Antenna Modeling and Reconstruction Accuracy of Time...

Hindawi Publishing CorporationInternational Journal of Biomedical ImagingVolume 2013 Article ID 343180 14 pageshttpdxdoiorg1011552013343180

Research ArticleAntenna Modeling and ReconstructionAccuracy of Time Domain-Based Image Reconstruction inMicrowave Tomography

Andreas Fhager1 Shantanu K Padhi2 Mikael Persson1 and John Howard3

1 Biomedical Engineering Division Department of Signal and Systems Chalmers University of Technology 41296 Gothenburg Sweden2 Curtin Institute of Radio Astronomy (CIRA) ICRAR Curtin University Perth WA 6102 Australia3 PRL Research School of Physics and Engineering Australian National University Canberra ACT 0200 Australia

Correspondence should be addressed to Andreas Fhager andreasfhagerchalmersse

Received 18 September 2012 Revised 13 January 2013 Accepted 21 January 2013

Academic Editor Kenji Suzuki

Copyright copy 2013 Andreas Fhager et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nonlinear microwave imaging heavily relies on an accurate numerical electromagnetic model of the antenna systemThe model isused to simulate scattering data that is compared to its measured counterpart in order to reconstruct the image In this paper anantenna system immersed in water is used to image different canonical objects in order to investigate the implication of modelingerrors on the final reconstruction using a time domain-based iterative inverse reconstruction algorithm and three-dimensionalFDTD modeling With the test objects immersed in a background of air and tap water respectively we have studied the impactof antenna modeling errors errors in the modeling of the background media and made a comparison with a two-dimensionalversion of the algorithm In conclusion even small modeling errors in the antennas can significantly alter the reconstructed imageSince the image reconstruction procedure is highly nonlinear general conclusions are very difficult to make In our case it meansthat with the antenna system immersed in water and using our present FDTD-based electromagnetic model the imaging resultsare improved if refraining from modeling the water-wall-air interface and instead just use a homogeneous background of water inthe model

1 Introduction

Microwave imaging has received significant attention in theresearch community during the last couple of decades as amodality that potentially could improve the diagnostics offor example breast cancer tumors Recent progress in thefield has been reviewed in [1] and [2] Today the research hascome to the stage where early clinical trials have been andare being performed [3ndash6]The results from the clinical workare promising but further development of the measurementsystems as well as of the image reconstruction algorithmsremains before the technique can be considered for dailyclinical practice

When performing microwave tomography the aim is toquantitatively reconstruct the dielectric parameters in theregion under test This involves solving a computationallychallenging nonlinear and ill-posed optimization problem

The image reconstruction algorithm utilizes measured datathat are compared against a corresponding numerical sim-ulation of the system and the dielectric profile is iterativelyupdated based on the difference between the simulation andthe measurement Even though this comparison requires arealistic numerical model for the best accuracy most of thepublishedworks have used 2Dmodels togetherwith a calibra-tion procedure to enable the comparison with experimentaldata Largely this can be attributed to a significant increasein the computational load when moving from 2D to three-dimensional (3D) modeling However the electromagneticscattering and consequently the reconstruction problemsare inherently a 3D problem Furthermore it is usually notpossible to create realistic antenna models in 2D except forline source antennas By using a 2Dmodel to solve the inversescattering problem inaccuracies will therefore inevitably beintroduced in the reconstructed image This problem has

2 International Journal of Biomedical Imaging

been identified by the research community and with theever increasing computational resources available these daysthe focus is now more and more turning to solving the full3D problem Recent works using 3D algorithm have beenreported in [7ndash11]

In this paper we show several examples where imageshave been reconstructed from scattering data in order todiscuss and illustrate the need for accurate modeling of theantenna system and its geometry to enable robust imagereconstruction The aim is also to get an understanding ofwhat accuracy we could realistically expect in the recon-struction and how it is affected by various modeling errorsExamples of targets placed in a surrounding of air are studiedand in an effort to approach more biologically relevantsettings we have studied examples where the antenna systemwas entirely immersed in water The examples studied in thispaper are entirely based on experimental measurement dataOur reconstruction algorithm described in [12] is based onFDTDmodeling to solve the forward scattering problem andthe adjoint Maxwellrsquos equations to compute gradients used inan iterative optimization procedure

The paper is organized as follows In Section 2 the theo-retical background of our work is described including FDTDmethods with theminimization procedureThe experimentalprototype is described in Section 3 In Section 4 the forwardmodeling is investigated and the corresponding imagingresults originating from experimental data are presented anddiscussed And finally the conclusions are drawn in Section 5

2 Theoretical Background

An iterative electromagnetic time-domain inversion algo-rithm has been used and applied in estimating the dielectricparameters of different test objects The foundation of thealgorithm is an electromagnetic solver based on the FDTDmethod [13] is used to numerically model the antennas andto simulate the field propagation inside the systemThe samesolver is used to compute the adjointMaxwell problemwhichis required for the gradient computation in the optimizationalgorithm The adjoint field is used extensively elsewhere invarious types of inverse problems see for example [14 15]The algoithm used here is described in [12] A correspoding2D version has also been described in detail in [16 17]

Our basic idea for solving the inverse electromagneticproblem is to use scattering measurements of widebandpulses for several transmitterreceiver combinations sur-rounding a region of interest and thereafter to compare themeasured data in the time domain with a correspondingnumerical simulation of the system In the first iteration onestarts with comparing the measurement with a simulation ofan empty antenna system Thus there is a difference betweenthe measured and the simulated data and this difference isused to update the dielectric properties inside the region ofinterest In this way the dielectric distribution is iterativelyrefined until the desired agreement between the simulatedand measured signals has been achieved The underlyingassumption for this approach is that as the difference between

the simulated and measured data is decreasing the recon-struction is also converging In other words the aim ofthe reconstruction procedure is to minimize the objectivefunctional 119865 defined as

