Reconstruction of magnetic source images using the Wiener filter and a multichannel magnetic imaging system

Reconstruction of magnetic source images using the Wiener filter and a multichannel
magnetic imaging system
J. A. Leyva-Cruz, E. S. Ferreira, M. S. R. Miltão, A. V. Andrade-Neto, A. S. Alves, J. C. Estrada, and M. E. Cano
Citation: Review of Scientific Instruments 85, 074701 (2014); doi: 10.1063/1.4884641
View online: http://dx.doi.org/10.1063/1.4884641
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200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

REVIEW OF SCIENTIFIC INSTRUMENTS 85, 074701 (2014)
Reconstruction of magnetic source images using the Wiener filter
and a multichannel magnetic imaging system
J. A. Leyva-Cruz,1
E. S. Ferreira,2
M. S. R. Miltão,1
A. V. Andrade-Neto,1
A. S. Alves,2
J. C. Estrada,3
and M. E. Cano3
1
Instrumentation Physics Lab, Department of Physics, Universidade Estadual de Feira de Santana,
44036-900 Feira de Santana, BA, Brazil
2
Materials Physics Lab, Department of Physics, Universidade Estadual de Feira de Santana,
44036-900 Feira de Santana, BA, Brazil
3
Centro Universitario de la Ciénega, Universidad de Guadalajara, Av. Universidad, 1115,
Ocotlán, JAL, CP.47810, Mexico
(Received 13 February 2014; accepted 9 June 2014; published online 3 July 2014)
A system for imaging magnetic surfaces using a magnetoresistive sensor array is developed. The
experimental setup is composed of a linear array of 12 sensors uniformly spaced, with sensitivity of
150 pT∗
Hz−1/2
at 1 Hz, and it is able to scan an area of (16 × 18) cm2
from a separation of 0.8 cm
of the sources with a resolution of 0.3 cm. Moreover, the point spread function of the multi-sensor
system is also studied, in order to characterize its transference function and to improve the quality
in the restoration of images. Furthermore, the images are generated by mapping the response of the
sensors due to the presence of phantoms constructed of iron oxide, which are magnetized by a pulse
of 80 mT. The magnetized phantoms are linearly scanned through the sensor array and the remanent
magnetic field is acquired and displayed in gray levels using a PC. The images of the magnetic sources
are reconstructed using two-dimensional generalized parametric Wiener filtering. Our results exhibit
a very good capability to determine the spatial distribution of magnetic field sources, which produce
magnetic fields of low intensity. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884641]
I. INTRODUCTION
The problem of obtaining the spatial distributions of elec-
tromagnetic properties inside an object has attracted much
interest in the last years and some solutions have been
published.1–10
The experimental measurements of magnetic
field maps of known sources is called the magnetic forward
problem (MFP). The reverse process to obtain a magnetic
source image (MSI) from the magnetic map image (MMI) is
called the magnetic inverse problem (MIP).
Several techniques have been published for imaging mag-
netic surfaces by using transducers which involve different
physics effects. For instance, the scanning systems with Hall
probes possess theoretical field sensitivity 2 nT/
√
Hz,11–13
and
recently smallest sensors with dimensions of ∼ 50 nm exhibit
a low magnetic field sensitivity of 8.0 × 10−5
T/
√
Hz with
a nanometer-scale spatial resolution. Although, the field sen-
sitivity of a typical giant magneto impedance sensor (GMI)
can reach a value as high as 500%/Oe,14
however, their
large size required for the sensing element restricts its spatial
resolution.15
Nevertheless, only superconducting quantum in-
terference device (SQUIDs) and AMR devices have the po-
tential to localize buried and non-visual field sources such as
defects in small electronic pieces,16–18
magnetic field sources
in biological environments19–27
, and other applications in geo-
physical survey28–30
or nondestructive evaluation.31–37
The
AMR sensors have been used for scanning with resolution
below 500 nm and sensitivity to detect currents as low as
50 nA (indeed this fact has been used with advantage for in-
tegrated circuit mapping). The latter, together with their rel-
atively low cost, ease of implementation, and their ability to
detect very small magnetic fields, give AMR devices signifi-
cant advantages over other magnetic imaging techniques such
as SQUIDs, Hall sensors, GMI sensors, or Magnetic Force
Microscopes. For their part, SQUID scanners provide an ex-
treme sensitivity 10 fT/
√
Hz with a spatial resolution of about
30 μm, but they bear the main disadvantage of operating at
cryogenic temperatures, at least 77 K.31
Furthermore, the generalized Wiener parametric filter has
been employed by Moreira et al.,38
in order to develop an AC
bio-susceptometer imaging system with pickup coils. In fact,
the viability of this device is tested by studying the images of
a set of iron oxide phantoms, but also is realized a previous
analysis of the point spread function (PSF) to solve the MIP.
