algebraic approach for robust localization with heterogeneous information

5334 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 10, OCTOBER 2013

Algebraic Approach for Robust Localization withHeterogeneous Information

Davide Macagnano, Member, IEEE,and Giuseppe Thadeu Freitas de Abreu, Senior Member, IEEE

Abstract—We offer a redesigned form of the classical multi-dimensional scaling (C-MDS) algorithm suitable to handle thelocalization of multiple sources under line-of-sight (LOS) andnon-line-of-sight (NLOS) conditions. To do so we propose tomodify the kernel matrix used in the MDS algorithm to allow forboth distance and angle information to be processed algebraically(without iteration) and simultaneously. In so doing we also showthat the new formulation overcomes two well known limitations ofthe C-MDS approach, namely the propagation error problem andthe possibility to weight the dissimilarities used as measurementinformation, including, for the case of binary weights, the dataerasure problem. Due to the increased size of the proposededge kernel matrix KE used in the algorithm, the Nystromapproximation is applied to reduce the overall computationalcomplexity to few matrix multiplications. Range only scenariosare also dealt with by approximating the matrix KE. Simulationsin range-angle as well as range-only scenarios demonstrate thesuperiority of our solution under both LOS and NLOS conditionsversus semidefinite programming (SDP) formulations of theproblem specifically designed to exploit the heterogeneity of theinformation available.

Index Terms—Localization, multidimensional scaling (MDS),heterogeneous networks, range-based measurements, angle-basedmeasurements.

I. INTRODUCTION

IN the last decade the need for highly personalized infor-mation services has thriven the creation of novel location

based services (LBS), contributing to make accurate locationinformation anywhere anytime to become a strategic assetfor companies operating in the information technology (IT)business.

All this interest from companies stimulated a surge ofinterest from the research community. Considerable progresshas been made on various aspects of the problem includingadvanced multilateration techniques [1], synchronization prob-lems [2], increasing accuracy by cooperative schemes [3] androbust ranging in the presence of non-line-of-sight (NLOS)conditions and interference, sophisticated optimization tech-niques capable to exploit geometric and statistical informationcharacterizing the scenario of interest [4], [5] and distributedsolution [6], [7]. In addition to contributions of such nature, it

Manuscript received February 17, 2013; revised May 30, 2013; acceptedJuly 15, 2013. The associate editor coordinating the review of this paper andapproving it for publication was W. Choi.

Parts of this work were presented at the 7-th IEEE WCNC in 2007 and atthe 45-th IEEE ASILOMAR Conference on Signals, Systems and Computersin November 2011.

D. Macagnano is with the Centre for Wireless Communications, Depart-ment of Communications Engineering, University of Oulu, P.O.Box 4500,90014-FIN (e-mail: [email protected]).

G. Abreu is with the School of Eng. & Science - SES, Department Elec.Eng. and Computer Science, Jacobs University Bremen, 28759-Germany (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2013.090313.130301

has also been shown that due to its nonparametric formulation,the multidimensional scaling (MDS) can be an advantageoustechnique compared to more conventional methods especiallywhen the targets mobility cannot be simply and reliablymodeled a priori [8], [9]. MDS-based solution applied to thelocalization problem typically rely on range dissimilarities thatcan be estimated either on the bases of time of arrival (TOA) orreceived signal strength (RSS) measurements. Due to the factthat RSS-based ranging is highly subject to environmental con-ditions such as fading, shadowing and antenna distortion, time-based ranging methods are particularly attractive, especiallysince the advent of Ultra-Wideband Impulse Radio (UWB-IR)[10].

Another type of measurement used in localization systemsis the angle information obtained by estimating the angle ofarrival (AOA) of signals at different receivers.

While AOA estimates can be accurately and inexpensivelyperformed with a low-frequecy narrowband signal even un-der NLOS conditions, provided that the multipath in theenvironment is not too dense [11], TOA-based ranging canbe performed at low cost even in dense multipath environ-ments, provided that line-of-sight (LOS) conditions exist. Asa consequence, AOA and TOA measurements can be seen ascomplimentary information in which the errors affecting theobservations are due to entirely different factors. Moreover, asshown in [4, pp. 217] recent systems are able to make both,angle and range observations simultaneously available, whichin turn calls for algorithms that can exploit this heterogeneityin the information to enhance the localization accuracy.

The challenge is therefore to design a localization algo-rithms that can jointly and efficiently exploit angle and ranginginformation. This paper proposes one such algorithm whereangle and ranging information are processed jointly in anatural and straightforward manner. Our contribution is aMDS-based formulation of the localization problem, wherethe entities are the vectors pointing from a source to another,and the measure of dissimilarity their inner product.

