Sampling and reconstruction schemes for biomagnetic sensor arrays

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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 47 (2002) N239–N248 PII: S0031-9155(02)36007-X

NOTE

Sampling and reconstruction schemes for biomagneticsensor arrays

Adele Naddeo1, Stefania Della Penna2, Ciro Nappi3, Emanuele Vardaci1

and Vittorio Pizzella2

1 Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II”, Napoli, Italy2 Istituto di Tecnologie Avanzate Biomediche, Universita di Chieti “G. D’Annunzio”,

Chieti Scalo, Italy3 CNR-Istituto di Cibernetica “Eduardo Caianiello,” Pozzuoli (NA), Italy

E-mail: adele.naddeo@na.infn.it

Received 19 April 2002

Published 5 September 2002

Online at stacks.iop.org/PMB/47/N239

AbstractIn this paper we generalize the approach of Ahonen et al (1993 IEEE Trans.

Biomed. Eng. 40 859–69) to two-dimensional non-uniform sampling. The

focus is on two main topics: (1) searching for the optimal sensor configuration

on a planar measurement surface; and (2) reconstructing the magnetic field

(a continuous function) from a discrete set of data points recorded with a finite

number of sensors. A reconstruction formula for Bz is derived in the framework

of the multidimensional Papoulis generalized sampling expansion (Papoulis

A 1977 IEEE Trans. Circuits Syst. 24 652–4, Cheung K F 1993 Advanced

Topics in Shannon Sampling and Interpolation Theory (New York: Springer)

pp 85–119) in a particular case. Application of these considerations to the

design of biomagnetic sensor arrays is also discussed.

1. Introduction

In the last few years many research groups from all over the world have developed large

multichannel arrays for magnetoencephalography(Hamalainen et al 1993, Pizzella et al 2001,

Neuromag Ltd. 1997, Yoshida et al 2000, Della Penna et al 2000, Pasquarelli 2000)4,5. These

systems are designed to cover the whole brain, so the key in the optimization is to know the

most efficient geometry of the pick-up coil of each SQUID sensor, and how to dispose such

sensors on the measurement surface to get the optimal reconstruction of components of the

magnetic field outside the skull and the location of the current sources in the human brain (in

the least-squares sense).

4 For a description and specifications of all 4D neuroimaging systems see webpage: www.4dneuroimaging.com.5 For a description and specifications of all CTF Systems Inc. systems see webpage: www.ctf.com.

0031-9155/02/180239+10$30.00 © 2002 IOP Publishing Ltd Printed in the UK N239

N240 A Naddeo et al

In this paper we assume the simplest geometry for the pick-up coil of each sensor, a square

magnetometer, and a planar measurement surface but the sampling points on this surface are

not regularly spaced. So we discuss the overall geometry and the minimum density of such

a sensor array in a more difficult and more general context than Ahonen et al (1993), i.e.

for a non-uniform sampling grid. In section 2 we sketch briefly the pattern and the spatial

frequency content of biomagnetic fields and then we introduce the Shannon sampling theorem

(Shannon 1949) and the interplay between white noise and aliasing. The Papoulis generalized

sampling expansion (Papoulis 1977) and its multidimensional extension (Cheung 1993) are

briefly reviewed in section 3, then we apply these tools to the design of our non-uniform sensor

array (Naddeo 2001). Finally, in section 4 a few comments are made and some applications

are discussed, then a new proposal of a measurement module for a MEG multichannel system

is presented (Naddeo 2001).

2. Pattern and spatial frequency content of biomagnetic fields

The first step to carry forward the optimization of a biomagnetic sensor array is to build up a

scheme for the faithful reconstruction of a continuous field from a discrete and finite set of its

values on the measurement surface. This problem is a general one; to solve it a useful starting

point is the Shannon sampling theorem (Shannon 1949) which we rewrite, following Ahonen

et al, in a form suitable to deal with our case: the distance between two adjacent samples

must be smaller than half of the shortest spatial wavelength present in the detected quantity

(the radial component Bz of the magnetic field). If the intersensor distance is consistent with

this constraint, then there is no aliasing and we can reconstruct the magnetic field on the

measurement surface. In the following sections we shall see how this request can be cast in

such a form to be employed in our particular case.

In addition to the Nyquist frequency there is another critical spatial frequency (Ahonen

1993), fcr, which we must consider when we make measurements with such systems; this

frequency depends on the signal-to-noise ratio and above its value the spectrum of Bz falls

below the white noise background, taken uncorrelated and of instrumental origin. So it is a

nonsense to try to avoid aliasing at spatial frequencies higher than fcr, which can be seen as

the ultimate limit for every intersensor spacing (Ahonen 1993).