In other researches Cano et al.,39,40
have shown the
suitability of determine magnetic image maps of phantoms,
which are transported under an array of 16 magnetic AMR
sensors with precision of 0.1 μT and the maximum scan-
ning area of 15.5 × 8 cm2
. Later their setup was replaced
by an XY scanner to obtain images of magnetic susceptibil-
ity. The system is composed of a mobile array of three AMR
sensors increasing the resolution to 10 nT and solves the in-
verse problem using the Fourier filtering method, but after a
long scanning time. However, with these procedures it is not
possible to obtain the density of magnetic sources inside the
phantoms. This determination is an important task because it
gives punctual information about the inside of the samples,
but it is necessarily a difficult technical work in the character-
ization of the scanning system. As a continuation, in this work
is developed the instrumentation for acquiring weak mag-
netic maps of magnetized phantoms using an array of very
0034-6748/2014/85(7)/074701/11/$30.00 © 2014 AIP Publishing LLC85, 074701-1
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

074701-2 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)
FIG. 1. The schematic diagram of the magnetic imaging system consists of a
magnetoresistive sensor multichannel array, a computer controlled x-y stage
sample positioning, data acquisition (DAQ), and analysis capability.
sensitive AMR sensors. Additionally is presented the recon-
struction of the magnetic sources using the spatial Wiener fil-
tering to solve the MIP. Indeed, to optimize the quality in the
imaging restoration is taken into account the dimensions of
the sensors, to determine the PSF and the optimal resolution
employing the Rayleigh’s criterions. This imaging method
represents a best alternative in applications where determin-
ing the local concentration of weak magnetic sources is re-
quired and it is a powerful tool on the scale of centimeters.
II. INSTRUMENTATION
A. Magnetic imaging system
The magnetic imaging system consists of a motorized
platform for transporting the samples on a sensing unit, which
is composed of a static array of very sensitive AMR sensors
distributed along a straight line. A total of 12 AMR sensors
Honeywell (HMC-1001) are used in this work. The device
is capable of scanning planar samples with sizes up to 16
× 18 cm2
and a noise density of 150 pT∗
Hz−1/2
at 1 Hz. Fig-
ure 1 shows a schematic diagram of the experimental setup,
where is shown the sensor unit (see the close up of two sen-
sors), the platform transporting a phantom and a PC during an
imaging procedure. The distance between the geometric cen-
ter of two sensors is 1.5 cm and the separation between the
platform and the line sensors is z = 0.8 cm.
The signal of the sensors are filtered using operational
amplifiers and passive low-pass filters with corner frequency
in 10 Hz, to obtain a fixed gain of 70 dB for each channel.
All the electronic components are integrated circuits of very
low noise and the magnetic sensor array is supplied with (9
± 0.01) V using a set of batteries. An increase of sensitivity
can be realized by substituting the single sensors with gra-
diometers composed by two sensors differentially amplified,
which is very useful to remove the background noise.41,42
Par-
ticularly, to detect AC magnetic sources, the sensitivity of the
system can be increased an order of magnitude by using a
lock-in amplifier, but this application is restricted to work at a
fixed frequency.38,43
The voltage signals are acquired using a PCI-6034E DAQ
card from National Instrument with 16 analog inputs (AI), 16
bits of resolution, a maximum sampling rate of 200 kS/s, and
one AI is assigned for each sensor. The automatic acquisition
of magnetic maps is carried out by the synchronous control of
a stepper motor, which is composed of a mechanism to move
the phantom on the array and an electronically powered stage.
Both, the scanning and data acquisition parameters are con-
trolled by the user through the computer software developed
using LabVIEW.
To diminish the noise due to high frequency artifacts in
the signals, the measurements are oversampled and averaged
by the data acquisition software. Additionally the platform is
mechanically well connected to the sample-scanning system
to avoid vibrations.