In a Graph-Theoretical setting, the technique proposed in[12] translates to modeling the network as an undirectedgraph Gη,N (X, �V,D), embedded in a η-dimensional space,where the entities are its N vertices V, and the dissimilaritiesits weights D. Conversely, in our formulation, the networkis modeled as a complete oriented graph (also know as atournament), where the entities are the edges V of the graph,and the dissimilarity metric their inner product.1

In allusion to its construction, the Gram Kernel used in the

1In our context edge orientations are arbitrary and necessary only so as toallow for the definition of an edge inner product in similar terms to that ofelementary vector algebra.

1536-1276/13$31.00 c© 2013 IEEE

MACAGNANO and ABREU: ALGEBRAIC APPROACH FOR ROBUST LOCALIZATION WITH HETEROGENEOUS INFORMATION 5335

algorithm is referred to as the Edge Gram Kernel. A fortunateproperty of the Edge Gram Kernel is that its elements onlydepends on one angle and a single pair of distance estimates.

In contrast, the Gram Kernel used in the C-MDS is com-puted by applying a double-centering transformation [13]–[15]onto the Euclidean distance matrix (EDM), such that eachof its elements depends on all the distance measurements. Inshort, the Edge Gram Kernel is, by construction, far morerobust to the propagation of measurement errors, which mayhave devastating effects on the performance of the MDS-basedlocalization algorithm subject to noisy and biased distanceestimates [16].

The new MDS formulation is referred to as edge multidi-mensional scaling (E-MDS), and offers the following advan-tages over standard MDS solutions [9]. First, it allows a unifiedprocessing of angle and ranging information. Second, it leadsto a simplified kernel structure that is resilient to the errorpropagation problem and third it allows to weight the singlerange information used to infer the sources’ locations.2 Sincethe proposed algorithm requires the eigen-decomposition ofa kernel matrix whose size increases quadratically with thenumber of nodes in the network it is shown how to incorporatethe Nystrom approximation [18] to lower the computationalcomplexity to few matrix multiplication only.

The remainder of this article is as follows. After introducingthe necessary notation, the classical multidimensional scaling(C-MDS) and the closed form solution to the STRAIN costfunction are presented in Section II-A.

Sections II-B and II-C show how to exploit either completerange information or partial range and angle informationby means of the C-MDS algorithm. Section III introducesthe E-MDS framework and shows how distance and angleinformation can be processed algebraically (without iteration)and simultaneously. The same section discusses the robustnessof the solution to erroneous angle information as well as thepossibility to weight the single dissimilarities to cope eitherwith the so called erasure problem, namely partially connectednetworks.

Section IV explains how to cope with the complexityproblem encountered in the E-MDS formulation by meansof a spectral approximation of the kernel matrix used in thealgorithm. Moreover it is also explained how to extend therange of applicability of the technique to range only multi-source scenario by approximating the angle part of the edgekernel directly from the set of observed distances. Results forboth a single and multi-source localization scenarios underboth LOS and NLOS are are provided in Section V. Finalremarks are offered in Section VI.

II. THE MDS PROBLEM OVER STRAIN

Let X ∈ RN×η denote the matrix containing the set ofCartesian coordinates for N points in an η-dimensional space,then the Euclidean distance between the two generic points xi

and xj is given by

dij = dji �(

η∑k=1

(xik − xjk)2

)12

. (1)

2As shown in [17] weighting strategies can substantially improve localiza-tion estimates even in presence of scarce ranging information.

Equation (1) can be written as

d2ij = xi · xTi + xj · xT

j − 2xi · xTj , (2)

which is the well known cosine-law [14].3

It follows that the generic squared distance d2ij is related tothe triangle whose vertices are the points xi, xj and the origin.Also let the squared Euclidean distance matrix (EDM) D besuch that [D]ij = d2ij , ∀(i, j) = ({1, · · · , N}, {1, · · · , N}),yielding to

D(X) = D =

⎡⎢⎢⎢⎢⎢⎣

0 d21,2 d21,3 · · · d21,Nd22,1 0 d22,3 · · · d22,Nd23,1 d23,2 0 · · · d23,N

......

. . .. . .

...d2N,1 d2N,2 d2N,3 · · · 0

⎤⎥⎥⎥⎥⎥⎦ . (4)

While a detailed description of the properties related to Dcan be found in [13], [14] it is important to realized that beingD a symmetric matrix with zero diagonal elements, only theM =

(N2

)= N(N − 1)/2 entries of either its upper or lower

triangular part are informative.Moreover, since any rigid transformation of X results in the

same EDM D, algorithms that rely exclusively on D to inferX can only find the solution up to an isometry in the embed-ding space [14]. This uncertainty can be however resolved bythe Procrustes transformation detailed in Appendix A whichis used to map the solution into the reference system in use[19].

