1

Radio Tomographic Imaging with WirelessNetworks

Joey Wilson and Neal PatwariThe University of Utah

Abstract—Radio Tomographic Imaging (RTI) is an emerging technology for imaging the attenuation caused by physical objects inwireless networks. This paper presents a linear model for using received signal strength (RSS) measurements to obtain images ofmoving objects. Noise models are investigated based on real measurements of a deployed RTI system. Mean-squared error (MSE)bounds on image accuracy are derived, which are used to calculate the accuracy of an RTI system for a given node geometry. Theill-posedness of RTI is discussed, and Tikhonov regularization is used to derive an image estimator. Experimental results of an RTIexperiment with 28 nodes deployed around a 441 square foot area are presented.

Index Terms—Wireless, Sensor Networks, Inverse Filtering, Linear Systems, Applications

F

1 INTRODUCTION

WHEN an object moves into the area of a wirelessnetwork, links which pass through that object

will experience shadowing losses. This paper exploresin detail the use of shadowing losses on links betweenmany pairs of nodes in a wireless network to imagethe attenuation of objects within the network area. Werefer to this problem as radio tomographic imaging (RTI),as depicted in Fig. 1.

RTI may be useful in emergencies, rescue operations,and security breaches, since the objects being imagedneed not carry an electronic device. Using the images totrack humans moving through a building, for example,provides a basis for new applications in security systemsand “smart” buildings.

The reduction in costs for radio frequency integratedcircuits (RFICs) and advances in peer-to-peer data net-working have made realistic the use of hundreds orthousands of simple radio devices in a single RTI de-ployment. Since the relative cost of such devices is low,large RTI networks are possible in applications that maybe otherwise impractical.

Radio tomography draws from the concepts of twowell-known and widely used types of imaging systems.First, radar systems transmit RF probes and receiveechoes caused by the objects in an environment [1]. A de-lay between transmission and reception indicates a dis-tance to a scatterer. Phased array radars also compute anangle of bearing. Such systems image an object in spacebased on reflection and scattering. Secondly, computedtomography (CT) methods in medical and geophysicalimaging systems use signal measurements along manydifferent paths through a medium. The measurementsalong the paths are used to compute an estimate of thespatial field of the transmission parameters throughoutthe medium [2]. RTI is also a transmission-based imag-

Fig. 1. An illustration of an RTI network. Each nodebroadcasts to the others, creating many projections thatcan be used to reconstruct an image of objects inside thenetwork area.

ing method which measures signal strengths on manydifferent paths through a medium, but similar to radarsystems, it does so at radio frequencies. It also faces twosignificant challenges:• The system discussed in this paper measures only

signal strength. No information about the phase orthe time-domain of a signal is available.

• The use of RF, as opposed to much higher frequencyEM waves (e.g., x-rays), introduces significant non-line-of-sight (NLOS) propagation in the transmis-sion measurements. Signals in standard commercialwireless bands do not travel in just the line-of-sight(LOS) path, and instead propagate in many pathsfrom a transmitter to a receiver.

2

1.1 ApplicationsDespite the difficulties of using RF, there is a majoradvantage: RF signals can travel through obstructionssuch as walls, trees, and smoke, while optical or infraredimaging systems cannot. RF imaging will also workin the dark, where video cameras will fail. Even forapplications where video cameras could work, privacyconcerns may prevent their deployment. An RTI systemprovides current images of the location of people andtheir movements, but cannot be used to identify a per-son.

One main future application of RTI is to reduce injuryfor correctional and law enforcement officers; many areinjured each year because they lack the ability to de-tect and track offenders through building walls [3]. Byshowing the locations of people within a building duringhostage situations, building fires, or other emergencies,RTI can help law enforcement and emergency respon-ders to know where they should focus their attention.

Another application is in automatic monitoring andcontrol in “smart” homes and buildings. Some buildingcontrol systems detect motion in a room and use itto control lighting, heating, air conditioning, and evennoise cancellation [4]. RTI systems can further determinehow many people are in a room and where they arelocated, providing more precise control.

RTI has application in security and monitoring sys-tems for indoor and outdoor areas. For example, mostexisting security systems are trip-wire based or camera-based. Trip-wire systems detect when people cross aboundary, but do not track them when they are withinthe area. Cameras are ineffective in the dark and havelimited view angles. An RTI system could serve both asa trip-wire, alerting when intruders enter into an area,and a tracking system to follow their movements.

1.2 Related WorkRF-based imaging has been dominated in the commer-cial realm by ultra-wideband (UWB) based through-the-wall (TTW) imaging devices from companies like TimeDomain [5], Cambridge Consultants [6], and CameroTech [7]. These companies have developed productsusing a phased array of radars that transmit UWBpulses and then measure echoes to estimate a range andbearing. These devices are accurate close to the device,but inherently suffer from accuracy and noise issues atlong range due to mono-static radar scattering losses andlarge bandwidths. Some initial attempts [8] allow 2-4 ofthese high-complexity devices to collaborate to improvecoverage.

