Wireless Tomography in Noisy Environments Using Machine Learning

956 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 52, NO. 2, FEBRUARY 2014

Wireless Tomography in Noisy EnvironmentsUsing Machine Learning

Zhen Hu, Shujie Hou, Student Member, IEEE, Michael C. Wicks, Fellow, IEEE,and Robert C. Qiu, Senior Member, IEEE

Abstract— This paper, one in a continuing series, describes anew initiative in wireless tomography. Our goal is to combinetwo technologies: wireless communication and radio frequencytomography, for the close-in remote sensing. The hybrid system,including wireless communication devices for wireless tomogra-phy is proposed in this paper. Noise reduction, modified standardphase reconstruction, and imaging are exploited sequentiallyto perform wireless tomography in noisy environments. Theperformance given in this paper illustrates the significance andprospect of wireless tomography. The contributions of this paperare threefold: 1) the hybrid system provides a strong and flexibleinfrastructure for wireless tomography; 2) machine learning,especially nonlinear dimensionality reduction, is explored to exe-cute noise reduction and combat the nonlinear noise effect; and3) modified standard phase reconstruction is well achieved usingthe de-noised amplitude-only total fields from the simple sensorsand the received accurate full-data total fields from the advancedsensors. Experimental data provided by the Institute Fresnel inMarseille, France are used to demonstrate the concept of wirelesstomography and validate the corresponding algorithms.

Index Terms— Imaging, machine learning, noise reduction,phase reconstruction, wireless tomography.

I. INTRODUCTION

AS SMART phones are widely used, there will be apotential large-scale wireless communication network

deployed around the world [1]. Currently, we are interestedin embedding remote sensing into this kind of large-scalewireless communication network. A new project called cog-nitive radio network as sensors has been launched [2]–[6].However, wireless communication devices, components, andnodes are not specifically designed for remote sensing [1].They do not meet the high-accuracy measurement requirementof remote sensing. For example, radio frequency (RF) tomog-raphy or inverse scattering requires accurate phase informationto perform imaging. Accurate phase information is hard to

Manuscript received April 18, 2011; revised December 2, 2012; acceptedJanuary 2, 2013. Date of publication March 13, 2013; date of current versionDecember 12, 2013. This work was supported in part by the National ScienceFoundation under Grant ECCS-0901420 and Grant ECCS-0821658, and theOffice of Naval Research under Grant N00010-10-1-0810 and Grant N00014-11-1-0006.

Z. Hu, S. Hou, and R. C. Qiu are with the Department of Electrical andComputer Engineering, Cognitive Radio Insitute, Center for ManufacturingResearch, Tennessee Technological University, Cookeville, TN 38505 USA(e-mail: [email protected]; [email protected]; [email protected]).

M. C. Wicks is with the Sensor Systems Division, University of Day-ton Research Institute, Dayton, OH 45469 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2013.2245904

obtain using wireless communication devices, especially in thepresence of noise [1].

Wireless tomography was first proposed in [7]. Wirelesstomography, which combines wireless communication andRF tomography, gives a novel approach to remote sensing.Incoherent tomography, coherent tomography, and self-coherent tomography have been summarized and comparedin [7]. Machine learning and waveform diversity have beenexplored in the context of wireless tomography to improvethe systems performance [8]. Potential applications of timereversal and compressive sensing to wireless tomography havebeen discussed in [7] and [8].

When wireless communication devices instead of sophisti-cated equipment are exploited to perform wireless tomography,accurate phase information of the received total fields is hardto obtain. Thus, self-coherent tomography is proposed, whichhas two steps [7]. First, phase reconstruction is achieved usingthe received amplitude-only total fields. Second, the standardinverse scattering algorithms, e.g., Born iterative method,distorted Born iterative method, contrast source inversion[9]–[12], and so on, are used for data analysis and imaging.

However, the complex working situation and dynamic radioenvironment still cause many challenges for wireless tomog-raphy. For real-world applications, the effect of noise cannotbe ignored. Previous literature on RF tomography or inversescattering rarely addressed the noise issue. The correspondingalgorithms were assumed to work in the extremely high signal-to-noise ratio (SNR) region. With the consideration of noise,noise reduction should be executed before phase reconstructionin wireless tomography, which means the de-noised amplitude-only total fields can be obtained from the large-scale receivednoise-polluted data. Otherwise, the poor performance of phasereconstruction in the presence of noise will cause severeimaging degradation.