119865 (120598120590) = int

119879

0

119872

sum

119898=1

119873

sum

119899=1

1003816100381610038161003816E119898119899 (120598 120590 119905) minus E119898

119898119899(119905)1003816100381610038161003816

2119889119905 (1)

where E119898119899(120598 120590 119905) is the calculated field from the computa-

tional model and E119898119898119899(119905) is the corresponding measured data

when antenna number 119898 has been used as transmitter andantenna 119899 as receiver119872 is the number of transmitters 119873 isthe number of receivers and 119879 is the duration of the pulse Ina 2DTM-mode formulation only one spatial field componentis used in (1) but for the 3D formulation it is necessary toinclude all three spatial components

In search for the minimum of the objective functional itis differentiated with respect to the dielectric components bya first order perturbation analysis In this way the Frechetderivatives with respect to the conductivity and the permit-tivity of the functional are used to define gradients in everygrid point inside the region The gradients are used withthe conjugate-gradient method together with the successiveparabolic interpolation line search to minimize the objectivefunctional The reconstruction procedure is then iteratedwith the objective functional as a measure to monitor theconvergence and to determine when the reconstruction iscompleted Usually the minimization procedure convergeswithin 10ndash20 iterations

In the 2D simulations it is not possible to construct arealistic antenna model but the transmitter is only modeledwith a hard point source in which the field strength isprescribed at the source position At the receiver locationsthe field values are sampled directly from the correspondingE-field component in the grid By contrast the 3D algorithmallows for realistic antenna models We are using the thin-wire approximation to model the monopoles [18] and theRVS with 50Ω impedance to model the feed at the transmit-ting receiving and inactive antennas [19 20] Furthermorethe ground plane in our experimental system is modeled asa perfect electric conductor that is the corresponding fieldcomponents in the FDTD grid are set to zero The wallsof the tank have dielectric properties close to those of airand are therefore neglected in the numerical model Outsidethe antenna system the computational grid is truncated bythe CPML absorbing boundary condition implying that theregion outside the antenna array is treated as open emptyspace In the case where the tank is filled with a liquid ithas been modeled as a cubic volume with side lengths equalto the inner measures of the tank and a height equal to thelevel of the liquid Even in this case we have not modeled thetankmaterial but instead treated everything outside as emptyspace and terminated the computation domain with CPMLA more detailed discussion and validation of the FDTDantenna arraymodel with a comparison against experimentaldata measured with the antennas in air can be found in ourprevious work [21]

As alreadymentioned the solution of the inverse problemheavily relies on the comparison between the measured and

International Journal of Biomedical Imaging 3

the simulated scattering data To compensate for systematicmodeling errors a calibration procedure of themeasured datais used such that

119864119898

cal (119891) =119878119898

scat (119891)

119878119898

ref (119891)119864119904

ref (119891) (2)

119878119898

scat is the measured reflection and transmission coefficientsof the test object 119878119898ref(119891) is a reference measurement ofan empty system and 119864119904ref(119891) is a corresponding referencesimulation Finally 119864119898cal(119891) is the calibrated data used forcomparing the FDTD simulations in the reconstructionprocess This calibration procedure has been applied to themeasured data for both the 2D and the 3D imaging examplesin this paper

In our experimental prototype the antennas are posi-tioned in a plane The possibility to accurately reconstructout-of-plane objects is thus very limited to do so it wouldbe necessary also to make additional measurements outsidethe antenna plane To allow imaging with the 3D algorithmof a test object with finite height we implemented a heuristicpseudo-3D technique that assumed constant properties of thetest object as a function of height 119911 above the ground planeThe gradients computed in the grid cell plane immediatelyabove the ground plane were copied upwards to the heightof the test object This method needs a priori informationabout the height of the reconstructed target Since thereconstruction problem is both nonlinear and ill-posed theresulting image strongly depends on the adopted regulariza-tion technique the initialization of the reconstruction andalso the spectral content of the pulse Here we used the sametechniques as described in [12] to overcome these challenges

3 Experimental Setup

Themeasurement strategy is to measure the multistatic scat-tering matrix at a large number of frequencies and to use thatdata to generate a time-domain pulse via an inverse Fouriertransformation In the experimental system 20 monopoleantennas each of length 195mm and diameter 08mm arearranged evenly distributed on a circle with radius 100mmThe circle of antennas is centered on a square ground planewith side length 250mm mounted at the bottom of a tankmade of 1 cm thick perspex sheets with inner measures 350 times350mm2 To measure the multistatic matrix each antenna isoperated as a transmitter as well as a receiverThemicrowavemeasurements aremadewith network analyzerAgilent E8362B PNA which is a two-port network analyzer To fullycontrol the experiment a 232 switch multiplexer moduleCytec CXM128-S-W is used to automatically connect anddisconnect the different combinations of antenna pairs to thenetwork analyzer Figure 1 shows a photograph of the antennaarray

4 Results and Discussion

In our previous publication [21] a detailed studywasmade ofthe FDTDmodeling compared to measured data of an emptysystem and the accuracy of the modeling was verified One

(a) The measurement system

(b) The antenna array

Figure 1 (a)The system consists of a VNA a switch and an antennaarray (b) Closeup of the antenna array placed inside a tank Themonopoles are seen mounted in a circle over the ground plane Theentire antenna system is mounted inside a tank made of perspexsheets

of the aims of the present paper is to study how errors inthe antenna model for example the monopole length of theFDTD model impact the reconstruction To do so we studythe antenna modeling and image reconstruction both whenthe tank is empty and when it is filled with water The reasonwhy it is interesting to study the antenna system immersed inwater is that water is a good model for the matching mediumthat has to be used when applying microwave tomography toimaging the interior of the human body Without a matchingmedium the majority of the irradiated energy would bereflected from the skin thereby never penetrating into thebody and producing useful data

To enable a quantitative evaluation of the accuracy of thereconstructed images we have adopted the relative squarederror of the image and for the permittivity image it isdefined in (3) and analogously for the conductivity imageThe integration of the relative squared error is made over thereconstruction domainΩ where 119903 lt 119877rd

120575 =

intΩ

10038161003816100381610038161003816120598original minus 120598reconstructed

10038161003816100381610038161003816

2

119889119878

intΩ

10038161003816100381610038161003816120598original minus 120598background

10038161003816100381610038161003816

2

119889119878

(3)

