1

Continuous Wavelet Transform Based FrequencyDispersion Compensation Method for

Electromagnetic Time-Reversal ImagingAmmar M. Abduljabbar, Student Member, IEEE, Mehmet E. Yavuz, Fumie Costen, Senior Member, IEEE,

Ryutaro Himeno, Hideo Yokota

Abstract—The invariance of wave equations in lossless mediaallows the Time Reversal (TR) technique to spatio-temporallyrefocus back-propagated signals in a given ultrawideband imag-ing scenario. However, the existence of dispersion and loss in thepropagation medium breaks this invariance and the resultant TRfocusing exhibits frequency and propagation duration dependentdegradation. We propose an algorithm based on the continuouswavelet transform that tackles this degradation to improve focus-ing resolution under such conditions. The developed algorithmhas been successfully applied to the scenario for localization oflung cancer.

Index Terms—Time reversal (TR), continuous wavelet trans-form (CWT), dispersive media, attenuation compensation.

I. INTRODUCTION

LOCALIZATION of targets behind structures or withinobstructed media is of great interest in many microwave

imaging applications such as detection of landmines [1],explosives [2], subsurface imaging [3] or tumors in the body[4] (e.g. breast [5], brain [6] and lung [7].)

A recent method in the microwave imaging which hasbeen introduced to detect and localize objects is the TimeReversal (TR) technique [8]. In a lossless medium, TR statesthat for every wave component propagating away from asource point following a certain path, there is a correspondingtime-reversed wave that can trace the same path back to theoriginal point of the source due to invariance of Maxwell’sequations to time. Conventional TR technique uses a set ofantennas, called TR Array (TRA), to receive, record and time-reverse the reflected signal from the object(s). Time-reversedsignals at each TRA antenna are sent back to the mediumto focus around the original object location. One importantproperty of TR is that it achieves super-resolution by utilizingthe multipath propagation in the medium as discussed in[9], [10], [11]. On the other hand, as with other microwaveimaging techniques, TR suffers from dispersion and lossesin the propagating medium [12], [13] where TR invariance

A. M. Abduljabbar, and F. Costen are with the School of Electricaland Electronic Engineering, The University of Manchester, U.K. (email:fumie.costen@manchester.ac.uk)

M. E. Yavuz, Hillsboro, Oregon, USA. (email: yavuz.5@osu.edu)R. Himeno is Advanced Center for Computing and Communication,

RIKEN, Saitama, JapanH. Yokota and F. Costen are with the Image Processing Research Team,

Center for Advanced Photonics, RIKEN, Saitama, Japan.Color versions of the figures in this paper are available online at

http://ieeexplore.ieee.org. Additional research data supporting this publicationare available from http://dx.doi.org/ repository at 10.15127/1.306001.

of the wave equations is broken. While the additional phaseshift undergone due to dispersion can be compensated by theTR process itself [12], attenuation which affects the signalsin both the forward and backward propagation stages arenot compensated. This attenuation ultimately degrades thefocusing resolution of the localization. The earliest works oncompensation of such attenuation involved a uniform absorp-tion model over the entire bandwidth for acoustic propagationin skull [14], multipaths in ocean communication [15] andat a single frequency with the so-called decomposition ofthe TR operator method [16]. A more recent compensationtechnique using the Short-Time Fourier Transform (STFT)method takes frequency dependency into account and triesto improve the resolution focusing of the TR technique indispersive and lossy media [12]. Although this method provedto be a good candidate for compensation, the inverse filtersutilized in the method also amplify unwanted oscillationsand perturbations in the received signals, which affect theperformance of TR technique. STFT-based method of [12]produced the inverse filters by comparing the solution ofthe wave equation in the applied dispersive medium againsta corresponding non-dispersive test medium by transmittinga pulse and observing it at various propagation distancesfor both media. The process of creating the inverse filtersis empiric and time consuming. Later, the method in [12]is extended by adding a threshold approach to overcomethe amplification of the noise in the inverse filter in [17].Different window types and lengths are applied in [17] toobtain the optimum settings for the STFT technique. However,the work only applied in dispersive and homogeneous mediawhich do not accurately represent the real-life applications.The inverse filters in [12] and [17] need the non-dispersiveversion of the propagation media to create the inverse filterwhich might be unknown in real-life scenarios. The STFTmethod uses a fixed time window length which provides alower resolution in determining attenuation when comparedwith other size-adjustable window methods [18], [19]. Inthis paper, we propose a compensation method that usesadaptive-window scheme based on the Continuous WaveletTransform (CWT). Our compensation method uses an inversefilter in the wavelet domain to overcome the attenuation anddispersion in the electromagnetic (EM) wave propagation dueto the dispersive medium. Our method uses adjustable-lengthwindows by applying long time-windows at low frequenciesand short time-windows at high frequencies [20]. Therefore

2

we do not need optimization of the time-window length aswas carried out in [17]. The proposed inverse filters needonly the complex permittivity of one dominant medium atthe center frequency to create the inverse filter. creation ofinverse filters in the STFT method requires more computationcompared to those in the CWT, hence another advantage ofCWT over STFT. To the best knowledge of the authors, suchan adaptive compensation algorithm has not been consideredelsewhere before. Therefore introduction of such an algorithmis considered as the main contribution of this paper. Numericalsimulations have been chosen to validate the algorithm. Theoutline of the paper is as follows: First, the details of theproposed compensation method is described in Section II.Section III and Section IV provide the radio environmentsettings and the analysis of the numerical simulations for thecanonical test and practical application respectively. Section Vpresents the conclusions.

