Reconstruction of magnetic source images using the Wiener filter and a multichannelmagnetic imaging systemJ. 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 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Resolution evaluation of MR images reconstructed by iterative thresholding algorithms for compressed sensing Med. Phys. 39, 4328 (2012); 10.1118/1.4728223 Use of vehicle magnetic signatures for position estimation Appl. Phys. Lett. 99, 134101 (2011); 10.1063/1.3639274 High sensitivity magnetic imaging using an array of spins in diamond Rev. Sci. Instrum. 81, 043705 (2010); 10.1063/1.3385689 In situ detection of single micron-sized magnetic beads using magnetic tunnel junction sensors Appl. Phys. Lett. 86, 253901 (2005); 10.1063/1.1952582 Blind reconstruction of x-ray penumbral images Rev. Sci. Instrum. 69, 1966 (1998); 10.1063/1.1148881

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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 074701 (2014)

Reconstruction of magnetic source images using the Wiener filterand 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

1Instrumentation Physics Lab, Department of Physics, Universidade Estadual de Feira de Santana,44036-900 Feira de Santana, BA, Brazil2Materials Physics Lab, Department of Physics, Universidade Estadual de Feira de Santana,44036-900 Feira de Santana, BA, Brazil3Centro 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. Theexperimental setup is composed of a linear array of 12 sensors uniformly spaced, with sensitivity of150 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 cmof the sources with a resolution of 0.3 cm. Moreover, the point spread function of the multi-sensorsystem is also studied, in order to characterize its transference function and to improve the qualityin the restoration of images. Furthermore, the images are generated by mapping the response of thesensors due to the presence of phantoms constructed of iron oxide, which are magnetized by a pulseof 80 mT. The magnetized phantoms are linearly scanned through the sensor array and the remanentmagnetic field is acquired and displayed in gray levels using a PC. The images of the magnetic sourcesare reconstructed using two-dimensional generalized parametric Wiener filtering. Our results exhibita very good capability to determine the spatial distribution of magnetic field sources, which producemagnetic 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 muchinterest in the last years and some solutions have beenpublished.1–10 The experimental measurements of magneticfield maps of known sources is called the magnetic forwardproblem (MFP). The reverse process to obtain a magneticsource image (MSI) from the magnetic map image (MMI) iscalled the magnetic inverse problem (MIP).

Several techniques have been published for imaging mag-netic surfaces by using transducers which involve differentphysics effects. For instance, the scanning systems with Hallprobes possess theoretical field sensitivity 2 nT/

√Hz,11–13 and

recently smallest sensors with dimensions of ∼ 50 nm exhibita 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, theirlarge size required for the sensing element restricts its spatialresolution.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 asdefects in small electronic pieces,16–18 magnetic field sourcesin biological environments19–27, and other applications in geo-physical survey28–30 or nondestructive evaluation.31–37 TheAMR sensors have been used for scanning with resolutionbelow 500 nm and sensitivity to detect currents as low as50 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 suchas SQUIDs, Hall sensors, GMI sensors, or Magnetic ForceMicroscopes. 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 atcryogenic temperatures, at least 77 K.31

Furthermore, the generalized Wiener parametric filter hasbeen employed by Moreira et al.,38 in order to develop an ACbio-susceptometer imaging system with pickup coils. In fact,the viability of this device is tested by studying the images ofa set of iron oxide phantoms, but also is realized a previousanalysis of the point spread function (PSF) to solve the MIP.