Even with the aid of noise reduction, phase reconstructioncannot be well achieved [13]. Hence, the standard phasereconstruction [7], [14], [15], should be redesigned based onoptimization theory. More relevant and accurate informationabout scattered fields should be incorporated into the standardphase reconstruction [7], [14], [15]. This paper proposes ahybrid system for wireless tomography. Some sensors in thesystem are advanced and sophisticated. These sensors candirectly provide accurate full-data (both amplitude and phaseinformation) total fields even in the presence of noise. Othersensors are cheap and simple, e.g., wireless communicationdevices. Those simple sensors, on the other hand, can onlyget the noise-polluted amplitude-only data for the received

0196-2892 © 2013 IEEE

HU et al.: WIRELESS TOMOGRAPHY IN NOISY ENVIRONMENTS USING MACHINE LEARNING 957

Fig. 1. System diagram for wireless tomography.

total fields. Kernel principal component analysis (PCA), whichis a nonlinear dimensionality reduction, will be used to per-form noise reduction. Then, the accurate full-data total fieldsobtained by the advanced sensors will be exploited for phasereconstruction to retrieve the full-data scattered fields for thosesimple sensors. In this paper, we assume all incident fieldsare perfectly known. Single-frequency wireless tomography isconsidered. Meanwhile, the perfect wireless communicationlinks are provided.

The key steps of novel wireless tomography in noisy envi-ronments can be summarized as follows.

1) Noise reduction using kernel PCA.2) Modified standard phase reconstruction for the simple

sensors using the information from the advanced sen-sors.

3) Imaging.

The system diagram of wireless tomography is shown in Fig. 1compared with that of the self-coherent tomography shown inFig. 2.

The rest of this paper is organized as follows. In Section II,the hybrid system for wireless tomography is presented.Sections III, IV, and V will describe the theories for noisereduction, modified standard phase reconstruction, and imag-ing, respectively. Performance of wireless tomography will beprovided in Section VI, followed by some remarks given inSection VII.

II. HYBRID SYSTEM

The hybrid system for wireless tomography is shown inFig. 3. Some sensors in the system are advanced and sophis-ticated, while other sensors are low-cost and simple. Due tothe energy limitation of the simple sensor, the data obtainedby the simple sensor will be sent to the nearest advancedsensor. In this way, the lifetime of the simple sensor can beincreased. Then the data obtained by the advanced sensorsand received from the simple sensors will be transferred to

Fig. 2. System diagram for self-coherent tomography.

Fig. 3. Hybrid system for wireless tomography.

the central controller of the hybrid system to perform signalprocessing for wireless tomography.

Assume the imaging area � is a circle plane with radiusof a, which encloses the target. The measurement domain� is a circle line with radius of b. There are total N + Lsensors on the circle line, in which N sensors are simple withangles of θn, n = 1, 2, . . . , N and L sensors are advancedwith angles of θl , l = 1, 2, . . . , L. How to determine the ratioof the simple sensors to the advanced sensors as well as theoptimal deployment is out of scope of this paper. The sourcedomain is also a circle line with radius of c. There are I totalsources with angles of θi , i = 1, 2, . . . , I . �, �, and the sourcedomain are concentric.


Horn antennas are used for all the sources and sensors,which keep the excitation as close as possible to transversemagnetic (TM) polarization [16]. 2-D near field diffrac-tion tomography is considered. The background medium isassumed to be homogeneous with dielectric permittivity of εb

and magnetic permittivity of μ0. When the i th source soundsthe imaging area using a single-tone signal with frequencyof ω, the scalar and nonlinear state equation can be writtenas [7], [14], [15]

Etot,i (r) = Einc,i (r) + k2∫

�G(r − r′)χ(r′)Etot,i (r′)dr′,

r ∈ � (1)

where1) k is the wavenumber given as

k = ω√

μ0εb; (2)

2) G(r − r′) is the 2-D free space Green’s function givenas [17], [18]

G(r − r′) = −√−1

4H (2)

0 (k∣∣r − r′∣∣) (3)

in which H (2)0 is the Hankel function of the zero order

and the second kind;3) χ (r) is the contrast function given as

χ (r) = εr (r) − 1; (4)

4) target’s dielectric permittivity is εr (r) εb;5) the conductivity of the target is not considered in this

paper;6) Einc,i (r) and Etot,i (r) are incident and total fields eval-

uated in �, respectively.Based on the state equation, the corresponding measurement

equation for the lth advanced sensor is

Etot,i (θl) = Einc,i (θl) + Ed,i(θl)

= Einc,i (θl) + k2∫

�G(r − r′)χ(r′)Etot,i (r′)dr′

(5)

where Einc,i (θl), Ed,i(θl), and Etot,i (θl) are the direct incidentfield, scattered field, and total field measured by the lthadvanced sensor r = (b cos (θl) , b sin (θl)) ∈ �, respectively.

However, those simple sensors can only get noise-pollutedamplitude-only data for the received total fields. Thus, themeasurement equation for the nth simple sensor is

xi (θn) = ∣∣Etot,i (θn) + ni (θn)∣∣ (6)

where Etot,i (θn) is the true total field and ni (θn) is thecomplex-valued background noise the real and imaginary partsof which both follow Gaussian distribution with zero meanand the same variance. Though ni (θn) is the well knownGaussian noise, due to the nonlinear operation in (6), the noiseeffect will be more severe than that caused by additive whiteGaussian noise (AWGN).