A. The STRAIN Formulation

Let DN (η) be the space of N × N distance matricesgenerated by N points in an η dimensional space and letδij to express the dissimilarity [19] between xi and xj . Nowconsider the squared dissimilarity matrix Δ ∈ R[N×N ] suchthat [Δ]ij = δ2ij and its transformation

K(Δ, a) = −J(a)T ·Δ · J(a), (5a)

J(a) � 1√2

[I− 1N · aT

]T, (5b)

where 1N ∈ RN×1 is a vector whose entries are all one anda ∈ RN×1 is a signed distribution of N terms such thatN∑i=1

ai = 1 and whose values influence the origin used to

represent X [13].4

Schoenberg’s embedding theorem for Euclidean distancematrices [21] states that a dissimilarity matrix Δ ∈ DN (η)if and only if K(Δ, a) ∈ SN (η), where SN (η) is the space ofsymmetric positive semi-definite (PSD) matrices with rank atmost η.5

From the above it follows that the metric scaling problem[19] can be seen as the minimization of the distance between

3Let θijbe the angle seen from the origin between the points xi and xj ,then equation (2) is equivalent to

cos(θij) =d2i + d2j − d2ij

2didj. (3)

4Notice that equation (5a) is simply the generalization of equation (2), i.e.the cosine-law, to all points of X [20].

5Equation (5) is a linear mapping from DN (η) to SN (η).


the closed convex set of dissimilarity matrices and the set ofsymmetric positive matrices [22], yielding to the followingoptimization problem

minG

‖K (Δ, a)−G‖2F , (6)

s.t. Δ ∈ EN

G ∈ GN

where EN is the closed convex set of possible dissimilaritymatrices and GN a subset of SN .

In the special case of a single dissimilarity matrix, namelyEN = {Δ}, the optimization problem in equation (6) modifiesinto

minG

‖K (Δ, a)−G‖2F , (7)

s.t. G ∈ SN (η),

where the objective to be minimized is the so called STRAINcost function.

As shown in [23] and further generalized in [22], anattractive feature of STRAIN is that its global solution can becomputed explicitly. Moreover, from Schoenberg’s embeddingtheorem D(X) = Δ if and only if K(Δ, a) is a Gram matrix(i.e, X ·XT), from which it follows that X can be obtainedby matrix factorization of a PSD matrix.

Let Ka = K(Δ, a) be a kernel matrix constructed ac-cording to equation (5) for a particular value of a, then thecoordinate matrix X can be recovered – up to similaritytransformation6 – by

X = [Ua]1:N,1:η · [Λa]� 1

21:η,1:η , (8)

where (Λa,Ua) is the eigen-pair for Ka containing the de-creasingly ordered eigenvalues and their corresponding eigen-vectors respectively, and where �m denotes the m-th element-wise power.

Some important generalizations of STRAIN are discussedin [22] while studies on the robustness of STRAIN to perturba-tions are offered in [24] where it is shown that in the presenceof dissimilarities corrupted by zero mean Gaussian noiseSTRAIN is comparable to a sub-optimal, in the maximumlikelihood (ML)-sense, cost function called SSTRESS.

In the particular case of a = 1N/N , then kernel matrixG ∈ SN (η) can be mapped into D ∈ DN (η) by the followinglinear operator

T (G) � 1N · diag(G)T + diag(G) · 1TN − 2G, (9)

where diag(·) is a function that when applied to a matrixreturns vector whose elements are identical to the diagonalof the matrix given as argument, while when applied to avector it returns a diagonal matrix whose nonzero entries arethe one in the original vector.

Also notice that the inverse mapping from D to G isperformed using the operatorK(Δ, a) defined in equation (5a)[25].

6Similarity transformation includes scaling, rotation and shift and is recov-ered by Procrustes transformation [19].

B. Complete Distance Information

Assume all the M pairwise distances between the N pointsin X as known and consider the double-centred Euclideankernel constructed from equation (5a) setting a = 1

N · 1N as[14]

K = −JT ·D · J, (10)

where

J =1√2

[IN − 1

N · 1N1TN

]. (11)

Then the C-MDS algorithm recovers the estimated config-uration of points X from K using equation (8). This solutionrepresents is exact provided that D ∈ DN (η) and it isknown to be the best approximation of K obtained under theconstraint that rank(K) = η [23]. In practical localizationapplications, however, one can only obtain a dissimilaritymatrix Δ, corrupted by noise and bias7, with the obviousconsequence that no exact solution is possible. However,under this realistic scenarios a measure of the quality of theapproximation obtained by the spectral truncation in equation(8) can be obtained by value of the STRAIN cost function,namely

CT = ‖X · XT −K(Δ, a)‖2F.

