Advertisement

Dual Signal Subspace Projection (DSSP): A Powerful Algorithm for Interference Removal and Selective Detection of Deep Sources

  • Kensuke SekiharaEmail author
  • Srikantan S. Nagarajan
Living reference work entry

Abstract

MEG signals are often contaminated with interference that can be of considerable magnitude compared with the signals of interest. One such example is large artifacts from a brain stimulation device. Quite a few algorithms have been developed to deal with such interference, but they often rely on the availability of separate noise measurements. This chapter describes a novel algorithm that can remove overlapping interference without requiring such separate noise measurements. The algorithm is based on twofold definitions of the signal subspace in the spatial and time-domains. Since the algorithm makes use of this duality, it is named the dual signal subspace projection (DSSP). The algorithm consists of three steps: de-signaling, estimation of the time-domain interference subspace, and time-domain signal space projection (SSP). The first de-signaling step removes the signal of interest from the sensor data by applying the spatial-domain SSP algorithm. The second step estimates interference subspace in the time-domain by computing the intersection between the row spaces of the two modified data matrices obtained with and without de-signaling. The third step implements the time-domain SSP to remove interference from the data. The DSSP algorithm is extended for selective detection of a deep source by suppressing interference from superficial sources; the extended version is called the beamspace DSSP (bDSSP). To demonstrate the effectiveness of these algorithms, results of experiments in which the DSSP algorithm was applied to MEG data measured from patients with an implanted vagus nerve stimulation device are presented, as well as results of phantom experiments conducted to show the validity of the bDSSP algorithm. Comparison with the spatiotemporal signal space separation (tSSS) algorithm is also discussed.

1 Introduction

MEG signals are often contaminated with interference that can be of considerable magnitude compared with the signals of interest. One striking example of such cases is MEG recordings obtained from patients with epilepsy who have an implanted vagus nerve stimulation (VNS) device. In such recordings, artifacts from the stimulator and the lead wires can completely contaminate the recordings such that it is extremely difficult to see interictal epileptiform activity or stimulus evoked responses such as patient’s primary sensory responses. We will show such examples from our experiments in Sect. 6.

Although quite a few algorithms have been developed for removal of overlapping interference from MEG sensor data, these algorithms often rely on the availability of separate measurements that capture the statistical properties of the interference. Therefore, if such separate measurements are unavailable, as in the case of VNS artifacts, then the existing algorithms will not be effective for removing overlapped interferences.

This chapter describes a novel algorithm that can remove overlapping interference without requiring separate noise measurements. The algorithm is based on the two kinds of signal subspaces, namely, the spatial-domain signal subspace and the time-domain signal subspace. Since the algorithm makes use of this duality, it is named the dual signal subspace projection (DSSP) algorithm (Sekihara et al. 2016; Sekihara and Nagarajan 2017). This chapter provides a comprehensive review on the DSSP algorithm. we explain, in detail, how the DSSP algorithm estimates the time-domain interference subspace, of which basis vectors are used to implement the time-domain SSP for interference removal.

This chapter also describes an extension of the DSSP algorithm to selective detection of a deep source by suppressing interference from superficial sources. The extended version of the algorithm is called the beamspace DSSP (bDSSP). The algorithm is intended to overcome the well-recognized weakness of MEG in detecting deep brain activity. Thus, the proposed bDSSP algorithm can be a powerful tool in neuroscience studies of the physiological function of midbrain structures because many studies require accurate localization of physiological and pathophysiological activity in deep brain regions.

This chapter is organized as follows. The signal subspaces are introduced in Sect. 2, and details of the DSSP algorithm are provided in Sect. 3. We describe the bDSSP algorithm in Sect. 4. Results of computer simulations that validate the DSSP algorithm are presented in Sect. 5. Section 6.1 presents results of applying the DSSP algorithm to MEG data measured from patients with an implanted VNS device. Phantom experiments to test the validity of the bDSSP algorithm are presented in Sect. 6.2. Comparison with the spatiotemporal signal space separation (tSSS) algorithm is provided in Sect. 7. Results are summarized in Sect. 8.

2 Signal Subspaces in the Spatial and Time Domains

2.1 Sensor Array Measurements

Biomagnetic measurement is conducted using a sensor array, which simultaneously measures the signal with multiple sensors. Let us define the measurement of the mth sensor at time t as ym(t). The measurement from the whole sensor array is expressed as a column vector y(t): y(t) = [y1(t), y2(t), …, yM(t)]T. Here, M is the number of sensors, and the superscript T indicates the matrix transpose. Throughout this paper, plain italics indicate scalars, lower-case boldface italics indicate vectors, and upper-case boldface italics indicate matrices. The location in the three-dimensional space is represented by r: r = (x, y, z). The source magnitude at r and time t is denoted as a scalar s(r, t). The source vector is denoted s(r, t), and the source orientation is denoted η = [ηx, ηy, ηz]T. We thus have the relationship: s(r, t) = s(r, t)η.

Let us assume that a unit magnitude source exists at r. When this unit magnitude source is directed in the x, y, and z directions, the outputs of the mth sensor are respectively denoted by \(l^x_m ( \boldsymbol { r } )\), \(l^y_m ( \boldsymbol { r } )\), and \(l^z_m ( \boldsymbol { r } )\). Let us define an M × 3 matrix L(r) whose mth row is equal to \([l^x_m ( \boldsymbol { r } ), l^y_m ( \boldsymbol { r } ),l^z_m ( \boldsymbol { r } )]\). This matrix L(r), referred to as the lead field matrix, represents the sensitivity of the sensor array at r. When the unit magnitude source at r is oriented in the η direction, the outputs of the sensor array are expressed as l(r) = L(r)η. This column vector l(r), referred to as the lead field vector, represents the sensitivity of the sensor array in the direction of η at the location r.