In other researches Cano et al.,39, 40 have shown thesuitability of determine magnetic image maps of phantoms,which are transported under an array of 16 magnetic AMRsensors with precision of 0.1 μT and the maximum scan-ning area of 15.5 × 8 cm2. Later their setup was replacedby an XY scanner to obtain images of magnetic susceptibil-ity. The system is composed of a mobile array of three AMRsensors increasing the resolution to 10 nT and solves the in-verse problem using the Fourier filtering method, but after along scanning time. However, with these procedures it is notpossible to obtain the density of magnetic sources inside thephantoms. This determination is an important task because itgives 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 workis developed the instrumentation for acquiring weak mag-netic maps of magnetized phantoms using an array of very

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074701-2 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

FIG. 1. The schematic diagram of the magnetic imaging system consists of amagnetoresistive sensor multichannel array, a computer controlled x-y stagesample 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 theimaging restoration is taken into account the dimensions ofthe sensors, to determine the PSF and the optimal resolutionemploying the Rayleigh’s criterions. This imaging methodrepresents 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 motorizedplatform for transporting the samples on a sensing unit, whichis composed of a static array of very sensitive AMR sensorsdistributed along a straight line. A total of 12 AMR sensorsHoneywell (HMC-1001) are used in this work. The deviceis 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 animaging procedure. The distance between the geometric cen-ter of two sensors is 1.5 cm and the separation between theplatform and the line sensors is z = 0.8 cm.

The signal of the sensors are filtered using operationalamplifiers and passive low-pass filters with corner frequencyin 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 sensitivitycan 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 thesystem can be increased an order of magnitude by using alock-in amplifier, but this application is restricted to work at afixed frequency.38, 43

The voltage signals are acquired using a PCI-6034E DAQcard from National Instrument with 16 analog inputs (AI), 16bits of resolution, a maximum sampling rate of 200 kS/s, andone AI is assigned for each sensor. The automatic acquisitionof magnetic maps is carried out by the synchronous control ofa stepper motor, which is composed of a mechanism to movethe 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 developedusing LabVIEW.

To diminish the noise due to high frequency artifacts inthe signals, the measurements are oversampled and averagedby the data acquisition software. Additionally the platform ismechanically well connected to the sample-scanning systemto 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 PSFwith 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 experimentalMMI Bz(x,y) (with the noise η(x,y) superimposed) and theirreconstructed MSI CFe3O4 (x ′, y ′) considered as the output/input of the sensors array, respectively. From this point ofview, Figure 2(b) shows the relationship between the planeof the imaging where the MMI is obtained, the one of themeasurement process, and finally the plane of the magneticfield source. When the magnetic sources are known, using theBiot-Savart law is possible to solve the MFP. But if the MMIsare unknown, an inverse transference function is required, inthis 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) themathematical and physical concepts concerning to MFP and MIP, respectively.

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074701-3 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

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 magneticinverse problems do not have a unique solution, due mainlyto the ill-posed nature of the problem. But this can be avoidedif the direction of the magnetization is known. To solve this,the samples are magnetized only in the z-direction. Figure 3shows the coordinate system attached to magneto resistivesensors 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-independentmagnetization distribution of Mz(x′, y′) with dipolar momentdmz(x′, y′) extended in two-dimensions, the thickness of thesample is ε. Indeed, only the z-component of the magneticfield 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 magneticfield dBz(x, y) produced by dmz(x′, y′) in each sensor is givenby 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 permeabilityof the empty space.

After some physical considerations and algebraic trans-formations, we can obtain the magnetic field detected by thesensors 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 )Bexcz , which depends

on the density ρFe3O4 , the magnetic permeability μ ofthe phantom and the magnetic field intensity of a pulsedmagnetizer 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 voltageis related to magnetic field as Eq. (4)

V (x, y) = ϒBz (x, y) , (4)

where ϒ = 1 × 106 (V/T ) is the proportionality factor, theirinverse ϒ−1 is the sensors calibration factor.

B. The spectral response and PSF of the imagingsystem

The spectral response of the imaging system is studiedconsidering the MMI in the frequency space. In fact, thefrequency spectrum of MMI is the product of the magneticsource (frequency spectrum of the phantom) and the spatialresponse of the sensors. As the spatial frequency is limited bythe physical dimensions of the phantom, then the smallest de-tail within the phantom determines the shape and value of themaximum MSI cutoff frequency. The minimum spatial cutofffrequency 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 frequencyof the sensors depends of their dimensions (Lx = 0.137 cm

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074701-4 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

FIG. 4. Geometrical details of the ferromagnetic phantom (PhFive) and someparameter related to discrete sampling process. The sample and scanning areadimensions 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. Dueto the finite size of the sensors and the attenuation of themagnetic field to a distance z from the phantom, the linearmagnetometers array works as a spatial low-pass filters array.Thus, in the scan direction is obtained information only forlower frequencies than the maximum cutoff frequency of themagnetometer.