Thus, the goal of noise reduction and phase reconstructionis to retrieve the accurate full-data scattered fields for thosesimple sensors, i.e., Ed,i(θn), n = 1, 2, . . . , N , when the i thsource sounds the target. This procedure will be repeated foreach of the I sources.

III. NOISE REDUCTION

The task of noise reduction is to obtain the de-noised data∣∣∣Etot,i (θn)∣∣∣ from the noise-polluted data xi (θn). When the i th

source sounds the target, the vector representation of (6) canbe simplified as

x = |Etot + n| (7)

where

x =

⎡⎢⎢⎢⎣

xi (θ1)xi (θ2)

...xi (θN )

⎤⎥⎥⎥⎦ (8)

Etot =

⎡⎢⎢⎢⎣

Etot,i (θ1)Etot,i (θ2)

...Etot,i (θN )

⎤⎥⎥⎥⎦ (9)

and

n =

⎡⎢⎢⎢⎣

ni (θ1)ni (θ2)

...ni (θN )

⎤⎥⎥⎥⎦ . (10)

In this paper,∣∣∣Etot,i (θn)

∣∣∣ is related to the de-noised data

corresponding to the true data∣∣Etot,i (θn)

∣∣ , n = 1, 2, . . . , N .x constructs one sample vector in the dataset for noise

reduction. The whole dataset can be achieved by executinga large number of measurements,

xm = |Etot + nm | , m = 1, 2, . . . , M (11)

where M is the number of measurements (sample vectors) andnm, m = 1, 2, . . . , M for different measurements are assumedto be independent.

Hence, a set of sample vectors xm, m = 1, 2, . . . , M corre-sponding to the noise-polluted amplitude-only total fields forthe simple sensors are obtained. If the full-data scattered fieldsare directly reconstructed from the noise-polluted amplitude-only total fields, the reconstructed values will be far from thetrue values. In order to solve this critical problem in wirelesstomography, this paper proposes to perform noise reductionbefore phase reconstruction. Based on the observation thatsignal always lies in a low-dimensional space of the high-dimensional data, machine learning, especially dimensionalityreduction, will be used to execute noise reduction.

Dimensionality reduction [19]–[23] tries to find a low-dimensional embedding of the high-dimensional data.PCA [24] is the best-known linear dimensionality reductionmethod. PCA works well for the high-dimensional data withlinear relationship, but always fails in a nonlinear situation.PCA can be extended to the nonlinear counterpart by using akernel [25]–[28], called kernel PCA [29].

Assume the original dataset contains M sample vectors,i.e., xm ∈ RN , m = 1, 2, . . . , M . The corresponding samplevector of xm after dimensionality reduction is ym ∈ RK ,m = 1, 2, . . . , M where K � N .

Kernel PCA uses the kernel function k(xmi , xm j ) = ϕ(xmi )·ϕ(xm j ) to implicitly map the original data into a feature


space F, where ϕ is the mapping function and · representsinner product. In F, PCA can be applied.

If k is a valid kernel function, the kernel matrix

K =

⎡⎢⎢⎢⎣

k (x1, x1) k (x1, x2) · · · k (x1, xM )

k (x2, x1) k (x2, x2) · · · k (x2, xM )...

......

...

k (xM , x1) k (xM , x2) · · · k (xM , xM )

⎤⎥⎥⎥⎦ (12)

must be a positive semi-definite matrix [30], [31].Assume the mean of ϕ(xm), m = 1, 2, . . . , M is zero,

1

M

M∑m=1

ϕ(xm) = 0. (13)

The covariance matrix of ϕ(xm), m = 1, 2, . . . , M is

CF = 1

M

M∑m=1

ϕ(xm)ϕ(xm)T . (14)

In order to apply PCA in F, the eigenvectors vFm of CF

are needed. The eigenvectors vFm of CF lie in the span of

ϕ(x j ), j = 1, 2, . . . , M , i.e., [29]

vFm =

M∑j=1

α jmϕ(x j ). (15)

It has been proven that α1,α2, . . . ,αM are eigenvectors ofkernel matrix K [29] and α jm are componentwise elements ofαm ,

αm =

⎡⎢⎢⎢⎣

α1m

α2m...

αMm

⎤⎥⎥⎥⎦ . (16)

The procedure of kernel PCA can be summarized into thefollowing six steps.

1) Choose a kernel function k.2) Compute kernel matrix K.3) Obtain the eigenvalues λK

1 ≥ λK2 ≥ · · · ≥ λK

M and thecorresponding eigenvectors α1,α2, . . . ,αM by diagonal-izing K.