In comparison, in this paper we discuss using dozensto hundreds of low-capability collaborating nodes,which measure transmission rather than scattering andreflection. Further, UWB uses an extremely wide RFbandwidth, which may limit its application to emer-gency and military applications. RTI is capable of usingradios with relatively small bandwidths.

To emphasize the small required bandwidth comparedto UWB, some relevant research is being called “ultra-narrowband” (UNB) radar [9], [10], [11]. These systemspropose using narrowband transmitters and receiversdeployed around an area to image the environmentwithin that area. Measurements are phase-synchronousat the multiple nodes around the area. Such techniqueshave been applied to detect and locate objects buried un-der ground using what is effectively a synthetic aperturearray of ground-penetrating radars [12]. Experimentshave been reported which measure a static environmentwhile moving one transmitter or one receiver [11], andmeasure a static object on a rotating table in an anechoicchamber in order to simulate an array of transmittersand receivers at many different angles [11], [12], [9].

Multiple-input-multiple-output (MIMO) radar is anotheremerging field that takes advantage of multiple trans-mitters and receivers to locate objects within a spatialarea [13]. In this framework, signals are transmitted intothe area of interest, objects scatter the signal, and thereflections are measured at each receiver. The scatteringobjects create a channel matrix which is comparable tothe channel matrix in MIMO communication theory. RTIdiffers from MIMO radar in the same way that it differsfrom traditional radar. Instead of measuring reflections,RTI uses the shadowing caused by objects as a basis forimage reconstruction.

Recent research has also used measurements of sig-nal strength on 802.11 WiFi links to detect and locatea person’s location. Experiments in [14] demonstratethe capability of a detector based on signal strengthmeasurements determine the location of a person whois not carrying an electronic device. In this case, thesystem is trained by a person standing at pre-definedpositions, and RSS measurements are recorded at eachlocation. When the system is in use, RSS measurementsare compared with the known training data, and the bestposition is selected from a list.

Our approach is not based on point-wise detection.Instead, we use tomographic methods to estimate animage of the change in the attenuation as a function ofspace, and use the image estimate for the purposes ofindicating the position of a moving object.

1.3 OverviewSection 2 presents a linear model relating RSS mea-surements to the change in attenuation occurring ina network area, and investigates statistics for noise indynamic multipath environments. Section 3 describesan error bound on image estimation for a given nodegeometry. This is useful to determine which areas ofa network can be accurately imaged for a given set ofnode locations. Section 4 discusses the ill-posedness ofRTI, and derives a regularized solution for obtainingan attenuation image. Section 5 describes the setup ofan actual RTI experiment, the resultant images, and adiscussion of the effect of parameters on the accuracy ofthe images.

3

2 MODEL

2.1 Linear FormulationWhen wireless nodes communicate, the radio signalspass through the physical area of the network. Objectswithin the area absorb, reflect, diffract, or scatter someof the transmitted power. The goal of an RTI system isto determine an image vector of dimension RN that de-scribes the amount of radio power attenuation occurringdue to physical objects within N voxels of a networkregion. Since voxel locations are known, RTI allows oneto know where attenuation in a network is occurring,and therefore, where objects are located.

If K is the number of nodes in the RTI network, thenthe total number of unique two-way links is M = K2−K

2 .Any pair of nodes is counted as a link, whether ornot communication actually occurs between them. Thesignal strength yi(t) of a particular link i at time t isdependent on:• Pi: Transmitted power in dB.• Si(t): Shadowing loss in dB due to objects that

attenuate the signal.• Fi(t): Fading loss in dB that occurs from construc-

tive and destructive interference of narrow-bandsignals in multipath environments.

• Li: Static losses in dB due to distance, antennapatterns, device inconsistencies, etc.

• νi(t): Measurement noise.Mathematically, the received signal strength is describedas

yi(t) = Pi − Li − Si(t)− Fi(t)− νi(t) (1)

The shadowing loss Si(t) can be approximated asa sum of attenuation that occurs in each voxel. Sincethe contribution of each voxel to the attenuation of alink is different for each link, a weighting is applied.Mathematically, this is described for a single link as

Si(t) =N∑j=1

wijxj(t). (2)

where xj(t) is the attenuation occurring in voxel j attime t, and wij is the weighting of pixel j for link i. Ifa link does not “cross” a particular voxel, that voxel isremoved by using a weight of zero. For example, Fig.2 is an illustration of how a direct LOS link might beweighted in a non-scattering environment.