41 Reconstruction of a Single Dielectric Target in Air Withthe purpose to study how the reconstructed image is affectedby errors in the length of the monopole antenna model wefirst studied a single dielectric target in an otherwise emptyantenna array The imaging situation was the same as in ourprevious publication [12] that is a single target made of


sunflower oil surrounded by air The dielectric properties ofthe target were 120598

119903= 27 and 120590 = 0015 Sm at 23 GHz It

was shaped like a cylinder with diameter 56mm and height20mm and was placed at 14mm offset in the y direction fromthe center of the antenna array It had constant propertiesalong its height in the z direction and thus it is only necessaryto show a cross-sectional slice of the dielectric profile TheFDTD model data is summarized in Table 1 A cylindricalvolume of height 20mm and radius 119877rd = 90mm centeredin the antenna array was used as the reconstruction domaintogether with the pseudo-3D approach described earlier Toinvestigate the accuracy of the reconstructed image withrespect to the modeling different lengths of the monopoleswere used in the numerical electromagnetic model Thelength of the monopoles in the antenna system is 195mmbut images were reconstructed modeling the length as 16 1820 22 and 24mm respectively This resulted in a changein the computed resonance frequency and the associatedsignal strength The reconstruction results together with anillustration of the original dielectric distribution are shown inFigure 2 The results obtained with 20mm monopole lengthin Figures 2(d) and 2(j) are identical to what was publishedearlier [12] and represent a scenario where the numericalmodeling at the given grid size is as close as possible to theexperimental system As a measure of reconstruction accu-racy the relative error as defined in (3) has been calculatedfor each reconstructed image and it is shown in the figurebelow the respective reconstructionThe reconstructed objectpermittivities are on average about 120598

119903= 21 22 24 27 30

respectively for the variousmonopole lengthsTheminimumvalue is 125 smaller and the maximum value is 25 largerthan what was obtained for the 20mmmonopole lengthTherecontructions of the conductivity are however not accuratebut it is clearly evident that the artifacts increase the more themonopole lengths deviate from the real valueThe reasonwhythe reconstruction is less accurate is that the difference in theimaginary part of the complex dielectric permittivity betweenthe object and background is only a fraction in comparisonwith the difference in the real part Reconstruction errors inthe real part therefore overwhelm attempts to reconstruct theimaginary part and consequently the accuracy is reducedCompared to the original dielectric profile however themostaccurate reconstruction of the permittivity is obtained for the18mmmonopole This reflects the deviaton of half a grid cellbetween the 20mm antenna model and the precise length ofthe monopole with the real length being 195mm But alsothere is a tolerance in the cutting of the monopoles of about05mm Numerical uncertainties in the FDTD solution anderrors in the dielectric measurement of the sunflower oil areother reasons

In Figure 3 the functional values of the reconstructionshave been plotted In all cases the starting values have beennormalized to one Firstly these plots illustrate the conver-gence of the reconstruction process but it also shows that thelowest functional value was obtained for the reconstructionwith the 20mm monopole A nice illustration of the ill-posedness of the problem is that the difference betweenthe 20mm and the 22mm case is on the verge of beingnegligible even if the difference in the reconstructed image

Table 1 Specifications of the 3D FDTD modeling and reconstruc-tion parameters for the sun flower oil target in an otherwise emptyantenna system

FDTD grid 149 times 149 times 38Grid size length 2mmCPML 7 layersPulse center frequency 23 GHzPulse FWHM bandwidth 23 GHzWater level No water in tankBackground properties 120598

119903= 10 120590 = 00 Sm

Antenna model Thin wireFeed model 50Ω RVS

is certainly not Another conclusion from this result is thata model error in the antenna length of only one grid cellresulted in a considerable change in the reconstructed imageIn summary these results clearly show the need of using anaccurate antenna length in order to maximize the accuracyof the reconstructed image

42 Evaluation of the Forward Simulation with the AntennaArray Immersed in Water In this section we study thesituation when the antenna array tank was filled withtap water having dielectric properties 120598

119903= 775 120590 =

005 Sm at 05GHz We also present a comparison betweenmeasured data and corresponding computed reflection andtransmission coefficients By replacing the air in the tankwith water we further approach biologically relevant imagingscenarios With the aim to study how modeling errors affectthe reconstructed images we first studied how the forwardmodeling of the antenna system was affected by differentantenna modeling errors

421 Full 3D Model The experimental situation was suchthat the tank was filled with ordinary tap water up to alevel of 50mm above the ground plane In the FDTD modelmeasured dielectric values of the water at 05 GHz were usedas this was in the center of the frequency spectrum usedfor the imaging In the FDTD modeling care was also takento represent the physical reality as accurately as possibleand the corresponding settings are summarized in Table 2Modeling errors were introduced by varying some of theparameters in this table Unfortunately it is not viable toshow scattering data for all antenna combinations but insteadonly a few representative cases are shown Measured andsimulated reflection and transmission coefficients betweentwo adjacent antennas using the model parameters fromTable 2 are shown in Figure 4 As can be seen the agreementbetween the calculated and the measured data is very closeThe calculated resonant frequency is 070GHz comparedwith the measured resonant frequency of 067GHz Thereare ripples with approximately the same magnitude bothin calculated and measured data and where some ripplesagree with each other and some do not The details aboutthese ripples are further discussed in the following sectionsThe measured transmission coefficients are on average below


1

15

2

25

3

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(a) Original permittivity

1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Permittivity 16mm 120575 = 019

1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)

(c) Permittivity 18mm 120575 = 015

1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)

(d) Permittivity 20mm 120575 = 018

1

15

2

25

3

0 50

minus50

0

50119884

(mm

)

minus50

119883 (mm)

(e) Permittivity 22mm 120575 = 023

1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(f) Permittivity 24mm 120575 = 029

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(g) Original conductivity

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(h) Conductivity 16mm 120575 = 38

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(i) Conductivity 18mm 120575 = 10

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(j) Conductivity 20mm 120575 = 10

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(k) Conductivity 22mm 120575 = 15

minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)