4) Normalize vFk by [29]

αk = αk√λK

k

(17)

5) Constitute the basis of a subspace in F from the nor-malized eigenvectors vF

k , k = 1, 2, . . . , K6) Compute the projection of xm on vF

k , k = 1, 2, . . . , Kby

ykm = vFk · ϕ(xm)

=M∑

j=1

α j kk(x j , xm). (18)

Hence, ym corresponding to xm can be written as,

ym =

⎡⎢⎢⎢⎣

y1m

y2m...

yK m

⎤⎥⎥⎥⎦ . (19)

So far the mean of ϕ(xm), m = 1, 2, . . . , M has beenassumed to be zero. In fact, the data with zero mean in thefeature space are

ϕ(xm) − 1

M

M∑m=1

ϕ(xm). (20)

The kernel matrix for the centering data can be derived as [29]

K = HKH (21)

in which H is the so-called centering matrix defined as,

H = I − 1

M11T (22)

where I is an identity matrix, 1 = (1, 1, . . . , 1)T is a M × 1vector, and T represents transpose operator.

In order to apply kernel PCA for noise reduction, the pre-image xm (in the original space) of ym (in the feature space)will be calculated if xm is to be de-noised. The distance-constraint-based method used here for noise reduction exploresthe distance relationship between the original space and thefeature space for some specific kernels [32]. It tries to findthe distance between xm and x j , j = 1, 2, . . . , M once thedistance between ym and ϕ(x j ) is known. d(xm1, xm2) is usedto represent distance between two vectors xm1 and xm2 .

The squared distance between ym and ϕ(x j ) can be derivedas [33],

d2(ym, ϕ(x j ))

= (kxm + 1

M K1 − 2kx j

)THT MH

(kxm − 1

M K1)

+ 1M2 1T K1 + K j j − 2

M 1T kx j (23)

where

kxm =

⎡⎢⎢⎢⎣

k(xm, x1)

k(xm, x2)...

k(xm, xM )

⎤⎥⎥⎥⎦ (24)

K j j = k(x j , x j ) (25)

and

M =K∑

k=1

1

λkαk α

Tk (26)

in which λk and αk are the kth largest eigenvalue andcorresponding column eigenvector of K, respectively.

By making use of the distance relationship between theoriginal space and the feature space [32], d2(xm, x j ), j =1, 2, . . . , M can be obtained. In this paper, the radial basiskernel defined in (27) with parameter γ = 2 will be used

k(xm1, xm2) = exp(−γ∥∥xm1 − xm2

∥∥2). (27)

Then d2(xm, x j ), j = 1, 2, . . . , M will be expressed as [33]

d2(xm, x j ) = − 1

γlog

(1

2(k(xm, xm) + k(x j , x j )

− d2(ym, ϕ(x j )))

). (28)


These distances (in the original space) are exploited toconstrain the embedding of the pre-image xm [33], [34]. Onlythe local neighborhood structure needs to be preserved, sotwenty nearest neighbors are selected to determine the locationof xm [33].

Assume these twenty neighbors are x j1, x j2, . . . , x j20 .Define

X = [x j1 x j2 . . . x j20] (29)

and perform singular value decomposition (SVD) to XHas [33]

XH = U ∧ VT (30)

= UZ (31)

where H is the centering matrix defined in (22) with a size of20 × 20 and Z = [z1 z2 . . . z20].

Define

d20 =

⎡⎢⎢⎢⎣

‖z1‖22

‖z2‖22

...

‖z20‖22

⎤⎥⎥⎥⎦ (32)

and

d2 =

⎡⎢⎢⎢⎣

d2(xm, x j1)

d2(xm, x j2)...

d2(xm, x j20)

⎤⎥⎥⎥⎦ . (33)

Hence, xm can be calculated as [33]

xm = −1

2U�−1VT (d2 − d2

0) + 1

20

20∑i=1

x ji . (34)

Finally, the de-noised data∣∣∣Etot

∣∣∣ = xm can be achieved with

the componentwise element∣∣∣Etot,i (θn)

∣∣∣.IV. MODIFIED STANDARD PHASE RECONSTRUCTION

In diffraction tomography or inverse scattering, phase recon-struction means reconstructing the full-data scattered fieldsfrom the full-data incident fields and the amplitude-onlytotal fields [7], [14], [15]. Mathematically speaking, phasereconstruction tries to reconstruct Ed,i(θn) from Einc,i (θn) and∣∣Etot,i (θn)

∣∣2.

In order to solve the phase reconstruction problem,a sequence of nonlinear equations will be built. However,

B[Ed,i(θn)] = ∣∣Etot,i (θn)∣∣2 − ∣∣Einc,i (θn)

∣∣2

= (Ed,i(θn) + Einc,i (θn)

) (Ed,i(θn) + Einc,i (θn)

)∗ − ∣∣Einc,i (θn)∣∣2

= ∣∣Ed,i (θn)∣∣2 + (

Ed,i(θn)) (

Einc,i (θn))∗ + (

Einc,i (θn)) (

Ed,i (θn))∗

= ∣∣Ed,i (θn)∣∣2 + 2Re[Ed,i(θn)Einc,i (θn)

∗] (35)

∣∣Etot,i (θn)∣∣2 − ∣∣Einc,i (θn)

∣∣2

=∣∣∣∣∣

ka∑q=−ka

(xq + j yq)e jqθn

∣∣∣∣∣2

+ 2Re

[(

ka∑q=−ka

(xq + j yq)e jqθn)Einc,i (θn)∗]

, n = 1, 2, . . . , N(37)

if the number of variables to be solved is similar to or largerthan the number of equations, the problem will be underdeter-mined, which will make the solution non unique and sensitiveto disturbance. Thus, we should find the key variables,which are called degrees of freedom, inside the phasereconstruction problem to reduce the number of variables tobe solved.