Imaging only the changing attenuation greatly simpli-fies the problem, since all static losses can be removedover time. The change in RSS 4yi from time ta to tb is

4yi ≡ yi(tb)− yi(ta)= Si(tb)− Si(ta) + Fi(tb)− Fi(ta)

+νi(tb)− νi(ta), (3)

which can be written as

4yi =N∑j=1

wij4xj + ni, (4)

Fig. 2. An illustration of a single link in an RTI network thattravels in a direct LOS path. The signal is shadowed byobjects as it crosses the area of the network in a particularpath. The darkened voxels represent the image areas thathave a non-zero weighting for this particular link.

where the noise is the grouping of fading and measure-ment noise

ni = Fi(tb)− Fi(ta) + νi(tb)− νi(ta) (5)

and4xj = xj(tb)− xj(ta) (6)

is the difference in attenuation at pixel j from time ta totb.

If all links in the network are considered simultane-ously, the system of RSS equations can be described inmatrix form as

4y = W4x + n (7)

where

4y = [4y1,4y2, ...,4yM ]T

4x = [4x1,4x2, ...,4xN ]T

n = [n1, n2, ..., nM ]T

[W]i,j = wij (8)

In summary, 4y is the vector of length M all linkdifference RSS measurements, n is a noise vector, and4x is the attenuation image to be estimated. W is theweighting matrix of dimension M×N , with each columnrepresenting a single voxel, and each row describing theweighting of each voxel for that particular link. Eachvariable is measured in decibels (dB).

To simplify the notation used throughout the rest ofthis paper, x and y are used in place of 4x and 4y,respectively.

4

2.2 Weight ModelIf knowledge of an environment were available, onecould estimate the weights {wij}j for link i which re-flected the spatial extent of multiple paths between trans-mitter and receiver. Perhaps calibration measurementscould aid in estimation of the linear transformation W.However, with no site-specific information, we requirea statistical model that describes the linear effect of theattenuation field on the path loss for each link.

An ellipsoid with foci at each node location can beused as a method for determining the weighting for eachlink in the network [15]. If a particular voxel falls outsidethe ellipsoid, the weighting for that voxel is set to zero. Ifa particular voxel is within the LOS path determined bythe ellipsoid, its weight is set to be inversely proportionalto the square root of the link distance. Intuitively, longerlinks will provide less information about the attenua-tion in voxels that they cross. When link distances arevery long, the signals reflect and defract around theobstructions. A link with a distance of only a few feetwill experience more change in RSS when an obstructionoccurs than a link with a length of hundreds of feet.

Past studies have shown that the variance of linkshadowing does not change with distance. In accordancewith these studies, dividing by the square root of the linkdistance ensures that the voxel weighting takes this intoaccount [16]. The weighting is described mathematicallyas

wij =1√d

{1 if dij(1) + dij(2) < d+ λ0 otherwise (9)

where d is the distance between the two nodes, dij(1)and dij(2) are the distances from the center of voxel jto the two node locations for link i, and λ is a tunableparameter describing the width of the ellipse.

The width parameter λ is typically set very low inRTI, such that it is essentially the same as using the LOSmodel as depicted in Fig. 2. The use of an ellipsoid isprimarily used to simplify the process of determiningwhich voxels fall along the LOS path.

2.3 NoiseTo complete the model of (7), the statistics of thenoise vector n in (7) must be examined. Here, noise iscaused by time-varying measurement miscalibration ofthe receiver, by the contribution of thermal noise to themeasured receiver signal strength, and time-variationsin the multipath channel not caused by changes tothe attenuation experienced by the line-of-sight path.If these contributions are constant with time, then thecalibration (when moving attenuator existed in the field)would have been able to establish it as the baseline.Time variation in RSS measurement when no movingattenuator is blocking the line-of-sight path is “noise”for an RTI system.

Past studies have considered the time-variation ofRSS in fixed radio links. In particular, the work andmeasurements of Bultitude [17] were used to design

indoor fixed wireless communications systems whichperiodically experienced fading due to motion in thearea of the link. Bultitude found that RSS experiencedintervals of significant fading which were caused byhuman motion in and around the area. Most of the time,the measured RSS vary slowly around a nearly constantmean, what we call a non-fading interval. When in afading interval, RSS varies up to 10 dB higher and 20 dBbelow the non-fading interval mean, with a distributionthat can be characterized as a Rician distribution [17].Other measurements find temporal fading statistics moreclosely match a log-normal distribution [18]. The fading/ non-fading interval process can be modeled as a two-state Markov chain [19], which alternates between a low-variance and high-variance distribution. Over all time,measurements show a two-part mixture distribution forthe RSS on a fixed link.

In linear terms, we could model this data as a mixtureof two Rician distributions as in [17]; we could alsomodel it as a mixture of log-normal terms as suggestedby results in [18]. We note that the logarithm of a Ricianrandom variable is often similar in distribution to thelog-normal, perhaps a cause of disagreement betweenmeasurement studies. We choose to use the log-normalmixture model for simplicity; in the (dB) scale, this is atwo-part Gaussian mixture model:

fni(u) =∑

k∈{1,2}

pk√2πσ2

k

exp[− u2

2σ2k

], (10)

where pk is the probability of part k, p2 = 1−p1, σ2k is the

variance of part k, and fni(u) is the probability density

function of the noise random variable ni. Without lossof generality, we let σ2 > σ1 so that part 2 is the highervariance component of the mixture.