(l) Conductivity 24mm 120575 = 33

Figure 2 Reconstructed images of the sunflower oil target using monopole lengths 16 18 20 22 and 24mm respectively Both permittivityand conductivity images are shown and in (a) and (g) the original dielectric distribution has been illustratedThe original target had 120598

119903= 27

and 120590 = 0015 Sm at 23 GHz and had diameter 56mm

minus15 dB compared to the calculated transmission coefficientbelow minus12 dB

422 Hard Source FeedModel Due to its simplicity the hardsource model is very appealing to use in FDTD simulationsHowever it is not as accurate as the RVS when modeling

the monopole feed To quantify the errors associated witha hard source we investigate its applicability to model themonopole feed In the first example we replaced the RVSfeed model with a hard source In the second example weused the hard source and completely removed the modelingof the rest of the monopole The use of a hard source alsoimplied that we could not use the transmitting antenna in the


0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm

Figure 3 Normalized functional values as a function of the iterationnumber for the reconstructions of the sunflower target

Table 2 Specifications of the 3D FDTDmodeling when the antennasystem was filled with water

FDTD grid 179 times 179 times 35Grid size length 2mmCPML 7 layersPulse center frequency 05GHzPulse FWHM bandwidth 05GHzWater level in tank 50mmWater properties 120598

119903= 775 120590 = 005 Sm

Background properties 120598119903= 10 120590 = 00 Sm


gradient computation as we do not calculate the reflectioncoefficient with the hard source model Calculation of thereflection coefficient requires knowledge of the reflectedwave but since the E-field is directly set in the source cellno update of the field will be made due to reflected wavesInstead we used only transmission data for the followingreconstructions As an illustration of the impact of the hardsource on the simulated scattering data 119878

21for adjacent

antennas has been plotted in Figure 5 For convenience thesame measured and simulated data for the full 3D model asin Figure 4 has also been replotted in the same graph As canbe seen in the first example the RVS feed model improvesthe transmission coefficient data over the hard source modeland in the second example the system becomes very lossydue to the inaccuratemodel and therefore the deviation fromthe measured data increases

423 Open Water Model Solving the reconstruction prob-lem is a computationally very demanding problem and onestrategy to reduce the simulation time is to reduce the size

02 04 06 08 1Frequency (GHz)

minus20

minus15

minus10

minus5

0

11987811

(dB)

(a) Reflection coefficient 11987811


MeasurementSimulation

minus40

minus30

minus20

minus10

0

11987821

(dB)

(b) Transmission coefficient 11987821

Figure 4 Comparison between measured and simulated reflection(11987811) and transmission (119878

21) coefficients over the frequency range of

interest for the imaging examples

of the computational domainTherefore we have investigatedthe need for modeling the exact volume of the water in thetank and instead modeled the water as a background directlyterminated by CPML The CPML is absorbing any outgoingwave and the result inside the computation domain is thesame as if the simulation was made in an infinitely largespace In this FDTD simulation the entire computationalgrid was therefore assigned the properties of water and asthere were no need to model the water-air interface in thecomputational domain the CPML could be moved closer tothe antennas and thus the computational domain reduced InFigure 6 the corresponding reflection 119878

11 and transmission

11987821 coefficients have been plotted For comparison the figures

also contain the measured data and the data simulated with



MeasurementSimulation full 3D model

Simulation monopole-hard feedSimulation hard source only

minus80

minus60

minus40

minus20

0

11987821

(dB)

Figure 5 Measurement and simulations of the transmission coeffi-cients 119878

21between two adjacent antennas For ease of comparison

the transmission data from Figure 4 has been kept and plottedtogether with the case when (1) The RVS has been replaced by ahard source and (2) the wire model has been removed and only ahard source has been used as transmitter

the full 3D FDTD model from Figure 4 The 11987811

data showhow the computed resonance frequency has increased about05GHz and we can also see how the fast varying ripple dueto the water-air interface reflections has vanished Similarchanges can also be seen in the 119878

21data

424 2D Model We have also made a comparison with a2D computational modelThe reason is that the computationtime is significantly reduced compared to the 3D caseTherefore a good understanding is desired of when the 2Dapproximation is applicable for imaging In this section weshow computed transmission data obtained with a 2Dmodeland also here two examples are considered The first is themodeling of the volume of water in the tank as a 350times350mmsquare in the 2D FDTD grid In Table 3 the correspondingFDTD model parameters are summarized In analogy withthe 3D open water background model the second examplewas a 2D homogeneous background of water terminatedwithCPML In Figure 7 transmission data is plotted for thesetwo cases and again the measured data and the full 3Dmodel data are shown For the simulation data with the 2Dsquare tank model the deviation from the measured datais of similar magnitude as for the case as with the hardsource feed in Figure 5 at least around the center frequency05GHz However the amplitude of the ripple is larger andthe deviation ismuchmore significantThe fast varying rippleis clearly seen and is primarily due to the reflections fromthe water-air interface in the model For the simulation in thehomogeneous background this ripple is vanished but in thiscase the computed data do not showmuch resemblance at allto the measured data


minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)


MeasurementSimulation full 3D modelSimulation open water model


minus40

minus30

minus20

minus10

0

11987821

(dB)


Figure 6 (a) Reflection coefficients with the antennas modeled inan infinite water background For comparison the originalmeasuredand simulated data is shown (b) Corresponding transmissioncoefficients for two adjacent antennas

43 Reconstruction of Targets Immersed in Water In thefollowing section we show the reconstructed images fromexperimental scattering data Two different target modelswere used and reconstructed using the various antennaarray models discussed above The aim was to investigatethe impact on the reconstruction caused by the differentmodeling introduced in Section 42

We have performed image reconstruction of targetsimmersed into ordinary tap water In comparison to air aFDTD simulation of water ismore time consuming due to thehigher permittivity and the corresponding need for shortertime stepping in the FDTD algorithm To save some com-putation time the reconstructions were therefore made using



FDTD grid 179 times 179Grid size length 2mmCPML 7 layersPulse center frequency 05GHzPulse FWHM bandwidth 05GHzWater properties 120598

119903= 775 120590 = 005 Sm


Transmitting antenna model Hard sourceReceiving antenna model Field is sampled



2D simulation with tank model2D simulation open water model

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)