The operator related to the scattered field is given in (35),shown at the bottom of the page, [14], in which ∗ stands forconjugate operator.

It has been thoroughly investigated [35] that the scatteredfield can be accurately represented with 2ka Fourier harmon-ics, which have the finite degrees of freedom. Thus,

Ed,i(θn) =ka∑

q=−ka

cqe jqθn

=ka∑

q=−ka

(xq + j yq)ejqθn n = 1, 2, . . . , N (36)

in which cq = xq + j yq . The Fourier harmonic coefficientsxq and yq are the actual variables to be retrieved. N nonlinearequations in (37), shown at the bottom of the page, will beestablished with the known Einc,i (θn) and

∣∣Etot,i (θn)∣∣2.

In order to make the problem overdetermined, the numberof variables which is 4ka + 2 should be less than N . Aslong as the optimal xq and yq are obtained, Ed,i(θn) can becalculated by (36).

In this paper, the standard phase reconstruction meansretrieving the full-data scattered fields directly from the noise-polluted amplitude-only total fields defined in (6) for thosesimple sensors.

However, in wireless tomography, even after noise reduc-tion,

∣∣Etot,i (θn)∣∣ cannot be exactly obtained.

∣∣∣Etot,i (θn)∣∣∣ will be

used instead of∣∣Etot,i (θn)

∣∣. The slight deviation of∣∣∣Etot,i (θn)

∣∣∣from

∣∣Etot,i (θn)∣∣ will cause the reconstructed scattered fields

to be far from the true scattered fields.Thus, we should regularize xq and yq to force the recon-

structed scattered fields to approach the true scattered fields.In order to achieve this goal, the accurate full-data scatteredfields provided by the advanced sensors will be explored toset up L linear equations

Etot,i (θl) − Einc,i (θl) =ka∑

q=−ka

(xq + j yq

)e jqθl

l = 1, 2, . . . , L . (38)


Finally, the MATLAB function fsolve will be adopted to givethe solution to the established hybrid equation set with Nnonlinear equations defined in (37) in top of next page andL linear equations defined in (38).

V. IMAGING

Born iterative method [36] will be exploited to performcoherent tomography imaging. Imaging of the target with theclose-in sensors [37]–[39] shown in Fig. 3 will be taken intoaccount.

All the sources take turns to sound the target. Theadvanced sensors can measure the accurate full-data scat-tered fields directly, while the scattered fields from the sim-ple sensors will be reconstructed by noise reduction andmodified standard phase reconstruction mentioned before.After all the sources sound the target, the multistaticdata matrix E corresponding to scattered fields can beobtained.

Generally, assume the imaging area is discretized into Scells with the same area of . The location of each cell isrs, s = 1, 2, . . . , S and the contrast of each cell is χs, s =1, 2, . . . , S. The sensor location is β j , j = 1, 2, . . . , N +L. The source location is γ i , i = 1, 2, . . . , I . In thispaper

β j = (b cos

(θ j

), b sin

(θ j

))(39)

and

γ i = (c cos (θi ) , c sin (θi)). (40)

Based on the measurement equation shown in (5), E j i isexpressed as

E j i =S∑

s=1

χs G(β j − rs

)Etot,i (rs)(k

2 ) (41)

where E j i represents the scattered field measured by the j thsensor when the i th source sounds the target and the total fieldEtot,i (rs) can be calculated from the state equation shownin (1) when the i th source sounds the target.

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

G(β1 − r1

)Etot,1 (r1)(k2 ) · · · G

(β1 − rS

)Etot,1 (rS)(k2 )

G(β2 − r1

)Etot,1 (r1)(k2 ) · · · G

(β2 − rS

)Etot,1 (rS)(k2 )

......

...

G(β(N+L) − r1

)Etot,1 (r1)(k2 ) · · · G

(β(N+L) − rS

)Etot,1 (rS)(k2 )

G(β1 − r1

)Etot,2 (r1)(k2 ) · · · G

(β1 − rS

)Etot,2 (rS)(k2 )

......

...

G(β(N+L) − r1

)Etot,I (r1)(k2 ) · · · G

(β(N+L) − rS

)Etot,I (rS)(k2 )

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (44)

A =

⎡⎢⎢⎢⎣

1 −χ2G (r1 − r2)(k2

) · · · −χS G (r1 − rS)(k2

)−χ1G (r2 − r1)

(k2

)1 · · · −χS G (r2 − rS)

(k2

)...

......

...

