Recent advances in the analysis of biomagnetic signals

sd.tmu.ac.jp

Recent advances in the analysis of biomagnetic signals

The Mind and Brain III :Audition, Language, Communication

Dubrovnik, Croatia, April, 2003

Neuromagnetic Source Reconstruction and

Inverse Modeling

Part I: Introduction to adaptive spatial filter techniques

Kensuke Sekihara

Tokyo Metropolitan Institute of Technology


This talk:

•formulates the neuromagnetic source reconstruction

problem using spatial filters.

introduces non-adaptive and adaptive spatial filter

techniques.

•focuses on the adaptive spatial filter technique (adaptive

beamformer).


Magnetoencephalography

(Neuromagnetic measurements)

•can provide a high temporal resolution.

•cannot provide (adequate) information

on the source spatial configuration.


Efficient numerical algorithms for estimating source

configuration are need to be developed.

(Source localization problems)


Source localization problem

•Dipole modeling approach

•Image reconstruction approach

Tomographic reconstruction

Spatial filter


Tomographic reconstruction

•Assume pixel grids in the region of interest.

•Assume a source at each grid.

•Estimate the moment of each source by

least-squares fitting to the measured data.


Spatial filter technique

•Form spatial filter weight wr ( ) that focuses the

sensitivity of the sensor array at a small area at r.

•Scan this focused area over the region of interest

to obtain source reconstruction.

Focused region

r


Right posterior tibial nerve stimulation

measured by a 37-channel sensor array

Hashimoto et al., NeuroReport 2001


7 mm


Right median nerve stimulation

measured by a 160-channel whole-head sensor array

Hashimoto et al., J. Clinical Neurophysiology submitted for publication


Definitions

⎡b () t

1 ⎤

⎢ ⎥

⎢b () t

2 ⎥

• data vector: b(t) = ⎢



⎢ ⎥

⎢b

() t ⎥

⎣ M ⎦

b () t

1

b () t

2 b () t

M

T

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

• source magnitude: s( r, t)

• source orientation: η( r,t ) = [ ηx( r,t ), ηy( r,t ),

ηz(

r,t)] T


x

x l x l 3

2

l 1

l M

x

l 1

y l 2

y l 3

y

l M

y

z

l l z z

l

2 3

1

l M

z

z

s x

y

z

s y

y

z

s z

y

x

x

|s x

|=|s y

|=|s z

|=1

x

Lead field vector for the source orientation η()

r

x y z

⎡l1 () r l1() r l1()

r ⎤



⎡η


x y z

x()

r

⎢l2() r l2() r l2()

r ⎥

⎢ ⎥

Lr ( ) = ⎢

⎥, lr ( ) = Lr ( ) ⎢ηy

( r)





⎢ ⎥

⎢ x y z

⎢η


() () () ⎥

()


z

r

l ⎦


M

r lM r lM

r


Basic relationship

b () t = ( )( )

j ∫ l r s r,t dr

j

or

b() t = ∫ L( r)( s r,t)

dr

Problem of source localization:

Estimate sr ( , t) from the measurement b( t)

Estimate


Spatial filter

⎡b () t ⎤

1

⎢ ⎥ M

T

sr ˆ( ,) t = w ()() rbt = [ w (), r …, w ()] r

w () ()

1

⎢ ⎥ = ∑

M

m

r bm

t

m = 1

⎢ ⎥

⎢b

() t


⇑ ⎣ M ⎥⎦

estimate of sr (,) t weight vector


How to evaluate an appropriateness of the weight ?

b = ∫ L( r) s( r)

dr

T

} → ˆ sr ( ) = ( ) ( ') ( ')

d '

T

∫ w rLr sr r

ˆs( r)= w ( r)

b


R(

r, r ')


Resolution kernel

sr ˆ( ) = ∫ R( rr , ') sr ( ') dr'

(neglecting the explicit time notation)


Non-adaptive weight

wr ( )

is data independent

Adaptive weight

wr ( )

is data dependent


Data-independent (non-adaptive) weight

minimum-norm estimate (Hamalainen and Ilmoniemi)

The weight

wr ( ) is obtained by

[ R − − ] 2

min ∫ ( rr , ') δ ( r r′ ) dr′


T T 1

w ( r) L ( r) G −

T

= , where Gij ,

= ∫li( r) lj

( r)

dr


Gram matrix

T −1

Inverse solution: sr ˆ( ) = L( rG ) b


This is erroneous


Gram matrix G is usually calculated by introducing pixel grid

b = ∫ L( r) s( r) dr = ∑ L( r ) s( r)

N

j = 1

⎡sr

( ) ⎤

1

⎢ ⎥

= ⎡Lr ( ), ,

Lr ( ) ⎤ ⎢ ⎥ = Ls

⎣ 1

⎦ ⎢ ⎥

L

( )

N ⎢sr

⎣ N ⎥


s

j

j

N N N

N

r j

Therefore

G

=

L L

N

T

N

and

w T ( r ) = L T ( r )( L L T )

N

N

−1


Resolution kernel for non-adaptive (minimum-norm) method


One example

•Auditory-evoked field were measured using 148-channel

whole-head sensor array (Magnes 2500).