In the following we assume a white noise level such as to fulfil the previous request, so

we concentrate on aliasing. An essential step forward is the calculus of the magnetic field due

to the model source assumed on a planar measurement surface and the analysis of its spatial

frequency content.

2.1. Spatial distribution of biomagnetic fields

The biomagnetic inverse problem, in general, does not have a unique solution; we can constrain

it to have a unique solution only if we choose a suitable source model and volume conductor

model. Extending the concept of the equivalent current dipole model (Hamalainen 1993), we

see that all magnetic activity of the examined tissue can be modelled as a sum of multiple

simultaneous or sequentially activated current dipoles; thus the conclusions found for a current

dipole source are true for a generic electrophysiological activity. Also the analysis of the spatial

frequency content of biomagnetic fields reduces to that of the dipolar field.

So, let us consider a current dipole �Q ≡ (Qx,Qy , 0), pointing along the straight line of

equation y = x and located at −→r0 ≡ (0, 0, d), inside a semi-infinite conducting medium with

z < 0; the formula (Sarvas 1987)

�B(�r) =µ0

4πK2· [ �Q × �a · −→ez

�∇K − K−→ez × �Q] (1)

Sampling and reconstruction schemes for biomagnetic sensor arrays N241

Figure 1. Vertical component of the magnetic field, Bz, on a plane at a distance z = D due to a

current dipole �Q embedded in a semi-infinite conducting medium.

where K = a(a + �a · −→ez ), �a = �r − −→r0 , a = |�a|, �∇K = (2 + a−1 �a · −→ez )�a + a−→ez , gives us the

magnetic field of such a dipole on a planar measurement surface z = D. In the particular

case we are interested in, the neuromagnetic one, if we take into account the thicknesses

of the dewar walls, the scalp and the skull, it is sufficient to assume a dipole depth d =−0.005 m and to compute the vertical component of the magnetic field, Bz, on the plane

z = D, where D = 3 cm. Also we choose Qx = Qy = 10 nA m, which are plausible

values for an equivalent current dipole, so formula (1) gives the spatial distribution shown in

figure 1.

2.2. Spatial frequency content of biomagnetic fields

Formula (1) gives us the componentBz of the magnetic field, so we are ready for the calculation

of its spatial Fourier transform. If we fix the height z of the measurement plane above the

plane z = 0, Bz becomes a function of x and y alone, so we have to compute a two-dimensional

spatial Fourier transform. We do it numerically, using a two-dimensional fast Fourier transform

(FFT) algorithm (Press et al 1986); the absolute value of the Fourier transformF(Bz) is shown

in figure 2, with kx and ky denoting the coordinate axes of the momentum space.

We see clearly that Bz(x, y) is a band-limited function and the support S of F(Bz) in the

reciprocal space is isotropic, i.e. a circle of radius W = FN where FN is the Nyquist frequency.

It can also be shown (Ahonen 1993) that this limiting value becomes higher if we diminish the

distance source-measurement surface, so the highest spatial frequencies in Bz(x, y) are due

to the shallowest sources. In this paper we are not interested in the exact evaluation of this

frequency, but only in the fact that there is a Nyquist frequency, so we can apply a Shannon-like

sampling theorem.

In the following we focus on a particular configuration of sampling points on the

measurement plane, a non-uniform one, so we need a generalization of Shannon sampling

N242 A Naddeo et al

Figure 2. The absolute value of the two-dimensional Fourier transform F(Bz) of Bz ; the two-peak

structure is due to numerical computation.

theorem. In the next section we briefly review such a generalization and then we apply it to

our case.

3. Non-uniform sampling and reconstruction of biomagnetic fields

In this section we start from a regular but non-uniform array of sampling points on the

measurement plane, as shown in figure 3. In each point we suppose to have the centre

of a planar, square magnetometer coupled to a dc-SQUID (Pizzella 2001) for the detection

of the normal component of the magnetic field Bz. We also know the spatial frequency

content of Bz, i.e. the Nyquist frequency FN. The advantage of this non-uniform grid relies

on the better versatility of the sensor. Indeed, it is possible to combine the output of the

single magnetometers of the 2 × 2 blocks shown in figure 3 to obtain planar gradiometers

with constant baseline. On the other hand, a uniform grid allows this only for a planar

sensor geometry, whereas in a spherical geometry the magnetometers are not co-planar and

the baseline is not constant. As a matter of fact, the superior sensitivity of magnetometers

sometimes vanishes on account of a possible large noise source (e.g. pacemaker, implanted

electrode). In this case the use of planar gradiometers may improve the overall performances

of the biomagnetic sensor.