−χ1G (rS − r1)(k2

) −χ2G (rS − r2)(k2

) · · · 1

⎤⎥⎥⎥⎦ . (47)

Define e ∈ C I (N+L)×1 as e( j+(i−1)(N+L))1 = E j i

e =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E11E21...

E(N+L)1E12...

E(N+L)I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (42)

Define χ as χ s1 = χs ,

χ =

⎡⎢⎢⎢⎣

χ1

χ2...

χS

⎤⎥⎥⎥⎦ . (43)

Define B ∈ C I (N+L)×S as B( j+(i−1)(N+L))s =G

(β j − rs

)Etot,i (rs)(k2 ), i.e., (44), shown at the bottom

of the page.Define U ∈ C S×I as Usi = Etot,i (rs), i.e.,

U =

⎡⎢⎢⎢⎣

Etot,1 (r1) Etot,2 (r1) · · · Etot,I (r1)

Etot,1 (r2) Etot,2 (r2) · · · Etot,I (r2)...

......

...

Etot,1 (rS) Etot,2 (rS) · · · Etot,I (rS)

⎤⎥⎥⎥⎦ . (45)

Define G ∈ C S×I as Gsi = G(rs − γ i

)

G =

⎡⎢⎢⎢⎣

G(r1 − γ 1

)G

(r1 − γ 2

) · · · G(r1 − γ I

)G

(r2 − γ 1

)G

(r2 − γ 2

) · · · G(r2 − γ I

)...

......

...

G(rS − γ 1

)G

(rS − γ 2

) · · · G(rS − γ I

)

⎤⎥⎥⎥⎦ . (46)

Define A ∈ C S×S as (47), shown at the bottom of the page.Based on the state equation shown in (1)

AU = G. (48)

Born iterative method is used to get χ by the followingsteps [9], [17], [36], [40]–[42]:

1) set χ (0) to be zero; t = −1;2) t = t + 1; get B(t) based on (44)–(48) using χ (t)


Fig. 4. Hybrid target called FoamDielInt in French data [16].

3) solve the inverse problem in (49) by Tikhonov regular-ization to get χ (t+1;)

e = B(t)χ (t+1); (49)

4) if χ converges, Born iterative method is stopped; other-wise Born iterative method goes to Step 2.

VI. PERFORMANCE

Experimental data provided by the Institute Fresnel in Mar-seille, France are used to demonstrate the concept of wirelesstomography and verify the corresponding algorithms. The freespace experimental setup and measurement precision of dataare mentioned in [16]. These experimental data have beenwidely used to test the novel algorithms for inverse scatteringand RF tomography [43]–[47].

First, the data corresponding to the hybrid target calledFoamDielInt are exploited. TM polarization is measured. Thishybrid target shown in Fig. 4 consists of a foam cylinder(SAITEC SBF 300) with diameter of 80 mm and relativepermittivity of 1.45 ± 0.15 as well as a plastic cylinder(berylon) with diameter of 31 mm and relative permittivityof 3 ± 0.3 [16]. The frequency of the sounding signal is2 GHz. The imaging area, i.e., the investigated domain, is150 × 150 mm. The imaging resolution is 5 mm. The numberof sources is eight and the number of total sensors is 240. Thedistance between each source and the center of the imagingarea is 1.67 m. The distance between the center of the imagingarea and each sensor is also 1.67 m. The source positions aretaken with a step of 45° from 0° [16]. The sensor positionsare taken with a step of 1° from 60° [16]. If all the sensors areadvanced, the imaging result using accurate full-data scatteredfields is shown in Fig. 5.

For verifying the concept of wireless tomography in thepresence of noise, the man-made noises are added into theexperimental data. The number of measurements M is 200.SNR is equal to 15 dB. The number of advanced sensorsfor each source is two. These advanced sensors are uniformlydeployed among all the sensors. The reconstructed amplitudeinformation and phase information using the proposed phase

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.061

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Fig. 5. Imaging result for FoamDielInt using accurate full-data scatteredfields.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Radian

Am

plitu

de

TrueProposed Phase ReconstructionStandard Phase Reconstruction

Fig. 6. Reconstructed amplitude information when the number of theadvanced sensors for each source is two and SNR is 15 dB. (Proposedphase reconstruction includes noise reduction and modified standard phasereconstruction for the simple sensors using the information from the advancedsensors.)

reconstruction, including noise reduction and modified stan-dard phase reconstruction for the simple sensors are shown inFigs. 6 and 7, respectively, when the first source sounds the tar-get. The reconstructed full-data scattered fields for the simplesensors by the proposed phase reconstruction approach the truescattered fields. The imaging result of wireless tomography isshown in Fig. 8. If the Born iterative method is applied to theaccurate full-data scattered fields measured by two advancedsensors for each source, the imaging result is shown in Fig. 9.If self-coherent tomography is applied to the noise-pollutedamplitude-only total fields measured by simple sensors, Borniterative method cannot converge to get the imaging result.Hence, with the aid of a small amount of advanced sensors,the imaging performance of wireless tomography in noisyenvironments is acceptable and promising. Wireless tomogra-phy explores a priori information from advanced sensors andthe state-of-the-art mathematics, e.g., machine learning andoptimization theory, to extract knowledge from the large-scale


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−4

−3

−2

−1

0

1

2

3

Radian

Pha

seTrueProposed Phase ReconstructionStandard Phase Reconstruction

Fig. 7. Reconstructed phase information when the number of the advancedsensors for each source is two and SNR is 15 dB. (Proposed phase recon-struction includes noise reduction and modified standard phase reconstructionfor the simple sensors using the information from the advanced sensors.)