III. THEORETICAL BACKGROUND AND METHODS
A. Magnetic inverse problem
The scanning system can be represented by a linear, dis-
crete, and shift-invariant system, characterized by their PSF
with transfer function h(x − x , y − y , z − z ). This func-
tion is given by the output-to-input signal ration. Figures 2(a)
and 2(b) show a schematic diagram summarizing the mathe-
matical and physical concepts concerning to MFP and MIP,
respectively. In Figure 2(a) are represented the experimental
MMI Bz(x,y) (with the noise η(x,y) superimposed) and their
reconstructed MSI CFe3O4
(x , y ) considered as the output/
input of the sensors array, respectively. From this point of
view, Figure 2(b) shows the relationship between the plane
of the imaging where the MMI is obtained, the one of the
measurement process, and finally the plane of the magnetic
field source. When the magnetic sources are known, using the
Biot-Savart law is possible to solve the MFP. But if the MMIs
are unknown, an inverse transference function is required, in
this case to solve the MIP.
FIG. 2. (a) Schematic representation the experimental MMI Bz(x,y) and the MSI ˜CFe3O4 (x , y ), considered as the output/input of the sensors array; and (b) the
mathematical and physical concepts concerning to MFP and MIP, respectively.
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

FIG. 3. Coordinate system attached to the AMR sensors, with spacing inter-sensor of 1.5 cm and the PhFive phantom placed in z = 0.8 cm from the sensors.
The extended magnetic source is restricted to the x -y plane and the AMR sensors measure the z component of the magnetic fields Bz (x, y, ε).
According to Tan et al.,44
most of 3D general magnetic
inverse problems do not have a unique solution, due mainly
to the ill-posed nature of the problem. But this can be avoided
if the direction of the magnetization is known. To solve this,
the samples are magnetized only in the z-direction. Figure 3
shows the coordinate system attached to magneto resistive
sensors on a magnetized sample and the contribution of an el-
ement of volume dV containing the ferromagnetic material.
The excitation magnetic field Bexc
z induces a z-independent
magnetization distribution of Mz(x , y ) with dipolar moment
dmz(x , y ) extended in two-dimensions, the thickness of the
sample is ε. Indeed, only the z-component of the magnetic
field is relevant.
In the measurement plane XY, the MMI is obtained scan-
ning the sample below and close to the sensors. Consider-
ing this measurement in z-direction, the differential magnetic
field dBz(x, y) produced by dmz(x , y ) in each sensor is given
by the two-dimensional convolution integral kernel, Eq. (1).45
dBz (x, y) = (μ0/4π)
2 (z − ε)2
− [(x − x )2
+ (y − y )2
]
[(x − x )2 + (y − y )2 + (z − ε)2
]5/2
× dmz(x , y ), (1)
where μ0 = 4π × 10−7
TmA−1
is the magnetic permeability
of the empty space.
After some physical considerations and algebraic trans-
formations, we can obtain the magnetic field detected by the
sensors Bz(x, y) for all the area, which obeys Eq. (2):
Bz (x, y)
= ξ
Ymáx
Ymín
Xmáx
Xmín
2 (z − ε)2
− [(x − x )2
+ (y − y )2
]
[(x − x )2 + (y − y )2 + (z − ε)2]5/2
× CFe3O4
(x , y )dx dy , (2)
where CFe3O4
(x , y ) is the concentration of ferromag-
netic particles. Moreover, ξ is a constant function
ξ = (μ0/4π)(μ/μ0 − 1)(ε/ρFe3O4
)Bexc
z , which depends
on the density ρFe3O4
, the magnetic permeability μ of
the phantom and the magnetic field intensity of a pulsed
magnetizer system Bexc
z .
The Eq. (2) can be discretized following Eq. (3):
Bz (x, y) = ξ
ymáx
y=ymín
xmáx
x =xmín
×
2 (z − ε)2
− [(x − x )2
+ (y − y )2
]
[(x − x )2 + (y − y )2 + (z − ε)2]5/2
× CFe3O4
(x , y) x y . (3)
Using the sensitivity of the detectors and taking into ac-
count the gain in the amplification stage, the detected voltage
is related to magnetic field as Eq. (4)
V (x, y) = ϒBz (x, y) , (4)
where ϒ = 1 × 106
(V/T ) is the proportionality factor, their
inverse ϒ−1
is the sensors calibration factor.
