Dual signal subspace projection (DSSP)
algorithm
Kensuke Sekihara1,2 and Srikantan S. Nagarajan3
1Tokyo Medical and Dental University
2Signal Analysis Inc.
3University of California, San Francisco

I am talking about the dual signal subspace projection (DSSP)
algorithm. This algorithm can remove interference an order-ofmagnitude greater than the signal without requiring separate noise
measurements.
My talk:• reviews the spatial-domain and time-domain signal subspaces,• explains the DSSP algorithm,• presents results of applications on removal of VNS artifacts,• compares the DSSP algorithm with the tSSS algorithm.

Conventional (spatial-domain) signal subspacey(t )Data model: =data vectoryS (t ) + ε

signal vector
QKey relationship: yS (t ) = ∑ sq (t )lqq =1

noise vector
lead field vector at the q-thsource location and orientation.

Signal vector is expressed as the linear combination of these lead field vectors.
yS (t ) ∈ span {l1, , lQ } =ES
Signal vector belongs to the span of these lead field vectors, which is definedas the signal subspace ES .

We can define the signal subspace in the time-domain as well.

Time-domain signal subspace
We assume time-series measurement at time samples:
Time course of the q-th source:sq (t1 ), , sq (tK )Source time-course (row) vector:
span {s1, , sQ } can be defined as the time-domain signal subspace,which is denoted as K S : K S = span {s1, , sQ } .

Sekihara, K., & Nagarajan, S. S. (2017). Subspace-based interference removal methods for a
multichannel biomagnetic sensor array. Journal of Neural Engineering, 14(5), 051001.

Let us introduce matrix-based representation.
Data are measured at K time samples: t1, t2 , , tK
Data matrix:

B = [ y(t1 ), , y(tK )]Signal matrix: BS= [ yS (t1 ), , yS (tK )]Data model
vector representation=y(t ) yS (t ) + ε

matrix representation=B BS + Bε

We have symmetric relationships between time- and spatial-domain signal subspaces.

Each column ofColumn space of BS = ES

Each row of

Row space of BS = K S

Signal space projection (SSP) is a well known algorithm for interference removal.

There are two-types of SSP algorithms
Data model: B = BS + BI + Bε
Spatial-domain SSP

Compute a projector PI from the column space of BI .
Compute ( I − PI )B to remove BI .
Time-domain SSP

Compute a projector Π I from the row space of BI .
Compute B( I − Π I ) to remove BI .
DSSP makes use of these two types of SSPs

Structure of the DSSP algorithm
DSSP
Pseudo-signal subspace
projectorbDSSP

Poster # M-084

Interference overlapped data
Step#1De-signaling by spatial-domain SSP

Beam space projectortSSS
SSS internal subspace(pseudo) projector

Step#2Estimation of time-domain interferencesubspace
Step#3
Interference removal by time-domain SSP

Interference removed data

Step#1: De-signaling (signal nulling)

Remove signal components from the data: B = BS + BI + Bε
If the projector onto the signal subspace PS is known, de-signaling
is attained by spatial domain SSP, such that:( I − PS )B = ( I − PS )BS + ( I − PS )BI + ( I − PS )Bε= ( I − PS )BI + ( I − PS )Bε
However, because PS is unknown, we must use somethingthat can substitute PS .

DSSP uses the pseudo-signal subspace projector.

Pseudo-signal subspace projectorrj

Augmented voxel lead field matrix over source space
LV =  L(r1 ),, L(rN ) 
SVD of LV

approximates the signal subspace, whereare leading ξ singular values.
T =  u , , u   u ,, u  ,Defining the pseudo-signal subspace projector as Pξ 1ξ 1
 = B approximately holds, and de-signaling ( I − P)B =relationship: PB
0 can be achieved.
SSSStep#2: Estimation of time-domain interference subspace
De-signaling step creates two kinds of these modified data matrices: = PB + PB + PBPBεSI( I − P )B = ( I − P )BI + ( I − P )Bε
We thus have these relationships:

Row space of PB
Row space of ( I − P )B⊂ KS + KI + Kε⊂

Time-domain interferencesubspace is the commonsubspace.

K I + K ε'

After some mathematical arguments, we can show: )  (row space of ( I − P )B)K I ≈ (row space of PBThe basis vectors of KI are obtained as the basis vectors of this intersection.
Time-domain SSP is then implemented for interference removal in Step#3.

Computer simulation
Source time courses

interference

Interference time courses

Signal magnetic field

Signal plus sensor noise• 275 CTF sensor array
Signal, sensor noise plus interference• signal to sensor noise ratio:4• Interference to signal ratio: 100

Results of source localization
Source image with no interference:
Ground truth
Signal plus sensor noise

Signal, sensor noise plus interference

Source image from DSSP results

Source image from interferenceoverlapped sensor data

Somatosensory data from a patient with a VNS device
LD1 tactile stimulation

RD1 tactile stimulation

Original
sensor data

DSSP results

Sourceimage fromDSSP results

Contralateral SI cortices were reconstructed from DSSP results.
Without DSSP, a single strong fake source was reconstructed near the center of the head for the bothcases.