II. COMPENSATION OF DISPERSIVE ATTENUATION INHUMAN TISSUES

A dispersive and lossy medium acts as a low-pass filter forsignals propagating through it. Thus, for compensation of sucheffects, inverse filters are needed. The real part of the mediumdielectric permittivity ε causes a phase shift to the travelingwaves during the forward propagation of the TR process.Thanks to the TR concept, this phase shift is inherentlycorrected in the backward propagation due to the fact thatthe signals in the backward propagation are phase-conjugatedcoherently along all bandwidth [21]. On the other hand, theimaginary part of ε causes inevitable signal attenuation. Theattenuation is a function of time (in other words, duration thesignal has traveled in the dispersive medium) and frequency.Note that the more the signal propagates in a dispersivemedium the more it is attenuated (duration dependency). Simi-larly, for certain wave packet that has traveled in the dispersivemedium, different frequency components undergo differentattenuations (frequency dependency). Therefore the proposedmethod uses CWT technique which inherently incorporatesboth time and frequency variations as CWT suits well forthis kind of problem. The wavelet function that we used forthe CWT method is Morlet wavelet [22]. Morlet wavelet isconsidered in [19] as the proper choice for analyzing thepropagated wave in attenuating and dispersive media. Themother wavelet of Morlet in time domain [23] is

Ψ0[n] =1

4√πefΨn∆te−

(n∆t)2

2 (1)

where n is the time index, ∆t is the time step in secondsand fΨ is the central frequency of the mother wavelet. Notethat without loss of generality, we have chosen to work withdiscrete signals instead of the continuous ones, hence thenotation of Ψ0[n] instead of Ψ0(t). Ψ0[n] is expressed inscaled-frequency domain as

Ψ0(ajωk) =1

4√πe−

(ajωk−fΨ)2

2 (2)

where aj is the dimensionless scaling factor of aj = a02j∆j

for j = 0, 1, . . . , J − 1, a0 is the smallest scale, ∆j is the

scale step, J is the largest scale J =⌈ 1

∆jlog2(

N∆t

a0)⌉

,⌈ ⌉

represents the ceiling function, N is the sampling points inthe time domain signal, k is frequency index and the angularfrequency ωk is [23]

ωk =

0 k = 1

2πkN∆t 1 < k ≤ N

2 + 1− 2πkN∆t

N2 + 1 < k ≤ N

for k = 1, 2, . . . , N . a0 is dimensionless and it is set tothe value of ∆t and ∆j is set to 0.025 based on the valuechosen in [23]. Both Ψ0[n] in (1) and Ψ0(ajωk) in (2) arecomplex. In order for the analyzing function Ψ0[n] to be calledwavelet it needs to be admissible [24]. Admissibility is one ofthe necessary conditions of the wavelet transform and it isachieved when [24]

limN→∞

N∑n=−N

Ψ0[n] = 0 (3)

and

Ψ0(ajωk) =

{1

4√πe−

(ajωk−fΨ)2

2 ωk > 0

0 ωk ≤ 0.

We added white Gaussian noise to each observed signal x[n]at each TRA antenna. The average power of the additive noiseis 20% of that of x[n] which is chosen to mimic the systemnoise based on [25]. Our compensation method starts withconversion of the observed signal x[n] in time domain to thewavelet domain. The CWT of x[n] is the convolution of thewave function Ψ∗( naj ) and x[n] which is equivalent to theinverse Fourier transform of the product of Ψ∗(ajωk) andx[k] in the frequency domain where ∗ represents the complexconjugate, Ψ( naj ) is

Ψ

(n

aj

)=

√∆t

ajΨ0

(n

aj

)(4)

and Ψ(ajωk) is

Ψ(ajωk) =

√2πaj∆t

Ψ0(ajωk). (5)

Ψ(ajωk) represents the normalized Ψ0(ajωk) to guaranteethat the CWTs at each aj are comparable to each other andto the CWTs of other time series [23]. Ψ(ajωk) satisfies

N∑k=1

| Ψ(ajωk) |2= N. (6)

The CWT for the observed signal x[n] at each TRA antennais

X[n, aj ] =

N∑k=1

(N∑i=1

x[i]e−2πN ikΨ∗(ajωk)

)e

2πN nk. (7)

The CWT method applies a long window (large aj) at thelower end of the spectrum (low frequency) and a short window(small aj) at the higher end of the spectrum (high frequency)to provide a perfect filter for time-dependent implementationsin terms of time and frequency localization. Furthermore

3

the CWT method provides us with an improved separationbetween the unwanted noise and our signal [19]. Next step isapplying our inverse filter to X[n, aj ] in the wavelet domain tocompensate the attenuation caused by the dispersive medium.In order to derive the inverse filter, we need to calculate theattenuation. The solution of Maxwell equations for a planewave propagating in a dispersive medium [26] is

E[n] = e2πf ·n∆te−2πf√εµd[n] (8)

where f is the frequency of interest, d[n] is the distancebetween the excitation and the observation, µ = µ0µr is thepermeability of the medium, µ0 is the free space permeabilityand µr is the relative permeability which is set to 1 as a non-magnetically dispersive medium is assumed, ε = ε0εr is thepermittivity of the medium, ε0 is the free space permittivity,εr = ε∞ +

εS − ε∞1 + 2πfτD

− σ

2πfε0is the complex relative

permittivity of medium, ε∞ is the optical relative permittivity,εS is the static relative permittivity, τD is the relaxation time,σ is the static conductivity.

