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

## 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 *m*th sensor at time *t* as *y*_{m}(*t*). The measurement from the whole sensor array is expressed as a column vector * y*(

*t*):

*(*

**y***t*) = [

*y*

_{1}(

*t*),

*y*

_{2}(

*t*), …,

*y*

_{M}(

*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

*and time*

**r***t*is denoted as a scalar

*s*(

*,*

**r***t*). The source vector is denoted

*(*

**s***,*

**r***t*), and the source orientation is denoted

*= [*

**η***η*

_{x},

*η*

_{y},

*η*

_{z}]

^{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

*m*th 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***) whose*

**r***m*th row is equal to \([l^x_m ( \boldsymbol { r } ), l^y_m ( \boldsymbol { r } ),l^z_m ( \boldsymbol { r } )]\). This matrix

*(*

**L***), 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*

**r***direction, the outputs of the sensor array are expressed as*

**η***(*

**l***) =*

**r***(*

**L***)*

**r***. This column vector*

**η***(*

**l***), referred to as the lead field vector, represents the sensitivity of the sensor array in the direction of*

**r***at the location*

**η***.*

**r***(*

**y***t*) are expressed as the sum of the signal component

**y**_{S}(

*t*) and the noise

*:*

**ε**

**y**_{S}(

*t*) is called the signal vector, which is expressed as

*M*× 1 random vector

*represents additive sensor noise, which is assumed to obey the normal distribution:*

**ε***is the identity matrix and*

**I***ϱ*

^{2}is the variance of the sensor noise.

*(*

**y***t*

_{1}), …,

*(*

**y***t*

_{K}), 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

*as*

**B***(*

**y***t*

_{j}) is denoted

**y**_{j}for simplicity. We also define a matrix of the signal vector such that

*j*th column of

**B**_{S}is denoted \( \boldsymbol {y }^S_j\). This

**B**_{S}is called the signal matrix in this paper. Then, the data model in Eq. (1) is expressed in a matrix form as

**B**_{ε}is the noise matrix whose

*j*th column is equal to the noise vector

*at time*

**ε***t*

_{j}.

### 2.2 Signal Subspace in the Spatial Domain

*Q*discrete sources exist. Their locations are denoted by

**r**_{1}, …,

**r**_{Q}, their orientations by

**η**_{1}, …,

**η**_{Q}, and their magnitudes by

*s*

_{1}(

*t*), …,

*s*

_{Q}(

*t*). Then, the source distribution is expressed as

*δ*(

*) indicates the delta function. Substituting the equation above into Eq. (2), the signal vector*

**r**

**y**_{S}(

*t*) is expressed as

**l**_{q}represents the lead field vector of the

*q*th source obtained such that

**l**_{q}=

*(*

**L**

**r**_{q})

**η**_{q}. 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.)

**y**_{S}is expressed as a linear combination of the lead field vectors

**l**_{1}, ⋯ ,

**l**_{Q}. That is, the signal vector

**y**_{S}lies within a subspace spanned by

**l**_{1}, ⋯ ,

**l**_{Q}. The subspace spanned by the source lead field vectors

**l**_{1}, ⋯ ,

**l**_{Q}is defined as the signal subspace (Sekihara et al. 2000), which is denoted by \( \mathcal {E}_S \), i.e.,

### 2.3 Signal Subspace in the Time Domain

**s**_{q}consisting of the time course of the

*q*th source such that

*q*th source. We then prove that a row of the signal matrix

**B**_{S}is expressed as a linear combination of the time course vectors,

**s**_{1}, …,

**s**_{Q}. We assume, in this paper, that the source time course vectors

**s**_{q}(

*q*= 1, …,

*Q*) are linearly independent. Substituting Eq. (8) into Eq. (5), the following relationship is obtained:

**l**_{q}: \( \boldsymbol {l} _q=[l_q^1,\ldots ,l_q^M ]^T\). Denoting the

*j*th row vector of

**B**_{S}by \( \boldsymbol {\beta } _j^S\), Eq. (12) shows that

**s**_{q}(

*q*= 1, …,

*Q*). That is, we have

*K*>

*Q*, the column space of

**B**_{S}is equal to the spatial-domain signal subspace, i.e.,

*M*>

*Q*, the row space of

**B**_{S}is equal to the time-domain signal subspace, i.e.,

### 2.4 Interference Removal Using Signal Subspace Projection (SSP)

#### 2.4.1 Spatial-Domain SSP

**y**_{I}(

*t*) represents the interference overlapped on the signal vector

**y**_{S}(

*t*). We define the interference matrix

**B**_{I}as

**P**_{I}, 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

**P**_{I}

**B**_{I}=

**B**_{I}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

**P**_{I}

**B**_{S}, 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

**Π**_{I}, 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

**B**_{I}

**Π**_{I}=

**B**_{I}is used. The method of removing the interference

**B**_{I}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

**B**_{S}

**Π**_{I}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

**B**_{S}from the sensor data

*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.*

**B**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

*̆*

**P**_{S}(obtained in Eq. (47)) and

*−*

**I***̆*

**P**_{S}to the data matrix

*to create two kinds of data matrices:*

**B***−*

**I***̆*

**P**_{S}projects out the signal matrix

**B**_{S}because

*̆*

**P**_{S}

**B**_{S}=

**B**_{S}holds. According to Sekihara and Nagarajan (2017), the following relationships hold:

**ψ**_{1}, …,

**ψ**_{r}are obtained, we can compute the projector onto the intersection

*such that*

**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:*

**Π**

**ψ**_{1}, …,

**ψ**_{r}, 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

**B**_{deep}indicates the signal magnetic field generated from a deep source and

**B**_{sup}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

**u**_{1}, …,

**u**_{P}by setting the local source space as a small region just covering the ROI. The basis vectors

**u**_{1}, …,

**u**_{P}are derived in a manner described in Sect. I.2.

*, 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 \( \mathcal {K}_{{\mathrm {sup}}} \).*

**B**

**Π**_{b}, 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

## 5 Computer Simulation Validating the DSSP Algorithm

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.

*was generated by adding the interference matrix*

**B**

**B**_{I}onto the signal matrix

**B**_{S}with the interference-to-signal ratio (ISR), defined as ∥

**B**_{I}∥∕∥

**B**_{S}∥, 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.

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

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 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

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 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.

*t*= 1200 are shown in (b).

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, *z*_{ori}, is equal to *z*_{ori} = 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, *z*_{ori} 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 *z*_{ori} = 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 *z*_{ori} = 8.25 cm are shown in Fig. 9c, and those with *z*_{ori} = 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 *z*_{ori} = 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.

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