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.

Appendix

1.1 I.1 Pseudo-Signal Subspace Projector

The DSSP algorithm uses the so-called pseudo-signal subspace projector (Sekihara et al. 2016) for projecting out the signal from the sensor data. To derive it, voxels are defined over the source space , in which the voxel locations are denoted r₁, …, r_N. The augmented leadfield matrix over these voxel locations is defined as

$$\displaystyle \begin{aligned} \boldsymbol{F}=[ \boldsymbol{L} ( \boldsymbol{ r } _1), \ldots, \boldsymbol{L} ( \boldsymbol{ r } _N)], \end{aligned} $$

(44)

and the pseudo-signal subspace $\breve {\mathcal {E}}_S$ is defined such that

$$\displaystyle \begin{aligned} \breve{\mathcal{E}}_S = \mathop{\mathrm{csp}} ( \boldsymbol{F} ). \end{aligned} $$

(45)

If the voxel interval is sufficiently small and voxel discretization errors are negligible, we have the relationship $\breve {\mathcal {E}}_S \supset \mathcal {E}_S$ where $\mathcal {E}_S$ indicates the true signal subspace. Therefore, a vector contained in the signal subspace is also contained in the pseudo-signal subspace.

Let us derive the orthonormal basis vectors of the pseudo-signal subspace. To do so, we compute the singular value decomposition of F:

$$\displaystyle \begin{aligned} \boldsymbol{F}=\sum_{j=1}^M \lambda_j \boldsymbol{e} _j \boldsymbol{f} ^T_j. \end{aligned} $$

(46)

If the singular values λ₁, …, λ_τ are distinctively large and other singular values λ_τ+1, …, λ_M are nearly equal to zero, the leading τ singular vectors e₁, …, e_τ form orthonormal basis vectors of the pseudo-signal subspace $\breve {\mathcal {E}}_S$ (Ipsen 2009). Thus, the projector onto $\breve {\mathcal {E}}_S$ is obtained using

$$\displaystyle \begin{aligned} \breve{\boldsymbol{P}}_S = [ \boldsymbol{e} _1,\ldots, \boldsymbol{e} _{\tau}][ \boldsymbol{e} _1,\ldots, \boldsymbol{e} _{\tau}]^T. \end{aligned} $$

(47)

Note that, since $\breve {\mathcal {E}}_S \supset \mathcal {E}_S$, the orthogonal projector (I −P̆_S) removes the signal vector , i.e., (I −P̆_S)y_S(t) = (I −P̆_S)B_S = 0.

1.2 I.2 Beamspace Processing and Beamspace Basis Vectors

Beamspace processing refers to a signal processing algorithm used for data-dimensionality reduction. Such data-dimensionality reduction is achieved by projecting the data vector onto a low-dimensional subspace. In other words, beamspace methods look for known basis vectors u₁, …, u_P that represent an M × 1 data vector y(t), where the number of basis vectors P is smaller than the dimension of the data vector M. If y(t) is expressed using a linear combination of a set of known P basis vectors such that

$$\displaystyle \begin{aligned} \boldsymbol{y }(t) \approx \sum_{j=1}^P c_j (t) \boldsymbol{u} _j , \end{aligned} $$

(48)

the sensor measurements y₁(t), y₂(t), …, y_M(t) can be represented by only P coefficients c₁(t), …, c_P(t). Since we assume P < M, the data dimension is reduced from M to P in Eq. (48).

The problem here is how to find basis vectors u₁, …, u_P which satisfy the relationship in Eq. (48). A method of deriving the basis vectors based on the prior knowledge of signal source locations has been proposed in Rodríguez-Rivera et al. (2006). In this proposed method, the augmented lead field matrix $\bar {\boldsymbol {F}}$ is defined over a local region that just contains the signal sources. The voxels are defined over this local region and the voxel locations are denoted $\bar {\boldsymbol { r }} _1,\ldots , \bar {\boldsymbol { r }} _{\bar {N}}$. The augmented leadfield matrix over these voxel locations is expressed as

$$\displaystyle \begin{aligned} \bar{\boldsymbol{F}}=[ \boldsymbol{L} ( \bar{\boldsymbol{ r }} _1), \ldots, \boldsymbol{L} ( \bar{\boldsymbol{ r }} _{\bar{N}})], \end{aligned} $$