(8) can be rewritten as

E[n] = e2πf ·n∆te−2πf√εr

d[n]C

= e2πf ·n∆te−2πf<[√εr ]

d[n]C e2πf=[

√εr ]d[n]C

, e2πf ·n∆t ·Θ · Γ(9)

where C is the speed of light C = 1√ε0µ0

in the vacuum,

<[√εr] and =[

√εr] are the real and the imaginary parts of√

εr, Γ is the attenuation defined as

Γ[n, f ] = e2πf=[√εr ]

d[n]C (10)

and Θ is the phase shift defined as

Θ[n, f ] = e−2πf<[√εr ]

d[n]C . (11)

Our filter is obtained from (10) and (11) as

H[n, aj ] =1

Θ[n, aj ] · Γ[n, aj ]

= e2π fcaj

<[√εr ]

d[n]C e−2π fcaj

=[√εr ]

d[n]C

(12)

where aj is a scaling factor which controls the actual fre-quency f(, fc

aj) of the filter in order to have a variable window

size [19] and fc is the central frequency of x[n]. H[n, aj ]uses εr of the dominant tissue in the propagating media at fc.H[n, aj ] is an exponential function which amplifies the noisein compensated signal and thus causes instability. ThereforeH should be stabilized [27] as

Hs[n, aj ] =Γ[n, aj ]

Γ[n, aj ]2 + T· 1

Θ[n, aj ](13)

where Hs is the stabilized filter and T is the stabilizationfactor. The system is stable with a fixed value of T = 10−3

for all our simulations. Alternatively we can adaptively set Tas [19]

T =σ2g

σ2s

(14)

where σg and σs are the variances for the noise and the signalrespectively. By applying Hs[n, aj ] to X[n, aj ], we obtain the

FFT IFFTX[n, aj]x[n]

Y [n, aj]

Continous Wavelet Transform(CWT)

Inverse Continous WaveletTransform (ICWT)

Xc[n]C−1δ

x[k]

ℜ[Y ]

Ψ∗[ajωk]

Hs[n, aj]

1√aj

Σj = 1

J

Fig. 1. Compensation method for the dispersive attenuation in the waveletdomain where ⊗ means multiplication in frequency domain and correspondsto convolution in time domain.

stabilized compensated wave Y [n, aj ] in the wavelet domainas

Y [n, aj ] = X[n, aj ]Hs[n, aj ]. (15)

To avoid the amplification of the high frequency noise,Hs[n, aj ] is set to 1 for 0 ≤ j ≤ J

2 (short window)which affects the higher part of the spectrum. Finally thecompensated signal Xc[n] is obtained by applying the inversecontinuous wavelet transform (ICWT) to Y [n, aj ] as [23]

Xc[n] =1

Cδ

J∑j=1

<[Y [n, aj ]]√aj

(16)

where Cδ is a constant for each wavelet function. It iscalculated as

Cδ =

J∑j=1

<[Xδ[aj ]]√aj

(17)

for the Morlet wavelet where Xδ is the CWT of a deltafunction (δ) as

Xδ[aj ] =1

N

N∑k=1

Ψ∗(ajωk). (18)

The parameters for the CWT and the ICWT were selectedbased on the theory in [23]. Our proposed method is depictedin Fig. 1.

III. CANONICAL TEST

We expect that the proposed method improves the localiza-tion and the resolution of the conventional TR imaging becausethe inverse filters should compensate the losses caused by thedispersive media. We evaluate the accuracy and the resolutionof our method in a random medium [28] to increase thecomplexity of the propagating medium. The random mediumis of Gaussian distribution and the spatial random relativepermittivity is defined as

εrnd[x, y, f ] = εr[x, y, f ] +G[x, y] (19)

where εr is calculated using the one-pole Debye [29] pa-rameters of the muscle tissue [30][31], [x, y] is the Cartesiancoordinates in the finite difference time domain (FDTD) space,

4

00

150

30

60

90

120

TRA aperture

PML

PML

PM

L

PM

L

Distance(×∆x)

Distance

(×∆y)

15030 60 90 120

Scatterer

(PEC)

Random

Dispersive

Medium

Fig. 2. The simulations setups.

G is a function of space defining the random fluctuation onεrnd[x, y] and is calculated as

G[x, y] =1

RxRy

Rx∑u=1

Ry∑v=1

(√√√√ Rx∑x=1

Ry∑y=1

L[x, y]e−2π( uxRx+ vy

Ry)

·Rx∑x=1

Ry∑y=1

R · e−2π( uxRx+ vyRy

)

)e2π( uxRx+ vy

Ry)

(20)

where Rx × Ry is the 2D FDTD size, [u, v] is the Cartesiancoordinates in the spatial frequency domain, R is a Gaussianrandom number with zero mean and probability density func-

tion of 1√2πη

e−ζ22η , ζ is a random variable, η is the variance

and L is the correlation function between εr at two differentspatial points [x1, y1] and [x2, y2] given by a Gaussian functionas

L[x, y] = ηe

(− |x1−x2|

2+|y1−y2|2

l2s

)(21)

where ls is the transverse correlation length. Gaussian functionis chosen for its generalization and mathematical proper-ties [28]. The random media is characterized by ls and η. Wechose muscle tissue because it is one of the most dispersivetissues in the human body.