The outputs of the sensor array y(t) are expressed as the sum of the signal component yS(t) and the noise ε:
$$\displaystyle \begin{aligned} \boldsymbol{y} (t)= \boldsymbol{y }_S(t)+ \boldsymbol{\varepsilon}. \end{aligned} $$
(1)
In Eq. (1), yS(t) is called the signal vector, which is expressed as
$$\displaystyle \begin{aligned} \boldsymbol{y }_S (t) =\int_{\Omega} \boldsymbol{L} ( \boldsymbol{ r } ) \boldsymbol{s } ( \boldsymbol{ r } ,t) \, d \boldsymbol{ r } , \end{aligned} $$
(2)
where the integral on the right-hand side is carried out over a three-dimensional volume Ω where signal sources of interest exist. This Ω is called the source space. In Eq. (1), an M × 1 random vector ε represents additive sensor noise, which is assumed to obey the normal distribution:
$$\displaystyle \begin{aligned} p(\boldsymbol{ \varepsilon} )= \mathcal{N} (\boldsymbol{ \varepsilon} | \boldsymbol{ 0 } ,\varrho^2 \boldsymbol{ I } ), \end{aligned} $$
(3)
where I is the identity matrix and ϱ2 is the variance of the sensor noise.
We denote the time series outputs of a sensor array y(t1), …, y(tK), where K is the total number of measured time points. It is assumed that K > M in this paper. We define the measured data matrix B as
$$\displaystyle \begin{aligned} \boldsymbol{B} =[ \boldsymbol{y }(t_1),\ldots, \boldsymbol{y }(t_K)]=[ \boldsymbol{y }_1,\ldots, \boldsymbol{y }_K], \end{aligned} $$
(4)
where y(tj) is denoted yj for simplicity. We also define a matrix of the signal vector such that
$$\displaystyle \begin{aligned} \boldsymbol{B}_S =[ \boldsymbol{y }_S(t_1),\ldots, \boldsymbol{y }_S(t_K)]=[ \boldsymbol{y }^S_1,\ldots, \boldsymbol{y }^S_K], \end{aligned} $$
(5)
where the jth column of BS is denoted \( \boldsymbol {y }^S_j\). This BS is called the signal matrix in this paper. Then, the data model in Eq. (1) is expressed in a matrix form as
$$\displaystyle \begin{aligned} \boldsymbol{B} = \boldsymbol{B}_S + \boldsymbol{B}_{\boldsymbol{ \varepsilon}} , \end{aligned} $$
(6)
where Bε is the noise matrix whose jth column is equal to the noise vector ε at time tj.

2.2 Signal Subspace in the Spatial Domain

Let us assume that a total of Q discrete sources exist. Their locations are denoted by r1, …, rQ, their orientations by η1, …, ηQ, and their magnitudes by s1(t), …, sQ(t). Then, the source distribution is expressed as
$$\displaystyle \begin{aligned} \boldsymbol{s } (\boldsymbol{ r }, t) = \sum_{q=1}^Q s_q(t) \boldsymbol{\eta} _q \delta( \boldsymbol{ r } - \boldsymbol{ r } _q), \end{aligned} $$
(7)
where δ(r) indicates the delta function. Substituting the equation above into Eq. (2), the signal vector yS(t) is expressed as
$$\displaystyle \begin{aligned} \boldsymbol{y }_S (t) = \int_{\Omega} \boldsymbol{L} ( \boldsymbol{ r } ) \sum_{q=1}^Q s_q(t) \boldsymbol{\eta} _q \delta( \boldsymbol{ r } - \boldsymbol{ r } _q) \, d \boldsymbol{ r } = \sum_{q=1}^Q s_q(t) \boldsymbol{l}_q , \end{aligned} $$
(8)
where lq represents the lead field vector of the qth source obtained such that lq = L(rq)ηq. We assume that the number of sources Q is smaller than the number of sensors, i.e., Q < M. This assumption is referred to as the low-rank signal assumption (Paulraj et al. 1993; Sekihara et al. 2000; Sekihara and Nagarajan 2008), and we hold this assumption throughout the paper. (Since we assume that K > M, the assumption K > M > Q holds throughout the paper.)
Equation (8) claims that the signal vector yS is expressed as a linear combination of the lead field vectors l1, ⋯ , lQ. That is, the signal vector yS lies within a subspace spanned by l1, ⋯ , lQ. The subspace spanned by the source lead field vectors l1, ⋯ , lQ is defined as the signal subspace (Sekihara et al. 2000), which is denoted by \( \mathcal {E}_S \), i.e.,
$$\displaystyle \begin{aligned} \mathcal{E}_S = \mathop{\mathrm{csp}} ( \, [ \boldsymbol{l} _1,\cdots, \boldsymbol{l} _Q] \, ). \end{aligned} $$
(9)
Here, the notation \( \mathop {\mathrm {csp}} (\cdot )\) indicates the column space of a matrix within the parentheses. Equation (8) indicates the relationship,
$$\displaystyle \begin{aligned} \boldsymbol{y }_S (t) \in \mathcal{E}_S . \end{aligned} $$
(10)
The signal vector lies within the signal subspace, which is the subspace formed by all possible signal vectors (Paulraj et al. 1993).