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 stepfunction is localized on a rectangle with sides (Kc

x,Kcy ). These

results confirm that each AMR sensors in the MSI acts as aspatial low-pass filter with cutoff frequency justly in (2π /Lx,2π /Ly). In agreement with Roth et al.,46 this is an importantfeature because it ensures that the major part of the frequen-cies content in the MSI of the sampled phantom is localizedin a limited bandwidth. The values of kxzero establish the max-imum limits for spatial sampling only in x-direction, becausein y-direction the sampling increment is fixed in �y = 1.5 cm(see Figure 4). This is an important feature because it ensuresthat the major part of the frequencies content in the MSI of thesampled phantom is localized in a limited bandwidth, in the Ydirection 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):

Ksx

>= 2Kcx = 2 (2π/Lx) = 91.7 samples/cm. (6)

During the acquisition process of MMI, all the signalswith frequencies above Ks

x will be considered as noise. Withthis consideration, the spatial frequency content of the MMIis attenuated, minimizing the possibility of aliasing error.

In order to solve the MIP, the two-dimensional fastFourier transform (2D-FFT) of the measured MMI must be

divided by the step function (kx, ky) before reconstructingthe MSI, to consider the magnetic field averaging on the areaof the sensors. Then, the magnetic field measured by the arrayof AMR due to an extended source CFe3O4 (x ′, y ′) may also berewritten 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 convolutionof CFe3O4 (x ′, y ′) with their PSF, in this sense the MMI is givenin Eq. (8):

Bz (x, y) = h(x − x, y − y ′, z − ε) ⊗ CFe3O4 (x ′, y ′). (8)

The concentration CFe3O4 (x ′, y ′) is discretized using theDirac Function CFe3O4 (x ′, y ′) = C0δ(x ′ − x, y ′ − y, ε − z),where C0 is a punctual magnetic charge. So the PSF can berewritten 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 )Bexcz is

a new constant function.Obtaining the discrete 2D-FFT (�) of both sides of

Eq. (9) to apply the theorem of the discrete convolution, isreached 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 frequencyspace and (xp′ , yp′ ) are the dimensions of a magnetic pointphantom in X-Y directions, respectively. This phantom is nec-essary to determine experimentally and theoretically the PSFof the imaging system.

Regarding Eq. (5), the Dirac delta function properties�{C0δz(x′ − x, y′ − y)} = C0 and dealing with the analyticalexpression for the 2D-FFT of the Green’s function40 Gz(kx,

ky, z − ε) = (μ0/4π )e−kZ(1 − e−kε), where k =√

k2x + 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 importantto find the impulse response of our imaging system trough

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074701-5 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

the PSF or the OTF, because then is possible to calculate theresponse 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 distributionof magnetic particles CFe3O4 (x ′, y ′) into the sample, it is alsonecessary 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 theoriginal MSI, as Eq. (13):

CFe3O4 (kx ′ , ky ′ ) = H−1z (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 electronicinstrumentation. Thus, it is important to investigate the noisepresent in the imaging process, making necessary the studyof the feasibility of using different methods which permit tomanipulate 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 inversefilter acquires plausible values. Therefore, in this work, theinverse filter H−1

γ (kx − kx ′ , ky − ky ′ ) = ( 1Hz(kx−kx′ ,ky−ky′ ) ) for

|Hz| > γ ; and H−1γ (kx − kx ′ , ky − ky ′ ) = 1

γ( 1Hz(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 levelof 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 bewritten as Eq. (14)

Wα,β,γ

Wiener(Kx,Ky)

=⎧⎨⎩ 1

1 + [α10−S(Kx ,Ky )

10 ]H−1γ [Kx − Kx ′ ,Ky − Ky ′ ]

⎫⎬⎭

β

×H−1γ [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 powerspectra 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−1γ (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→ Hα,β,γ

Wiener(kx, ky) → 0. The α-parameter controls thelevel 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. Highervalues result in more aggressive filtering; in this case the de-convolution can be referred as an over-filtering process. Inthis work is used β = 1 because the quality of the imageswas not sensitive to this parameter. From mathematical pointof view, the Wiener deconvolution can be expressed applyingthe Wiener filter, Hα,β,γ

Wiener(kx, ky) to the spectral MMI to obtainthe MSI in the frequency space, Crest

Fe3O4(Kx ′ ,Ky ′ ), this is given

in Eq. (15)

CrestFe3O4

(kx, ky) = Wα,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)

CrestFe3O4

(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) betweenthe maximum value of the reconstructed image and the initialconcentration (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 witha larger phantom, with uniform concentration of magnetizedparticles:

MSD = |Ci − CrestFe3O4

(x ′, y ′)MAXIMUM |2C2

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 withVaseline gel distributed in thin layer of plastic figure withthe “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 mixtureCi = 80.00 mg/cm3 and the phantom is magnetized with auniform pulse of 50 ms and intensity Bexc

z = 80 mT in the z-direction. This pulse is generated with an array of Helmholtzcoils of 1 m in diameter.

The magnetized sample is fixed on the motorized plat-form positioned below the magnetic sensor array. As shownin Figure 1, the phantom is maintained far from the motor sys-tem to avoid electromagnetic noise and the distance betweenthe phantom and the sensors is z = 0.8 cm. The scanningof 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 parametersrelated to the scan process. As is shown in Figure 4, the dataare acquired in X-direction with an increment �x = 0.1 cm

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074701-6 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

at a velocity of 0.32 cm/s. The spatial sampling is ksx = 2000

S/cm, with sampling frequency of fs = 16.6 kS/s per channeland it is synchronized with the scanning of the sample. Thesevalues are higher than the limit imposed by the dimension andthe positioning of the sensor in Y-direction, to avoid aliasingerrors.

Also, a punctual phantom (C0) is constructed using thesame composite of the PhFive phantom, depositing the mixin a tiny cylinder with diameter and height of xp = yp= 0.4 cm. The PSF is determined obtaining their MMI and theexperimental OTF is determined through the application ofthe 2D-FFT. Moreover, the theoretical OTF is directly calcu-lated from Eq. (11) and their corresponding theoretical PSF isobtained 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 magneticnoise images

In the measuring process of very weak signals, it is notpossible to avoid the influence of the noise from differentsources and this is traduced in an offset on the signals. In theexperiments the offset is corrected via software similarly toCano 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 magnetoresistivesensor array.

Due to the ill-posed nature of the problem, the directsolution 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 multichannelarray measurement system, used in the filtering process for a separation of z = 0.8 cm.

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074701-7 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

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 aremost precisely used due to noise content normally presentin the measured MMI. Because the solution of the MIP is

better using these procedures, we can convert the ill-posedinto well-posed problems. The environmental magnetic noiseimage 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.

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074701-8 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

procedure but without phantom. The maximum value ob-tained is approximately 10 nT and this value matches withother studies.40, 49

B. The spatial resolution of the MSI

The spatial resolution of the MSI can be defined as themeasure of the ability of an imaging system to separate theMMI of two punctual sources or objects. In this experiment

is used an optical analogy with the Rayleigh’s resolution tothe magnetic images. In this method, two punctual magneticimages separated by a distance are resolved if there exists anintersection point with relative intensity of 60%, in compari-son with the maximum peak of the magnetic field. When theseconditions are satisfied the distance between the point sourcesis 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).

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074701-9 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

material of the PhFive phantom) mixed with 20 ml of Vase-line gel, in a cylindrical phantom with a concentration about10 mg/cm3 of iron oxide. Both sources are built with a radiusof 0.25 cm and height of 0.5 cm. The magnetized phantomsare placed over an acrylic plate where we can adjust the sepa-ration between them; also the distance between the phantomsand the sensor array is z = 0.8 cm. In Figures 6(b)–6(d) aredisplayed their MMIs obtained with our setup for differentdistance between them about 0.1, 0.3, 0.8, and 1.5 cm. Allthe images are interpolated using a bi-cubic function to 256× 256 pixels. Figure 6(c) shows the results, the best resolutionof the magnetic imaging system, which is about 0.3 cm. Theseresults are in agreement with some studies related with thespatial resolution analysis from magnetic images.49, 50 Sincethe spatial resolution of the AMR scanner depends directlyon the dimensions of the AMR sensor,14, 19 in this case theresolution is 1/3 of the length L. For this reason, to increasethe spatial resolution following correctly the Rayleigh´s cri-terion, the use of smaller sensors is needed. For instance, toreach 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 signalproduced from the phantom. The S/N required in the mea-surements is approximately S/N = 70 dB, in order to allow adeconvolution process of acceptable quality.

In the first experiment analysis, we did not find signifi-cant differences when the solution of the MIP was carried outusing the theoretical or experimental PSF, respectively. Theinversion 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 tocontrol the poles and additive noise, respectively, during the

deconvolution process. Figure 7 illustrates the analysis of theMSD versus α, each point on the curves corresponding to aMMI. The quality of the reconstruction method is better whenusing 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 exhibita loss of spatial resolution and a decrease of the magnitudeof the restored images, which implicates a bad quality in therestoration because the MSD ≈ 1. In these cases ,the restoredand real MSI are very different. In contrast, for low values ofα, 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 avalue of α = 100, and 1000, as shown in Figures 8(e) and8(f). In those situations, the deconvolution process began toaffect the filtering MSI and the high noise suppression is con-verted to over filtering process and the MSI loses the spatialresolution, 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 andthe reconstructed MSI of the phantom respectively using γ 2.The restored image shows a good quality and high reductionof noise, and as a consequence is obtained a better spatialresolution compared to MMI, in this cases the MSD > 0.1.The best reconstructed MSI for high quality performance ofdeconvolution process is observed for αf = 0.135. From ananalysis of this image, we can conclude that the restored MSIhas 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 smalldifferences between their magnitudes and the peak value ofthe ferromagnetic particles concentration, distributed into themagnetic phantom.

Finally, another phantom “Sixlines” is realized using anacrylic table with six straight lines (recorded using a millingmachine) 0.2 cm deep, 0.3 cm wide and 9 cm long, into whichthe 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 anoise level α = 0.135. The mean amplitude value of the ferromagnetic particles concentration is about 79.96 mg/cm3.

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074701-10 Leyva-Cruz et al. Rev. Sci. Instrum. 85, 074701 (2014)

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 areobtained using the Wiener method adjusted with the best pa-rameters determined previously αf, γ 2, and β. The images ofthe magnetic map and the reconstructed one are displayed inFigures 10(a) and 9(b), respectively; these straight lines couldsimulate currents flowing inside of conductors or electronicboards. In this case, the maximum amplitude determined inthe 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 Filteringmethod, from two-dimensional magnetic maps measured bya setup of 12-channels of AMR sensors. The magnetic mapsare obtained from a separation z = 0.8 cm at room tempera-ture and the experimental spatial resolution for the imagingsystems is 0.3 cm. The procedure for obtaining the recon-structed MSI produces a small reduction of the additive noiseand increases the stability of the solution because the use ofWiener Filter. The amplitude and the spatial resolution of re-constructed images are modified by the filter parameters. Thiswork illustrates the importance of knowing the PSF and thefilter parameters to improve the quality of the restored image.This technique can be extended to solve the inverse problemof 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 reviewingthe language of the paper, and also thank CNPq and CLAFfor financial support.

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