Subspace-based interference removal methods for a multichannel biomagnetic sensor array

Abstract

Objective. In biomagnetic signal processing, the theory of the signal subspace has been applied to removing interfering magnetic fields, and a representative algorithm is the signal space projection algorithm, in which the signal/interference subspace is defined in the spatial domain as the span of signal/interference-source lead field vectors. This paper extends the notion of this conventional (spatial domain) signal subspace by introducing a new definition of signal subspace in the time domain. Approach. It defines the time-domain signal subspace as the span of row vectors that contain the source time course values. This definition leads to symmetric relationships between the time-domain and the conventional (spatial-domain) signal subspaces. As a review, this article shows that the notion of the time-domain signal subspace provides useful insights over existing interference removal methods from a unified perspective. Main results and significance. Using the time-domain signal subspace, it is possible to interpret a number of interference removal methods as the time domain signal space projection. Such methods include adaptive noise canceling, sensor noise suppression, the common temporal subspace projection, the spatio-temporal signal space separation, and the recently-proposed dual signal subspace projection. Our analysis using the notion of the time domain signal space projection reveals implicit assumptions these methods rely on, and shows that the difference between these methods results only from the manner of deriving the interference subspace. Numerical examples that illustrate the results of our arguments are provided.

Beamspace dual signal space projection (bDSSP): a method for selective detection of deep sources in MEG measurements

“…A detailed explanation of the DSSP algorithm in the context of the time-domain signal subspace can be found in [27]. The DSSP algorithm applies ${\breve{\mathit{P}}}_{\text{S}}$ and $\mathit{I}-{\breve{\mathit{P}}}_{\text{S}}$ to the data matrix B to create two kinds of data matrices: $${\breve{\mathrm{bold-italicP}}}_{\text{S}}\mathit{B}={\mathrm{bold-italicB}}_{\text{S}}+{\breve{\mathrm{bold-italicP}}}_{\text{S}}{\mathrm{bold-italicB}}_{\text{I}}+{\breve{\mathrm{bold-italicP}}}_{\text{S}}{\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }},$$ $$\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_{\text{S}}\right)\mathit{B}=\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_{\text{S}}\right){\mathrm{bold-italicB}}_{\text{I}}+\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_{\text{S}}\right){\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }}.$$ To derive equations (10) and (11), we use ${\breve{\mathrm{bold-italicP}}}_{\text{S}}{\mathrm{bold-italicB}}_{\text{S}}={\mathrm{bold-italicB}}_{\text{S}}$ and $\left(\mathit{I}-{\breve{\mathit{P}}}_{\text{S}}\right){\mathrm{bold-italicB}}_{\text{S}}=0$.…”

“…Let us use the notation rsp( X ) to indicate the row space of a matrix X . Then, according to the arguments in [27], the following relationships hold: $$\text{rsp}\left({\breve{\mathrm{bold-italicP}}}_{\text{S}}\mathit{B}\right)\subset \text{rsp}\left({\mathrm{bold-italicB}}_{\text{S}}\right)+\text{rsp}\left({\breve{\mathit{P}}}_{\text{S}}{\mathrm{bold-italicB}}_{\text{I}}\right)+\text{rsp}\left({\breve{\mathit{P}}}_{\text{S}}{\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }}\right),$$ $$\text{rsp}\left(\left(\mathit{I}-{\breve{\mathit{P}}}_{\text{S}}\right)\mathit{B}\right)\subset \text{rsp}\left(\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_S\right){\mathrm{bold-italicB}}_{\text{I}}\right)+\text{rsp}\left(\left(\mathit{I}-{\breve{\mathit{P}}}_S\right){\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }}\right).$$ Since the relationships, $\text{rsp}\left({\breve{\mathit{P}}}_S{\mathrm{bold-italicB}}_{\text{I}}\right)={\mathrm{scriptK}}_{\text{I}}$ and $\text{rsp}\left(\mathit{I}-{\breve{\mathit{P}}}_{\text{S}}\right){\mathrm{bold-italicB}}_{\text{I}}={\mathrm{scriptK}}_{\text{I}}$, hold, equations (12) and (13) lead to the relationships: $$\text{rsp}\left({\breve{\mathrm{bold-italicP}}}_{\text{S}}\mathit{B}\right)\subset {\mathrm{scriptK}}_{\text{S}}+{\mathrm{scriptK}}_{\text{I}}+{\breve{\mathrm{scriptK}}}_{\epsilon },$$ $$\text{rsp}\left(\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_{\text{S}}\right)\mathit{B}\right)\subset {\mathrm{scriptK}}_{\text{I}}+{\breve{\mathrm{scriptK}}}_{}^{}$$…”