2.3 Signal Subspace in the Time Domain

We then define the signal subspace in the time-domain. To do so, we define a row vector sq consisting of the time course of the qth source such that
$$\displaystyle \begin{aligned} \boldsymbol{s }_q=[s_q(t_1),\ldots,s_q(t_K)], \end{aligned} $$
(11)
which we call the time course vector of the qth source. We then prove that a row of the signal matrix BS is expressed as a linear combination of the time course vectors, s1, …, sQ. We assume, in this paper, that the source time course vectors sq (q = 1, …, Q) are linearly independent. Substituting Eq. (8) into Eq. (5), the following relationship is obtained:
$$\displaystyle \begin{aligned} \boldsymbol{B}_S &=\left[ \sum_{q=1}^Q s_q (t_1) \boldsymbol{l} _q ,\ldots, \sum_{q=1}^Q s_q (t_K) \boldsymbol{l} _q \right] = \left[ \begin{array}{c} \sum_{q=1}^Q [ s_q (t_1),\ldots,s_q (t_K) ] l_q^1\\ \vdots \\ \sum_{q=1}^Q [ s_q (t_1),\ldots,s_q (t_K) ] l_q^M \end{array} \right]\\ &= \left[ \begin{array}{c} \sum_{q=1}^Q l_q^1 \boldsymbol{s }_q \\ \vdots \\ \sum_{q=1}^Q l_q^M \boldsymbol{s }_q \end{array} \right] , \end{aligned} $$
(12)
where \(l_q^1,\ldots ,l_q^M\) are the elements of the lead field vector lq: \( \boldsymbol {l} _q=[l_q^1,\ldots ,l_q^M ]^T\). Denoting the jth row vector of BS by \( \boldsymbol {\beta } _j^S\), Eq. (12) shows that
$$\displaystyle \begin{aligned} \boldsymbol{\beta} _j^S=\sum_{q=1}^Q l_q^j \boldsymbol{s }_q. \end{aligned} $$
(13)
This equation indicates that a row vector of the signal matrix, \( \boldsymbol {\beta } _j^S\), is expressed as a linear combination of sq (q = 1, …, Q). That is, we have
$$\displaystyle \begin{aligned} \boldsymbol{\beta} _j^S \in \mathop{\mathrm{rsp}} ( \, [ \boldsymbol{s } ^{T}_1,\ldots,\boldsymbol{s } ^{T}_Q ]^T \, ), \end{aligned} $$
(14)
where the notation \( \mathop {\mathrm {rsp}} (\cdot )\) indicates a row space of a matrix in the parentheses.
Analogous to Eqs. (9) and (10), it is reasonable to define \( \mathop {\mathrm {rsp}} ( \, [ \boldsymbol {s } ^{T}_1,\ldots ,\boldsymbol {s } ^{T}_Q ]^T \, )\) as the signal subspace in the time-domain, which is denoted \( \mathcal {K}_S\), i.e.,
$$\displaystyle \begin{aligned} \mathcal{K}_S= \mathop{\mathrm{rsp}} ( \, [ \boldsymbol{s } ^{T}_1,\ldots,\boldsymbol{s } ^{T}_Q ]^T). \end{aligned} $$
(15)
By defining the time-domain signal subspace this way, we can derive symmetric relationships between the time-domain signal subspace and the spatial-domain signal subspace. That is, we have already shown the relationships:
$$\displaystyle \begin{aligned} & \mbox{column of }\boldsymbol{B}_S: \, \boldsymbol{y }^S_j \in \mathcal{E}_S , \end{aligned} $$
(16)
$$\displaystyle \begin{aligned} & \mbox{row of }\boldsymbol{B}_S: \, \boldsymbol{\beta} _j^S \in \mathcal{K}_S. \end{aligned} $$
(17)
We can show that, with the assumption K > Q, the column space of BS is equal to the spatial-domain signal subspace, i.e.,
$$\displaystyle \begin{aligned} \mathcal{E}_S = \mathop{\mathrm{csp}} ( \boldsymbol{B}_S ). \end{aligned} $$
(18)
With the assumption M > Q, the row space of BS is equal to the time-domain signal subspace, i.e.,
$$\displaystyle \begin{aligned} \mathcal{K}_S= \mathop{\mathrm{rsp}} ( \boldsymbol{B}_S ). \end{aligned} $$
(19)
The proofs of Eqs. (18) and (19) are presented in Sekihara and Nagarajan (2017).

2.4 Interference Removal Using Signal Subspace Projection (SSP)

2.4.1 Spatial-Domain SSP

The SSP is an algorithm intended to remove the interference overlapped onto the signal magnetic field (Uusitalo and Ilmoniemi 1997; Nolte and Curio 1999). (Although the signal subspace projection is customarily called the signal space projection, we keep the term signal subspace in accordance with the correct terminology in linear algebra.) The algorithm is based on the theory of signal subspace. The measurement model is
$$\displaystyle \begin{aligned} \boldsymbol{y } (t) = \boldsymbol{y }_S(t)+ \boldsymbol{y }_I(t) +\boldsymbol{ \varepsilon} , \end{aligned} $$
(20)
where yI(t) represents the interference overlapped on the signal vector yS(t). We define the interference matrix BI as
$$\displaystyle \begin{aligned} \boldsymbol{B}_I =[ \boldsymbol{y }_I (t_1),\ldots, \boldsymbol{y }_I (t_K)]. \end{aligned} $$
(21)
Then, the data model in Eq. (20) is expressed as
$$\displaystyle \begin{aligned} \boldsymbol{B} = \boldsymbol{B}_S + \boldsymbol{B}_I + \boldsymbol{B}_{\boldsymbol{ \varepsilon}} . \end{aligned} $$
(22)
Using Eq. (18), the (spatial-domain) interference subspace is estimated as the column space of the interference matrix, such that,
$$\displaystyle \begin{aligned} \mathcal{E}_I = \mathop{\mathrm{csp}} ( \boldsymbol{B}_I ). \end{aligned} $$
(23)
Once basis vectors of \( \mathcal {E}_I \) can be obtained, the projector onto the interference subspace, PI, can be formulated. (These basis vectors can usually be estimated by applying the singular-value decomposition to the data matrix of interference-only data.) We then perform the interference removal by projecting the data matrix onto the subspace orthogonal to the interference subspace \( \mathcal {E}_I \), i.e., the estimated signal matrix \( \widehat {\boldsymbol {B}}_S \) is given by
$$\displaystyle \begin{aligned} \boldsymbol{B}_S =( \boldsymbol{ I } - \boldsymbol{P}_I ) \boldsymbol{B} = ( \boldsymbol{ I } - \boldsymbol{P}_I ) \boldsymbol{B}_S + ( \boldsymbol{ I } - \boldsymbol{P}_I ) \boldsymbol{B}_{\boldsymbol{ \varepsilon}} = \boldsymbol{B}_S - \boldsymbol{P}_I \boldsymbol{B}_S + ( \boldsymbol{ I } - \boldsymbol{P}_I ) \boldsymbol{B}_{\boldsymbol{ \varepsilon}} . \end{aligned} $$
(24)
The relationship PIBI = BI is used here. The method of interference removal based on Eq. (24) is called SSP. The influence of SSP on the signal component is evaluated by PIBS, which is the second term on the right-hand side of Eq. (24). This term is small when orthogonality of lead field vectors between signal and interference sources is high. However, if this orthogonality is low, the second term becomes large, and the SSP algorithm causes signal distortion. The SSP algorithm can also be implemented in the time-domain, as discussed below.

2.4.2 Time-Domain SSP

Using Eq. (19), the time-domain interference subspace is estimated as the row space of the interference matrix, such that
$$\displaystyle \begin{aligned} \mathcal{K}_I= \mathop{\mathrm{rsp}} ( \boldsymbol{B}_I ). \end{aligned} $$
(25)
Once basis vectors of \( \mathcal {K}_I\) can be obtained, the projector onto the interference subspace, ΠI, can be formulated. We then perform the interference removal by projecting the data matrix onto the subspace orthogonal to the time-domain interference subspace \( \mathcal {K}_I\), i.e., the estimated signal matrix \( \widehat {\boldsymbol {B}}_S \) is given by
$$\displaystyle \begin{aligned} \widehat{\boldsymbol{B}}_S = \boldsymbol{B} ( \boldsymbol{ I } - \boldsymbol{\varPi}_I )= ( \boldsymbol{B}_S + \boldsymbol{B}_I + \boldsymbol{B}_{\boldsymbol{\varepsilon}} ) ( \boldsymbol{ I } - \boldsymbol{\varPi}_I )= \boldsymbol{B}_S - \boldsymbol{B}_S \boldsymbol{\varPi}_I + \boldsymbol{B}_{\boldsymbol{\varepsilon}} ( \boldsymbol{ I } - \boldsymbol{\varPi}_I ), \end{aligned} $$
(26)
where the relationship BIΠI = BI is used. The method of removing the interference BI based on Eq. (26) is referred to as the time-domain signal subspace projection (time-domain SSP). The influence of the time domain SSP on the signal component is assessed by the second term BSΠI on the right-hand side of Eq. (26). This term becomes small when the correlations between the time courses of the signal and interference sources are small. This can be considered an advantage of the time-domain SSP over the spatial-domain SSP. This is because in many real-life applications, the time courses of the signal and interference sources are expected to differ significantly. However, in the spatial-domain, the orthogonality of lead field vectors between signal and interference sources may not be so high.