B. The spectral response and PSF of the imaging
system
The spectral response of the imaging system is studied
considering the MMI in the frequency space. In fact, the
frequency spectrum of MMI is the product of the magnetic
source (frequency spectrum of the phantom) and the spatial
response of the sensors. As the spatial frequency is limited by
the physical dimensions of the phantom, then the smallest de-
tail within the phantom determines the shape and value of the
maximum MSI cutoff frequency. The minimum spatial cutoff
frequency of the AMR sensor is determined by the dimen-
sions of the scan area Xscan = 16.0 cm and Yscan = 18.0 cm,
this is displayed in Figure 4. The maximum spatial frequency
of the sensors depends of their dimensions (Lx = 0.137 cm
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

FIG. 4. Geometrical details of the ferromagnetic phantom (PhFive) and some
parameter related to discrete sampling process. The sample and scanning area
dimensions are (6.0 × 8.0) cm2; and Xscan = 16.0 cm and Yscan = 18.0 cm,
respectively.
and Ly = 0.9883 cm) and the distance of the sample. Due
to the finite size of the sensors and the attenuation of the
magnetic field to a distance z from the phantom, the linear
magnetometers array works as a spatial low-pass filters array.
Thus, in the scan direction is obtained information only for
lower frequencies than the maximum cutoff frequency of the
magnetometer.
The filter function can also be accomplished in the fre-
quency space using the Fourier transform of the step function,
(kx, ky) given by Eq. (5).42
(kx, ky) = (sin (kxLx/2) /kxLx/2)
∗ (sin(kyLy/2)/kyLy/2). (5)
Considering that the first set of zeros (kxzero, kyzero) of the step
function is localized on a rectangle with sides (Kc
x, Kc
y ). These
results confirm that each AMR sensors in the MSI acts as a
spatial low-pass filter with cutoff frequency justly in (2π/Lx,
2π/Ly). In agreement with Roth et al.,46
this is an important
feature because it ensures that the major part of the frequen-
cies content in the MSI of the sampled phantom is localized
in a limited bandwidth. The values of kxzero establish the max-
imum limits for spatial sampling only in x-direction, because
in y-direction the sampling increment is fixed in y = 1.5 cm
(see Figure 4). This is an important feature because it ensures
that the major part of the frequencies content in the MSI of the
sampled phantom is localized in a limited bandwidth, in the Y
direction set to Ks
y = π/ y = 2.0 samples/cm. Furthermore,
the spatial sampling used to record the MMI in X direction,
must be in agreement with the Nyquist theorem to avoid alias-
ing effects; accordingly, the following sampling frequency Ks
x
will be obtained with Eq. (6):
Ks
x
>
= 2Kc
x = 2 (2π/Lx) = 91.7 samples/cm. (6)
During the acquisition process of MMI, all the signals
with frequencies above Ks
x will be considered as noise. With
this consideration, the spatial frequency content of the MMI
is attenuated, minimizing the possibility of aliasing error.
In order to solve the MIP, the two-dimensional fast
Fourier transform (2D-FFT) of the measured MMI must be
divided by the step function (kx, ky) before reconstructing
the MSI, to consider the magnetic field averaging on the area
of the sensors. Then, the magnetic field measured by the array
of AMR due to an extended source CFe3O4
(x , y ) may also be
rewritten in the real space using deconvolution Eq. (7):
Bz(x, y) = ξ (kx, ky)
⊗
2 (z − ε)2
− [(x − x )2
+ (y − y)2
]
[(x − x )2 + (y − y )2 + (z − ε)2]5/2
⊗ CFe3O4
(x , y ) x y . (7)
The imaging system will be modeled as the convolution
of CFe3O4
(x , y ) with their PSF, in this sense the MMI is given
in Eq. (8):
Bz (x, y) = h(x − x, y − y , z − ε) ⊗ CFe3O4
(x , y ). (8)
The concentration CFe3O4
(x , y ) is discretized using the
Dirac Function CFe3O4
(x , y ) = C0δ(x − x, y − y, ε − z),
where C0 is a punctual magnetic charge. So the PSF can be
rewritten using the discretized Green’s function of Eq. (9).42
h(x − x , y − y , z − ε)
= ℘ (x, y) ⊗ z(x − x , y − y , z − ε)
⊗ CFe3O4
(x , y ) x y , (9)
where z(x − x , y − y ) = (μ0/4π)[ 2 (z−ε)2
−[(x−x )2
+(y−y)2
]
[(x−x )2+(y−y )2+(z−ε)2]5/2 ]
is the Green’s function and ℘ = (μ/μ0 − 1)(ε/ρFe3O4
)Bexc
z is
a new constant function.