To gain insight into the effect that corrupted dissimilaritieshave in the C-MDS algorithm, let ai be the i-th element of a,then using equation (9) it follows that

[K]ij =−1

2

⎛⎝[D]ij−

∑i

ai[D]ij−∑j

aj[D]ij+∑i

∑j

aiaj [D]ij

⎞⎠.

(12)

From the above it is clear that the Euclidean kernel Kused in the C-MDS solution is structured in such a way thateach one of its elements depends on the errors of the entiremeasurement set D simultaneously, resulting in an error-propagation-prone algorithm [26].

C. Partial Heterogeneous Information

Section II-A showed that through equation (5) the dissim-ilarity matrix D can be transformed into the inner productmatrix K. Using the definition of inner product between pointsin Rη then the ij-th element of K is expressed as

[KD]i,j � 〈xi;xj〉 = didj cos(θi,j). (13)

It follows that the matrix KD can be written as

KD = ΘD ⊗(dN · dT

N

), (14)

where ⊗ denotes the Hadamard product, dN a vector con-taining the distances from the origin to the N points in Xand

ΘD =

⎡⎣ 1 · · · cos(θ1,N )

.... . .

...cos(θN,1) · · · 1

⎤⎦ , (15)

includes all the angles amongst the M edges in X as seenfrom the origin.

7As discussed later, erasures may also occur.


Using the definition of inner product in equation (14)then X can be recovered from the kernel matrix KD whichis obtained from a subset of all possible range and anglemeasurements. A by-product of this feature is that errorpropagation is significantly smaller since, as shown in equation(13), the kernel matrix is computed (element-by-element),straight from distance and angle estimates. Also notice that thekernel KD only exploits a subset of the entire range and angleinformation, namely only N out of M range observations andM out of the

(M2

)angles potentially available.

Moreover, to be of any practical use the kernel KD needs tobe centered at the i-th point of X from where those distancesand angles can be measured. The resulting kernel matrix K(i)

is then equivalent to the one obtained starting from the fullset of all mutual distances between the points in X andcomputed using equation (5) where the centering vector ais set to be zero everywhere except in correspondence of thei-th position.8 Once the new kernel matrix K(i) is available,then X can be recovered applying equation (8) followed byProcrustes transformation.

III. FULL HETEROGENEOUS INFORMATION

Although the kernel matrix K(i) allows to use angle anddistance information jointly, it only includes a subsets of allpossible measurements amongst the points in X. In what fol-lows an extension of the C-MDS algorithm that can potentiallyexploit all mutual distance and angle information measurablebetween all possible pairs of points in X is proposed.

Let associate to the network configuration the completeoriented graph Gη,N (X, �V,D), where X is the set of Nvertices in the η dimensional space and whose coordinateof the i-th vertex is given by the corresponding row of X,�V = {�vn,m} as the set of edges with arbitrary, but unique,orientations (vectors) and D the corresponding weights. Forconvenience, the orientation of all edges is chosen so thatthe M elements of �V are ordered progressively, i.e., �V ={�v1,2,�v1,3, · · · ,�vN−1,N}. To simplify the notation, the mu-tually exclusive pairs of indexes (n,m), with n < m, arerelabeled by a different number i and let V ∈ R

M×η the edge-matrix containing the coordinated for all M edges belongingto the graph Gη,N (X, �V,D) such that the edge correspondingto the pair of points (xn,xm) can be denoted simply by,

vi=(xm−xn)=[(xm,1−xn,1), · · ·, (xm,η−xn,η)]T. (16)

Next, define the dissimilarity metric between the i-th andj-th edges as the inner product,

[KE]i,j � 〈vi;vj〉 = 〈(xm − xn); (xq − xp)〉= dn,mdp,q cos(θi,j). (17)

where the set of all dissimilarity measures [KE]i,j , correspond-ing to all pairs of edges in the graph, can be convenientlyassembled into the edge-kernel

KE = 〈[v1, · · · ,vM ]; [v1, · · · ,vM ]T〉

= V ·VT =

⎡⎣ 〈v1;v1〉 · · · 〈v1;vM 〉

.... . .

...〈vM ;v1〉 · · · 〈vM ;vM 〉

⎤⎦ . (18)

8This is a generalization of the kernel matrix obtained using the full-rank

skinny Schoenberg auxiliary matrix J =1√2[−1, IN−1]

T ∈ RN×N−1,

which centers the reference system in the first point of X [14].