3 Dual Signal Subspace Projection Algorithm

3.1 Structure of the Algorithm

The structure of the DSSP algorithm is shown in Fig. 1. As shown here, the algorithm consists of three steps: de-signaling, estimation of time-domain interference subspace, and interference removal by time-domain SSP. The DSSP algorithm assumes the data model in Eq. (22). The first step projects out the signal magnetic field BS from the sensor data B by applying the spatial-domain SSP algorithm, that is, by projecting the data matrix B onto the subspace orthogonal to the signal subspace, namely, the noise subspace. This procedure is referred to as de-signaling in this paper. The projector used for de-signaling is called the de-signaling projector.
Fig. 1

Conceptual sketch of the DSSP and related algorithms

Theoretically, an ideal de-signaling projector is the noise subspace projector. However, since the signal and noise subspaces and the projectors onto them are unknown, we must use something that can substitute for these projectors. The DSSP algorithm uses the pseudo-signal subspace projector described in Sect. I.1. The well-known tSSS algorithm also shares the structure in Fig. 1. However, it uses a different de-signaling projector, which is the projector onto the SSS internal subspace (Taulu and Simola 2006; Sekihara and Nagarajan 2017). The DSSP and tSSS algorithms differ in only their manner of approximating the signal subspace projector, but this one difference causes a considerable difference in their performances, which will be discussed in Sect. 7.

In the DSSP algorithm, the second step estimates the time-domain interference subspace by computing the intersection between the row spaces of the two kinds of modified data matrices obtained with and without de-signaling. The third step implements the time-domain SSP by using the projector onto the subspace orthogonal to the interference subspace. These steps are explained in the subsection below.

3.2 The Interference Subspace Estimation and Interference Removal

The DSSP algorithm uses the projector onto the pseudo-signal subspace described in Sect. I.1 as the de-signaling projector, i.e., the algorithm applies P̆S (obtained in Eq. (47)) and I −P̆S to the data matrix B to create two kinds of data matrices:
$$\displaystyle \begin{aligned} & \breve{\boldsymbol{P}}_S \boldsymbol{B} = \boldsymbol{B}_S + \breve{\boldsymbol{P}}_S \boldsymbol{B}_I + \breve{\boldsymbol{P}}_S \boldsymbol{B}_{\boldsymbol{\varepsilon}} , \end{aligned} $$
(27)
$$\displaystyle \begin{aligned} &( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B} =( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B}_I + ( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B}_{\boldsymbol{\varepsilon}} . {} \end{aligned} $$
(28)
The projector I −P̆S projects out the signal matrix BS because P̆SBS = BS holds. According to Sekihara and Nagarajan (2017), the following relationships hold:
$$\displaystyle \begin{aligned} & \mathop{\mathrm{rsp}} ( \breve{\boldsymbol{P}}_S \boldsymbol{B} ) \subset \mathop{\mathrm{rsp}} ( \boldsymbol{B}_S ) + \mathop{\mathrm{rsp}} ( \breve{\boldsymbol{P}}_S \boldsymbol{B}_I ) + \mathop{\mathrm{rsp}} ( \breve{\boldsymbol{P}}_S \boldsymbol{B}_{\boldsymbol{\varepsilon}} ), \end{aligned} $$
(29)
$$\displaystyle \begin{aligned} & \mathop{\mathrm{rsp}} ( ( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B} ) \subset \mathop{\mathrm{rsp}} ( ( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B}_I ) + \mathop{\mathrm{rsp}} (( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B}_{\boldsymbol{\varepsilon}} ). {} \end{aligned} $$
(30)
Since \( \mathop {\mathrm {rsp}} ( \breve {\boldsymbol {P}}_S \boldsymbol {B}_I )= \mathcal {K}_I\) and \( \mathop {\mathrm {rsp}} ( ( \boldsymbol { I } - \breve {\boldsymbol {P}}_S ) \boldsymbol {B}_I = \mathcal {K}_I\), hold, Eqs. (29) and (30) lead to
$$\displaystyle \begin{aligned} & \mathop{\mathrm{rsp}} (\breve{\boldsymbol{P}}_S \boldsymbol{B} ) \subset \mathcal{K}_S + \mathcal{K}_I + \breve{\mathcal{K}}_{\varepsilon}, \end{aligned} $$
(31)
$$\displaystyle \begin{aligned} & \mathop{\mathrm{rsp}} ((\boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B} ) \subset \mathcal{K}_I + \breve{\mathcal{K}}_{\varepsilon}^{\prime} , {} \end{aligned} $$
(32)
where \( \mathcal {K}_S\) and \( \mathcal {K}_I\) respectively indicate the time-domain signal and interference subspaces. Here, we use the notations \( \mathop {\mathrm {rsp}} ( \breve {\boldsymbol {P}}_S \boldsymbol {B}_{\boldsymbol {\varepsilon }} )= \breve {\mathcal {K}}_\varepsilon \) and \( \mathop {\mathrm {rsp}} (( \boldsymbol { I } - \breve {\boldsymbol {P}}_S ) \boldsymbol {B}_{\boldsymbol {\varepsilon }} )= \breve {\mathcal {K}}_{\varepsilon }^{\prime }\).
The DSSP algorithm estimates the interference subspace \( \mathcal {K}_I\) by computing the intersection between \( \mathop {\mathrm {rsp}} ( \breve {\boldsymbol {P}}_S \boldsymbol {B} )\) and \( \mathop {\mathrm {rsp}} ( ( \boldsymbol { I } - \breve {\boldsymbol {P}}_S ) \boldsymbol {B} )\). Using Eqs. (31) and (32), we can finally derive the relationship (Sekihara and Nagarajan 2017):
$$\displaystyle \begin{aligned} \mathcal{K}_I \supset \mathop{\mathrm{rsp}} ( \breve{\boldsymbol{P}}_S \boldsymbol{B} ) \cap \mathop{\mathrm{rsp}} ( ( \boldsymbol{ I } - \breve{\boldsymbol{P}}_S ) \boldsymbol{B} ). \end{aligned} $$
(33)
The equation above shows that the intersection between \( \mathop {\mathrm {rsp}} ( \breve {\boldsymbol {P}}_S \boldsymbol {B} )\) and \( \mathop {\mathrm {rsp}} ( ( \boldsymbol { I } - \breve {\boldsymbol {P}}_S ) \boldsymbol {B} )\) forms a subset of the interference subspace \( \mathcal {K}_I\). The basis vectors of the intersection can be derived using the algorithm described in Sect. I.3 in the Appendix. Once the orthonormal basis vectors of the intersection ψ1, …, ψr are obtained, we can compute the projector onto the intersection Π such that
$$\displaystyle \begin{aligned} \boldsymbol{\varPi}=[ \boldsymbol{\psi} _1,\ldots, \boldsymbol{\psi} _r] [ \boldsymbol{\psi} _1,\ldots, \boldsymbol{\psi} _r] ^T. \end{aligned} $$
(34)
Using this Π as the projector onto the interference subspace \( \mathcal {K}_I\), the interference removal is achieved and the signal matrix is estimated by applying the time-domain SSP:
$$\displaystyle \begin{aligned} \widehat{\boldsymbol{B}}_S = \boldsymbol{B} ( \boldsymbol{ I } - \boldsymbol{\varPi}) = \boldsymbol{B} ( \boldsymbol{ I } -[ \boldsymbol{\psi} _1,\ldots, \boldsymbol{\psi} _r] [ \boldsymbol{\psi} _1,\ldots, \boldsymbol{\psi} _r] ^T). \end{aligned} $$
(35)
The method of removing the interference in a manner described above is called the DSSP algorithm (Sekihara et al. 2016). Note that since the basis vectors of the intersection, ψ1, …, ψr, span only a subset of the interference subspace \( \mathcal {K}_I\), this method cannot perfectly remove interferences. However, when the intersection \( \mathop {\mathrm {rsp}} ( \breve {\boldsymbol {P}}_S \boldsymbol {B} ) \cap \mathop {\mathrm {rsp}} ( ( \boldsymbol { I } - \breve {\boldsymbol {P}}_S ) \boldsymbol {B} )\) is a reasonable approximation of \( \mathcal {K}_I\), interferences can effectively be removed by the DSSP algorithm.