A. Radio Environment Settings

The simulation setup is depicted in Fig. 2. The simulationuses the TMz polarized 2D FDTD method. The FDTD spaceis 150 × 150 uniformly sampled with the spatial step of∆s , ∆x = ∆y = 1mm. The time step ∆t is set to2.36ps. The CFS-PML absorbing boundary conditions [32]are used with 9-cell layer to mimic open space settings. ATRA composed of 15 transceivers antennas are equally spacedfrom each other and located parallel to the x-axis. The TRAhas the aperture of 70∆s. The first TRA antenna is locatedat (40∆x,25∆y) and the 15th is located at (110∆x,25∆y). Around perfect electric conductor (PEC) scatterer with a 2mmradius is centered at (75∆x,120∆y). The 8th antenna of theTRA transmits a Gaussian pulse modulated at 3 GHz coveringfrom 1.72 GHz to 4.2 GHz.

ℜ[ǫr]

Distance

(×∆y)

Distance(×∆x)

Max

Min

Conductivity

Distance

(×∆y)

Distance(×∆x)

Max

Min

(a) (ls, η) = (4∆s, 0.14ε∞)with minimum and maximum

values of 15.04 and 42.16respectively.

(b) (ls, η) = (4∆s, 0.14ε∞)with minimum and maximumvalues of 0.38 [S/m] and 1.13

[S/m] respectively.ℜ[ǫr]

Distance

(×∆y)

Distance(×∆x)

Max

Min

Conductivity

Distance

(×∆y)

Distance(×∆x)

Max

Min

(c) (ls, η) = (10∆s, 0.14ε∞)with minimum and maximum

values of 16.88 and 39.51respectively.

(d) (ls, η) = (10∆s, 0.14ε∞)with minimum and maximumvalues of 0.44 [S/m] and 1.06

