Adaptive beamformer source reconstruction: Our recent developments toward spatio-temporal reconstruction of brain activities Kensuke Sekihara 1 , Srikantan S. Nagayajan 2 1 Department of Engineering, Tokyo Metropolitan Institute of Technology 2 Biomagnetic Imaging Laboratory, University of California, San Francisco ISBET 2003 Santa Fe, November, 2003
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Adaptive beamformer source reconstruction:
Our recent developments toward spatio-temporal reconstruction of brain activities
Kensuke Sekihara1, Srikantan S. Nagayajan2
1Department of Engineering, Tokyo Metropolitan Institute of Technology2Biomagnetic Imaging Laboratory, University of California, San Francisco
ISBET 2003Santa Fe, November, 2003
Right median nerve stimulationmeasured by a 160-channel whole-head sensor array
Hashimoto et al., “Muscle afferent inputs from the hand activate human cerebellum sequentially
through parallel and climbing fiber systems”, Clin. Neurophysiol. Nov;114, pp.2107-17, 2003 .
Right posterior tibial nerve stimulation
measured by a 37-channel sensor array
Hashimoto et al., “Serial activation of distinct cytoarchitectonic areas of the human {SI} cortex after
Lead field vector for the source orientation ( )rη
Adaptive spatial filter
1
11
( )
(̂ , ) ( ) ( ) [ ( ), , ( )] ( ) ( )
( )
MTM m m
m
M
b t
s t t w w w r b t
b t=
⎡ ⎤⎢ ⎥
= = =⎢ ⎥ ∑⎢ ⎥⎢ ⎥⎣ ⎦
r w r b r r…
1
1
( )( )( ) ( )
TT
T
−
−⇒ =l r Rw rl r R l r
Minimum-variance beamformer
21
1(̂ , )( ) ( )Ts t −=r
l r R l r
subject to ( ) 1min T =T
ww Rw w l r
Spatial filter
Following problems arise when applying minimum-variance beamformer to MEG/EEG source reconstruction.
(1) Output SNR degradation.
(2) Vector source detection.
(3) Statistical significance evaluation.
Minimum-variance beamformer is very sensitive to errors in forward modeling or errors in sample covariance matrix.
Because such errors are inevitable in neuromagnetic measurements, minimum-variance beamformer generally provides noisy spatio-temporal reconstruction results.
Introducing eigenspace projection
⇓
⇓
Output SNR degradation.
and
1
1 1
0 0
0 00
, [ , , | , , ]0
0 0
0 0
ST TP P P M
NS N
M
+
⋅⎡ ⎤⎢ ⎥
⋅⎢ ⎥⎡ ⎤⎢ ⎥= = =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⋅
⎢ ⎥⎢ ⎥⋅⎣ ⎦
ΛΛ
… …R U U U U U e e e eE E
λ
λ
λ
Output SNR ∝+
2[ ( ) ( )][ ( ) ( ) ]
TS
TS
TN
l r l rl r l r
Γε Γ εΓ
error in estimating ( )l r
Even when is small, may not be small,because noise level eigenvalueλ +≈ ←22 /
TN
TN p j
ε ε εε ε ε
ΓΓ
Also, −= =1 1T TS S S S N N
−Ν ΝΓ Ε Λ Ε Γ Ε Λ Ε,
Some definitions:
Eigenspace projection
Extension to eigenspace projection beamformer
where or , ,TS S x y zµw E E wµ µ= =
∝2[ ( ) ( )]
[ ( ) ( )]
TS
TS
l r l rl r l r
ΓΓ
Output SNR
Output SNR ∝+
2[ ( ) ( )][ ( ) ( ) ]
TS
TS
TN
l r l rl r l r
Γε Γ εΓ
⇓
(non-eigenspace projected)
(eigenspace projected)
The error term arises from the noise subspace component of ( ).TNε Γ ε w r
Application to 37-channel auditory-somatosensory recordingeigenspace-projection results
Application to 37-channel auditory-somatosensory recordingNon-eigenspace projected results
The minimum-variance beamformer formulation should be extended to incorporate the vector nature of sources.
The electromagnetic sources are three dimensional vectors.
Two-types of extensions has been proposed: scalar and vector formulations.
⇓
Vector source detection
=1 1
1 1
( , ) ( )( , )( , ) ( , ) ( ) ( )
T T TT
T T T
− −
− −=l r η R η L r Rw r η
l r η R l r η η L r R L r η
Scalar MV beamformer formulation
uses a single weight vector, but it depends not only on but also on .r η
S. E. Robinson et al., Recent Advances in Biomagnetism, Tohoku University Press, 1999
1 1 1[ ( ), ( ), ( )] [ ( ) ( )] ( )T T Tx y z
− − −=w r w r w r L r R L r L r R
Vector MV beamformer formulation
uses three weight vectors which detect , , and source components. x y z
M. E. Spencer et al., 26th Annual Asilomer Conference on Signals, Systems, and Computers, 1992B. D. van Veen et al., IEEE Trans. Biomed. Eng., 1997
max max21
11
1(̂ , )( ) ( )
1( ( ) ( ) )min
T T
T T
min
s t
γ
−
−−
=
⎡ ⎤= =⎢ ⎥⎣ ⎦
η η
η
rη L r R L r η
η L r R L r η
minimum eigenvalue of 1: [ ( ) ( )]Tminγ −L r R L r
Scalar formulation:
max22
1 1
ˆ ˆ ˆ ˆmax( , ) [ ( ), ( ), ( )]
1max[ [ ( ) ( )] ]min
x y z
T T
s t s s s
γ− −
=
= =
ηη
η
r r r r η
η L r R L r η
Vector formulation:
Two types of formulations give the same output power.
Output power
Asymptotic output SNR
2
2 2200
ˆ ˆ ˆ[ ( ), ( ), ( )] 1max( )( )
x y zoptV
min
s s sZ
ασσ= =
η
r r r ηr
W r η
max12 2
2 2 1 220 00
(̂ , , ) 1 ( ) ( ) 1( ) min( ) ( )( , )
T Topt
nT T
miS
s tZ
ασ σσ
−−
−
⎡ ⎤= = =⎢ ⎥
⎣ ⎦η η
r η η L r R L r ηrη L r R L r ηw r η
Scalar beamformer
Vector beamformer
minimum eigenvalue of 11 2: ( ) ( ) ( ) ( )T T
minα−− −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦L r R L r L r R L r
Two types of formulations give the same asymptotic output SNR.