4 Beamspace Dual Signal Subspace Projection (bDSSP) Algorithm

Use of the beamspace basis vectors for de-signaling leads to a novel algorithm that can selectively detect signals from a deep source by suppressing interference from superficial sources (Sekihara et al. 2018). Basics of beamspace processing is presented in Sect. I.2 in the Appendix. The data model for the bDSSP algorithm is expressed as
$$\displaystyle \begin{aligned} \boldsymbol{B} = \boldsymbol{B}_{{\mathrm{deep}}} + \boldsymbol{B}_{{\mathrm{sup}}} + \boldsymbol{B}_{\boldsymbol{\varepsilon}} , \end{aligned} $$
(36)
where Bdeep indicates the signal magnetic field generated from a deep source and Bsup the signal magnetic field from superficial sources. (We assume that the target deep source is a single source for simplicity.) This algorithm requires that a user sets a predetermined region of interest (ROI) so that it covers the target deep source location. In other words, a prerequisite of this algorithm is that an approximate location of the deep source be known. (This prerequisite should not be a strong limitation to the algorithm application, because a hypothesis about the target deep source usually exists when a brain deep region is investigated.) The algorithm then computes the beamspace basis vectors u1, …, uP by setting the local source space as a small region just covering the ROI. The basis vectors u1, …, uP are derived in a manner described in Sect. I.2.
The beamspace projector Open image in new window is then derived as By multiplying Open image in new window and Open image in new window with the data matrix B, we obtain Here, we use Open image in new window. We can then derive (Sekihara et al. 2018) where \( \mathcal {K}_{{\mathrm {deep}}} \) and \( \mathcal {K}_{{\mathrm {sup}}} \) are the time-domain signal subspaces of the deep and superficial sources, respectively. We also use the notations Open image in new window and Open image in new window. Using Eqs. (40) and (41), We can finally derive The equation above indicates that the intersection between the row spaces of Open image in new window and Open image in new window forms a subset of \( \mathcal {K}_{{\mathrm {sup}}} \).
The orthonormal basis set of the intersection, Open image in new window, and the projector onto this intersection, Πb, can be obtained using the procedure described in Sect. I.3. The signal from the deep source is then estimated by applying the time-domain SSP, such that
$$\displaystyle \begin{aligned} \widehat{\boldsymbol{B}}_{{\mathrm{deep}}} = \boldsymbol{B} ( \boldsymbol{ I } - \boldsymbol{\varPi}_{b}). \end{aligned} $$
(43)

5 Computer Simulation Validating the DSSP Algorithm

To show the validity of the DSSP algorithm, we present results of a computer simulation of MEG measurements (Sekihara et al. 2016). The sensor alignment of the 275-channel whole-head sensor array from the Omega™ neuromagnetometer (VMS Medtech, Coquitlam, Canada) was used. The coordinate system and source-sensor configuration used in the computer simulation are depicted in Fig. 2a. We assumed a single source at a location shown by a square in this figure; its location was assumed to be in the right parietal area near the primary somatosensory cortex. We also put a single interference source just outside the sensor array (50 cm from the center of the array), although it is not shown. This source generated simulated interference caused from some types of brain stimulators.
Fig. 2

(a) The coordinate system, sensor layout, and the location of the signal source used in the computer simulation in Sect. 5. The square indicates the location of the signal source, and small dots indicate locations of sensors. (b) Computer-generated sensor time courses. The time courses of the signal magnetic field (plus sensor noise) are shown in the top panel, and the time courses of the interference magnetic field are shown in the bottom panel. The ordinate indicates the normalized relative intensity, and the abscissa indicates the time points

The signal source is assumed to have an exponentially dumped sinusoid time course, and the interference source is assumed to have a low-pass-filtered random time course. To generate the magnetic fields, signal-source activity was projected onto the sensor time courses through the lead field, which is obtained using the homogeneous spherical head model (Sarvas 1987). Spatiotemporal data with 2400 time points were generated. In Fig. 2b, the time courses of the signal magnetic field (plus sensor noise) are shown in the top panel. The time courses of the interference magnetic field are shown in the bottom panel.