Past radio link measurements have not distinguishedbetween motion which shadows the line-of-sight path(the signal in RTI), and motion which does not shadowthe line-of-sight path (the noise) [17], [18], [19], [14]. Toinvestigate the statistics of RTI noise, we present exper-imental tests which measure the time-varying statisticsof links during motion which does not obstruct a link.

To collect experimental samples of noise, we set up 28nodes in an indoor office area empty of people. Whilethe nodes are transmitting and measuring RSS on eachpairwise link, people move around the outside of theperimeter of the deployment area. In no case did themotion of a person obstruct the LOS path of any link.From each link, about 66,000 measurements were taken.For example, consider the data on a typical link, thelink (3, 20). The temporal fading plot in Figure 3(a)shows similar results to [17], with alternating periods ofheavy fading and low fading. During low fading, datais confined within a range of 2-3 dB around -84 dBm.During high fading, variations at ± 10 dB from the meanoccur. The histogram shown in Fig. 3(b) correspondinglyshows a mixture of one high-variance and one low-variance distribution.

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Fig. 3. Temporal fading on link (3, 20) during non-obstructing motion, showing (a) time plot and (b) histogram.

We also summarize the measured data on all(282

)links.

The mean was removed from each link’s data, and thedata was merged. Fig. 4(a) is a quantile-quantile plotcomparing the removed-mean RSS measurements with aGaussian distribution N (0, σ2

d), where σ2d is the empirical

variance of the measurements. The PDF is approximatedby a Gaussian within ±2.5 quantiles.

As described, the data seems to follow a mixturedistribution. From measured data, we estimate the mix-ture parameters with an expectation-maximization (EM)algorithm [20], and the results are shown in Table 1. Fig.4(b) is a quantile-quantile plot comparing the removed-mean RSS measurements with a mixture model with thestated parameters.

Parameter Valueσ1 0.971σ2 3.003p1 0.548p2 0.452

TABLE 1Gaussian-Mixture Noise Model Parameters Estimated

From Measurements

3 ERROR BOUND3.1 DerivationThis section presents a lower bound on estimation errorfor the linear model (7) under the noise model discussedin Section 2.3. The estimation error vector is defined asε = x̂− x, and the error correlation matrix is

Rε = E[εεT]. (11)

A well-known result in estimation theory known as theMSE, Bayesian or Van Trees bound states that the errorcorrelation matrix is bounded by

Rε ≥ (JD + JP )−1 = J−1 (12)

where the inequality indicates that the matrix Rε − J−1

is positive semi-definite [21]. The matrix

JD = E[{∇x[lnP (y|x)]} {∇x[lnP (y|x)]}T ] (13)

is known as the Fisher information matrix and representsthe information obtained from the data measurements.The matrix

JP = E[{∇x[lnP (x)]} {∇x[lnP (x)]}T ] (14)

represents the information obtained from a priori knowl-edge about the random parameters.

We assume that the noise components n =[n1, . . . , nM ]T are independent and identically dis-tributed as two-component zero-mean Gaussian mixturerandom variables as in (10). The noise is independentbecause we assume nodes are placed at distances largerthan the coherence distance of the indoor fading channel.

From (13), we can derive that JD is given by [22, Eqn10],

JD = γWTW

where γ =∫ ∞−∞

[f ′ni(u)]2

fni(u)

du (15)

and f ′ni(u) is the derivative of fni

(u) with respect to u.When p2 = 0, that is, the distribution of ni is purelyGaussian, γ reverts to 1/σ2

1 , one over the variance ofthe distribution. For two-component Gaussian mixtures,we compute γ in (15) from numerical integration. Forexample, for the Gaussian-mixture model parameterscalculated from the measurement experiment, as givenin Table 1, we find γ = 0.548.

To calculate JP , the prior image distribution P (x) mustbe known or assumed. One possibility is to assume thatx is a zero-mean Gaussian random field with covariancematrix Cx. Then

P (x) =1√

(2π)N |Cx|e−

12 (xT C−1

x x) (16)

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−20

−15

−10

−5

0

5

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Mea

sure

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ata

Qua

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−10 −5 0 5 10

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Gaussian Mixture Quantile

Mea

sure

dD

ata

Qua

ntile

(b) Mixture Model

Fig. 4. Quantile-quantile plots comparing measured RSS data with Gaussian and Mixture distributions.

Plugging (16) into (14) results in

JP = C−1x . (17)

These derivations of JD and JP lead to the linear MSEbound for RTI

Rε ≥ (γWTW + C−1x )−1. (18)

An important result of the bound in (18) comes fromthe following property [21]

E[(x− x̂)2i ] ≥ (γWTW + C−1x )−1

ii = J−1ii (19)

where E[(x− x̂)2i ] represents the mean-squared-error forpixel i. In other words, the diagonal elements of J−1

are the lower bounds on the mean-squared-error for thecorresponding pixels.