The non-uniform sampling grid shown in figure 3 needs a suitable generalization of

Shannon sampling theorem, so we first introduce Papoulis’ GSE (Papoulis 1977) and its

multidimensional extension (Cheung 1993), and then apply it to our problem.

3.1. GSE and multidimensional GSE: a note

In 1977 Papoulis (1977) showed that a band-limited signal can be exactly reconstructed from

samples of the response of m linear time-invariant (LTI) systems if the sampling rate is 1/m

times the Nyquist rate. So, in the framework of a ‘m-order’ GSE we get the continuous

Sampling and reconstruction schemes for biomagnetic sensor arrays N243

Figure 3. Configuration of the sampling points of the component Bz on the measurement surface.

function f (x) univocally from m sets of samples {{gk(nT )}|k = 0, . . . ,m − 1}, where each

signal gk(x) is the response of an LTI system, hk(x), to the input f (x):

gk(x) = hk(x) ∗ f (x) k = 0, . . . ,m − 1. (2)

The sampling step T is such that T = mTN where TN is the Nyquist value, so the sampling rate

of each gk(x) is 1/m times the Nyquist rate: the whole rate is the Nyquist rate. If Hk(ν) is the

Fourier transform of hk(x), putting m = 1 and H1(ν) = 1 we recover the ‘classic’ Shannon

sampling theorem (Shannon 1949). Also in this case we can write a reconstruction formula

Shannon-like (Papoulis 1977) for the signal f (x):

f (x) =

m−1∑

k=0

+∞∑

n=−∞

gk(nT )yk(x − nT ) (3)

where we have an interpolation function yk(x) for each set of samples.

We clearly see that the power of Papoulis GSE lies in the freedom of choosing the m LTI

systems; so we can apply this generalized sampling scheme to a wide class of problems.

The last remark concerns the multidimensional extension of Papoulis’ approach (Cheung

1993) which we present in great detail in the following, applied to our case. Also in this

framework, the signal f (�r) in input to L LTI systems can be reconstructed by an interpolation

formula such as (3), i.e.

f (�r) =

L−1∑

i=0

∑

�n

gi( V g �n)yi(�r − V g �n). (4)

Now we are ready to discuss the non-uniform sampling of biomagnetic fields.

3.2. Sampling and reconstruction of biomagnetic fields on a square grid

We start from the regular but non-uniform sampling grid shown in figure 3: in each point

we have a measure of the component Bz which is a band-limited function with Nyquist

frequency FN. The aim of this section is to show (Naddeo 2001) that we can reconstruct the

normal component Bz of biomagnetic field from its non-uniform samples, i.e. from the values

Bz1, Bz2, Bz3 and Bz4 recorded by the magnetometers 1, 2, 3, 4 of each 4-bunch (see figure 3).

N244 A Naddeo et al

1

4

Bz1

2

3

Bz2

Bz3Bz4

Bz

h0

h1

h2

h3

Bz1

Bz2

Bz3

Bz4

Figure 4. Equivalence between a ‘4-order’ two-dimensional GSE and the 4-bunch of the given

non-uniform sampling grid.

ky

C0 S

kx

ky

kx

SC

0

(a) (b)

Figure 5. (a) Basic unit cell C0 and support S in the reciprocal space K2—square sampling

geometry; (b) Hexagonal sampling geometry (Cheung 1993).

The following remark is crucial: we can consider the four values Bz1, Bz2, Bz3 and Bz4 as

outputs of four LTI systems, i.e. as response to the input signal Bz, so the 4-bunch of samples

implements a ‘4-order’ two-dimensional GSE as shown in figure 4 (Naddeo 2001).

Now, each value Bzi, i = 1, . . . , 4, is evenly sampled on the measurement plane with a

sampling step Ti ≡ TN = 12FN

; also, the support S of F(Bz) in the reciprocal space K2 is

isotropic, i.e. a circle with radius W = FN . The simplest sampling geometry we can conceive

in this context is fixed through the diagonal sampling matrix

V =

( 12W

0

0 12W

)

. (5)

It corresponds to a square lattice with a square basic unit cell C0, 2W -sided, such that the

support S of F(Bz) is entirely embedded in it, as shown in figure 5(a). This condition is a

crucial one for the sampling scheme just outlined to be well defined (Cheung 1993).