[S/m] respectively.Fig. 3. The distribution of εr for Fig. 2 at 3 GHz.

`````````(ls, η)Resolution Rt

Rc

RsRc

(4∆s, 0.14ε∞) 3.13 3.76(10∆s, 0.14ε∞) 1.51 3.34

TABLE ITHE SPATIAL RESOLUTION RATIO.

B. Numerical Results

Fig. 3 shows the example of the spatial distribution of εrfor Fig. 2 at 3 GHz when ls is set to 4∆s and 10∆s. Fig. 4(a), (b) and (c) show |Ez| in the xy-plane with conventionalTR, with the method in [12] and with our compensationmethod. Fig. 5 (a) and (b) show the cross-section of Fig. 4at y = 120∆y with (ls, η) = (4∆s, 0.14ε∞) and the one with(ls, η) = (10∆s, 0.14ε∞) respectively. Rc = (x4 − x3)∆x,Rs = (x5 − x2)∆x and Rt = (x6 − x1)∆x representthe resolution of the spatial focusing with the proposedcompensation method, the method in [12] and without anycompensation method respectively where xi (1 ≤ i ≤ 6)are on the cross-range and |Ez(xi)| = 0.5 for 1 ≤ i ≤ 6where Ez is normalized and 0.5 is the cross-range at halfmaximum to calculate the resolution of the spatial focusing.

The spatial resolution of the proposed method is(RsRc

=

)3.76

(Fig. 5(a)),3.34 (Fig. 5(b)) times higher than the compensation

method in [12] and(RtRc

=

)3.13 (Fig. 5(a)),1.51 (Fig. 5(b))

times higher than without using any compensation methodas summarized in Table I. Fig. 6 shows the cross-section

5

0.48

1

Distance

(×∆y)

Distance(×∆x)

0

20

20 60

60

100

100

140

140

0.13

Normalised |Ez(x, y)|

(a) Conventional TR.

0.61

1

Distance

(×∆y)

Distance(×∆x)

0

20

20 60

60

100

100

140

140

0.29

0.08

Normalised |Ez(x, y)|

(b) TR with STFT.

0.5

1

Distance

(×∆y)

Distance(×∆x)

0

20

20 60

60

100

100

140

140

0.16

Normalised |Ez(x, y)|

(c) TR with CWT.

Fig. 4. The spatial distribution of |Ez | in the xy-plane for the canonicaltest with conventional TR (without applying any compensation method),with the method in [12] (STFT) and with our compensation method (CWT)when (ls, η) = (4∆s, 0.14ε∞). The scatterer and the TRA antennas arerepresented by “◦” and “x” respectively. ∆x = ∆y = 1mm.

at y = 120∆y with our compensation method for the cases(ls, η) = (4∆s, 0.14ε∞), (10∆s, 0.14ε∞) and (4∆s, 0.05ε∞).TR techniques utilize the multipaths components generated bythe random media to achieve more accurate focusing resolutionthan in homogeneous media. Both the high correlation lengthls such as (10∆s, 0.14ε∞) and the low variance η such as(4∆s, 0.05ε∞) reduce the randomness of the medium as bothcases result in reduction of the overall fluctuations of the per-mittivity of the propagation medium, hence its scattererness.Reducing the randomness decreases the multipath componentswhich lowers the resolution of the TR focusing.

0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120 140

CWT

x1x2 x3 x4 x5x6

Rc

Rt

Rs

Norm

alize

d|E

z|[

V/m]

Distance(×∆x)

TR

STFT

(a) (ls, η) = (4∆s, 0.14ε∞).

0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120 140

CWT

TR

STFT

x1x2 x3 x4 x5x6

Rc

Rt

Rs

Norm

alize

d|E

z|[

V/m]

Distance(×∆x)

(b) (ls, η) = (10∆s, 0.14ε∞).

Fig. 5. The cross-section of the normalized |Ez | distribution in Fig. 4 aty = 120∆y in the FDTD space after applying CWT, STFT and withoutapplying any compensation method.

0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120 140

(4∆s, 0.14ǫ∞)

(10∆s, 0.14ǫ∞)

(4∆s, 0.05ǫ∞)

Distance(×∆x)

Norm

alize

d|E

z|[

V/m]

Scatterer

Center

Fig. 6. The cross-section of the normalized |Ez | distribution at y =120∆y in the FDTD space after applying CWT for the cases (ls, η) =(4∆s, 0.14ε∞), (10∆s, 0.14ε∞) and (4∆s, 0.05ε∞).

6

Distance

(×∆y)

Distance(×∆x)

50 150 250

50

450

150

250

350

tumor

(a)

(b)

Fig. 7. The human phantom with 5mm radius round tumor. The TRA antennasare represented by “|”. ∆x = ∆y = ∆z = 1mm.

When the computational times between our proposedmethod and original STFT method of [12] are compared,it is not merely comparison between the STFT and CWToperations but the whole algorithm implementation startingfrom the filter generations. The way [12] used to produce theinverse filter took the major part of the simulation time becausethe creation of the inverse filters in [12] needs two FDTDcalculations while our approach with CWT does not requireany FDTD calculations. Applying different FDTD sizes withdifferent simulation times shows that the proposed method is1.8 times faster than the STFT method in [12] and [17] inaverage.

IV. PRACTICAL APPLICATION

Lung cancer is identified as one of the deadliest cancers af-fecting people [33], [34]. Early-stage lung cancer is difficult todetect because the tumor is small and its electromagnetic prop-erties are close to the parameters of the healthy lung. We applythe TR technique with our proposed compensation method tolocate an early-stage tumor in a digital human phantom (DHP).The DHP was provided by RIKEN (Saitama,Japan) undernon-disclosure agreement between RIKEN and the Universityof Manchester. The usage was approved by RIKEN ethicalcommittee. The DHP has 1 mm resolution and contains 52segmented tissue. We fitted the one-pole Debye parametersof human tissues [35] using the measurement provided by[36], [37]. The Debye media parameters for human tissues arepresented in [31]. The purpose of this numerical simulation isto evaluate the accuracy and the resolution of our method inthe human phantom medium.

A. Radio Environment Setting

Fig. 7 shows the DHP with an early stage tumor repre-sented by a round scatterer with a 5mm radius centered at(115∆x,285∆y,95∆z). The FDTD space is 265× 490× 290including the CFS-PML cells. The 102 TRA z−directed 14mm dipole antennas [38] are placed in the fat tissue assumingthat we use the implantable antennas [39], [40], [41]. The hu-man tissues are represented using the one-pole Debye model.