3.2 Spatial Covariance Model

Previous work has shown that an exponential functionis useful in approximating the spatial covariance of anattenuation field [23], [16]. The exponential covariance isa close approximation to the covariance that results frommodeling the spatial attenuation as a Poisson process, acommon assumption for random placement of objectsin space. Applying this model, the a priori covariancematrix Cx is generated by

[Cx]kl = σ2xe−dkl/δc , (20)

where dkl is the distance from pixel k to pixel l, δc isa “space constant” correlation parameter, and σ2

x is thevariance at each pixel.

The exponential spatial covariance model is appealingdue to its simplicity and low number of parameters.Other models based on different distributions of attenu-ating objects could also be utilized.

3.3 Example Error Bounds

The bound in (19) provides a theoretical basis for deter-mining the accuracy of an image over the network area.The node locations affect which pixels are accuratelyestimated, and which are not. To visualize how the nodelocations affect the accuracy of the image estimation,three examples are provided in Fig. 5. Table 2 shows theparameters of the normalized ellipse weighting modelthat were used to generate these bounds.

Parameter Value Description∆p .1 Pixel width (m)λ .007 Width of weighting ellipse in (9) (m)δc 1.3 Pixel correlation constant in (20) (m)σ2

x .1 Pixel variance in (20) (dB)2

γ .5483 Bound parameter in (19)

TABLE 2Reconstruction parameters used to generate MSE

bound surfaces shown in Fig 5.

As seen in the surfaces of Fig. 5, voxels that arecrossed by many links have a higher accuracy thanvoxels that are rarely or never crossed. The voxels inthe corners of the square deployment, the sides of thefront-back deployment, and the low-density areas in therandom deployment, are crossed only by a few links.In some voxels, no links cross at all, and the boundsurface is limited only by the covariance of the priorstatistics. The known covariance of the image has theeffect of smoothing the bound surface, since knowledgeof the attenuation of a voxel is statistically related to itsneighbors.

3.4 Effect of Node Density

The node density plays a key role in the accuracy of anRTI result. Imaging can be expected to be more accuratein areas where nodes are placed closely together than inareas where nodes are spaced at large distances. When

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Fig. 5. MSE bound surface plots for a square, front-back, and random node deployments. These plots were generatedusing the normalized elliptical weight model with a Gaussian image prior.

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0.1 0.2 0.3 0.4 0.5Node Density (nodes/sq. foot)

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Fig. 6. The lower bound on average MSE vs. node densityfor three RTI network geometries.

many links pass through a particular area, more RSSinformation can be used to reconstruct the attenuationoccurring in that area. This has the effect of averagingout noise and other corruptions in the measurements.Furthermore, when links are close together, the RSSinformation is more concentrated on the voxels that arecrossed. This is due to the weighting function that isinversely proportional to the square root of the linkdistance.

To illustrate the effect of node density on the MSEbound, Fig. 6 shows the lower bound on the averageMSE over all voxels for the three deployment geometriesshown in Fig. 5 as the density is increased. For eachpoint on the curves, the bound surface is calculated, thenaveraged over all voxels. The parameters are equal tothose used previously in Table 2. Each geometry containsthe same number of nodes for each point on the curve,and is deployed around the same area. In the squaregeometry, nodes are placed uniformly around a squarearea. In the front-back geometry, the same number ofnodes are placed along two sides of the square, resultingin the same number of nodes per square foot. In the ran-dom geometry, the same number of nodes are randomlyplaced throughout the square.

In all three cases shown in Fig. 6, the lower boundon average MSE for each deployment decreases rapidlywith increasing node density. The square geometry out-performs the others, due to the fact that the entire areaof the square is surrounded by nodes. There are veryfew voxels that are not crossed by at least a few links,and many short links exist that cross the corners of thesquare. The random geometry performs the worst outof the three when density is low, largely due to the factthat in a random deployment, many voxels will not becrossed by any links. As density increases, the randomdeployment out-performs the front-back geometry be-cause nodes are closer together, and the density is such

that very few areas contain voxels that are not crossedby at least some links.

4 IMAGE RECONSTRUCTION

4.1 Ill-posed Inverse ProblemLinear models for many physical problems, includingRTI, take the form of

y = Wx + n (21)

where y ∈ RM is measured data, W ∈ RM×N is a transfermatrix of the model parameters x ∈ RN , and n ∈ RM isa measurement noise vector. When estimating an imagefrom measurement data, it is common to search for asolution that is optimal in the least-squared-error sense.

xLS = arg minx||Wx− y||22 (22)

In other words, the least-squares solution minimizes thenoise energy required to fit the measured data to themodel. The least-square solution can be obtained bysetting the gradient of (22) equal to zero, resulting in

xLS = (WTW)−1WTy (23)

which is only valid if W is full-rank. This is not the casein an RTI system.