“…Using equations (14) and (15), we can finally derive the relationship [27]: $${\mathrm{scriptK}}_{\text{I}}\supset \text{rsp}\left({\breve{\mathrm{bold-italicP}}}_{\text{S}}\mathit{B}\right)\cap \text{rsp}\left(\left(\mathit{I}-{\breve{\mathrm{bold-italicP}}}_{\text{S}}\right)\mathit{B}\right).$$ The equation above shows that the intersection between $\text{rsp}\left({\breve{\mathit{P}}}_S\mathit{B}\right)$ and $\text{rsp}\left(\left(\mathit{I}-{\breve{\mathit{P}}}_{\text{S}}\right)\mathit{B}\right)$ forms a subset of the interference subspace ${\mathrm{scriptK}}_{\text{I}}$. Once the orthonormal basis vectors of the intersection ψ 1 …, ψ r are obtained, we can compute the projector onto the intersection Π isc such that $${\mathrm{bold-italic\Pi }}_{isc}=\left[{\psi }_1,\dots ,{\psi }_r\right]{\left[{\psi }_1,\dots ,{\psi }_r\right]}^T.$$…”

“…Then, by using exactly the same derivations as in [27], we can derive $$\text{rsp}\left({\mathrm{bold-italicP}}_{\text{deep}}\mathit{B}\right)\subset {\mathrm{scriptK}}_{\text{sup}}+{\mathrm{scriptK}}_{\text{deep}}+{\overline{\mathcal{K}}}_{\epsilon },$$ $$\text{rsp}\left(\left(\mathit{I}-{\mathrm{bold-italicP}}_{\text{deep}}\right)\mathit{B}\right)\subset {\mathrm{scriptK}}_{\text{sup}}+{\overline{\mathcal{K}}}_{\epsilon }^{\prime },$$ where ${\mathrm{scriptK}}_{\text{deep}}$ and ${\mathrm{scriptK}}_{\text{sup}}$ are the time-domain signal subspaces of the deep and superficial sources, respectively. We also use the notations: $\text{rsp}\left({\mathrm{bold-italicP}}_{\text{deep}}{\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }}\right)={\overline{\mathcal{K}}}_{\epsilon }$ and $\text{rsp}\left(\left(\mathit{I}-{\mathrm{bold-italicP}}_{\text{deep}}\right){\mathrm{bold-italicB}}_{\mathrm{bold-italic\epsilon }}\right)={\overline{\mathcal{K}}}_{\epsilon }^{\prime }$.…”

“…The orthonormal basis set of the intersection, rsp( P deep B ) ∩ rsp(( I − P deep ) B ), can be obtained using the procedure presented in [24, 27, 28]. Denoting these orthonormal basis vectors by ϕ 1, … , ϕ r , the projector onto the intersection is obtained as $${\mathrm{bold-italic\Pi }}_{isc}=\left[{\varphi }_1,\dots ,{\varphi }_r\right]{\left[{\varphi }_1,\dots ,{\varphi }_r\right]}^T.$$ Using this Π isc as the projector onto the signal subspace ${\mathrm{scriptK}}_{\text{sup}}$, the signal from the deep source is estimated by projecting the rows of the data matrix onto the direction orthogonal to ${\mathrm{scriptK}}_{\text{sup}}$, such that $${\hat{\mathit{B}}}_{\text{deep}}=\mathit{B}\left(\mathit{I}-{\mathrm{bold-italic\Pi }}_{isc}\right)=\mathit{B}\left(\mathit{I}-\left[{\varphi }_1,\dots ,{\varphi }_r\right]{\left[{\varphi }_1,\dots ,{\varphi }_r\right]}^T\right).$$ Note that since the basis vectors ϕ 1 ,…, ϕ r span only a part of ${\mathrm{scriptK}}_{\text{sup}}$, the orthogonal projection on the right-hand side of equation (29) cannot perfectly remove B sup .…”