The data matrix B was generated by adding the interference matrix BI onto the signal matrix BS with the interference-to-signal ratio (ISR), defined as ∥BI∥∕∥BS∥, equal to 10. The resultant sensor time courses are shown in the upper panel of Fig. 3a. Since the interference magnetic field is 10 times stronger than the signal magnetic field, these sensor time courses are dominated by the interference magnetic field. We set the source space to a region that covers the whole brain, and the augmented lead field matrix was computed. We then applied the DSSP algorithm, and resultant interference-removed sensor time courses are shown in the bottom panel in Fig. 3a, which shows that the interference is nearly completely removed.
Fig. 3

(a) Top panel: Interference-overlapped sensor time courses in which the interference-to-signal ratio (ISR) is equal to 10. Bottom panel: Sensor time courses of the DSSP interference removal results. The ordinate indicates the normalized relative intensity, and the abscissa indicates the time points. (b) Maps of the magnetic field at t = 1200 overlaid onto a deformed sensor layout. Here, small dots indicate locations of sensors. The top left, top right, and bottom left panels respectively show the map of the original signal magnetic field, the map of the interference-overlapped sensor outputs, and the map of the DSSP interference-removal results

The maps of the magnetic field at t = 1200 are shown in Fig. 3b. The top left panel shows the map of the original signal magnetic field, which serves as the ground truth. The top right panel shows the map of the interference-overlapped sensor data, and the bottom left panel shows the map of the DSSP interference-removed results. We can see that the map of the DSSP results is almost the same as that of the original signal magnetic field, demonstrating that the DSSP algorithm can remove the interference without introducing signal distortion. Here, the signal distortion indicates the distortion in the spatial-domain. In this case, the signal distortion in the time-domain is very small because the time course of signal source is very different from that of interference source.

6 Experiments

6.1 Experiments Using MEG Data from Patients with an Implanted VNS Device

6.1.1 Somatosensory Data

We present here results of applying the DSSP algorithm to MEG data measured from patients with a VNS device (Sekihara et al. 2016). Such MEG data contain huge interference generated from the stimulator located near the patient’s chest area. The measurements were conducted using the 275-channel whole-head sensor array of the Omega™ neuromagnetometer. In Fig. 4a, the somatosensory MEG sensor data measured with tactile stimulation applied to the patient’s left index finger are shown. Since tactile stimulation was used, a large peak should be observed around a latency of 50 ms. However, such a peak is not observed due to the presence of interference from the VNS device. The sensor time course with the interference removed by the DSSP algorithm is shown in Fig. 4b. Here, a peak around the latency of 50 ms is clearly observed.
Fig. 4

Results of interference-removal experiments using somatosensory MEG obtained from a patient with a VNS device. (a) Original sensor time courses when the tactile stimulus was applied to the patient’s left index finger. (b) Sensor time courses obtained by applying the DSSP algorithm to the sensor data in (a). (c) Results of source localization obtained using the DSSP-processed sensor time courses in (b). The relative voxel intensity is color-coded according to the color bar and overlaid onto the patient’s MRI. The crosshair indicates the point with the maximum reconstruction intensity, and three cross-sectional MR images at this point are shown

The source localization results are shown in Fig. 4c. We applied a sparse Bayes (Champagne) algorithm (Wipf et al. 2010; Sekihara and Nagarajan 2015) to the interference-removed data in (b). Here, the source activity is localized near the primary somatosensory area in the contralateral, right hemisphere. These reconstruction results show that the DSSP algorithm reduced the influence of the VNS device and enables mapping of the primary somatosensory cortex. It should be noted that, without interference removal, a source was localized outside the subject’s skull, although these results are not shown here.

6.1.2 Epilepsy Data

The DSSP algorithm was next applied to epilepsy data measured from patients with an implanted VNS device. These MEG data were recorded in a frequency band of 0–75 Hz. Two cases are shown in Figs. 5 and 6. In both of these figures, original sensor time courses of selected MEG channels are shown in Figs. 5a and 6a where VNS artifacts with partial periodicity, low frequency, and high amplitude can be seen. Results of removing the artifacts are shown in Figs. 5b and 6b. Here, we can see that this periodic feature existing in the original sensor data was greatly diminished. Consequently, some spikes that are not discernible in the original MEG sensor time courses become visible. In this study, to evaluate the algorithm performance, we checked whether spikes can be identified at the same locations where spikes were identified in simultaneous EEG recordings. In these figures, the red and green vertical lines indicate the locations of spikes identified in the EEG recordings. Note that here spikes were visually identified by experts – a certified EEG technologist and a clinical neurophysiologist – and the results were confirmed by a board-certified clinical neurophysiologist and epileptologist.
Fig. 5

Results of applying the DSSP algorithm to epilepsy data that were measured from a patient with a VNS device. (a) Original sensor time courses of selected MEG channels in which VNS artifacts are evident. (b) DSSP-processed sensor time courses. The red and green vertical lines indicate the locations of spikes identified in simultaneous EEG recordings. (c) Source reconstruction results obtained using the DSSP-processed sensor data in (b). (d) Time course of the voxel indicated by the crosshair in (c)

Fig. 6

Another set of results of applying the DSSP algorithm to epilepsy data that were measured from a patient with a VNS device. (a) Original sensor time courses of selected MEG channels in which VNS artifacts are evident. (b) DSSP-processed sensor time courses. The red vertical line indicates the location of spike identified in simultaneous EEG recordings. (c) Source reconstruction results obtained using the DSSP processed sensor data in (b). (d) Time course of the voxel indicated by the crosshair in (c)

The results of source localization using the sparse Bayesian algorithm are shown in Figs. 5c and 6c, and the voxel time course at the voxels indicated by the crosshairs in Figs. 5c and 6c are shown in Figs. 5d and 6d. In both cases, the sources were localized near plausible brain areas that are in agreement with these patients’ presumable epileptogenic zones suggested by other clinical tests. Also, in these results, smoothed spike-like voxel time courses were obtained. It should be noted that applying Champagne to the original MEG recordings in Figs. 5a and 6a resulted in localization failure in both cases, either no strong activation could be found or the activity was localized to obviously wrong locations (e.g., near or outside of the skull). The performance of the DSSP algorithm for VNS artifact removal has been evaluated via a retrospective cohort study of more than 40 patients with a VNS device. Details of this study will be published elsewhere (Cai et al. 2019).