We can divide each side of C0 into l = 2 identical partitions, so we get L = 2N=2 = 4

identical subcells, all geometrically congruent to the basic unit cell but with dimensions

scaled down by the same factor l = 2. Such new geometry, fixed through the square subcells

Cgi, i = 1, . . . , 4,W -sided, is implemented by the sampling matrix

V g = lV = 2V =

(

1W

0

0 1W

)

(6)

Sampling and reconstruction schemes for biomagnetic sensor arrays N245

so the corresponding periodicity matrix in the reciprocal space is

U g =1

lU =

1

2U =

1

2

(

2W 0

0 2W

)

=

(

W 0

0 W

)

. (7)

If we assume as a reference the subcell Cg0 located on the lower left-hand vertex of the unit

cell C0

�V e = −1

2

2∑

i=1

�bi = −1

2

{(

2W

0

)

+

(

0

2W

)}

= −1

2

(

2W

2W

)

=

(

−W

−W

)

(8)

we get the following region in the reciprocal space K2:

Cg0 ≡ {(k1, k2)|−W � k1, k2 < 0} (9)

where we have put kx = k1 and ky = k2. In order to fix the notation, in the following k1 and k2

denote the coordinate axes in K2 while x1 and x2 denote the coordinate axes in R2.

The four LTI systems which we introduce to give rise to the four sets of samples

Bzi, i = 1, . . . , 4, see figure 4, are defined through the following transfer functions:

H0(k1, k2) = ej2π(k1d11+k2d12) H1(k1, k2) = ej2π(k1d21+k2d22)

H2(k1, k2) = ej2π(k1d31+k2d32) H3(k1, k2) = ej2π(k1d41+k2d42)(10)

where di1 and di2, i = 1, . . . , 4, give us the offset of the sampling locations of each sample set

from the centre of the 4-bunch. If �k ∈ Cg0, the Fourier spectrum of the sampled function from

the output of the lth LTI system is

GBzl(�k) =1

|V g|

3∑

i=0

Hl(�k + Ug �q i)FBzi(�k) l = 0, 1, 2, 3 (11)

where FBzi(�k) = FBzi(�k + Ug �q i) is the ith partition of FBz(�k) shifted to Cg0. If Cg0 is the

reference subcell, we can express the ith subcell as

Cgi ≡ {�k|�k ∈ (Cg0 ⊕ Ug �q i)} i = 0, . . . , 3. (12)

�q i , which defines the position of Cgi , is a two-dimensional vector of integers ranging from 0

to 1, i.e. it is a binary representation of the integer i:

i =

1∑

p=0

qi,p2p = qi,0 + 2qi,1 i = 0, . . . , 3. (13)

So we can write explicitly the 4 × 4 matrix Hl(�k + Ug �q i) of formula (11):

H =

ej2π(k1d11+k2d12) ej2π((k1+W)d11+k2d12) ej2π(k1d11+(k2+W)d12) ej2π((k1+W)d11+(k2+W)d12)

ej2π(k1d21+k2d22) ej2π((k1+W)d21+k2d22) ej2π(k1d21+(k2+W)d22) ej2π((k1+W)d21+(k2+W)d22)

ej2π(k1d31+k2d32) ej2π((k1+W)d31+k2d32) ej2π(k1d31+(k2+W)d32) ej2π((k1+W)d31+(k2+W)d32)

ej2π(k1d41+k2d42) ej2π((k1+W)d41+k2d42) ej2π(k1d41+(k2+W)d42) ej2π((k1+W)d41+(k2+W)d42)

(14)

and, thus, H T .

Formula (11) becomes

G �Bz =1

|V g|H F �Bz (15)

where G �Bz =(

GBz0(�k), GBz1(�k), . . . , GBz3(�k))T

and F �Bz =(

FBz0(�k), FBz1(�k), . . . ,

FBz3(�k))T

.

N246 A Naddeo et al

Then we define the carrier vector �E ≡ (1 ej2πWx1 ej2πWx2 ej2πW(x1+x2)), so from the

following matrix equation:

H T �Y = �E (16)

we get the solution:

�Y = (H T )−1 �E. (17)

Given �Y , the four interpolation functions are obtained by

yi(x1, x2) = | V g|

∫

Cg0

Yi(�k, �r) ej2π �kT �r d�k

=1

W 2

∫ 0

−W

∫ 0

−W

Yi(k1, k2, x1, x2) ej2π(k1x1+k2x2) dk1 dk2 (18)

and our signal Bz is restored by formula (4), which we rewrite as

Bz(�r) =

3∑

i=0

∑

�n

Bzi(V g �n)yi(�r − V g �n). (19)