Figure 7 Transmission coefficients for the 2D FDTD model withthe tankmodel andwith the homogeneous background of water Forease of comparison both the measured data and the simulation withthe full 3D model are also shown

a multigrid technique where 10 iterations where performedon a (90 times 90 times 16) computational domain with grid sizelength 4mm The final reconstruction of the 101015840th iterationwas then taken as a starting point for 10 additional iterationson a (179 times 179 times 31) domain with grid size length 2mmThe electromagnetic pulse had center frequency 05GHz andFWHM bandwidth 05GHz The same pseudo-3D approachas previously described was used in a reconstruction domainof height 50mm and radius 119877rd = 80mm Two differenttargets were used in the reconstructions The first was madeof a mixture of deionized water and ethanol resulting inpermittivity 120598

119903= 57 and conductivity 120590 = 011 Sm at

the frequency 05GHz The mixture was filled up to height50mm in a thin-walled plastic cup with diameter 42mmand immersed into the water for the measurement Thesecond target was a plastic rod with permittivity 120598

119903asymp 25

conductivity 120590 asymp 0 Sm and diameter 15mm

431 Full 3D Model Two different scenarios were recon-structed the first with the center of the waterethanol target

positioned 39mm from the center point of the antenna arrayIn Figure 8 an illustration of the original target is showntogether with the reconstructed images

The second scenario that was reconstructed includedboth targets the waterethanol mixture and the plastic targetand these results are shown in Figure 9 In both cases we seethat in the permittivity both size and position of the targetshave been accurately reconstructedThe relative error for thissecond example with the two targets was computed to be120575 = 042 In the conductivitywe can perhaps see an indicationof a target in the appropriate positions but the image regionis also cluttered with other artifacts and comparison withthe relative error is notmeaningful Furthermore the absolutevalues of the permittivity are not in total agreement with theoriginal values especially not for the small plastic targetWiththe same arguments as for the previous object in air wherethe conductivity also was badly reconstructed it is not toexpect anything else in this case The difference between thebackground and the target made of waterethanol mixture isabout RΔ120598lowast

119903 = 20 in the real part but for the imaginary

part only IΔ120598lowast119903 = 2 For the plastic rod the difference is

RΔ120598lowast119903 = 75 andIΔ120598lowast

119903 = 2

432 Hard Source Feed Model To assess the question of howthe reconstructed image is affected by inaccuracies in theforward modeling we have taken the example with the twotargets the plastic rod and the waterethanol mixture andperformed reconstructions with distorted electromagneticmodels

Using the two altered antenna models with hard sourcefeed new images of the original targets from Figures 9(a)and 9(b) have been reconstructed and shown in Figure 10In (a) and (b) in this figure the RVS was replaced withthe hard source feed and in (c) and (d) the model of themonopole wire was completely removed and only the hardsource was used as transmitter model The relative errors inthe permittivity imageswere computed to be 120575 = 054 and 120575 =058 respectively One can clearly see that for the case whereonly the hard source was used to model the transmitter thedistortions of the reconstructed images are also the largestThis also corresponds to the case where the S-parameter datadeviate the most from the measured data in Figure 5 How-ever one should not believe that there exists a simple relationbetween the deviation between the measured and simulateddata on one hand and the errors in the reconstructed imageson the other hand It is instead a highly nonlinear relationand a balance between the measurement the reconstructionalgorithm the regularization themodeling accuracy and thecalibration procedure To summarize so far the results shownindicate that the accuracy of the reconstructed image is highlyinfluenced by the details in the antenna models and in thepropagation model So far we have also modeled the volumeof water according to measurements of the level in the tank

433 Open Water Model We have investigated the need ofactually modeling the exact extent of the water volume inthe tank For this experiment we used the RVS feed thin-wire monopole model and the algorithmic settings were


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(d) Full 3D model 120590 120575 = 29

Figure 8 (a and b) Original configuration of the waterethanol target immersed in water (c and d) Reconstructions using the 3D algorithm

otherwise identical to the previous examples but with thedielectric properties of water assigned to all the grid cellsin the computational domain which was then terminatedby seven layers of CPML The reconstruction can be seenin Figure 11 Interestingly the results are improved over thereconstructions made with the tank model and for examplethe absolute values of the objects are better estimated withthe 3D algorithm The improvement is confirmed by therelative error which was calculated to be 120575 = 036 forthe permittivity image a decrease of about 14 comparedto the reconstruction with the full 3D model in Figure 9These results might be a bit surprising and contradictory tothe idea that the better the forward model the better thereconstruction However we do not believe that it is the caseIf we examine the scattering data in Figure 6 carefully wesee that even if the data simulated with the numerical tankmodel contains a similar ripple of the same magnitude as themeasured data the agreement in the details is not perfectOn average one can however approximately estimate themodeling errors in the two cases to be of similarmagnitude incomparison to the measured data Even if we have carefully

created the model the numerical grid is ultimately limitingthe resolution and causing simulation errors A refined gridshould improve the modeling accuracy but that would alsoimply that the problem size increases beyond what can bepractically handled by our computer cluster due to memoryand simulation time requirements These results suggest thatwe in this situation are better off by using the simpleropen water model and relying on the calibration to resolvethe remaining discrepancy between the simulated and themeasured data This is not a result that is obvious to predictbut instead a result of the nonlinear property of the problemIn practice the optimal numerical model would thus haveto be determined from case to case in different imagingsituations and with different imaging systems

434 2D Model For comparison we have also performedimage reconstruction of the same targets using a 2D versionof the algorithm The first reconstruction made with thewater tank modeled as a square of the single target asshown in Figures 8(a) and 8(b) and the reconstructed images