(49)

and its singular value decomposition is given by

$$\displaystyle \begin{aligned} \bar{\boldsymbol{F}}=\sum_{j=1}^R \bar{\lambda} _j \bar{\boldsymbol{e}} _j \bar{\boldsymbol{e}} ^T_j. \end{aligned} $$

(50)

where $R=\min \{M, \bar {N} \}$. Let us assume that the leading $\bar {\tau }$ singular values $\bar {\lambda } _1,\ldots , \bar {\lambda } _{\bar {\tau }}$ are distinctively large, compared to the rest of the singular values $\bar {\lambda } _{\bar {\tau }+1},\ldots , \bar {\lambda } _R$. Then, the beamspace basis vectors u₁, …, u_P are obtained as the leading $\bar {\tau }$ singular vectors $\bar {\boldsymbol {e}} _1,\ldots , \bar {\boldsymbol {e}} _{\bar {\tau }}$ where P is equal to $\bar {\tau }$ .

1.3 I.3 Derivation of Basis Vectors that Span Intersection of Two Row Spaces

Let us assume that X and Y are low-rank data matrices. We define the basis vectors of $\mathop {\mathrm {rsp}} ( \boldsymbol {X} )$ as $\mathcal {S}_X=\{ \boldsymbol {x} _1,\ldots , \boldsymbol {x} _\mu \}$ where μ is the dimension of $\mathop {\mathrm {rsp}} ( \boldsymbol {X} )$ and the basis vectors of $\mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$ as $\mathcal {S}_Y=\{ \boldsymbol {y }_1,\ldots , \boldsymbol {y }_\nu \}$ where ν is the dimension of $\mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$. The procedure used to find a set of basis vectors of $\mathop {\mathrm {rsp}} ( \boldsymbol {X} ) \cap \mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$ is described below. The procedure is according to Golub and Van Loan (2012).

An orthonormal set of basis vectors 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 define matrices whose columns consist of the basis vectors such that

$$\displaystyle \begin{aligned} & \boldsymbol{U} =\left[ \boldsymbol{x}^{T} _1,\ldots, \boldsymbol{x}^{T} _\mu \right], \end{aligned} $$

(51)

$$\displaystyle \begin{aligned} & \boldsymbol{V} = \left[ \boldsymbol{y } ^{T}_1,\ldots, \boldsymbol{y } ^{T}_\nu \right]. \end{aligned} $$

(52)

The results of singular-value decomposition of a matrix U ^T V are expressed as

$$\displaystyle \begin{aligned} \boldsymbol{U} ^T \boldsymbol{V} = \boldsymbol{Q} \left[ \begin{array}{ccc} \cos{}(\theta_1) & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \ldots & \cos{}(\theta_\nu ) \end{array} \right] \boldsymbol{T}^T , \end{aligned} $$

(53)

where Q and T are matrices whose columns consist of singular vectors, and we assume that μ > ν. In Eq. (53), singular values of the matrix U ^T V are equal to the cosines of the principal angles between the two subspaces $\mathop {\mathrm {csp}} ([ \boldsymbol {x}^{T} _1,\ldots , \boldsymbol {x}^{T} _\mu ])$ ($= \mathop {\mathrm {rsp}} ( \boldsymbol {X} )$) and $\mathop {\mathrm {csp}} ([ \boldsymbol {y } ^{T}_1,\ldots , \boldsymbol {y } ^{T}_\nu ])$ ($= \mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$). The intersection has the property that the principal angles are equal to zero. Thus, by observing the relation

$$\displaystyle \begin{aligned}\cos{}(\theta_1)=\cos{}(\theta_2)=\cdots=\cos{}(\theta_r) \approx 1 > \cos{}(\theta_{r+1}) \ge \cdots \ge \cos{}(\theta_\nu), \end{aligned}$$

the dimension of $\mathop {\mathrm {csp}} ( \boldsymbol {U} ) \cap \mathop {\mathrm {csp}} ( \boldsymbol {V} )$, (namely, the dimension of $\mathop {\mathrm {rsp}} ( \boldsymbol {X} ) \cap \mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$) is determined to be r. The principal vectors are then obtained either as the first r columns of the matrix UQ or the first r columns of the matrix V T. Defining the first r columns of UQ as $\boldsymbol {\psi }^T _1,\ldots , \boldsymbol {\psi }^T _r$, the vectors ψ₁, …, ψ_r form an orthonormal basis set for the intersection $\mathop {\mathrm {rsp}} ( \boldsymbol {X} ) \cap \mathop {\mathrm {rsp}} ( \boldsymbol {Y} )$.