Obtaining the discrete 2D-FFT ( ) of both sides of
Eq. (9) to apply the theorem of the discrete convolution, is
reached the optical transfer function (OTF) of the measure-
ment system, in the frequency space, Eq. (10)
Hz(kx, ky) = ℘ { (x, y)} { z(x − x , y − y , z − ε)}
×{C0δz(x − x, y − y)} { x y }, (10)
where { x } = kx = 2π/xp and { x } = ky = 2π/yp
are the distance between two adjacent points in the frequency
space and (xp , yp ) are the dimensions of a magnetic point
phantom in X-Y directions, respectively. This phantom is nec-
essary to determine experimentally and theoretically the PSF
of the imaging system.
Regarding Eq. (5), the Dirac delta function properties
{C0δz(x − x, y − y)} = C0 and dealing with the analytical
expression for the 2D-FFT of the Green’s function40
Gz(kx,
ky, z − ε) = (μ0/4π)e−kZ
(1 − e−kε
), where k = k2
x + k2
y is
the total spatial frequency. Then the optical transfer function
(OTF) is given by Eq. (11)
Hz(kx, ky)
= ℘ sin
kxLx
2
kxLx
2
∗ sin
kyLy
2
kyLy
2
∗ (μ0/4π)e−kZ
(1 − e−kε
) kx ky C0. (11)
The punctual magnetic charge C0 is also very important
to find the impulse response of our imaging system trough
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

the PSF or the OTF, because then is possible to calculate the
response to any arbitrary input.
C. Solution of MIP using the spatial Wiener filtering
In the frequency space, the solution of the MIP of Eq. (8)
can be rewrite as Eq. (12):
Bz(kx, ky) = Hz(kx − kx , ky − ky )CFe3O4
(kx , ky ). (12)
Since the main aim of this work is to obtain the distribution
of magnetic particles CFe3O4
(x , y ) into the sample, it is also
necessary to find the inverse of the OTF. In this case, the func-
tion H−1
z (kx − kx , ky − ky ) is the inverse filter, which oper-
ates on the MMI in the frequency space to reconstruct the
original MSI, as Eq. (13):
CFe3O4
(kx , ky ) = H−1
z (kx − kx , ky − ky )Bz(kx, ky). (13)
The inverse filter is very susceptible to additive noise,
such as the electromagnetic one produced by the electronic
instrumentation. Thus, it is important to investigate the noise
present in the imaging process, making necessary the study
of the feasibility of using different methods which permit to
manipulate the noise inherent in the solution of MIP.47
A pos-
sibility to diminish the problem of noise sensitivity of the sys-
tem is to cutoff the frequency response of the filter to a thresh-
old value. For that we define a limit γ under which the inverse
filter acquires plausible values. Therefore, in this work, the
inverse filter H−1
γ (kx − kx , ky − ky ) = ( 1
Hz(kx −kx ,ky −ky )
) for
|Hz| > γ ; and H−1
γ (kx − kx , ky − ky ) = 1
γ
( 1
Hz(kx −kx ,ky −ky )
)
for |Hz| γ .
This is also called the truncated pseudo-inverse filter
and γ is the truncation parameter for controlling the level
of poles that could appear during the deconvolution process.
The deconvolution of the MMI involves a strong amplifica-
tion of high spatial frequency noise.48
For stationary signals,
the Wiener filter represents the mean square error-optimal lin-
ear filter for degraded images by additive noise49
and can be
written as Eq. (14)
W
α,β,γ
Wiener(Kx, Ky)
=
⎧
⎨
⎩
1
1 + [α10
−S(Kx ,Ky )
10 ]H−1
γ [Kx − Kx , Ky − Ky ]
⎫
⎬
⎭
β
×H−1
γ [Kx − Kx , Ky − Ky ], (14)
where the parameters α and β are real and S(kx, ky) = 10
× log10(
Pη(kx ,ky )
Pi (kx ,ky )
) is an expectation of the signal to noise ra-
tio (S/N). As we can see, this filter depends on the power
spectra of MMI Pi(kx, ky) and on the additive noise im-
age Pη(kx, ky), respectively. The (
Pη(kx ,ky )
Pi (kx ,ky )
) term can be in-
terpreted as 1/(S/N). If (
Pη(kx ,ky )
Pi (kx ,ky )
) ≈ 0, then the Wiener filter
becomes in H−1
γ (kx − kx , ky − ky ), that is the inverse filter
for the PSF. In contrast, if the signal is very weak (
Pη(kx ,ky )
Pi (kx ,ky )
)
≈ ∞
yields
→ H
α,β,γ
Wiener(kx, ky) → 0. The α-parameter controls the
level of the additive noise present in the measured MMI.