RTI is an ill-posed inverse problem, meaning thatsmall amounts of noise in measurement data are am-plified to the extent that results are meaningless. This isdue to very small singular values in the transfer matrixW that cause certain spectral components to grow out ofcontrol upon inversion. To see this, W is replaced by itssingular value decomposition (SVD):

W = UΣVT (24)

where U and V are unitary matrices, and Σ is a diagonalmatrix of singular values. Plugging (24) into (23), theleast squares solution can be written as

xLS = VΣ−1UTy =N∑i=1

1σi

uTi yvi (25)

where ui and vi are the ith columns of U and V,and σi is the ith diagonal element of Σ. It is evidentthat when singular values are zero or close to zero,the corresponding singular basis vectors are unboundedupon inversion.

The heuristic explanation for the ill-posedness of theRTI model lies in the fact that many pixels are estimatedfrom relatively few nodes. There are multiple possibleattenuation images that can lead to the same set ofmeasurement data. For example, assume a particularpixel is not crossed by any link in the network. Thiswould result in the same measurement data for everypossible attenuation value of that pixel, so inversion ofthe problem would be impossible.

Regularization involves introducing additional infor-mation into the mathematical cost model to handle theill-posedness. In some methods, a regularization term

9

J(x) is added to the minimization objective function ofthe original problem as

freg = f(x) + αJ(x), (26)

where α is the weighting parameter. Small values of αlead to solutions that fit the data, while large valuesfavor the solution that matches prior information.

Some regularization techniques follow from aBayesian approach, where a certain prior distributionis imposed on the model parameters. Other forms ofregularization modify or eliminate small singular valuesof the transfer matrix. An overview of regularizationand image reconstruction in general can be found in[24] and [25].

4.2 Tikhonov Regularization

In Tikhonov regularization [24], an energy term is addedto the least squares formulation, resulting in the objectivefunction

f(x) =12||Wx− y||2 + α||Qx||2, (27)

where Q is the Tikhonov matrix that enforces a solutionwith certain desired properties.

In this paper, we use a difference matrix approxi-mating the derivative operator as the Tikhonov matrixQ. By minimizing the energy found within the imagederivative, noise spikes are suppressed and a smoothimage is produced. This form of Tikhonov regularizationis known as H1 regularization.

Since the image is two dimensional, the regularizationshould include the derivatives in both the vertical andhorizontal directions. The matrix DX is the differenceoperator for the horizontal direction, and DY is thedifference operator for the vertical direction. The reg-ularized function can be written in this case as

f(x) =12||Wx− y||2 + α(||DXx||2 + ||DY x||2). (28)

Taking the derivative and setting equal to zero results inthe solution

x̂ = (WTW + α(DTXDX + DT

Y DY ))−1WTy. (29)

One major strength of Tikhonov regularization lies inthe fact that the solution is simply a linear transforma-tion Π of the measurement data.

Π = (WTW + α(DTXDX + DT

Y DY ))−1WT (30)x̂ = Πy (31)

Since the transformation does not depend on instan-taneous measurements, it can be pre-calculated, andthen applied for various measurements for fast imagereconstruction. This is very appealing for realtime RTIsystems that require frequent image updates [15], [26].

The total number of multiplications Nmult required totransform the measurements into the image is the total

number of voxels N times the number of unique linksM in the network

Nmult = NM =N(K2 −K)

2(32)

where K is the number of nodes in the network. Wesee that complexity increases linearly as the number ofvoxels increases, and quadratically as the number ofnodes in the network increases.

5 EXPERIMENTAL RESULTS

5.1 Physical Description of Experiment

A wireless peer-to-peer network containing 28 nodes isdeployed for the purpose of testing the capability of RTIto image changed attenuation. Each node is placed threefeet apart along the perimeter of a 21x21 foot square,surrounding a total area of 441 square feet. The networkis deployed on a grassy area approximately 15 feet awayfrom the Merrill Engineering Building at the Universityof Utah. Each radio is placed on a stand at three feet offthe ground.

The area surrounded by the nodes contains two treeswith a circumference of approximately three feet. Thenetwork is intentionally placed around the trees so thatstatic objects exist in the tested RTI system. RTI shouldonly image attenuation that has changed from the timeof calibration within the deployment area. Markers aremeasured and placed in 35 locations within the networkso that the humans’ locations are known and can beutilized in the subsequent error analysis. A map andphoto of the experiment are shown in Fig. 7.

The network is comprised of TelosB wireless nodesmade by Crossbow. Each node operates in the 2.4GHzfrequency band, and uses the IEEE 802.15.4 standardfor communication. A base station node listens to allnetwork traffic, then feeds the data to a laptop computervia a USB port for the processing of the images. Sincethe base station node is within range of all nodes,the latency of measurement retrieval to the laptop islow, on the order of a few milliseconds. If a multi-hopRTI network were to be deployed, this latency wouldcertainly increase.