From equation (17) it follows that the generic element[KE]ij is altered only by the errors affecting the i-th and thej-th edge and the corresponding angle. Moreover, differentlyfrom the kernel KD, the edge-kernel KE can exploit the entireset of angles and rage measurements potentially available.

The proposed algorithm results directly from the fact thatKE is a Gram matrix from which it follows that the V can berecovered from KE using equation (8). Namely, the key ideabehind the E-MDS algorithm is to solve the cost function inequation (6) for the kernel matrix associated to the graph ofX, namely to solve the scaling problem at hand over the setof edges of Gη,N (X, �V,D) rather than its set of vertices.

Being interested in the coordinates of the targets, the finalstep of the localization algorithm is to recover X from theretrieved vector matrix V. To this end, consider the followingsystem of linear equations, derived directly from equation (16),

C ·X = V, (19)

where the coefficient matrix C has the block upper-triangularstructure shown below

C =

⎡⎢⎢⎢⎣

1N−1×1 −IN−1×N−1

0N−2×1 1N−2×1 −IN−1×N−1

. . .. . .

. . .01×N−2 1 −1

⎤⎥⎥⎥⎦ . (20)

It is trivial to prove9 that rank(C) = N − 1, such thatthe system given in equation (19) is under-determined by oneequation. In fact, any partition of N−1 linearly independentrows of C has rank N−1.

In other words, if it is assumed that the absolute coordinatesof a single source (which without loss of generality can betaken to be the first) is known, a solution of the E-MDSproblem can be found by considering any of the followingreduced systems[

1 01×N−1

[C]N−1×N

]·[

x1

[X]N−1×η

]=

[x1

[V]N−1×η

], (21)

where [C]N−1×N denotes an N−1 row-partition of C withrank N − 1, while [X]N−1×η and [V]N−1×η denote thecorresponding N−1 row-partitions of X and V, respectively.

Consider the following expression for the inverse of blockmatrices [27],

[x y

z W

]−1

=

⎡⎣ (x−y·W−1 ·z) −x·y·(W−x·z·y)−1

−W−1 ·z·(x−y·W−1 ·z) (W−x·z·y)−1

⎤⎦ ,

(22)

where x is a scalar, y and z row- and column-vectors, and Wan invertible matrix.

It is clear from equation (22) that any of the linear systemsdescribed by equation (21) is determined, because W is anN−1 column-partition of [C]N−1×N , which has rank N− 1.

9Proof: Let Ci denote the i-th N − i row-partition of C, such that wemay write C =

[CT

1 · · · CTN−1

]T. Then, simply notice that for all

1 � i < N , the partition Ci+1 is given by the last N−i−1 rows of Ci,subtracted by its first row.


A particularly case is the one where [C]N−1×N is given bythe first N − 1 rows of C, yielding[

1 01×N−1

1N−1×1 −IN−1×N−1

]·XN×η=

[x1

VN−1×η

]. (23)

From equation (22) it is found that the coefficient matrixof this reduced linear system is invariable to the inverseoperation, so that the solution of the E-MDS becomes simply

XN×η=

[1 01×N−1

1N−1×1 −IN−1×N−1

]·[

x1

VN−1×η

]. (24)

The fact that only N − 1 rows of C suffice to solvethis reduced version of the E-MDS localization problem isobviously a direct consequence of the fact that there are onlyN−1 linearly independent vectors vi in V. This in turn impliesthat turn, means that the algorithm can be formulated over anN−1-by-N−1 reduced Edge Gram Kernel KE = [V]N−1×η ·[V]TN−1×η , where [V]N−1×η denotes the N−1 row-partition ofV. Also notice that this solution is equivalent to the scenariowith partial heterogeneous information discussed in II-C.

One benefit of the decomposition in equation (18) is itsability to exploit different sources of information, i.e. rangeand angle measurements, in a disjoint manner, contributing tomake the overall solution robust to the different sources oferrors. This can be better understood by expressing the edgekernel KE similarly to equation (14), namely

KE = ΘE ⊗(dM · dT

M

), (25)

where ΘE is the angle-kernel including all the mutual angles,or better saying its cosines, and dM ∈ RM×1 the vectorcontaining the corresponding lengths for all the edges inGη,N (X, �V ,D).