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Fig. 8. Imaging result of wireless tomography for FoamDielInt when thenumber of the advanced sensors for each source is two and SNR is 15 dB.

noisy data provided by simple sensors. Wireless tomographycan be treated as one realization of data-enabled science andengineering for “Big Data” in remote sensing.

The number of advanced sensors for each source isincreased to three. These advanced sensors are also uniformlydeployed among all the sensors. The reconstructed amplitudeinformation and phase information using the proposed phasereconstruction are shown in Figs. 10 and 11, respectively,when the first source sounds the target. The imaging result ofwireless tomography is shown in Fig. 12. If the Born iterativemethod is applied to the accurate full-data scattered fieldsmeasured from three advanced sensors for each source, theimaging result is shown in Fig. 13. If SNR is increased to20 dB, the imaging result of wireless tomography is shown inFig. 14.

Then, the data corresponding to the hybrid target calledFoamDielExt are considered. TM polarization is alsomeasured. This hybrid target consists of a foam cylinder

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Fig. 9. Imaging result for FoamDielInt when the Born iterative method isapplied to the accurate full-data scattered fields from two advanced sensorsfor each source.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Radian

Am

plitu

de

TrueProposed Phase ReconstructionStandard Phase Reconstruction

Fig. 10. Reconstructed amplitude information when the number of theadvanced sensors for each source is three and SNR is 15 dB. (Proposedphase reconstruction includes noise reduction and modified standard phasereconstruction for the simple sensors using the information from the advancedsensors.)

(SAITEC SBF 300) with diameter of 80 mm and relativepermittivity of 1.45 ± 0.15 as well as a plastic cylinder(berylon) with diameter of 31 mm and relative permittivityof 3 ± 0.3 [16]. The experiment setup for FoamDielExt is thesame as the experiment setup for FoamDielInt. The numberof advanced sensors for each source is three. SNR is equal to20 dB. The imaging result of wireless tomography is shown inFig. 15. If the Born iterative method is applied to the accuratefull-data scattered fields measured from three advanced sensorsfor each source, the imaging result is shown in Fig. 16.

Finally, the data corresponding to the hybrid targetcalled FoamTwinDiel are used. TM polarization is alsomeasured. This hybrid target consists of a foam cylinder(SAITEC SBF 300) with diameter of 80 mm and relativepermittivity of 1.45 ± 0.15 as well as two plastic cylinders(berylon) with diameter of 31 mm and relative permittivity of3±0.3 [16]. The number of sources is 18. The source positions


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5−4

−3

−2

−1

0

1

2

3

4

Radian

Pha

seTrueProposed Phase ReconstructionStandard Phase Reconstruction

Fig. 11. Reconstructed phase information when the number of the advancedsensors for each source is three and SNR is 15 dB. (Proposed phase recon-struction includes noise reduction and modified standard phase reconstructionfor the simple sensors using the information from the advanced sensors.)

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Fig. 12. Imaging result of wireless tomography for FoamDielInt when thenumber of the advanced sensors for each source is three and SNR is 15 dB.

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.8

1

1.2

1.4

1.6

1.8

Fig. 13. Imaging result for FoamDielInt when the Born iterative method isapplied to the accurate full-data scattered fields from three advanced sensorsfor each source.

are taken with a step of 20° from 0° [16]. The number ofadvanced sensors for each source is three. SNR is equal to20 dB. The imaging result of wireless tomography is shown inFig. 17. If the Born iterative method is applied to the accurate

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.061

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Fig. 14. Imaging result of wireless tomography for FoamDielInt when thenumber of the advanced sensors for each source is three and SNR is 20 dB.

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.5

1

1.5

2

2.5

Fig. 15. Imaging result of wireless tomography for FoamDielExt when thenumber of the advanced sensors for each source is three and SNR is 20 dB.

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.060.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Fig. 16. Imaging result for FoamDielExt when the Born iterative method isapplied to the accurate full-data scattered fields from three advanced sensorsfor each source.

full-data scattered fields measured from three advanced sensorsfor each source, the imaging result is shown in Fig. 18. In sum,wireless tomography gives more accurate imaging results.Target characteristics and locations can be easily identified.Due to the limited number of advanced sensors for eachsource, only using the accurate full-data scattered fields from


y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.060.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Fig. 17. Imaging result of wireless tomography for FoamTwinDiel when thenumber of the advanced sensors for each source is three and SNR is 20 dB.

y (m)

x (m

)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0

0.02

0.04

0.06 1

1.2

1.4

1.6

1.8

2

2.2

Fig. 18. Imaging result for FoamTwinDiel when the Born iterative method isapplied to the accurate full-data scattered fields from three advanced sensorsfor each source.

advanced sensors cannot provide full information to image thetarget.