20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29

Figure 9 (a and b) Original configuration of the waterethanol and plastic targets immersed in water (c and d) Reconstructions using the3D algorithm

are shown in Figures 12(a) and 12(b) The correspondingreconstruction of the scenario with two objects as shownin Figures 9(a) and 9(b) is shown in Figures 12(c) and12(d) In these reconstructions we hardly see any object atall Previous reconstructions made in air of sunflower oiltargets with the 2D algorithm published in [12] showed aserious distortion of the size and dielectric values of thetargets but a qualitatively correct image could usually beobtained The reason for the distortion of the reconstructedobjects could be attributed to 2D approximation errors as thetargets only had the same height as the monopoles It is abit surprising that the reconstructed images shown in thispaper of targets immersed in water hardly shows anythingat all as the electromagnetic waves now in fact should bebetter confined inside the layer constituted by the waterand therefore better conforming to the 2D approximationEnclosure is provided by the ground plane on the bottom andthe impedance step in the water-air interface However theseimages are representative for our results showing that robust

image reconstruction is not possible to achieve in water usingthis 2D algorithm The situation can be somewhat improvedby employing the open water modeling also in 2D that is wereplace the square water tank model in the background of airwith a modeling of a homogeneous background of water tosimulate an open domainThe corresponding reconstructionis plotted in Figure 13 In this reconstruction we clearly seethe two objects appearing where the large object containsa spurious hole and the dielectric values are not quite closeto the original values Even if the situation is improved thisreconstruction cannot be considered satisfactory A similarspurious hole in the reconstruction can be seen also in thereconstructions from shortmonopole lengths in Figure 2 andarises due to themodeling errors of the antenna Furthermoreassociated with every target object is an optimal spectralcontent that will produce an optimal image [17] and holesusually appear when the spectral content is moved towardshigher frequencies With these two causes for the inaccuracyin the image we have not been able to improve the outcome


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Thin-wire with hard feed 120598119903 120575 =054

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Thin-wire with hard feed 120590 120575 = 30

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Hard source transmitter 120598119903 120575 = 058

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Hard source transmitter 120590 120575 = 87

Figure 10 Reconstructions of the two target model from Figures 9(a) and 9(b) Here (a) and (b) show the reconstructed result when replacingthe RVS feed of the monopole with a hard source In (c) and (d) the results when removing the wire model of the antenna and using only ahard source as the transmitter model and on the receiver side the field was directly sampled in the corresponding grid point

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open water model 120598119903 120575 = 036

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open water model 120590 120575 = 71

Figure 11 For an open space of water (a) and (b) show the dielectric distribution reconstructed with the 3D model


50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36

Figure 12 Reconstructed images using the 2DmodelThe water tank was modeled as a square in a background medium consisting of air (a)and (b) Reconstruction of the original target from Figures 8(a) 8(b) 8(c) and 8(d) Reconstruction of the original target from Figures 9(a)and 9(b)

any further by tuning the parameters in the reconstructionalgorithm

5 Conclusion

We have shown successful reconstruction of single andmultiple targets immersed in water using a FDTD-based 3Dreconstruction algorithm By investigating various imagingscenarios we have also assessed the question of how theaccuracy in the numerical forward model affects the recon-structed image The results show that a realistic model of theantennas is necessary to achieve robust and reliable imagingThe same results also indicate that small modeling errorssuch as in the length of the monopole can have a clearlyevident impact on the resulting image For a simple antennasuch as a monopole the modeling is very simple and it is notdifficult to produce a reasonably accurate numerical modelusing the thin-wire approximation and the RVS However

there is always an inherent limitation due to the finite gridsize and when moving to more advanced antennas suchas patch antennas modeling accuracy will become a morechallenging task Accurate antenna modeling requires a fineFDTD grid but at the same time we must keep the grid ascoarse as possible to lower the simulation time and memoryrequirement Furthermore we have seen that when fillingthe tank containing the antenna system with normal tapwater the water-air interface causes significant reflectionsthat are clearly identified in the scattering data Despite thiseffect the accuracy of the reconstructed results is in factimproved when we instead use an open water model Thisresult shows that an apparent improvement in the antennamodeling and the corresponding computed scattering datais in fact not beneficial when it comes to imaging Onepossible explanation could be that when comparing differentmodels with similar modeling errors the nonlinearity of thereconstruction problem makes it impossible to predict theoutcome Instead the result will be strongly case dependent


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56

Figure 13 Reconstructed images using the open water 2DmodelThewater tank wasmodeled as a homogeneous background of water wherethe computation domain was terminated by CPML to model an open domain (a) and (b) Reconstruction of the original target from Figures9(a) and 9(b)

and to fully predict the result one would have to performdetailed studies of the particular imaging scenario

Acknowledgments

This work was supported in part by VINNOVA withinthe VINN Excellence Center Chase and in part by SSFwithin the Strategic Research Center Charmant and by theAustralian Research Council (ARC) Australia under GrantDP-0667064

References

[1] E C Fear P M Meaney and M A Stuchly ldquoMicrowaves forbreast cancer detectionrdquo IEEE Potentials vol 22 no 1 pp 12ndash18 2003

[2] N K Nikolova ldquoMicrowave imaging for breast cancerrdquo IEEEMicrowave Magazine vol 12 no 7 pp 78ndash94 2011

[3] M Klemm I Craddock J Leendertz A Preece and RBenjamin ldquoExperimental and clinical results of breast cancerdetection using UWB microwave radarrdquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium vol 1ndash9 pp 3014ndash3017 July 2008

[4] P M Meaney MW Fanning T Raynolds et al ldquoInitial clinicalexperience with microwave breast imaging in women withnormal mammographyrdquo Academic Radiology vol 14 no 2 pp207ndash218 2007

[5] S P Poplack T D TostesonWAWells et al ldquoElectromagneticbreast imaging results of a pilot study in women with abnormalmammogramsrdquo Radiology vol 243 no 2 pp 350ndash359 2007

[6] M Shere A Preece I Craddock J Leendertz and M KlemmldquoMultistatic radar first trials of a new breast imaging modalityrdquoBreast Cancer Research vol 11 supplement 2 article 05 2009

[7] Q Fang P M Meaney and K D Paulsen ldquoViable three-dimensional medical microwave tomography theory andnumerical experimentsrdquo IEEE Transactions on Antennas andPropagation vol 58 no 2 pp 449ndash458 2010

[8] T RubaeligkO S Kim andPMeincke ldquoComputational validationof a 3-D microwave imaging system for breast-cancer screen-ingrdquo IEEE Transactions on Antennas and Propagation vol 57no 7 pp 2105ndash2115 2009