Equation (25) shows how, by using the proposed algorithm,it is possible to exploit the two sources of information in analmost completely decoupled manner by first decomposing ΘE

into the eigen-pair (UΘE ,ΛΘE) and then finding the estimatededges as

V = diag(dM ) · [UΘE ]1:M,1:η · [ΛΘE ]� 1

21:η,1:η . (26)

Once V is computed by means of equation (26), then theestimated positions is obtained solving equation (19), namelyX = C† · V, followed by the procrustes transformationdetailed in Appendix A. In what follows the performance ofthe E-MDS technique described above are compared againstthe C-MDS one, which utilizes only distance information. Thesimulations are designed to provide insight on how muchaccuracy can be gained by utilizing angle information or,conversely, how much accuracy is needed from angle estimatesin order to reap better accuracies in comparison to using onlydistance measurements.

In each realization, a fully connected random network ofsensors Gaussianly distributed in the 2-dimensional Euclideanspace, with statistical center at the origin and a unitary spread(variance) is generated. All metric units are normalized and,therefore, shall be omitted henceforth.

Range estimations d, i.e., the noisy measurements ofthe distance between each pair of sensors, are modeled as

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5Performance versus Ranging Error

σd

RMSE

E-MDS

C-MDS

εθ = 12

εθ = 20

εθ = 30

εθ = 46

εθ = 77

(a) RMSE as a function of σd.

12 22 32 42 52 62 72 82 92 1020.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Performance versus Angle Error

εθ

RMSE

E-MDS

C-MDS

Increasingσd

σd = .4

σd = .35

σd = .3

σd = .25

σd = .2

σd = .15

σd = .1

(b) RMSE as a function of εθ .

Fig. 1. Performance of metric and E-MDS as a function of σd and εθ .

Gaussian-distributed random variables with the mean givenby the true distance d and a standard deviation σd related tothe ranging error affecting its measurement, i.e.,

pGauss(r; d, σd) �1

σ√2π· exp

(−(r − d)2

2σ2d

). (27)

The complete EDM sample corresponding to theseGaussian-distributed distance estimates are then fed to theC-MDS localization algorithm described in Section II-B.

In turn, angle estimation errors are modeled as randomprocesses distributed according to the Tikhonov probabilitydensity function (pdf) [28]

pT(θ; ρ) =1

2πI0(α)· eρ cos(θ), θ ∈ [−π, π], ρ � 0. (28)

The parameter ρ controls the shape of this pdf, such thatpT(θ; ρ) tends to a uniform distribution for ρ → 0, and to aDirac delta at 0 when ρ→∞. For a given ρ, we shall definethe angle error εθ as the limiting angle that encloses 90% of


the area below pT(θ; ρ), i.e.,

εθ = θL

∣∣∣∣θL∫

−θL

pT(θ; ρ) dθ = 0.9. (29)

The range and angle estimates perturbed according to theGaussian and Tikhonov distributions are then fed to theE-MDS localization algorithm described above.

The performance of the algorithms are measured by theFrobenius norm of the difference between the estimate coor-dinate vector X and the corresponding true coordinate vectorX, namely the root mean square error (RMSE), i.e.,

RMSE � 1√N· ‖X− X‖2. (30)

The algorithms are compared in Figure 1 and Figure 2.First, the estimation accuracies attained by both algorithms asfunctions of the ranging error are shown in Figure 1(a). In thecase of the E-MDS algorithm, the curves are parameterizedby the angle error εθ.

The plots show that the localization accuracy of the E-MDSalgorithm grows (almost linearly) with σd at a lower pace thanthat of the C-MDS in the interval of interest (of moderate rang-ing errors). In other words, it is found that angle information,even when not highly accurate, can significantly improve thelocalization accuracy.10

For example, in a scenario with σd = 0.2 and εθ ≈ 20◦,the E-MDS algorithm exhibits the same performed attained bythe C-MDS when the range estimation errors are inferior to0.1.

The fact that angle information, even if imperfect, can beutilized to increase the robustness versus ranging errors can befurther appreciated in Figure 1(b), where the localization accu-racy of the E-MDS algorithm as a function of AOA estimationerrors and parameterized by σd – varying from 0.1 to 0.4 – arecompared against the accuracy of the corresponding C-MDSsolution. As the plot illustrates, although the accuracies of bothalgorithms are affected by an increase on the range estimationerror, the relative performance degradation suffered by theE-MDS solution (for any given underlying εθ) is significantlylower than the one experienced by its classical counterpart.

Also notice that the mapping defined in equation (19) allowsto weight the rows of C and consequently the single entriesof D. For instance, by weighting to zero the rows of Ccorresponding to the missing entries in D then it is possibleto make the E-MDS algorithm robust to the distance erasuresproblem, which is know to be an extremely limiting factorfor the applicability of the C-MDS algorithm [9]. This isillustrated in Figure 2 where it is shown that, for distanceinformation corrupted by σd = 0.05 and different values of εθthe performance of the E-MDS are almost independent on thelevel of completeness of D. This robustness to of the algorithmversus missing measurements can be understood looking atthe single elements defined in equation (17) as well as thestructure of the Edge kernel KE. Indeed while equation (18)shows how each single entry of the kernel only depends on twodistances and the corresponding mutual angle it also illustrates

10This is in line with the theory which states that the Fisher informationprovided by independent measurement is the sum of the information fromsingle observations [29].