0.1

1

10

100

1 2 3 4 5

Frequency [GHz]

conductivity [S/m] lung tumorconductivity [S/m] healthy lung

lung tumorℜ[ǫr]healthy lungℜ[ǫr]

Fig. 8. Relative permittivity and the conductivity of the lung and the scatterer.

We shifted the frequency response of the healthy lung tissueto match the complex permittivity of the lung cancer tissuepresented in [42] and generated the Debye parameters for thelung tumor by data-fitting of the modified frequency response.Fig. 8 shows the relative permittivity and the conductivity ofthe lung and the tumor. The size and the Debye parameters ofthe tumor are set to mimic an early stage cancer. The antennaat (31∆x,281∆y,95∆z) transmits a Gaussian pulse modulatedat 3 GHz. The remaining simulation settings are the same asin Section III-A.

B. Numerical Results

Fig. 9 shows |Ez| on z = 95∆z plane within the 3D DHPwith conventional TR without applying any compensation,one with the method given in [12] (STFT) and one with ourmethod (CWT). Fig. 9 (b) shows that the method in [12] didnot improve the conventional TR imaging or locate the tumorcorrectly. The inverse filters utilized in [12] not only amplifythe signal of interest, but also the noise. This reduces thesignal to noise ratio (SNR) and in return lowers the accuracyof the localization. The proposed CWT filters compensate thelosses caused by the dispersive tissues. Therefore applicationof the proposed CWT method to the TR technique gives moreaccurate spatial focusing than both the conventional TR andthe method in [12].

V. CONCLUSION

The wave equation invariance of the time reversal techniqueis broken in human tissues due to their dispersive charac-teristics. Since the dispersive attenuation is both durationand frequency dependent, we have introduced CWT basedcompensation method that uses inverse filters in the waveletdomain to overcome this attenuation. The inherent time andfrequency variations in a wavelet decompose the recordedsignals into different time and frequency components to whichdifferent filters would be applied for compensation. CWT doesnot need optimization of the time-window length as is the casewith STFT. The proposed inverse filters need only the complexpermittivity at the center frequency of one dominant mediumto create the inverse filter as opposed to the whole wide-banddispersion characteristic in the STFT method [12]. ThereforeCWT technique can be applied in real-life scenarios whileSTFT method [12] [17] needs the non-dispersive version of the

7

Distance

(×∆y)

Distance(×∆x)50 150 250

450

350

250

150

50

Normalised |Ez(x, y)|1

0.16

0.5

0

(a) TR with CWT.

Distance

(×∆y)

Distance(×∆x)50 150 250

450

350

250

150

50

0.1

Normalised |Ez(x, y)|

0.3

0.6

1

0

(b) TR with STFT.

Distance

(×∆y)

Distance(×∆x)50 150 250

450

350

250

150

50

Normalised |Ez(x, y)|

0.3

0.6

1

0.1

0

(c) TR.

Fig. 9. The spatial distribution of |Ez | on z = 95∆z plane for the humanphantom with our compensation method, the method in [12] and without anycompensation method. The scatterer and the TRA antennas are representedby “◦” and “x” respectively. ∆x = ∆y = ∆z = 1mm. In order to present|Ez | inside the lung clearly, |Ez | on the skin, muscle and fat are removedfrom the figures.

propagation media to create the inverse filter (in real-life, suchan information might not be available readily). We applied ourcompensation method to the TR signals in different scenariosto examine how our method improved the TR imaging. Inthe canonical numerical simulation, our compensation methodimproved the spatial focusing of the TR technique. The TRfocusing around the object location is more accurate with ourproposed method than those with the method in [12] or withoutapplying any method (i.e. with conventional TR imaging).Furthermore, the resolution is 3.76 and 3.13 times higher withthe proposed method than with the method in [12] or withoutany compensation method respectively. The tumor is locatedaccurately after applying our proposed compensation method.An additional advantage of the CWT-based method is that itis 1.8 times faster than the STFT method in average.

VI. ACKNOWLEDGMENT

“Wavelet software was provided by C. Tor-rence and G. Compo, and is available at URL:http://paos.colorado.edu/research/wavelets/”. The base codeto produce Fig. 7 (a) was provided by one of our researchgroup members, Mr. K. Tekbas.

REFERENCES

[1] P. D. Gader, M. Mystkowski, and Y. Zhao, “Landmine detection withground penetrating radar using hidden markov models,” IEEE Trans.Geosci. Remote Sens., vol. 39, No. 6, pp. 1231–1244, June 2001.

[2] P. Pratihar and A. K. Yadav, “Detection techniques for human safetyfrom concealed weapon and harmful EDS,” International Review ofApplied Eng. Research., vol. 4, No. 1, pp. 71–76, 2014.

[3] M. E. Yavuz, A. E. Fouda, and F. L. Teixeira, “GPR signal enhancementusing sliding-window space-frequency matrices,” Progress In Electro-magnetics Research, vol. 145, pp. 1–10, 2014.

[4] W. Shao, B. Zhou, Z. Zheng, and G. Wang, “UWB microwave imagingfor breast tumor detection in inhomogeneous tissue,” Eng. in Med. andBiol. 27th Annual Conference, pp. 1496–1499, 2005.

[5] M. Hossain, A. Mohan, and M. Abedin, “Beamspace time-reversalmicrowave imaging for breast cancer detection,” IEEE Antennas WirelessPropag. Lett., vol. 12, pp. 241–244, 2013.

[6] N. Ray, B. Saha, and M. Graham Brown, “Locating brain tumors fromMR imagery using symmetry,” in 2007 Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers (ACSSC),pp. 224–228, Nov 2007.