[9] S Y Semenov A E Bulyshev A Abubakar et al ldquoMicrowave-tomographic imaging of the high dielectric-contrast objectsusing different image-reconstruction approachesrdquo IEEE Trans-actions on Microwave Theory and Techniques vol 53 no 7 pp2284ndash2293 2005

[10] D W Winters J D Shea P Kosmas B D van Veen and S CHagness ldquoThree-dimensional microwave breast imaging dis-persive dielectric properties estimation using patient-specificbasis functionsrdquo IEEE Transactions on Medical Imaging vol 28no 7 pp 969ndash981 2009

[11] Z Q Zhang and Q H Liu ldquoThree-dimensional nonlinearimage reconstruction formicrowave biomedical imagingrdquo IEEETransactions on Biomedical Engineering vol 51 no 3 pp 544ndash548 2004

[12] A Fhager S K Padhi and J Howard ldquo3D image reconstructionin microwave tomography using an efficient FDTD modelrdquoIEEEAntennas andWireless Propagation Letters vol 8 pp 1353ndash1356 2009

[13] A Taove and S C Hagness Computational ElectrodynamicsThe Finite-Difference Time-Domain Method Artech HouseBoston Mass USA 3rd edition 2005

[14] Q Fang P M Meaney S D Geimer S Y Streltsov and KD Paulsen ldquoMicrowave image reconstruction from 3D fieldscoupled to 2D parameter estimationrdquo IEEE Transactions onMedical Imaging vol 23 pp 475ndash484 2004

[15] C T Liauh R G Hills and R B Roemer ldquoComparison of theadjoint and influence coefficientmethods for solving the inversehyperthermia problemrdquo Journal of Biomechanical Engineeringvol 115 no 1 pp 63ndash71 1993

[16] A Fhager and M Persson ldquoComparison of two image recon-struction algorithms for microwave tomographyrdquo Radio Sci-ence vol 40 no 3 Article ID RS3017 2005

[17] A Fhager P Hashemzadeh and M Persson ldquoReconstructionquality and spectral content of an electromagnetic time-domain


inversion algorithmrdquo IEEE Transactions on Biomedical Engi-neering vol 53 no 8 pp 1594ndash1604 2006

[18] R M Makinen J S Juntunen and M A Kivikoski ldquoAnimproved thin-wire model for FDTDrdquo IEEE Transactions onMicrowave Theory and Techniques vol 50 no 5 pp 1245ndash12552002

[19] J Juntunen ldquoNote on the S-parameter and input impedanceextraction in antenna simulations using FDTDrdquoMicrowave andOptical Technology Letters vol 28 pp 8ndash11 2001

[20] O P M Pekonen and J Xu ldquoRigorous analysis of circuitparameter extraction from an FDTD simulation excited witha resistive voltage sourcerdquo Microwave and Optical TechnologyLetters vol 12 no 4 pp 205ndash210 1996

[21] S Nordebo A Fhager M Gustafsson and M Persson ldquoMea-sured antenna response of a proposed microwave tomographysystem using an efficient 3-D FDTDmodelrdquo IEEEAntennas andWireless Propagation Letters vol 7 pp 689ndash692 2008


been identified by the research community and with theever increasing computational resources available these daysthe focus is now more and more turning to solving the full3D problem Recent works using 3D algorithm have beenreported in [7ndash11]

In this paper we show several examples where imageshave been reconstructed from scattering data in order todiscuss and illustrate the need for accurate modeling of theantenna system and its geometry to enable robust imagereconstruction The aim is also to get an understanding ofwhat accuracy we could realistically expect in the recon-struction and how it is affected by various modeling errorsExamples of targets placed in a surrounding of air are studiedand in an effort to approach more biologically relevantsettings we have studied examples where the antenna systemwas entirely immersed in water The examples studied in thispaper are entirely based on experimental measurement dataOur reconstruction algorithm described in [12] is based onFDTDmodeling to solve the forward scattering problem andthe adjoint Maxwellrsquos equations to compute gradients used inan iterative optimization procedure

The paper is organized as follows In Section 2 the theo-retical background of our work is described including FDTDmethods with theminimization procedureThe experimentalprototype is described in Section 3 In Section 4 the forwardmodeling is investigated and the corresponding imagingresults originating from experimental data are presented anddiscussed And finally the conclusions are drawn in Section 5

2 Theoretical Background

An iterative electromagnetic time-domain inversion algo-rithm has been used and applied in estimating the dielectricparameters of different test objects The foundation of thealgorithm is an electromagnetic solver based on the FDTDmethod [13] is used to numerically model the antennas andto simulate the field propagation inside the systemThe samesolver is used to compute the adjointMaxwell problemwhichis required for the gradient computation in the optimizationalgorithm The adjoint field is used extensively elsewhere invarious types of inverse problems see for example [14 15]The algoithm used here is described in [12] A correspoding2D version has also been described in detail in [16 17]

Our basic idea for solving the inverse electromagneticproblem is to use scattering measurements of widebandpulses for several transmitterreceiver combinations sur-rounding a region of interest and thereafter to compare themeasured data in the time domain with a correspondingnumerical simulation of the system In the first iteration onestarts with comparing the measurement with a simulation ofan empty antenna system Thus there is a difference betweenthe measured and the simulated data and this difference isused to update the dielectric properties inside the region ofinterest In this way the dielectric distribution is iterativelyrefined until the desired agreement between the simulatedand measured signals has been achieved The underlyingassumption for this approach is that as the difference between

the simulated and measured data is decreasing the recon-struction is also converging In other words the aim ofthe reconstruction procedure is to minimize the objectivefunctional 119865 defined as

119865 (120598120590) = int

119879

0

119872

sum

119898=1

119873

sum

119899=1

1003816100381610038161003816E119898119899 (120598 120590 119905) minus E119898

119898119899(119905)1003816100381610038161003816

2119889119905 (1)

where E119898119899(120598 120590 119905) is the calculated field from the computa-

tional model and E119898119898119899(119905) is the corresponding measured data

when antenna number 119898 has been used as transmitter andantenna 119899 as receiver119872 is the number of transmitters 119873 isthe number of receivers and 119879 is the duration of the pulse Ina 2DTM-mode formulation only one spatial field componentis used in (1) but for the 3D formulation it is necessary toinclude all three spatial components

In search for the minimum of the objective functional itis differentiated with respect to the dielectric components bya first order perturbation analysis In this way the Frechetderivatives with respect to the conductivity and the permit-tivity of the functional are used to define gradients in everygrid point inside the region The gradients are used withthe conjugate-gradient method together with the successiveparabolic interpolation line search to minimize the objectivefunctional The reconstruction procedure is then iteratedwith the objective functional as a measure to monitor theconvergence and to determine when the reconstruction iscompleted Usually the minimization procedure convergeswithin 10ndash20 iterations

In the 2D simulations it is not possible to construct arealistic antenna model but the transmitter is only modeledwith a hard point source in which the field strength isprescribed at the source position At the receiver locationsthe field values are sampled directly from the correspondingE-field component in the grid By contrast the 3D algorithmallows for realistic antenna models We are using the thin-wire approximation to model the monopoles [18] and theRVS with 50Ω impedance to model the feed at the transmit-ting receiving and inactive antennas [19 20] Furthermorethe ground plane in our experimental system is modeled asa perfect electric conductor that is the corresponding fieldcomponents in the FDTD grid are set to zero The wallsof the tank have dielectric properties close to those of airand are therefore neglected in the numerical model Outsidethe antenna system the computational grid is truncated bythe CPML absorbing boundary condition implying that theregion outside the antenna array is treated as open emptyspace In the case where the tank is filled with a liquid ithas been modeled as a cubic volume with side lengths equalto the inner measures of the tank and a height equal to thelevel of the liquid Even in this case we have not modeled thetankmaterial but instead treated everything outside as emptyspace and terminated the computation domain with CPMLA more detailed discussion and validation of the FDTDantenna arraymodel with a comparison against experimentaldata measured with the antennas in air can be found in ourprevious work [21]

As alreadymentioned the solution of the inverse problemheavily relies on the comparison between the measured and



119864119898

cal (119891) =119878119898

scat (119891)

119878119898

ref (119891)119864119904

ref (119891) (2)

119878119898












120575 =

intΩ


10038161003816100381610038161003816

2

119889119878

intΩ


10038161003816100381610038161003816

2

119889119878

(3)




119903= 27 and 120590 = 0015 Sm at 23 GHz It


119903= 21 22 24 27 30





119903= 10 120590 = 00 Sm




119903= 775 120590 =




1

15

2

25

3

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50119884

(mm

)

minus50

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)



119903= 27






0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm




119903= 775 120590 = 005 Sm




21for adjacent




minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)






11 and transmission







minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References


























119864119898

cal (119891) =119878119898

scat (119891)

119878119898

ref (119891)119864119904

ref (119891) (2)

119878119898












120575 =

intΩ


10038161003816100381610038161003816

2

119889119878

intΩ


10038161003816100381610038161003816

2

119889119878

(3)




119903= 27 and 120590 = 0015 Sm at 23 GHz It


119903= 21 22 24 27 30





119903= 10 120590 = 00 Sm




119903= 775 120590 =




1

15

2

25

3

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50119884

(mm

)

minus50

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)



119903= 27






0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm




119903= 775 120590 = 005 Sm




21for adjacent




minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)






11 and transmission







minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References


























119903= 27 and 120590 = 0015 Sm at 23 GHz It


119903= 21 22 24 27 30





119903= 10 120590 = 00 Sm




119903= 775 120590 =




1

15

2

25

3

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50119884

(mm

)

minus50

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)



119903= 27






0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm




119903= 775 120590 = 005 Sm




21for adjacent




minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)






11 and transmission







minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























1

15

2

25

3

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50

119884(m

m)

minus50

119883 (mm)


1

15

2

25

3

0 50

minus50

0

50119884

(mm

)

minus50

119883 (mm)


1

15

2

25

3

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)


minus50

0

50

119884(m

m)

0 50minus50

0

001

002

003

004

005

119883 (mm)



119903= 27






0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm




119903= 775 120590 = 005 Sm




21for adjacent




minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)






11 and transmission







minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























0 5 10 15 2002

04

06

08

1

Iteration

Nor

mal

ised

func

tiona

l

16 mm18 mm20 mm

22 mm24 mm




119903= 775 120590 = 005 Sm




21for adjacent




minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)






11 and transmission







minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References




























minus80

minus60

minus40

minus20

0

11987821

(dB)






21data



minus25

minus20

minus15

minus10

minus5

0

11987811

(dB)




minus40

minus30

minus20

minus10

0

11987821

(dB)








119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References



























119903= 775 120590 = 005 Sm






minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

11987821

(dB)





119903asymp 25








119903 = 2





50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























50

60

70

minus50

0

50

minus50 0 50

(mm

)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(b) Original 120590

50

60

70

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 = 021

0

02

04

06

minus50

0

50

minus50 0 50

119884(m

m)

119883 (mm)

(d) Full 3D model 120590 120575 = 29






20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























20

40

60

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(a) Original 120598119903

0

02

04

06

minus50

minus50

0

0

50

50

119884(m

m)

119883 (mm)

(b) Original 120590

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) Full 3D model 120598119903 120575 =042

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) Full 3D model 120590 120575 = 29





20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)



20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)


0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)




50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























50

60

70

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) 2D model 120598119903 120575 = 088

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) 2D model 120590 120575 = 31

20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(c) 2D model 120598119903 120575 = 092

0

02

04

06

minus50 0 50

minus50

0

50119884

(mm

)

119883 (mm)

(d) 2D model 120590 120575 = 36



5 Conclusion




20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References

























20

40

60

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(a) Open 2D model 120598119903 120575 = 057

0

02

04

06

minus50 0 50

minus50

0

50

119884(m

m)

119883 (mm)

(b) Open 2D model 120590 120575 = 56



Acknowledgments


References
























Antenna Modeling and Reconstruction Accuracy of Time...

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Transcript of Antenna Modeling and Reconstruction Accuracy of Time...