6.2 Phantom Experiments Validating the bDSSP Algorithm

Experiments using an MEG phantom were performed to test the validity of the bDSSP algorithm. The phantom used in our experiments is shown in Fig. 7a. In this phantom, dipole sources consist of isosceles-triangular coils; these triangular coils generate magnetic fields expressed by the Sarvas formula (Sarvas 1987). Thus, the coils behave like dipole sources embedded in the spherical homogeneous conductor (Ilmoniemi et al. 1985; Oyama et al. 2015). A whole-head MEG system with a 160-channel sensor array (Uehara et al. 2003), installed at Applied Electronics Laboratory, Kanazawa Institute of Technology, Amaike, Kanazawa, Japan, was used to measure the phantom data. Figure 7b shows how the phantom was installed within the MEG sensor helmet.
Fig. 7

(a) Configuration of a dry phantom used in our experiments. Squares show the locations of dipole sources. Dipoles annotated by “Dipole pair” were used in the experiments. (b) Depiction showing how the disc-shaped phantom was installed inside the sensor helmet

The dipole sources shown with the annotation “Dipole pair” in Fig. 7a were used. These dipoles were 2-cm apart, and they were placed presumably near the parietal-lobe region. The superficial dipole was driven by an 11 Hz sinusoid with a current strength of 1.42 mA. The deep dipole was driven by an amplitude-modulated sinusoid in which the carrier frequency was 15 Hz and the modulation frequency was 1 Hz. The current strength to drive the deep dipole was 0.225 mA. The current values of the two dipoles were chosen in order for the magnetic field of the superficial dipole to have an intensity 16 times stronger than that of the deep dipole. Namely, the interference-to-signal ratio (ISR) was 16. The data were acquired for 2 s at a sampling frequency of 1 kHz.

The sensor time courses measured when only the superficial dipole was turned on are shown in Fig. 8a, and the time courses measured when only the deep dipole was turned on are shown in Fig. 8b. Adaptive beamformer source reconstruction was applied to these data sets. The image of the superficial dipole is shown in Fig. 8c, and the image of the deep dipole is shown in Fig. 8d. The sensor time courses (Fig. 8b) and the dipole location (Fig. 8d) of the deep source serve as the ground truth in the experiments described below.
Fig. 8

(a) Sensor time courses measured when only the superficial dipole was turned on. (b) Sensor time courses measured when only the deep dipole was turned on. (c) Source reconstruction results of the superficial dipole. (d) Source reconstruction results of the deep dipole. (e) Sensor time courses measured when the superficial and deep dipoles were simultaneously turned on. (f) Sensor time courses of the bDSSP results. (g) Source reconstruction results from the sensor data in (e). (h) Source reconstruction results from the bDSSP-processed sensor data in (f). These sensor time courses in (a), (b), (e), and (f) are normalized to each maximum field intensity, and the ordinate indicates the normalized values, and the abscissa indicates time (ms). In reconstruction results in (c), (d), (g), and (h), the left, middle, and right panels, respectively, show the axial, coronal, and sagittal projections of the three-dimensional reconstructed source distribution

The sensor time courses measured when the superficial and deep dipoles were simultaneously turned on are shown in Fig. 8e. In these sensor data, since the signal from the superficial dipole was 16 times stronger than the signal from the deep dipole, the sensor time courses were dominated by the signal from the superficial source. Results of source reconstruction from these sensor data are shown in Fig. 8g. Although the sensor data show only the dominant superficial dipole activity, the reconstruction results show both the superficial and deep dipoles. We then applied the bDSSP algorithm to detect the signal from the deep source. We set the local source space at a 1 cm-cubic region whose center was at the location of the deep dipole. The bDSSP algorithm was applied to the sensor data shown in Fig. 8e to extract the signal from the deep source. The resultant sensor time courses are shown in Fig. 8f, and source reconstruction results are shown in Fig. 8h. Comparison between these results and the ground truth in Fig. 8b, d demonstrates that the proposed bDSSP algorithm can extract the activity of a deep dipole from sensor data dominated by large interference from a superficial dipole.

7 Comparison with the tSSS Algorithm

As mentioned in Sect. 3.1, the DSSP and tSSS algorithms differ in their de-signaling projectors. The tSSS algorithm (Taulu and Simola 2006) uses the projector onto the SSS internal subspace, while the DSSP algorithm uses the projector onto the pseudo-signal subspace for removing the signal from the data. Since the tSSS algorithm uses the SSS basis vectors (Taulu and Kajola 2005), it cannot be applied to non-MEG applications in which an array of sensors are arranged on a flat (or nearly flat) plane. The DSSP algorithm has no such limitations. This is an obvious advantage of the DSSP over the tSSS algorithms. In fact, the DSSP algorithm has been used in the removal of stimulus-induced artifacts in functional spinal cord biomagnetic imaging (Sumiya et al. 2017), in which biomagnetic sensors are arranged on a nearly flat plane.

Even in MEG applications, the use of different projectors for de-signaling causes some performance differences between these two algorithms. Specifically, in the tSSS algorithm, the size and the location of the internal region depend on the choice of the origin for computing the SSS basis vectors. To show this, the tSSS algorithm was applied to the same computer-generated data used in Sect. 5. The results are shown in Fig. 9 where resultant sensor time courses are shown in (a) and the field maps at t = 1200 are shown in (b).
Fig. 9

Results of applying the tSSS algorithm to the same computer-generated data described in Sect. 5. (a) Sensor time courses of the tSSS results are shown. (b) Field maps at t = 1200 of the tSSS results are shown. The top panel in (a) and the left panel in (b) show the results of setting the origin z coordinate, zori, to 8.25 cm. The bottom panel in (a) and the right panel in (b) show the results of setting zori to 2.25 cm. (c) and (d) Internal and external components of the signal magnetic field used to derive the results in (a) and (b). Internal components are shown in the upper panels and the external components are shown in the lower panels. (c) zori was set to 8.25 cm. (d) zori was set to 2.25 cm

The top panel in Fig. 9a and the left panel in Fig. 9b show the results of setting the origin to 10 cm below the sensor located at the vertex of the helmet. This vertex sensor has the maximum z coordinate, which is z = 18.25 cm in the device coordinate used in the Omega™ neuromagnetometer. In this case, the z coordinate of the origin expressed in the device coordinate, zori, is equal to zori = 8.25 cm. The tSSS-processed sensor time courses are almost the same as the ground truth in Fig. 2b, and the tSSS-processed field map is nearly identical to the ground truth in Fig. 3b. These results indicate that tSSS algorithm effectively removed the interference in this case.

The bottom panel in Fig. 9a and the right panel in Fig. 9b show the results of setting the origin to 16 cm below the vertex sensor; that is, zori was set to 2.25 cm. Although the time courses in the bottom panel in Fig. 9a suggest that the interference was mostly removed, the comparison between the field map in the right panel in Fig. 9b and the ground truth (Fig. 3b) indicates that signal magnetic field was significantly distorted. This signal distortion can be explained if the signal source location was outside the internal region with the choice of zori = 2.25 cm, and the signal magnetic field has large external-subspace components.