To avoid network transmission collisions, a simpletoken passing protocol is used. Each node is assignedan ID number and programmed with a known orderof transmission. When a node transmits, each node thatreceives the transmission examines the sender identifi-cation number. The receiving nodes check to see if it istheir turn to transmit, and if not, they wait for the nextnode to transmit. If the next node does not transmit, orthe packet is corrupted, a timeout causes each receiver tomove to the next node in the schedule so that the cycleis not halted.

At the arrival of each packet to the laptop, the RTIprogram running on the laptop updates a link RSS mea-surement vector. At each update, the base station hearsfrom only one node in the network, so only RSS values

10

0 5 10 15 20X (feet)

0

5

10

15

20Y

(fe

et)

Nodes

Positions

Trees

Links

(a) Map (b) Photo

Fig. 7. The network geometry and links that correspond to Fig. 8 are illustrated in (a). (b) is a photo of the deployednetwork with an experimenter standing at location (3,9).

on links involving that particular node are updated.Each link’s RSS measurement is an average of the twodirectional links from i to j and j to i.

In this experiment, the system is calibrated by takingRSS measurements while the network is vacant frommoving objects. The RSS vector is averaged over a30 second period, which results in approximately 100RSS samples from each link. The calibration RSS vectorprovides a baseline against which all other RSS measure-ments are differenced, as discussed in Section 2. Othermethods of calibration could be used in situations whereit is impossible to keep the network vacant from movingobjects. For example, a single past measurement or asliding window average of RSS measurement historycould be used as the baseline.

5.2 Effect of Human ObstructionSince RTI is based on the assumption that objects shadowindividual links in a wireless network, it is helpful toexamine the effect of obstructions on a single link. InFig. 8, a human stands at position (9,9) and RSS measure-ments for each link are collected. These measurementsare compared with the calibration measurements thatwere taken when the network was vacant.

The top plot in Fig. 8 shows that a significant decreasein RSS, anywhere from 5 to 10 dB, is experienced bylink (0,18) to (18,0) as it travels through the obstruction.The middle plot shows that even though the link (9,0)to (9,21) passes through the tree, it still experiencessignificant loss when the human is present on the LOSpath. The bottom plot in the figure shows an exampleof a link that does not pass through the obstruction,resulting in very little difference in RSS.

In environments where links travel over long dis-tances, or when many objects block the direct LOSpath, we expect the effect of a human obstruction tobe lessened. In those cases, certain links may experi-ence losses, while others may not. Future research will

100 200 300 400 500 600 700 800 900 1000

�52�55�58�61�64�67

RS

S (

dB

m)

Link (0,18) to (18,0)

No obstruction

LOS obstruction

100 200 300 400 500 600 700 800 900 1000

�60�63�66�69�72�75

RS

S (

dB

m)

Link (0,9) to (21,9)

100 200 300 400 500 600 700 800 900 1000Samples

�53�56�59�62�65�68

RS

S (

dB

m)

Link (3,0) to (3,21)

Fig. 8. A comparison of the effect of human obstructionon three links. In the unobstructed case, the networkis vacant from human experimenters. In the obstructedcase, a human stands at coordinate (9,9).

investigate the effect of human obstruction on a link’sRSS when a link passes through walls or other majorstatic obstructions. This will be essential in making thetechnology practical for the future applications of RTI aspreviously discussed.

5.3 Cylindrical Human ModelTo assess the accuracy of RTI images, one must firstknow or assume the “true” attenuation field that is beingestimated. Since imaging the location of humans is theprimary goal of RTI, a model for the size, shape, and

11

(a) Cylindrical model image

0 5 10 15 20X (feet)

0

5

10

15

20

Y (

feet)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(b) RTI result

(c) Cylindrical model image (d) RTI result

Fig. 9. Images of attenuation in a wireless network where each human is modeled as a uniformly attenuating cylinderof radius RH = 1.3 feet. In (a) and (b), a human stands at coordinate (9,9) and the total squared error is ε = .021. In(c) and (d), two humans stand at coordinates (3,15) and (18,15) and the total squared error is ε = .036.

attenuation of the human body at the frequencies ofinterest would be required. This information is difficultto model, since it is dependent on body types, the planeof intersection, and other variables.

For simplicity, a human is modeled as a uniformlyattenuating cylinder with radius RH . In this case, the“true” image xc for a human positioned at location cHcan be described as

xcj ={

1 if ||xj − cH || < RH0 otherwise (33)

where xcj is the center location of voxel j.By scaling the image such that the maximum equals

one, resulting in the normalized image x̂N , we can define

the mean-squared error of the normalized image to be

ε =||xc − x̂N ||2

N(34)

where N is the number of voxels in the image.