40 50 60 70 80 90 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Performance versus Completeness

Completeness

RMSE

E-MDS

C-MDS

εθ = 12

εθ = 30

εθ = 49

εθ = 70

εθ = 93

Fig. 2. Robustness of the E-MDS versus incomplete information.

how the same range information is used redundantly in M−1combinations.

Moreover, another strong point of the E-MDS solutioncompared to the standard C-MDS, is that eventual knowledgeon the confidence of the single range measurements could beincorporated in the estimation process through the matrix C,for instance relying on techniques such as the one proposedin [17], [26].

IV. THE E-MDS IN LOCALIZATION SCENARIOS

In the form presented in Section III the E-MDS algorithmstill suffers the following limitations:

- increased complexity to eigen-decompose KE,- all the angles in ΘE are required.Indeed, being KE ∈ R

M×M , where M =(N2

), one

bottleneck of the algorithm is represented by the complex-ity required to eigen-decompose KE, which, in presence ofnetworks of sufficiently large size could become unbearable.

Fortunately, however, depending on the kind of scenario athand, different kind of solutions can be exploited to drasti-cally reduce the aforementioned computational requirements.With reference to the problem at hand we will consider thefollowing two cases, complete and structurally incomplete in-formation in the angle kernel ΘE in equation (25). Specifically,in the former case, although different kind of algorithms couldbe used to alleviate the computation associated with the eigen-decomposition of ΘE [27], in [9] it was shown that in dynamicscenarios its eigenspace can be tracked at a computational costthat grows only linearly with

√N . Nevertheless, while in this

scenario the computational complexity does not pose a signif-icant restriction on the applicability of the algorithm, a morestringent requirement comes from the underline assumptionthat, although with low accuracy, all the M mutual anglesamongst the edges in �V need to be known.11

Fortunately, however, in many scenarios of interest, e.g.cellular network like scenarios, it can be expected that only

11Notice that the incompleteness discussed in Section III refers to thedistance information, which in light of equation (25) it is equivalent to astructured incompleteness in ΘE.


a subset of nodes, e.g. the base stations, are equipped withthe hardware necessary to provide the AOA measurements.Under this assumption then the completeness of ΘE is limitedto those rows corresponding to the edges in which at least oneof the two vertices is an anchor node.

In presence of this well-structured incompleteness of ΘE,then it becomes possible to exploit a numerical approximationto the eigenfunction problem used in the context of kernelbased predictors which goes under the name of Nystromapproximation [18] to make the computational complexityrequired to recover V substantially lower.

Indeed using the Nystrom algorithm it is possible to ap-proximate the eigen-problem associated to KE with one ofsize η = rank(KE), and then to expand back the result [18].Let BA be a number of rows selected at random from thekernel matrix12, BT = M − BA the remaining edges inGη,N (X, �V,D) and without loss generality let’s assume thatthose are permuted to the first BA position of the the matrix,yielding

KE ≈[ABA×BA TBA×BT

TTBT×BA

TTBT×BA

·A−1BA×BA

·TBA×BT

]. (31)

Let V =[VT

A,VTT

]Tand A = UA ·ΛA ·UT

A, then using

equation (8) it follows that VA = [UA]BA×η · [ΛA]� 1

21:η , and

since T = VA ·VTT then

VT =(V−T

A ·T)T

=

([ΛA]

�(− 12 )

η×η · [UA]TBA×η ·T

)T

. (32)

Hence the matrix VT containing the edges referred to thetargets can be recovered from the eigen-decomposition of Aonly. Moreover, begin A the edge kernel associated to theanchors, then its eigen-space can be fed to the algorithm asan a priori information.

In light of the discussion above it becomes evident thatthis approach is particularly convenient in all those cases,such as the multitarget localization scenario here considered inwhich BA � BT. Indeed, due to the combinatorial relationshipexisting between the number of nodes and edges in a graph,when NT > NA then BT BA. It follows that the Nystromapproximation of the kernel KE only relies on the first BA

rows of KE. More importantly the edges associated to thetargets can be computed directly by means of equation (32),namely without involving any decomposition.