This can be confirmed in Fig. 9c, d, in which the internal and external components of the signal magnetic field are shown. In this figure, the internal components are shown in the upper panels and the external components in the lower panels. Results with zori = 8.25 cm are shown in Fig. 9c, and those with zori = 2.25 cm are shown in Fig. 9d. In Fig. 9c, there are almost no external components, but in Fig. 9d, a significant amount of the external components exist with zori = 2.25 cm. Therefore, the estimated time-domain interference subspace included these external components, and the time-domain SSP removed these external components, resulting in the distortion of the signal magnetic field.

8 Conclusion

This chapter provides a detailed review of the DSSP algorithm proposed to remove large interference that overlaps with biomagnetic data. We first provide a review of the spatial-domain and time-domain signal subspaces and then provide a thorough explanation of the DSSP algorithm, including the details of estimating the time-domain interference subspace. This paper also describes an extension of the DSSP algorithm, called the bDSSO algorithm, which has been developed for selective detection of a deep source by suppressing interference signals from superficial sources. A comparison with the tSSS algorithm is also discussed.

References

  1. Cai C, Xu J, Velmurugan J, Knowlton R, Sekihara K, Nagarajan SS, Kirsch H (2019) Evaluation of a dual signal subspace projection algorithm in magnetoencephalographic recordings from patients with intractable epilepsy and vagus nerve stimulators. NeuroImage 188:161–170CrossRefGoogle Scholar
  2. Golub GH, Van Loan CF (2012) Matrix computations, vol 3. The Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  3. Ilmoniemi R, Hämäläinen M, Knuutila J (1985) The forward and inverse problems in the spherical model. In: Weinberg H, Stroink G, Katila T (eds) Biomagnetism: applications and theory. Pergamon Press, New YorkGoogle Scholar
  4. Ipsen IC (2009) Numerical matrix analysis: linear systems and least squares. SIAM, PhiladelphiaCrossRefGoogle Scholar
  5. Nolte G, Curio G (1999) The effect of artifact rejection by signal-space projection on source localization accuracy in MEG measurements. IEEE Trans Biomed Eng 46(4):400–408CrossRefGoogle Scholar
  6. Oyama D, Adachi Y, Yumoto M, Hashimoto I, Uehara G (2015) Dry phantom for magnetoencephalography: configuration, calibration, and contribution. J Neurosci Methods 251:24–36CrossRefGoogle Scholar
  7. Paulraj A, Ottersten B, Roy R, Swindlehurst A, Xu G, Kailath T (1993) Subspace methods for directions-of-arrival estimation. In: Bose NK, Rao CR (eds) Handbook of statistics. Elsevier Science Publishers, Netherlands, pp 693–739Google Scholar
  8. Rodríguez-Rivera A, Baryshnikov BV, Van Veen BD, Wakai RT (2006) MEG and EEG source localization in beamspace. IEEE Trans Biomed Eng 53(3):430–441CrossRefGoogle Scholar
  9. Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys Med Biol 32:11–22CrossRefGoogle Scholar
  10. Sekihara K, Nagarajan SS (2008) Adaptive spatial filters for electromagnetic brain imaging. Springer, Berlin/HeidelbergGoogle Scholar
  11. Sekihara K, Nagarajan SS (2015) Electromagnetic brain imaging: a Bayesian perspective. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  12. Sekihara K, Nagarajan SS (2017) Subspace-based interference removal methods for a multichannel biomagnetic sensor array. J Neural Eng 14(5):051001CrossRefGoogle Scholar
  13. Sekihara K, Poeppel D, Marantz A, Miyashita Y (2000) Neuromagnetic inverse modeling: application of eigenstructure-based approaches to extracting cortical activities from MEG data. In: Image, language, brain: papers from the first mind articulation project symposium. MIT Press, p 197Google Scholar
  14. Sekihara K, Kawabata Y, Ushio S, Sumiya S, Kawabata S, Adachi Y, Nagarajan SS (2016) Dual signal subspace projection (DSSP): a novel algorithm for removing large interference in biomagnetic measurements. J Neural Eng 13(3):036007CrossRefGoogle Scholar
  15. Sekihara K, Adachi Y, Kubota HK, Cai C, Nagarajan SS (2018) Beamspace dual signal space projection (bDSSP): a method for selective detection of deep sources in MEG measurements. J Neural Eng 15(3):036026CrossRefGoogle Scholar
  16. Sumiya S, Kawabata S, Hoshino Y, Adachi Y, Sekihara K, Tomizawa S, Tomori M, Ishii S, Sakaki K, Ukegawa D et al (2017) Magnetospinography visualizes electrophysiological activity in the cervical spinal cord. Sci Rep 7(1):2192CrossRefGoogle Scholar
  17. Taulu S, Kajola M (2005) Presentation of electromagnetic multichannel data: the signal space separation method. J Appl Phys 97(12):124905CrossRefGoogle Scholar
  18. Taulu S, Simola J (2006) Spatiotemporal signal space separation method for rejecting nearby interference in MEG measurements. Phys Med Biol 51:1759–1768CrossRefGoogle Scholar
  19. Uehara G, Adachi Y, Kawai J, Shimogawara M, Higuchi M, Haruta Y, Ogata H, Hisashi K (2003) Multi-channel SQUID systems for biomagnetic measurement. IEICE Trans Electron 86(1):43–54Google Scholar
  20. Uusitalo M, Ilmoniemi R (1997) Signal-space projection method for separating MEG or EEG into components. Med Biol Eng Comput 35(2):135–140CrossRefGoogle Scholar
  21. Wipf DP, Owen JP, Attias HT, Sekihara K, Nagarajan SS (2010) Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG. NeuroImage 49:641–655CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Signal Analysis Inc.HachiojiJapan
  2. 2.Department of Advanced Technology in MedicineTokyo Medical and Dental UniversityBunkyo-kuJapan
  3. 3.Biomagnetic Imaging Laboratory, Department of Radiology and Biomedical ImagingUniversity of CaliforniaSan FranciscoUSA

Section editors and affiliations

  • Selma Supek
    • 1
  • Cheryl J. Aine
    • 2
    • 3
  1. 1.Faculty of Science, Department of PhysicsUniversity of ZagrebZagrebCroatia
  2. 2.The Mind Research Network ,AlbuquerqueUSA
  3. 3.School of Medicine Department of RadiologyUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations