acknowledgements -...

The organizers of this symposium express our deepest thanks toProf. Alec Marantz and MIT Department of Linguistics andPhilosophy for offering to use this lecture room for this symposium and for helping us to make various arrangements for this symposium.

We also thank VMS medtech for their courtesy in providing lunch and refreshments.

Acknowledgements

Comparison between adaptive and non-adaptive spatial filter performances

Kensuke Sekihara1, Maneesh Sahani2, Srikantan S. Nagarajan3

1Department of Engineering, Tokyo Metropolitan Institute of Technology2Keck Center for Integrative Neuroscience, University of California, San Francisco

3Department of Radiology, University of California, San Francisco

Spatial filtering in BiomagnetismSatellite Symposium of Biomag 2004Boston, August, 2004

This talk compares the adaptive spatial filters such as minimum-variance spatial filter with the minimum-norm-based tomographic reconstruction methods, by formulating them as non-adaptive spatial filters.

•Bias in the reconstructed source location in the absence or presence of noise.

•Spatial resolution.

•Influence of source correlation.

Performance measures:

( )

( ) data vector: (t) =

( )

1

2

M

b t

b t

b t

⎡ ⎤⎢ ⎥⎢ ⎥

• ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

b

Definitions

data covariance matrix: ( ) ( )Tb t t• =R b b

= [ , ,

source magnitude: ( , )

source orientation: ( ) ( ) ( ) ( )]

•

• Tx y z

s t

,t ,t ,t ,t

r

r r r rη η ηη

( ) 1b t( ) 2b t ( ) Mb t

y

x

sx

l1x l2

x l3x

lMx

z

y

x

l1z l2

z l3z lM

z

z sz

y

x

l1y l2

y l3y lM

y

z sy

|sx|=|sy|=|sz|=1

( ) ( ) ( )

( ) ( ) ( )( ) =

( ) ( ) ( )

1 1 1

2 2 2

( )

, ( ) ( ) ( )

( )

x y z

xx y z

y

zx y zM M M

l l l

l l l

l l l

η

η

η

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

r r rr

r r rL r l r L r r

rr r r

Sensor lead field

Spatial filter for bioelectromagnetic source reconstruction

2

( , ) ( , ) ( ) ( , ) ( )

ˆmax ( , ) max ( , ) ( , )

= ≈⇑

= =η η

r w r η b w r η b

η r w r η R w r η

T Topt

Topt b

s t t t

s t

Scalar spatial filter

ˆ ˆ ˆ ˆ[ ( ), ( ), ( )] ( , ) [ ( ), ( ), ( )] ( )T Tx y z x y zs s s s t t= =r r r η r w r w r w r b

Vector spatial filter

The spatial filter incorporate the 3D vector nature of sources

Adaptive spatial filter

Non-adaptive spatial filter

is data independent( )w r

is data dependent( )w r

( ) ( )1N N, ,⎡ ⎤= ⎣ ⎦L L r L r

Then TN N=G L L

Gram matrix is usually calculated by introducing pixel grid jG r

jr

Define Gram matrix =: ( ) ( )T d∫G G L r L r r

Define composite lead field matrixfor all pixel locations such that

Gram Matrix

Non-adaptive spatial filters

1( ) ( )−=w r G l r

Minimum-norm (Hamalainen)

Weight normalized minimum norm (Dale et al.)

1 2( ) ( )/ ( ) ( )T− −=w r G l r l r G l r

sLORETA (Pasucual-Marque)

1 1( ) ( )/ ( ) ( )T− −=w r G l r l r G l r

( ) ( ) opt=l r L r η

1 1( ) ( )/[ ( ) ( )]Tb b− −=w r R l r l r R l r subject to ( ) 1min T

b =⇐ T

ww R w w l r

Adaptive spatial filters

Minimum variance

Minimum variance with normalized lead field

1 1( ) ( )/[ ( ) ( )]Tb b− −=w r R l r l r R l r subject to ( ) ( )min T

b⇐ =T

ww R w w l r l r

Weight-normalized minimum variance (Borgiotti-Kaplan)

1 2( ) ( )/ ( ) ( )Tb b− −=w r R l r l r R l r subject to 1min T

b =⇐ T

ww R w w w

( ) ( )/ ( )=l r l r l r

Bias of estimated source locations

We first take a look at the bias of reconstructed source locations for these adaptive and non-adaptive spatial filters.

Resolution kernel

( ) ( ) ( ) ( ) ( ) ( )

( )= (( )

)T

T

' s ' d 's ' s ' d '

ˆ'

s,

= ∫→ = ∫

b l r r rr w r l r r r

r w br

rrR⇑

Resolution kernel analysis

1 1 1ˆ( ') ( ' ) ( ) ( , ') ( ' ) ' ( , )δ δ= − ⇒ = − =∫r r r s r r r r r r r rs dR R

No location bias Resolution kernel peaks at r1

1 1 1( , ) ( , )>R Rr r r r

⇔

A single source at 1r

The condition: 1 1 1( , ) ( , )>R Rr r r r

The source at has lead field vector 1 1: ( )=r f f L r η

Minimum-norm

Weight normalized minimum norm

sLORETA

1 1− −>fG f lG f ( ( ) )=l L r ηopt

1 1

2 2T T

− −

− −>

fG f lG ff G f l G l

1 1 1 2( )( ) ( )T− − −>fG f l G l lG f

Schwartz inequality valid because G is a positive definite matrix

Non-adaptive spatial filters

The condition: 1 1 1( , ) ( , )>R Rr r r r

Minimum-variance

Minimum variancewith normalized lead field

Weight-normalizedminimum variance

2

cos( , ) 11 [1 cos ( , )]α

<+ −

f l fl l f

Adaptive spatial filters

2

cos( , ) 11 [1 cos ( , )]α

<+ −

l fl f

2

cos( , ) 11 ( 2)[1 cos ( , )]α α

<+ + −

l fl f

022 21 0( / ) ( ), cos( , ) 1

T

Mα σ σ= > ≤ = ≤l f

f l ff l

148-channel sensor array

single sources

y (cm)z

(cm

)0-5 5

-5

-10

-15

Minimum norm(normalized leadfield)

Minimum norm

Weight normalized minimum norm

sLORERA

Minimum variance

Minimum variance(normalized lead field)

Reconstruction results of this single source

The cross mark indicates the source location

Effect of noise on location bias

Condition for no location bias:

2 22 2 2 21 1 1 0 1 1 1 0( , ) ( ) ( , ) ( )σ σ σ σ+ > +R Rr r w r r r w r

noise power, signal power, input SNR: 22 2 2 20 1 1 0: : ( / )σ σ α σ σ= f

sLORETA:

wh ere 2

1

2 1

1

2

2

[1 ( )/ ] cos ( , | )( )

( )1,[1 ( )/ ( , )]

r lf w r

rr

r rf ΩΩ α

Ω α−+

< =+

GR

Minimum variance with normalized lead field

2

1 11 [1 cos ( , )]α

<+ − l f

sLORETAMinimum-variance

(normalized lead field)

2

1 11 [1 cos ( , )]α

<+ − l f

(148)Mα =

4 (592)Mα =

8 (1184)Mα =

Spatial resolution comparison

When there is no location bias, the mail-lobe width of the kernelcan be a measure of spatial resolution.

Point spread function: 1 1 1( ) ( , )/ ( , )φ = R Rr r r r r

sLORETA: 11

1 1( ) cos( , | )( )

( )( )

T

T Tφ

−

− −

− ==l G f

f G f l Gf G

lr l

Minimum variance: 2[1 cos ( ,cos( , )( )

1 )]αφ =

+ − lff

lr

Because α is a large value, this part causes a rapid decay of the point spread function

⇑


Point spread functionz

(cm

)

y (cm)0-5 5

-5

-10

-15

Reconstruction experiments when two sources exist


(normalized lead field)Source distance

4.2cm

2.1cm

1.4cm

input SNR: Mα =

Source correlation influence for adaptive spatial filters

1

1

[ ]( ) ( )

[ ]pqT S

p qppS

−

−=R

w r l rR

source covariance matrix, :the element of 1 1: [ ] ( , )pqS SSp q− −R R R

( ) ( )Tpqp q δ=w r l r

When sources are correlated with the th source,Q p1

11

[ ]( , ) ( , ) ( , )

[ ]

Q pqSp qp q ppS

s t s t s t−

−=

= + ∑R

r r rR

spatial-filter output leakages from other correlated sources

(Sources are uncorrelated)

(Sources are partially correlated)

Signal cancellation

11 1 2

2

22 1 2

1

( , ) ( , ) ( ) ( , )

( , ) ( ) ( , ) ( , )

s t s t s t

s t s t s t

α µα

α µα

= −

= − +

r r r

r r r

⇓2 2 2

1 1

2 2 22 2

( , ) (1 ) ( , )

( , ) (1 ) ( , )

s t s t

s t s t

µ

µ

= −

= −

r r

r r

Source power decreases by a factor of 2(1 )µ−⇑

When two correlated sources exist

source correlation coefficient

Reconstruction experiments when two correlated sources exist


(normalized lead field)Correltioncoefficient

0

0.85

0.98

sLORETAMinimum-variance (normalized lead field)

Auditory somatosensory response

sLORETAMinimum-variance (normalized lead field)

Somatosensory response (right median nerve stimulation)

Summary•The resolution kernel analysis validates the previous findings that sLORETA has no location bias. Minimum-variance filter has no location bias, if it is used with normalized lead field.

•Noise may cause the location bias for sLORETA. Minimum-variance spatial filter with normalized lead field has no location bias even in the presence of noise.

•Minimum-variance filter generally has significantly higher spatial resolution than sLORETA.

•Performance of sLORETA is not affected by source correlation.

•High spatial resolution of the minimum-variance filter is demonstrated by the somatosensory and somatosensory-auditory applications.

Collaborators

Kanazawa Institute of TechnologyHuman Science Laboratory

Dr. Isao Hashimoto

University of MarylandLinguistics and Cognitive Neuroscience Laboratory

Dr. David Poeppel

Massachusetts Institute of Technology,Department of Linguistics and Philosophy

Dr. Alec Marantz

http://www.tmit.ac.jp/~sekihara/

Visit

The PDF version of this power-point presentation as well as PDFs of my recent publications are available.

Generate many random dipoles in a volume:

-4 < x < -1, 1< x <4-4 < y < 4

-10 < z < -2

Number of noise random dipoles: 200

The power of noise dipoles is set at

0.005 second source power

or 0.025 second source power

:NP

×

×

Time courses of the noise sources are incoherent to each other.

source planex=0

z

yx8cm

8cm

8cm


(normalized lead field)

No noise source

Noise source power:0.005

Noise source power:0.025

Violation of the low-rank signal assumption

acknowledgements -...

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