Stimulus: 1-kHz pure tone applied to subject’s left ear


The number of pixels: 12940 points

9

The condition number of G: ∼ 10


Property of the gram matrix G

G

= ∫ l ( r) l ( r)

dr

i,j i j

Biomagnetic instruments

X-ray computed tomography

Overlaps of sensor lead fields is large

G is poorly conditioned

G ≈

unit matrix


G is poorly conditioned

•Apply regularization when calculating G -1

Baysian-type approach

•Do not use G

min F : F


= b − Lsˆ

+


modified G

2 2

ˆ ( )


Hsˆ

T

T −1

s = HL LHL

N N

+ γ I b

Adaptive beamforming technique


Adaptive spatial filter

Minimum-variance beamformer

⎡b1()

t ⎤

⎢ ⎥ M

T

sˆ( r,) t = w ()() r b t = [ w1(), r …, wM()] r ⎢ ⎥ = ∑ wm() r bm()

t

⎢ ⎥

m=

1

⇑ ⎢bM

() t

⎣ ⎥⎦

weight vector

min T T

w Rw subject to

w l () r

=

1

w

ˆ( ,)

1

() r R l()

r

2

s r t =

T −1

l

T

⇒ w () r =

l

T −1

l () r R

r R l r

T −1

() ()


Assumption that source activities are uncorrelated

T

With constraint: w ( r ) l( r ) 1,

p p

=

T

2 2 T

w ( r ) Rw( r ) = s( r , t) + ∑ s( r, t) w ( r ) l(

r )

p p p q p q

q≠p


s( r , t) s( r, t) = 0 when p ≠ q

p

q

min

w

T

T

⎡⎣w ( rp) Rw( rp) ⎤⎦

⇒ w ( rp) l(

rq)=0,

q ≠ p

Therefore, this minimization gives the weight satisfying

w

T

( rp) l( rq) = 1 for p = q

= 0 for p ≠ q


Spatial filter technique

•Form spatial filter weight wr ( ) that focuses the

sensitivity of the sensor array at a small area at r.

Focused region

r


Adaptive beamformer sensitivity pattern: plot of wr (

0) lr ( )

The density of the colors is propotional to wr (

0) lr ( ).

wr ( ) lr ( ) > 0

0

pointing location r 0

wr ( ) lr

( ) < 01

0

wr ( ) lr ( ) = 0

0

The weight sets null-sensitivity at regions where sources exist.


Low-rank signal assumption

Consider a easiest case where we know locations and orientations

of all Q sources

weight

by solving

wr ( )(containing M unknowns)

1

a set of Q linear equations:

can be obtained

T

w ( r)( l r) = w ( r) l( r) + … + w ( r) l ( r)

=

1 1 1 1 1 1 M 1 M 1

T

w ( r)( l r) = w ( r) l ( r) + … + w ( r) l ( r)

=

1 2 1 1 1 2 M 1 M 2


T

w ( r)( l r ) = w ( r) l ( r ) + … + w ( r) l ( r ) =

1 Q 1 1 1 Q M 1 M Q

T

when Q > M, there is no solution for w ( r)

1

1

0

0


Low-rank signal

Number of sensors M > Number of sources Q

⎡λ1 0 ⋅ 0 ⎤



⎢ 0 0 ⋅ ⎥


⎥ ⎡Λ

S

0 ⎤

T

R = U λQ



⎥ U = U ⎢ ⎥ U

⎢ 0 Λ

N



⋅ 0 0 ⎥ ⎣ ⎦



⎢ 0 ⋅ 0 λ

M




T

U = [ e1, …, eQ | eQ+

1, …, eM] = [ ES | EN]


E E

S

N

T

− T

Γ = Ε Λ Ε and Γ = Ε Λ Ε ⇒ R = Γ + Γ

− 1 − 1 − 1 1 − 1 − 1 − 1

S S S S Ν N Ν N S Ν


Ε

Orthogonality principle

−1

lr ( ) = Γ lr ( ) = 0 at any source location r

T

N q N q q

Minimum-variance spatial filter output:

sˆ( r)