5.4 Example Images

Using the model and reconstruction algorithms de-scribed in Sections 2 and 4, we present some typicalimage results for humans standing inside the exper-imental RTI network. A human stands at coordinate(9,9) and RSS data is measured for a few seconds.The data is averaged for 10 samples per link, and thismeasurement differenced with the calibration data takenwhile the network is vacant. Figure 9 displays both the

12

“true” attenuation based on the cylindrical model, andthe RTI reconstruction using H1 regularization with theparameters listed in Table 3.

Parameter Value Description∆p .5 Pixel width (feet)λ .01 Width of weighting ellipse (feet)α 5 Regularization parameterRH 1.3 Human radius for cylindrical model (feet)

TABLE 3Image reconstruction parameters

Using the cylindrical human model with a radius ofRH = 1.3, the squared error for the single-human imagestanding at (9,9) was measured to be ε = .021. Thesquared error for the two-person image was measured tobe ε = .036. These error values are in general agreementwith the bounds derived in Section 3.

There are many areas in the images of Fig. 9 whereestimated attenuation is above zero, even where noobstruction exists. This is due to the fact that a humannot only attenuates a wireless signal, it reflects andscatters it. The simple LOS model used in this paperdoes not take into account the changes in RSS values dueto multipath caused by the obstructions being imaged.For example, a link may be bouncing off the human anddestructively interfering with itself on a path that doesnot cross through the obstruction, thus leading to errorin the estimated attenuation. Future research will seekto refine the weighting model used in RTI such that thismodeling error is lessened.

5.5 Effect of Parameters on Image AccuracyThe weighting and regularization parameters play animportant role in generating accurate RTI images. If theproblem is regularized too strongly, the resultant imagesmay be too smooth to provide a good indication ofobstruction boundaries. If the regularization parameteris set too low, noise may corrupt the results, making itdifficult to know if a bright spot is an obstruction ornoise.

Another parameter effecting the accuracy of an imageis the width of the weighting ellipse. If the ellipse is toowide, the detail of where attenuation is occurring withinthe network may be obscured. If the ellipse is too narrow,voxels that do in fact attenuate a link’s signal may notbe captured by the model. This may result in a loss ofinformation that degrades the final image quality.

In this paper, we empirically identify the parametersthat provide the most accurate images using the cylin-drical human model. For each parameter, images areformed from data measured while a human is standingat one of the known positions, as indicated in Fig.7(a). Such an image is formed for each of the possiblehuman positions shown in Fig. 7(a). The squared erroris calculated for each image, and averaged over theentire set. This is performed for a varying regularization

parameter, while the weighting ellipse parameter is heldconstant at λ = .1. Then, it is repeated for varying ellipseparameters while holding the regularization constant atα = 4.5. The resultant error curves are shown in Fig. 10.

The curves shown in Fig. 10 show that the choice ofregularization and weighting parameters is importantin obtaining accurate images. Future research possiblywill explore the automatic calculation and adjustment ofthese parameters. It should be noted that the error curvesand optimal values presented are dependent upon thepixel size used in generating the images. The generalshape of the curves, however, is similar for different pixelsizes. In this study, pixel size is held constant at .5 feetfor all experiments.

100 101 102

Regularization parameter �0.025

0.030

0.035

0.040

0.045

Ave

rag

e e

rror

10-2 10-1

Ellipse parameter �0.02

0.03

0.04

0.05

0.06

0.07

0.08A

vera

ge e

rror

Fig. 10. Error vs. parameter curves. In the first plot, theweighting ellipse width parameter is held constant at λ =.1 while the regularization parameter α is varied. In thesecond, α = 4.5 and the width of the weighting ellipse isvaried.

6 CONCLUSION

Radio tomographic imaging is a new and excitingmethod for imaging the attenuation of physical objectswith wireless networks operating at RF wavelengths.This paper discusses a basic model and image recon-struction technique that has low computational com-plexity. Experimental results show that RTI is capableof imaging the RF attenuation caused by humans indense wireless networks with inexpensive and standardhardware.

Future research will be important to make RTI real-istic in security, rescue, military, and other commercialapplications. First, new models and experiments mustbe developed for through-wall imaging. In this case,the shadowing and fading caused by many objects inthe environment may cause the LOS weighting modelto be inaccurate. New and possibly adaptive weightingmodels will need to be investigated and tested.

Wireless protocols, customized hardware, and signaldesign are also important for improving RTI. Protocols

13

that are capable of delivering low-latency RSS informa-tion for large networks will be essential when deployingthe technology over large areas. Antennas that direct theRF energy through an area may reduce the effects ofmultipath and increase the effect of human presence onsignal strength. Custom signals, perhaps taking advan-tage of frequency diversity may improve the quality ofRTI results.

Radio tomographic imaging may provide a low-costand flexible alternative to existing technologies likeultra-wideband radar. This would enable many appli-cations in the areas of security, search and rescue, po-lice/military, and others.

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