We can find an explicit form of the interpolation functions (18) if we keep in mind that the

matrix H T can be written asH T = H 1H 2, where

H 1 =

1 1 1 1

ej2πWd11 ej2πWd21 ej2πWd31 ej2πWd41

ej2πWd12 ej2πWd22 ej2πWd32 ej2πWd42

ej2πW(d11+d12) ej2πW(d21+d22) ej2πW(d31+d32) ej2πW(d41+d42)

(20)

and

H 2 =

ej2π(k1d11+k2d12) 0 0 0

0 ej2π(k1d21+k2d22) 0 0

0 0 ej2π(k1d31+k2d32) 0

0 0 0 ej2π(k1d41+k2d42)

. (21)

Given (H T )−1 = (H 1H 2)−1 = (H 2)

−1 ·(H 1)−1, let β ij, for i, j = 0, . . . , 3 be the (i + 1, j + 1)th

element of (H 1)−1, so (17) becomes

Yi(k1, k2, x1, x2) = e−j2π(k1di1+k2di2)

3∑

m=0

βimEm(x1, x2) (22)

where Em(x1, x2) is the (m + 1)th element of the vector �E.

The four functions (18) can be written as

yi(x1, x2) =βi0

W 2ejπW(−x1+di1−x2+di2)

sin[πW(x1 − di1)]

π(x1 − di1)

sin[πW(x2 − di2)]

π(x2 − di2)

+βi1

W 2ejπW(x1+di1−x2+di2)

sin[πW(x1 − di1)]

π(x1 − di1)

sin[πW(x2 − di2)]

π(x2 − di2)

+βi2

W 2ejπW(−x1+di1+x2+di2)

sin[πW(x1 − di1)]

π(x1 − di1)

sin[πW(x2 − di2)]

π(x2 − di2)

+βi3

W 2ejπW(x1+di1+x2+di2)

sin[πW(x1 − di1)]

π(x1 − di1)

sin[πW(x2 − di2)]

π(x2 − di2)

i = 0, . . . , 3 (23)

so we are ready to insert them in the reconstruction formula (19).

Sampling and reconstruction schemes for biomagnetic sensor arrays N247

Figure 6. Scheme of our MEG detector unit.

Also, given the value W from the analysis of the spatial frequency content of the

component Bz, the matrix V g in (6) defines a square sampling grid; the sampling step T = 1W

is equal to the spacing between the central points of the 4-bunches in figure 3. We emphasize

that T = 2TN, so the grid obtained gives us a larger sensor spacing without aliasing. We could

introduce another sampling geometry, an hexagonal one, and obtain a measurement array even

larger, but in this case the support S of F(Bz) is not entirely embedded in the basic unit cell

C0, as shown in figure 5(b); so the reconstruction scheme just outlined is not well defined

(Cheung 1993). In particular, it is impossible to recover the continuous function Bz directly

from the outputs Bzi( V g �n) through an interpolation formula like (4), so further generalizations

of this already non-simple approach would be needed (Cheung 1993). Then, the non-uniform

sampling grid we choose on the planar measurement surface is a square one.

4. Conclusions

The formal considerations just developed can be applied to the design of biomagnetic sensor

arrays. We see that, when the measurement points are arranged in a non-uniform pattern, i.e.

as 4-bunches on a planar surface, the square array is chosen: it allows the largest possible

sensor spacing without aliasing, leading to a well defined reconstruction scheme. Also it is

possible to recover a continuous function, i.e. the vertical component of biomagnetic field,

from a discrete set of its values recorded on the measurement surface. The quantitative results

we can obtain in the framework of the simplest source, volume conductor and measurement

surface model presented here are qualitatively valid for more realistic geometries, for example

a spherical volume conductor and a helmet-shaped measurement surface covering the whole

head as is in the case we are interested in, i.e. the design of a neuromagnetic sensor array.

In particular, the non-uniform sampling geometry just discussed leads us to a new proposal

of a complex detector unit for a multichannel neuromagnetometer. Our detector unit comprises

four square planar magnetometers, each having an area of about 100 mm2, lying on a square

support 3 cm sided, for the measurement of Bz, as shown in figure 6. The magnetometer area

is such as to fulfil a simple criterion: to be the best compromise between the lowest magnetic

field noise level and the best spatial resolution (Granata 2002). Also, with these magnetometer

dimensions, an array of about 500 sensors, that is a reasonable number for the next generation

neuromagnetic systems, is able to cover the whole head surface. This configuration allows

us realization of planar gradiometers through a suitable software coupling of the four pick-up

coils; so we also get the two derivatives ∂Bz

∂xand ∂Bz

∂y, adding more information which can be

useful for the solution of the inverse problem.

N248 A Naddeo et al

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