When this value is increased the noise is attenuated more ef-
fectively and allows us to adjust the aggressiveness of the fil-
ter. Standard Wiener method is obtained with α = 1. Higher
values result in more aggressive filtering; in this case the de-
convolution can be referred as an over-filtering process. In
this work is used β = 1 because the quality of the images
was not sensitive to this parameter. From mathematical point
of view, the Wiener deconvolution can be expressed applying
the Wiener filter, H
α,β,γ
Wiener(kx, ky) to the spectral MMI to obtain
the MSI in the frequency space, Crest
Fe3O4
(Kx , Ky ), this is given
in Eq. (15)
Crest
Fe3O4
(kx, ky) = W
α,β,γ
Wiener(kx, ky)Bz(kx, ky). (15)
This product is transformed back to provide filtered data.
Therefore, to determine the MSI reconstructed in real space,
is applied the two-dimensional inverse fast Fourier transform
(2D-IFFT) ( −1
) on the MSI obtained in the frequency do-
main, according to Eq. (16)
Crest
Fe3O4
(x , y ) = −1
Crest
Fe3O4
(kx, ky) . (16)
Finally, starting from an initial source concentration Ci,
we can to obtain the Mean Square Deviation (MSD) between
the maximum value of the reconstructed image and the initial
concentration (see Eq. (17)), this parameter indicate the qual-
ity in the restoration and the resolution of the images. The def-
inition of the MSD could be generalized if we compare with
a larger phantom, with uniform concentration of magnetized
particles:
MSD =
|Ci − ˜Crest
Fe3O4
(x , y )MAXIMUM |2
C2
i
. (17)
IV. EXPERIMENTS
A. Phantom preparations and experimental procedure
The prepared phantom is a composite of iron oxide Fe3O4
(Bayferrox) in powder presentation, which is mixed with
Vaseline gel distributed in thin layer of plastic figure with
the “number 5” shape (PhFive). The maximum diameter, den-
sity, and relative magnetic permeability of the magnetite par-
ticles are d = 125 μm, ρFe3O4
= 48 g/cm3
, and μ = 1900,
respectively. Using an analytical scale Mettler Toledo is ob-
tained a concentration of magnetic particles in the mixture
Ci = 80.00 mg/cm3
and the phantom is magnetized with a
uniform pulse of 50 ms and intensity Bexc
z = 80 mT in the z-
direction. This pulse is generated with an array of Helmholtz
coils of 1 m in diameter.
The magnetized sample is fixed on the motorized plat-
form positioned below the magnetic sensor array. As shown
in Figure 1, the phantom is maintained far from the motor sys-
tem to avoid electromagnetic noise and the distance between
the phantom and the sensors is z = 0.8 cm. The scanning
of the samples and data acquisition is performed using sub-
routines developed in LabVIEW. Figure 4 shows some geo-
metrical details of the PhFive phantom and some parameters
related to the scan process. As is shown in Figure 4, the data
are acquired in X-direction with an increment x = 0.1 cm
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

at a velocity of 0.32 cm/s. The spatial sampling is ks
x = 2000
S/cm, with sampling frequency of fs = 16.6 kS/s per channel
and it is synchronized with the scanning of the sample. These
values are higher than the limit imposed by the dimension and
the positioning of the sensor in Y-direction, to avoid aliasing
errors.
Also, a punctual phantom (C0) is constructed using the
same composite of the PhFive phantom, depositing the mix
in a tiny cylinder with diameter and height of xp = yp
= 0.4 cm. The PSF is determined obtaining their MMI and the
experimental OTF is determined through the application of
the 2D-FFT. Moreover, the theoretical OTF is directly calcu-
lated from Eq. (11) and their corresponding theoretical PSF is
obtained from their 2D-IFFT. The magnetic imaging process-
ing and visualization are performed offline, using the MAT-
LAB language.
