P}}$. In this example, the source is located on the y-axis (i.e., ${\boldsymbol{r}}=(0,y,0)$), and the configuration of the source space and the interference source is shown in the upper panel of figure 20. The power decrease ratio R versus the y coordinate, which is the distance of the interference source from the origin, is shown in the lower part of figure 20.

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The plot shows that the cutoff property of the pseudo signal subspace projector ${\boldsymbol{P}}$ is not sharp. It passes almost 100% of the signal power if the interference source is located within the source space ($y\leqslant 6$ cm) and attenuates only 5%–20% of the signal power when the source is located in the vicinity of the source space ($10\leqslant y\leqslant 50$ cm). The proposed DSSP algorithm first applies the pseudo subspace projector ${\boldsymbol{P}}$ to create ${{\boldsymbol{B}}}_{\mathrm{in}}$ and I −  P to create ${{\boldsymbol{B}}}_{\mathrm{out}}$. Both ${{\boldsymbol{B}}}_{\mathrm{in}}$ and ${{\boldsymbol{B}}}_{\mathrm{out}}$ should contain significant amount of the interference, because of this 'dull' cutoff of the pseudo signal subspace projector ${\boldsymbol{P}}$. The DSSP algorithm makes use of this property of ${\boldsymbol{P}}$ and detects the interference subspace by finding common components in the row spans of ${{\boldsymbol{B}}}_{\mathrm{in}}$ and ${{\boldsymbol{B}}}_{\mathrm{out}}$.

Here, we present a proof of equation (26) in its general form. That is, we define two subspaces, ${ \mathcal X }$ and ${ \mathcal Y }$, with basis vectors ${{\boldsymbol{x}}}_{1},\ldots ,{{\boldsymbol{x}}}_{D}$ and ${{\boldsymbol{y}}}_{1},\ldots ,{{\boldsymbol{y}}}_{{D}^{\prime }}$, respectively. We show that, if ${\boldsymbol{x}}\in { \mathcal X }=\mathrm{span}\{{{\boldsymbol{x}}}_{1},\ldots ,{{\boldsymbol{x}}}_{D}\}$ and ${\boldsymbol{y}}\in { \mathcal Y }=\mathrm{span}\{{{\boldsymbol{y}}}_{1},\ldots ,{{\boldsymbol{y}}}_{{D}^{\prime }}\}$, the relationship

$$\begin{eqnarray*}&&{\boldsymbol{z}}={\boldsymbol{x}}+{\boldsymbol{y}}\in { \mathcal X }\cup { \mathcal Y }\end{eqnarray*}$$

holds. The proof is straightforward. Since ${\boldsymbol{x}}\in \mathrm{span}\{{{\boldsymbol{x}}}_{1},\ldots ,{{\boldsymbol{x}}}_{D}\}$ and ${\boldsymbol{y}}\in \mathrm{span}\{{{\boldsymbol{y}}}_{1},\ldots ,{{\boldsymbol{y}}}_{{D}^{\prime }}\}$, we can write

$$\begin{eqnarray}&&{\boldsymbol{z}}={\boldsymbol{x}}+{\boldsymbol{y}}=\displaystyle \sum _{j=1}^{D}{c}_{j}{{\boldsymbol{x}}}_{j}+\displaystyle \sum _{j=1}^{{D}^{\prime }}{d}_{j}{{\boldsymbol{y}}}_{j}.\end{eqnarray} \tag{ 49 }$$

Therefore, we get

$$\begin{eqnarray}{\boldsymbol{z}} & \in & \mathrm{span}\{{{\boldsymbol{x}}}_{1},\ldots ,{{\boldsymbol{x}}}_{D},{{\boldsymbol{y}}}_{1},\ldots ,{{\boldsymbol{y}}}_{{D}^{\prime }}\}\\ & = & \mathrm{span}\{{{\boldsymbol{x}}}_{1},\ldots ,{{\boldsymbol{x}}}_{D}\}\cup \mathrm{span}\{{{\boldsymbol{y}}}_{1},\ldots ,{{\boldsymbol{y}}}_{{D}^{\prime }}\}\\ & = & { \mathcal X }\cup { \mathcal Y }.\end{eqnarray} \tag{ 50 }$$