[7] A. Zamani, S. Rezaeieh, and A. Abbosh, “Lung cancer detectionusing frequency-domain microwave imaging,” Electronics Lett., vol. 51,no. 10, pp. 740–741, 2015.

[8] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter,J. Thomas, and F. Wu, “Time-reversed acoustics,” Rep. Prog. Phys.,vol. 63, pp. 1933–1995, 2000.

[9] C.-X. Li, W. Xu, J.-L. Li, and X.-Y. Gong, “Time-Reversal detectionof multidimensional signals in underwater acoustics,” IEEE J. Ocean.Eng., vol. 36, pp. 60–70, Jan 2011.

[10] R. Solimene, G. Leone, and A. Dell’Aversano, “MUSIC algorithms forrebar detection,” J. Geophysics and Eng., vol. 10, No. 6, 2013.

[11] A. E. Fouda, F. L. Teixeira, and M. E. Yavuz, “Time-reversal techniquesfor MISO and MIMO wireless communication systems,” Radio Science,vol. 47, 2012.

[12] M. E. Yavuz and F. L. Teixeira, “Frequency dispersion compensation intime reversal techniques for UWB electromagnetic waves,” IEEE Geosci.Remote Sens. Lett., vol. 2, no. 2, pp. 233–237, April 2005.

[13] M. E. Yavuz and F. L. Teixeira, “Full time-domain DORT for ultra-wideband electromagnetic fields in dispersive, random inhomogeneousmedia,” IEEE Trans. Antennas Propag., vol. 54, pp. 2305–2315, Aug2006.

[14] M. Tanter, J. L. Thomas, and M. Fink, “Focusing and steering throughabsorbing and aberrating layers: Application to ultrasonic propagationthrough the skull,” J. Acoust. Soc. Am., vol. 103, pp. 2403–2410, May1998.

8

[15] T. Folegot, C. Prada, and M. Fink, “Resolution enhancement andseparation of reverberation from target echo with the time reversaloperator decomposition,” J. Acoust. Soc. Am., vol. 113, No. 6, pp. 3155–3160, 2003.

[16] M. Saillard, P. Vincent, and G. Micolau, “Reconstruction of buried ob-jects surrounded by small inhomogeneities,” Inverse Problems, vol. 16,pp. 1195–1208, 2000.

[17] A. M. Abduljabbar, M. E. Yavuz, F. Costen, R. Himeno, and H. Yokota,“Frequency dispersion compensation through variable window utiliza-tion in time reversal techniques for electromagnetic waves,” IEEE Trans.Antennas Propagat., vol. 64, pp. 3636–3639, Aug 2016.

[18] C. Reine, M. Van Der Baan, and R. Clark, “The robustness of seis-mic attenuation measurements using fixed- and variable-window time-frequency transforms,” Geophysics, vol. 74, No. 2, pp. WA123–WA135,2009.

[19] I. Braga and F. Moraes, “High-resolution gathers by inverse Q filteringin the wavelet domain,” Geophysics, vol. 78, No. 2, pp. V53–V61, 2013.

[20] I. Hostens, J. Seghers, A. Spaepen, and H. Ramon, “Validation of thewavelet spectral estimation technique in biceps brachii and brachioradi-alis fatigue assessment during prolonged low-level static and dynamiccontractions,” J. Electromyography and Kinesiology, vol. 14, no. 2,pp. 205 – 215, 2004.

[21] M. E. Yavuz and F. L. Teixeira, “Ultrawideband microwave sensing andimaging using time-reversal techniques: A review,” Remote Sens., vol. 1,no. 3, pp. 466–495, 2009.

[22] J. Morlet, G. Arens, E. Fourgeau, and D. Giard, “Wave propagation andsampling theory-Part I: Complex signal and scattering in multilayeredmedia,” Geophysics, vol. 47, No. 2, pp. 203–221, 1982.

[23] C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,”Bulletin of the American Meteorological Soc., vol. 79, No.1, pp. 61–78,1998.

[24] M. Farge, “Wavelet transforms and their applications to turbulence,”Annu. Rev. Fluid Mech, vol. 24, pp. 395–457, 1992.

[25] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “The multicell multiuserMIMO uplink with very large antenna arrays and a finite-dimensionalchannel,” IEEE Trans. Commun., vol. 61, pp. 2350–2361, June 2013.

[26] A. Taflove and S. Hagness, Computational Electromagnetics. The finite-difference Time-domain method. Artech House, 2005.

[27] Y. Wang, “Inverse Q-filter for seismic resolution enhancement,” Geo-physics, vol. 71, pp. V51–V60, 2006.

[28] C. Moss, F. Teixeira, Y. Yang, and J. A. Kong, “Finite-difference time-domain simulation of scattering from objects in continuous randommedia,” IEEE Trans. Geosci. Remote Sens., vol. 40, pp. 178–186, Jan2002.

[29] P. Debye, Polar Molecules. New York: Dover, 1929.[30] M. Abalenkovs, F. Costen, J.-P. Berenger, R. Himeno, H. Yokota, and

M. Fujii, “Huygens subgridding for 3-D frequency-dependent finite-difference time-domain method,” IEEE Trans. Antennas Propag., vol. 60,No. 9, pp. 4336–4344, September 2012.

[31] RIKEN, Wako Saitama, Japan, “Media parameters for the Debye relax-ation model.” http://cfd-duo.riken.jp/cbms-mp/. [Online; accessed 26-April-2016].

[32] J.-P. Berenger, “Numerical reflection from FDTD-PMLs: A comparisonof the split PML with the unsplit and CFS PMLs,” IEEE Trans. AntennasPropag., vol. 50, No. 3, pp. 258–265, 2002.

[33] W. Weber, Pharmacogenetics. Oxford University Press, Apr 2008.[34] N. Emaminejad, W. Qian, Y. Guan, M. Tan, Y. Qiu, H. Liu, and

B. Zheng, “Fusion of quantitative image and genomic biomarkers toimprove prognosis assessment of early stage lung cancer patients,” IEEETrans. Biomed. Eng., vol. 63, pp. 1034–1043, May 2016.

[35] T. Wuren, T. Takai, M. Fujii, and I. Sakagami, “Effective 2-debye-polefdtd model of electromagnetic interaction between whole human bodyand uwb radiation,” IEEE Microw. Wireless Compon. Lett., vol. 17,pp. 483–485, July 2007.

[36] C. Gabriel, S. Gabriel, and E. Corthout, “The dielectric properties ofbiological tissues: I. Literature survey,” Physics in Med. and Biol.,vol. 41, pp. 2231–2249, 1996.

[37] R. W. L. S. Gabriel and C. Gabriel, “The dielectric properties ofbiological tissues: II. Measurements in the frequency range 10 Hz to20 GHz,” Physics in Med. and Biol., vol. 41, pp. 2251–2269, 1996.