V. RESULTS AND DISCUSSIONS
A. Offset correction, PSF, OTF, and magnetic
noise images
In the measuring process of very weak signals, it is not
possible to avoid the influence of the noise from different
sources and this is traduced in an offset on the signals. In the
experiments the offset is corrected via software similarly to
Cano et al.,40
before the deconvolution process. Figures 5(a)
and 5(c) show the theoretical and experimental PSF images,
respectively. Also their corresponding OTF in logarithmic in-
tensity is displayed in Figures 5(b) and 5(d), respectively.
These functions are used to characterize the magnetoresistive
sensor array.
Due to the ill-posed nature of the problem, the direct
solution of MIP without control of the noise is not the best
FIG. 5. The theoretical (a) and experimental (c) PSF and their OTF images (logarithmic intensity) in (b) and (d), for the magnetoresistive sensor multichannel
array measurement system, used in the filtering process for a separation of z = 0.8 cm.
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

FIG. 6. Two-dimensional MMI measured from two dipolar point magnetic source phantoms, for analysis the spatial resolution of the MSI, with z = (a) 0.1 cm,
(b) 0.3 cm, (c) 0.8 cm, and (d) 1.5 cm. In (c) is shown the images for the best resolution of the MSI about 0.3 cm.
choice. The techniques known as regularization methods are
most precisely used due to noise content normally present
in the measured MMI. Because the solution of the MIP is
better using these procedures, we can convert the ill-posed
into well-posed problems. The environmental magnetic noise
image η(x,y) is measured in our lab, following the normal
FIG. 7. Analysis of the MSD versus α, using γ 1 and γ 2.
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

procedure but without phantom. The maximum value ob-
tained is approximately 10 nT and this value matches with
other studies.40,49
B. The spatial resolution of the MSI
The spatial resolution of the MSI can be defined as the
measure of the ability of an imaging system to separate the
MMI of two punctual sources or objects. In this experiment
is used an optical analogy with the Rayleigh’s resolution to
the magnetic images. In this method, two punctual magnetic
images separated by a distance are resolved if there exists an
intersection point with relative intensity of 60%, in compari-
son with the maximum peak of the magnetic field. When these
conditions are satisfied the distance between the point sources
is the resolution of the imaging system.
The experiment is carried out using two punctual mag-
netic sources made with 20 mg of Bayferrox (the same
FIG. 8. The reconstructed MSI images with γ 2 and α = 0 (a), 0.15 (b), 1 (c), 20 (d), 100 (e), and 1000 (f).
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

material of the PhFive phantom) mixed with 20 ml of Vase-
line gel, in a cylindrical phantom with a concentration about
10 mg/cm3
of iron oxide. Both sources are built with a radius
of 0.25 cm and height of 0.5 cm. The magnetized phantoms
are placed over an acrylic plate where we can adjust the sepa-
ration between them; also the distance between the phantoms
and the sensor array is z = 0.8 cm. In Figures 6(b)–6(d) are
displayed their MMIs obtained with our setup for different
distance between them about 0.1, 0.3, 0.8, and 1.5 cm. All
the images are interpolated using a bi-cubic function to 256
× 256 pixels. Figure 6(c) shows the results, the best resolution
of the magnetic imaging system, which is about 0.3 cm. These
results are in agreement with some studies related with the
spatial resolution analysis from magnetic images.49,50
Since
the spatial resolution of the AMR scanner depends directly
on the dimensions of the AMR sensor,14,19
in this case the
resolution is 1/3 of the length L. For this reason, to increase
the spatial resolution following correctly the Rayleigh´s cri-
terion, the use of smaller sensors is needed. For instance, to
reach a resolution of 500 μm could be L = 0.15 cm.
C. Measured MMI and reconstructed MSI
Using the Wiener filter in the deconvolution procedure,
a reconstructed MSI of the PhFive phantom is obtained. Re-
garding the experimental conditions cited above, it is possi-
ble to ensure a good noise rejection without loss of the signal
produced from the phantom. The S/N required in the mea-
surements is approximately S/N = 70 dB, in order to allow a
deconvolution process of acceptable quality.
In the first experiment analysis, we did not find signifi-
cant differences when the solution of the MIP was carried out
using the theoretical or experimental PSF, respectively. The
inversion process was done using different values of the addi-
tive noise and pseudo filter parameter γ .