VII. CONCLUSION

This paper followed the novel concept of wireless tomogra-phy and dealt with wireless tomography in noisy environmentsfor potential real-world applications. We would like to explorewireless tomography using currently available commercialwireless communication network or the next generation cog-nitive radio network. The bottleneck is to retrieve accuratephase information from noisy measurements when wirelesscommunication devices were used. This paper addressed thisfundamental issue. The experimental data were used to demon-strate the concept of wireless tomography and verified thecorresponding algorithms. From the simulation results, theimaging performance of wireless tomography is very promis-ing even in the presence of noise.

Eventually, we would like to take wireless tomography fromthe lab to the real-world situations, e.g., battle field, searchand rescue, surveillance, reconnaissance, and so on. For thelarge-scale deployment of sensors for wireless tomography,smart phone, universal software radio peripheral (USRP),or wireless open access research platform (WARP) can beused as simple sensors to perform sensing function and data

acquisition. We will continue this line of work and set upour own hybrid system to demonstrate the real-world wirelesstomography [48].

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their helpful comments and suggestions.

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Zhen Hu received the B.S. degree from theHuazhong University of Science and Technology,Wuhan, China, the M.S. degree from SoutheastUniversity, Nanjing, China, and the Ph.D. degree inelectrical engineering from Tennessee TechnologicalUniversity, Cookeville, USA, in 2004, 2007, and2010, respectively.

He is currently a Post-Doctoral Research Asso-ciate with the Center for Manufacturing Research,Tennessee Technological University. His currentresearch interests include system integration and

optimization for wireless communication, radars, and sensing.

Shujie Hou (S’11) received the B.S. degree from theDaqing Petroleum Institute, Daqing, China, and theM.S. degree inform the Dalian University of Tech-nology, Dalian, China, in 2006 and 2008, respec-tively. She is currently pursuing the Ph.D. degreein the Department of Electrical and Computer Engi-neering, Center for Manufacturing Research, Ten-nessee Technological University, Cookeville, USA.

Michael C. Wicks (S’81–M’89–SM’90–F’98)received the B.Sc. degree from Rensselaer Poly-technic Institute, Troy, NY, USA, in 1981, and theM.Sc. and Ph.D. degrees from Syracuse University,Syracuse, NY, in 1985 and 1995, respectively, all inelectrical engineering.

He retired from the Sensors Directorate of theAir Force Research Laboratory in 2011, and joinedthe University of Dayton Research Institute. Hehas authored or co-authored several papers in jour-nals and conferences. His current interests include

waveform diversity, radar signal processing, knowledge-based control, andradar systems engineering. He participates in numerous panels, committees,and working groups, including the IEEE, NATO, and other multinationalorganizations.

Robert Caiming Qiu (S’93–M’96–SM’01) receivedthe Ph.D. degree in electrical engineering from NewYork University (former Polytechnic University),Brooklyn, USA.

He is currently a Full Professor with the Depart-ment of Electrical and Computer Engineering, Cen-ter for Manufacturing Research, Tennessee Tech-nological University, Cookeville, USA, where hejoined as an Associate Professor in 2003 and becamea Full Professor in 2008. His current interests includewireless communication and networking, machine

learning, and smart grid technologies. He was the Founder-CEO and thePresident of Wiscom Technologies, Inc., which involved in the manufacturingand marketing of WCDMA chipsets. He was with GTE Labs, Inc. (nowVerizon), Waltham, MA, USA, and Bell Labs, Lucent, Whippany, NJ, USA.He has intensive research experience in wireless communications and network,machine learning, smart grid, digital signal processing, EM scattering, com-posite absorbing materials, RF microelectronics, UWB, underwater acoustics,and fiber optics. He has authored or co-authored over 50 papers in journals andconferences, and some book chapters. He has co-authored the book entitledCognitive Radio Communication and Networking: Principles and Practice(Wiley, 2012). He is a Guest Book Editor of Ultra-Wideband (UWB) WirelessCommunications (Wiley, 2005). He holds five patents.

He has contributed to 3GPP and IEEE standards bodies. In 1998, hedeveloped the first three courses on 3G for Bell Labs researchers. He wasan adjunct Professor with Polytechnic University, Brooklyn, NY, USA. Heis a Guest Editor of three special issues, including the IEEE JOURNAL

ON SELECTED AREAS IN COMMUNICATIONS, the IEEE TRANSACTIONS

ON VEHICULAR TECHNOLOGY, and the IEEE TRANSACTIONS ON SMART

GRID.

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Wireless Tomography in Noisy Environments Using Machine Learning