[38] Y. Chen and P. Kosmas, “Detection and localization of tissue malignancyusing contrast-enhanced microwave imaging: Exploring informationtheoretic criteria,” IEEE Trans. Biomed. Eng., vol. 59, pp. 766–776,March 2012.

[39] C. M. Furse and A. Chrysler, “A history & future of implantableantennas,” in 2014 IEEE Antennas and Propag. Soc. InternationalSymposium (APSURSI), pp. 527–528, July 2014.

[40] J. Kim and Y. Rahmat-Samii, “Implanted antennas inside a humanbody: simulations, designs, and characterizations,” IEEE Trans. Microw.Theory Tech., vol. 52, pp. 1934–1943, Aug 2004.

[41] S. Alamri, A. AlAmoudi, and R. Langley, “Gain enhancement ofimplanted antenna using lens and parasitic ring,” Electron. Lett., vol. 52,no. 10, pp. 800–801, 2016.

[42] A. Zamani, S. Rezaeieh, and A. Abbosh, “Lung cancer detection usingfrequency-domain microwave imaging,” Electron. Lett., vol. 51, no. 10,pp. 740–741, 2015.

Ammar M. Abduljabbar was born in Baghdad,Iraq, in 1985. He received the B.Sc. degree in com-puter engineering from the University of Baghdad,Baghdad, Iraq, in 2007, and the M.Sc. degree inwireless communication systems from Brunel Uni-versity, London, UK, in 2009. In 2013, he started hisPhD study in Electrical and Electronics Engineering,University of Manchester, UK. His current researchinterests include microwave imaging techniques andhigh performance computing.

Mehmet E. Yavuz received the B.S. degree from theMiddle East Technical University (METU), Ankara,Turkey in 2001, M.S. degree from the Ko University,Istanbul, Turkey in 2003 and the Ph.D. degree fromthe Ohio State University (OSU) in 2008, all inelectrical engineering. He has worked as a Princi-ple Scientist at the Halliburton Energy Services inHouston, TX, as a Visiting Assistant Professor at theElectrical Engineering Department of the AmericanUniversity of Sharjah in the U.A.E and as a ResearchScientist at Intel Corporation in Hillsboro, Oregon.

His current research interests include computational electromagnetics forindustrial applications, radar-based signal processing and microwave imagingand detection.

Fumie Costen (M’07) received the B.Sc. de-gree, the M.Sc. degree in electrical engineering andthe Ph.D. degree in Informatics, all from KyotoUniversity, Japan. From 1993 to 1997 she waswith Advanced Telecommunication Research Inter-national, Kyoto, where she was engaged in researchon direction-of-arrival estimation based on MultipleSIgnal Classification algorithm for 3-D laser micro-vision. She filed three patents from the research in1999 in Japan. She was invited to give 5 talks inSweden and Japan during 1996-2014. From 1998

to 2000, she was with Manchester Computing in the University of Manch-ester, U.K., where she was engaged in research on metacomputing and hasbeen a Lecturer since 2000. Her research interests include computationalelectromagnetics in such topics as a variety of the finite difference timedomain methods for microwave frequency range and high spatial resolutionand FDTD subgridding and boundary conditions. She filed a patent fromthe research on the boundary conditions in 2012 in the U.S.A. Her workextends to the hardware acceleration of the computation using general-purposecomputing on graphics processing units, Streaming Single Instruction MultipleData Extension and Advanced Vector eXtentions instructions. Dr. Costenreceived an ATR Excellence in Research Award in 1996 and a best paperaward from 8th International Conference on High Performance Computingand Networking Europe in 2000.

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Ryutaro Himeno received his Doctor of Engineer-ing degree from the University of Tokyo in 1988.In 1979, he joined Central Research Laboratories,Nissan Motor Co., Ltd., Yokosuka, Japan, wherehe has been engaged in the research of apply-ing Computational Fluid Dynamics analysis to thecar aerodynamic development. In 1998, he joinedRIKEN and is the director of Advanced Center forComputing and Communication and had been thedeputy program director of the Next GenerationComputational Science Research Program at RIKEN

till April, 2013. He is also a visiting professor at Hokkaido University, KobeUniversity and Tokyo Denki University. He currently studies ComputationalBioengineering, High Performance Computing and blood flows of humanbodies. He was a winner of Nikkei Science, Computer Visualization Contestin 2000 and Scientific Visualization Contest in 1996, and received JSMEComputational Mechanics Division Award in 1997 and JSME Youth EngineerAward in 1988. He has also received the Paper Award by NICOGRAPH in1993, Giga FLOPS Award by CRAY Research Inc. in 1990 and other awards.

Hideo Yokota received his Doctor of Engineer-ing degree from the University of Tokyo in 1999.In 1993, he joined Higuchi Ultimate MechatronicsProject, Kanagawa Academy of Science and Tech-nology, Kawasaki, Japan. In 1999, he joined RIKENand is the contract researcher of Computationalbiomechanics unit. 2004-2012 Bio-research Infras-tructure Construction Team Leader, VCAD systemresearch program, RIKEN. 2007-2012 Cell-scaleResearch and Development Team Leader,ResearchProgram for Computational Science, RIKEN. 2013-

present Image Processing Research Team Leader, Center for AdvancedPhotonics,RIKEN. He is also a visiting Professor at Hokkaido University,Kobe University, Tokai University and the Tokyo University of Agricultureand Technology. He is currently involved in biomedical imaging and imageprocessing to the biomedical simulation. Bioimaging Society, Best ImageAward, 2005. The Commendation for Science and Technology by the Ministerof Education, Culture, Sports, Science and Technology , Young ScientistsPrize in 2008.