As first evaluation for γ , the parameter α is ranged in α
= 0.1, 10, 50, 100, 500, 1000, 5000, 10 000, and 100 000 to
control the poles and additive noise, respectively, during the
deconvolution process. Figure 7 illustrates the analysis of the
MSD versus α, each point on the curves corresponding to a
MMI. The quality of the reconstruction method is better when
using γ 2 = 1.76 × 10−2
than γ 1 = 1.71 × 10−1
. Therefore,
the results obtained using γ 2 and higher values of α exhibit
a loss of spatial resolution and a decrease of the magnitude
of the restored images, which implicates a bad quality in the
restoration because the MSD ≈ 1. In these cases ,the restored
and real MSI are very different. In contrast, for low values of
α, the reconstruction can be considered to have higher quality,
in this case the MSD ≈ 0.
Following the analysis, Figures 8(a)–8(f) show the recon-
structed MSI using γ 2 for others values of α = 0, 0.15, 1, 20,
100, and 1000. The noise in the MSI is more reduced for a
value of α = 100, and 1000, as shown in Figures 8(e) and
8(f). In those situations, the deconvolution process began to
affect the filtering MSI and the high noise suppression is con-
verted to over filtering process and the MSI loses the spatial
resolution, decreasing their magnitude. In Figure 8(f), the im-
ages have a loss of the spatial resolution completely.
On the other hand, Figures 9(a)–9(b) show the MMI and
the reconstructed MSI of the phantom respectively using γ 2.
The restored image shows a good quality and high reduction
of noise, and as a consequence is obtained a better spatial
resolution compared to MMI, in this cases the MSD > 0.1.
The best reconstructed MSI for high quality performance of
deconvolution process is observed for αf = 0.135. From an
analysis of this image, we can conclude that the restored MSI
has a small spatial resolution improvement. The best filter-
ing image show that the maximum amplitude in ˜Crest
Fe3O4
(x , y )
is about 79.96 mg/cm3
and MSD ≈ 0.0005, indicating small
differences between their magnitudes and the peak value of
the ferromagnetic particles concentration, distributed into the
magnetic phantom.
Finally, another phantom “Sixlines” is realized using an
acrylic table with six straight lines (recorded using a milling
machine) 0.2 cm deep, 0.3 cm wide and 9 cm long, into which
the mixture of vaseline/magnetite is deposited. The magnetic
FIG. 9. (a) Measured MMI from PhFive magnetic phantom and (b) the reconstructed MSI, for a pseudo inverse filter parameter set to γ 2 = 1.76 × 10−2 and a
noise level α = 0.135. The mean amplitude value of the ferromagnetic particles concentration is about 79.96 mg/cm3.
200.128.81.8 On: Fri, 11 Jul 2014 00:14:22

FIG. 10. (a) Measured MMI from “Sixlines” magnetic phantom and (b) the reconstructed MSI, using the same parameters to reconstruct the PhFive phantom.
The mean amplitude value of the ferromagnetic particles concentration is about 0.82 mg/cm.
map of the phantom is measured and the magnetic sources are
obtained using the Wiener method adjusted with the best pa-
rameters determined previously αf, γ 2, and β. The images of
the magnetic map and the reconstructed one are displayed in
Figures 10(a) and 9(b), respectively; these straight lines could
simulate currents flowing inside of conductors or electronic
boards. In this case, the maximum amplitude determined in
the restored ˜Crest
Fe3O4
(x , y ) is about 0.82 mg/cm3
, this is 2.5%
bigger than the real concentration.
In conclusion, the initial results show the way to recon-
struct a magnetic image source using spatial Wiener Filtering
method, from two-dimensional magnetic maps measured by
a setup of 12-channels of AMR sensors. The magnetic maps
are obtained from a separation z = 0.8 cm at room tempera-
ture and the experimental spatial resolution for the imaging
systems is 0.3 cm. The procedure for obtaining the recon-
structed MSI produces a small reduction of the additive noise
and increases the stability of the solution because the use of
Wiener Filter. The amplitude and the spatial resolution of re-
constructed images are modified by the filter parameters. This
work illustrates the importance of knowing the PSF and the
filter parameters to improve the quality of the restored image.
This technique can be extended to solve the inverse problem
of any magnetized surface, and open new expectations for dif-
ferent applications in medical, electronic circuits, geophysics,
and other technological areas.
ACKNOWLEDGMENTS
Authors wish to thank Thomas M. Trent for reviewing
the language of the paper, and also thank CNPq and CLAF
for financial support.
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