2

1 1

= =

l r R l r l r l r l r l r

T −1 T − 1

T −1

() () () Γ

S

() + () ΓN

()

2

sˆ( r)

source location

small


T

l () r Γ l()

r

N


large


Non-adaptive spatial filter

Adaptive spatial filter


assumed source waveform




148-channel sensor array






latency (ms)



latency (ms)

-5

}

z (cm)

-10

-15

-5 0 5

y (cm)


Minimum-norm

reconstruction

Minimum-variance

spatial filter reconstruction

MUSIC reconstruction


Adaptive-beamformer techniques were originally developed in

the fields of array signal processing, including radar, sonar, and

seismic exploration.

Two major problems arise when applying minimumvariance

beamformer to MEG source localization.

(1) Vector source detection.

(2) Output SNR degradation.


Vector source detection

The neuromagnetic sources are three dimensional vectors.

The minimum-variance beamformer formulation

should be extended to incorporate the vector nature of

sources.


Two-types of extensions has been proposed: scalar and

vector formulations.


Scalar MV beamformer formulation

w

T

(, r η)

=

T −1 T T −1

l (, r η) R η L () r R

=

T −1 T T −1

l (, r η) R l(, r η) η L () r R L()

r η

The weight depends not only on r but also on η.

Vector MV beamformer formulation

[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

The three weight vectors detect x, y, and z components.

Calculation of the weight does not require η.


Scalar MV beamformer formulation

w

T

(, r η)

=

T −1 T T −1

l (, r η) R η L () r R

=

T −1 T T −1

l (, r η) R l(, r η) η L () r R L()

r η

The weight depends not only on r but also on η.

Vector MV beamformer formulation

[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

The three weight vectors detect x, y, and z components.

Calculation of the weight does not require η.


max

How to derive the optimum η in scalar formulation?


Choose η that gives the maximum power output

2 1

T T −1

sˆ( r,) t = max

=


( () ())


T T −1

min η L r R L r η

η L () r R L()

r η ⎢ ⎣

⎥ ⎦

η η η

T −1

T

Eigendecomposition: [ L ( r) R L( r)]

= ∑ γ v v , ( γ > γ > γ )


T T −1

min( η L ( r) R L( r) η)

= γ

3

η


minimum eigenvalue

3

j = 1

j j j

1 2 3

−1

max

η

2

sˆ( r,) t 1/γ 3

= =

S

opt


Vector beamformer formulation

Each weight vector, wx(), r wy(), r or wz()

r is obtained by using

the following multiple constraints.

T T T T

x x x x x y x z

min w Rw subject to w lre (, ) = 1, w lre (, ) = 0, w lre (, ) = 0

T T T T

y y y x y y y z

min w Rw subject to w lre (, ) = 0, w lre (, ) = 1, w lre (, ) = 0

T T T T

z z z x z y z z

min w Rw subject to w lre (, ) = 0, w lre (, ) = 0, w lre (, ) = 1

e , e , e : unit vectors in the , , directions.

x y z xyz


[ w ( r), w ( r), w ( r)] = [ L ( r) R L( r)] L ( r)

R

x y z

T T −1 −1 T −1

(The weight is not equal to the scalar weight with η = e , e ,or e .)

x y z

van Veen et al., 1996 Spencer et al., 1992


In vector formulation, η that gives the maximum power output

can be obtained using

max

η

2

2

T T −1 −1

= max

x y z

= max

η

η

sˆ(,) r t [ sˆ(), r sˆ(), r sˆ()] r η η [ L () r R L()]

r η

max

η


2

ˆ( r,) = 1/γ

3

=

s t S

opt

Either types of formulations attain S opt

when the beamformer

pointing direction is optimized.


Two types of formulations are mathematically equivalent.


148-channel sensor array

-5

1st

2nd

z (cm)

-10

3rd

(reconstruction region)

-15

scalar formulation

-5 0 5

y (cm)

vector formulation


Output SNR degradation.

Minimum-variance beamformer is very sensitive to errors in

forward modeling or errors in sample covariance matrix.


Because such errors are almost inevitable in neuromagnetic

measurements, minimum-variance beamformer generally

provides noisy spatio-temporal reconstruction results.