Removal of VNS artifacts from epilepsy data
Original
MEG Left

MEG Right

MEG Left

CleanedMEG Right

An epileptologist can identify several spikes in these cleaned data.

Example: Localization results of an epileptic spike retrieved by DSSP

Case 1Sparse Bayes source reconstruction (Champagne) algorithm was used.
Voxel time course

Source reconstruction results• With DSSP, localization was in plausible brain areas. The spike-like voxel timecourse was recovered.
• Without DSSP, an activity was localized to obviously wrong location (outside ofor near the skull).

Summary of
our study conducted to evaluate DSSP’s performance for epilepsy MEG data

Visually-identified spikes
Successfully-localized spikes

Data from 10 patients wereanalyzed with and withoutDSSPDSSP is a powerful tool to retrieve clinical information when large VNS artifacts exist.

Cai, C., Xu, J., Knowlton, R., Sekihara, K., Nagarajan, S.S., Kirsch, H. Evaluation of a dual signal subspace
projection algorithm in magnetoencephalographic recordings from patients with intractable epilepsy
and vagus nerve stimulators. submitted to NeuroImage.

Comparison between DSSP and tSSS algorithms
These algorithms differ only in the projectors used in Step#1
DSSP
Pseudo-signal subspace
projector

Interference overlapped dataStep#1
De-signaling by spatial-domain SSPtSSS
SSS internal subspace(pseudo) projector

Step#2Estimation of time-domain interferencesubspace

Step#3
Interference removal by time-domain SSP

Interference removed data

Difference in their performances
Non-MEG applications
tSSS cannot be applied to data taken from non-helmet-type sensor arrays.
DSSP can be and has been used in artifact removal in functional spinal cordbiomagnetic imaging that uses a flat sensor array.

MEG applications (using helmet sensor arrays)
Their overall performances are more or less the same, because theperformances of the de-signaling projectors are similar,.
There is one difference.
In tSSS, the de-signaling (internal) region can be smaller than the source space,and this fact requires users to choose the origin location with some care.

tSSS and DSSP de-signaling (internal) regions: Example with CTF sensor array
Plots of de-signaling gainφ =( I − P )BSBS

P: internal subspace projector (tSSS): pseudo signal subspace projector (DSSP)
OriginDSSPtSSS

source space widthtSSS de-signaling (internal) region

DSSP de-signaling region

De-signaling region for tSSS is a bit smaller than the source space.

Two cases where the origin is appropriately and inappropriately set
signal source

zori=8cm

Plot of de-signaling gain:zori=2cmφ=( I − Pint )BSBS

signal source
Origin location

zori=8cm

Signal source of interest
Internal components

External componentszori=2cm

zori=8cm

zori=2cm

Internal components

External components

signal source location andde-signaling region

If signal source is outside the internal region, the signal magnetic field has
external components.

What happens to the results of tSSS when signal magnetic field has external components
Interference-overlapped sensor data

ground truthDSSPtSSS: zori=8cm

tSSS: zori=2cm

ground truth

DSSP

tSSS:
zori=8cm
tSSS:
zori=2cm

tSSS results are distorted, if signal magnetic field has external components, because the algorithm considers
the signal external components to be a part of interference and time-domain SSP removes these components.

Related presentation
The beamspace DSSP (bDSSP) algorithm is presented in Poster #M-084 inPoster Session M2, 6:00—7:30PM on Monday.
The bDSSP algorithm is an extended version of DSSP.
I can selectively detect a deep source by suppressing stronger signals fromsuperficial sources.

DSSP algorithm

Related publications

Sekihara, K., Kawabata, Y., Ushio, S., Sumiya, S., Kawabata, S., Adachi, Y., & Nagarajan, S. S. (2016). Dual signal
subspace projection (DSSP): a novel algorithm for removing large interference in biomagneticmeasurements. Journal of Neural Engineering, 13(3), 036007.
Sekihara K. & Nagarajan S.S. Dual Signal Subspace Projection: A powerful Algorithm for Interference Removal andSelective Detection of Signal Sources, Book Chapter contribution to “Magnetoencephalography: From Signals to
Dynamic Cortical Networks, 2nd Edition”.

Time-Domain Signal Subspace
Sekihara, K., & Nagarajan, S. S. (2017). Subspace-based interference removal methods for a multichannel
biomagnetic sensor array. Journal of Neural Engineering, 14(5), 051001.

bDSSP algorithm
Sekihara, K., Adachi, Y., Kubota, H. K., Cai, C., & Nagarajan, S. S. (2018). Beamspace dual signal space projection(bDSSP): a method for selective detection of deep sources in MEG measurements. Journal of neuralengineering, 15(3), 036026.

Application to VNS artifact removalCai, C., Xu, J., Knowlton, R., Sekihara, K., Nagarajan, S.S., Kirsch, H. Evaluation of a dual signal subspace
projection algorithm in magnetoencephalographic recordings from patients with intractable epilepsy and vagus
nerve stimulators. submitted to NeuroImage.
QR codes for the links to download the reprints are available at Poster #M-084 in Poster Session M2,
6:00—7:30PM on Monday.

Thank you very much for your attention.