Introducing eigenspace projection


assumed source waveform

37-channel sensor array









latency (ms)

z (cm)

-5

-10

-15

}

three sources

reconstruction region

relative magnitude





latency (ms)

generated magnetic field

-5 0 5

y (cm)

Add very small amount of noise

to obtain the simulated MEG recordings


Spatio-tempotal reconstruction


Recall some definitions:

⎡λ1

0 ⋅ 0 ⎤



⎢0 0 ⋅ ⎥


⎥ ⎡ΛS

0 ⎤

T

T

R = U λP


⎥U = U ⎢ ⎥U , and U = [ e1, …, eP | eP+

1, …, eM]

⎢ 0 ΛN




⋅ 0 0 ⎥ ⎣ ⎦

ES

EN



⎢0 ⋅ 0 λM




Also,

Γ Ε Λ Ε , Γ Ε Λ Ε

−1 −1 T −1 −1

T

S

=

S S S Ν

=

N Ν N

Output SNR


[ l ( r) Γ l( r)]

T −1 2

S

T −2

T −2

[ l ( r) ΓS

l( r) + ε Γ ε ]

N

overall error in estimating lr ( )

T −2

Even when ε is small, ε Γ ε may not be small,

N

2 2 2

N

/ λp+

j

T −

because ε Γ ε ≈ ε

← noise level eigenvalue


Eigenspace projection

T −2

The error term ε Γ ε arises from the noise subspace component of

N

wr ().

Extension to eigenspace projection beamformer

w

T

= E E w , where µ = xy , or z

S S

µ µ

Output SNR ∝

Output SNR


[ l ( r) Γ l( r)]

T −1 2

S

T −2

T −2

[ l ( r) ΓS

l( r) + ε Γ ε ]

N


T

[ l ( r) Γ l( r)]

T

[ l ( r) Γ l( r)]

−1 2

S

−2

S

(non-eigenspace projected)

(eigenspace projected)


Spatio-tempotal reconstruction with eigen-space projection


Application to 37-channel auditory-somatosensory recording

eigenspace-projection results


Application to 37-channel auditory-somatosensory recording

Non-eigenspace projected results


Application to 37-channel auditory-somatosensory recording

eigenspace-projection results


Summary

•This talk reviews the application of adaptive beamformer to

reconstruction of brain activities, with some reference to the nonadaptive

methods.

•Two types of extensions, scalar and vector extensions to incorporate

the vector nature of the sources, are shown to be mathematically

equivalent.

•Eigenspace projection is shown to overcome the SNR degradation

problem.

•The implicit assumptions for the adaptive beamformer are that

sources are uncorrelated and that the signal is low rank. The cases

where these assumptions are invalidated will be discussed in Part II.


Summary

•This talk reviews the application of adaptive beamformer to

reconstruction of brain activities, with some reference to the nonadaptive

methods.

•Two types of extensions, scalar and vector extensions to incorporate

the vector nature of the sources, are shown to be mathematically

equivalent.

•Eigenspace projection is shown to overcome the SNR degradation

problem.

•The implicit assumptions for the adaptive beamformer are that

sources are uncorrelated and that the signal is low rank. The cases

where these assumptions are invalidated will be discussed in Part II.


Summary

•This talk reviews the application of adaptive beamformer to

reconstruction of brain activities, with some reference to the nonadaptive

methods.

•Two types of extensions, scalar and vector extensions to incorporate

the vector nature of the sources, are shown to be mathematically

equivalent.

•Eigenspace projection is shown to overcome the SNR degradation

problem.

•The implicit assumptions for the adaptive beamformer are that

sources are uncorrelated and that the signal is low rank. The cases

where these assumptions are invalidated will be discussed in Part II.


Summary

•This talk reviews the application of adaptive beamformer to

reconstruction of brain activities, with some reference to the nonadaptive

methods.

•Two types of extensions, scalar and vector extensions to incorporate

the vector nature of the sources, are shown to be mathematically

equivalent.

•Eigenspace projection is shown to overcome the SNR degradation

problem.

•The implicit assumptions for the adaptive beamformer are that

sources are uncorrelated and that the signal is low rank. The cases

where these assumptions are invalidated will be discussed in Part II.


Collaborators

University of California, San Francisco

Biomagnetic Imaging Laboratory

Dr. Srikantan S. Nagarajan

University of Maryland

Linguistics and Cognitive Neuroscience Laboratory

Dr. David Poeppel

Massachusetts Institute of Technology,

Department of Linguistics and Philosophy

Dr. Alec Marantz

Kanazawa Institute of Technology

Human Science Laboratory

Dr. Isao Hashimoto


Visit

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

The PDF version of this power-point presentation as

well as PDFs of the recent publications are available.

More magazines by this user
Similar magazines