A further implication of the Nystrom method can be under-stood by expressing the edge kernel KE similarly to equation(25). Indeed, using equation (31) an approximation of KE canbe constructed on the basis of

A = ΘA ⊗(dBA · dT

BA

), (33)

T = ΘT ⊗(dBT · dT

BT

), (34)

where ΘA and ΘT are respectively the matrices including allthe anchor-to-anchor and anchor-to-target angles and dBA ∈RBA×1 and dBT ∈ RBT×1 the vectors with the lengths for thecorresponding edges.

12Provided that the entries in A and T are correct, then the Nystromapproximation of KE is perfect [18].

Algorithm 1 E-MDS localization

1: Get dM ∈ RM×1 and ΘE ∈ RM×M ;

2: if ΘE is complete then

3: Factorize ΘE into (UΘE ΛΘE);4: Solve V ← diag(dM ) · [UΘE ]1:M,1:η ·

[ΛΘE ]� 1

21:η,1:η;

5: else6: Get (UΘA ,ΛΘA) and ΘT;7: Solve VT ← diag(dBT) ·(

[ΛΘA ]�(− 1

2 )η×η · [UΘA ]

TBA×η ·ΘT

)T

;

8: Compute V← [VTA, V

TT]

T;

9: end if

10: Map the edges in V into points: X← C† · V;

11: Use X to compute the Procrustes coefficients{α,FR, cD}

12: Solve X← α ·Y · FR + cD ⊗ 1N .

Since dBA = [dBA ,dBT ] then it is evident that the Nystromapproximation retains all range information in dM and onlydiscards part of the information in ΘE by neglecting themutual angles between the targets.13

Therefore, similarly to equation (26), the case of partialangle information can be rewritten as the solution of the edgesorientation problem rescaled appropriately, namely

VT = diag(dBT)·(Θ−T

A ·ΘT)T

= diag(dBT)·([ΛΘA ]

�(− 12 )

η×η ·[UΘA ]TBA×η ·ΘT

)T

,(35)

where ΘA = UΘA ·ΛΘA ·UTΘA

.The E-MDS solution applied to heterogeneous scenarios is

summarized in Algorithm 1, from which it evident how, atleast of the case of partial information, the most computation-ally demanding part is the computation of V which is obtainedby a mere matrix multiplication.

A. The E-MDS with Angle Completion

Although Section III already dealt with the problem ofeventual erasures in dM , the E-MDS approach still requiresthat all, or using the Nystrom approximation, that at least theangles in the first BA rows of ΘE are known.

Fortunately, however, in light of the robustness of theE-MDS method to erroneous angle information [30] it ispossible to either complete or even replace the angle matrixΘE by

ΘE =(VE · VT

E

)�

(dE · dT

E

), (36)

13Similarly to [9], generic scenarios in which a subset of the target nodesare able to measure some of the angles to their neighbors can be dealt withby combining the solution described in Section III together with the Nystromapproximation.


A1 A2

A3

A1 A2

A3

Fig. 3. Feasibility region Φ defined accordingly to equation (39) and (41)in presence of positive biases (left) and positive and negative biases (right).

where VE = V(X) is the matrix of edges constructed from aninitial estimates of X, � is the inverse of ⊗ and dE ∈ R

M×1+

is defined as

dE =(

diag(VE · VT

E

))� 12

. (37)

Once ΘE is computed, then the new edge kernel is obtainedby

KE = ΘE ⊗(dM · dT

M

). (38)

Unlike intuition may lead us to believe the angle informa-tion constructed from distances is in fact extrinsic information,and therefore, from an Information Theoretical stand pointfundamentally contributes to the potential reduction of local-ization errors. To elaborate, notice that angles, or better saying,their cosines in ΘE, are computed from ratios of distances (seeequation (3)), which are affected by independent and randomerrors.14 In view of such independence between distances andtheir derived angles, allied with the fact that the addition of anytype of extrinsic (independent) information to the estimatorleads to smaller Cramer Rao bounds [29], it follows that infact AOA obtained based on TOA measurement does provideadditional useful information to improve positioning accuracy.

Also notice that although reconstructed angles are indepen-dent from the distances from which they are calculated, theyare not fully uncorrelated amongst themselves. Indeed, eachof the random variables θij is in fact independent form thethe corresponding distances (di, dj). However, the pairs ofvariates (θij , θik) do exhibit correlation between them. Suchcorrelation is not present, of course, if (θij and θik) areindependently measured as assumed in Section III. Secondly,although no correlation exists amongst θij , di and dj , ifθij is reconstructed form di and dj (as well as dij ), thevariance of θij is determined by the variances of the noisydistances from which it is determined. In contrast, when θij’sare independently measured, their variances depend only onthe quality of the corresponding estimator. Therefore, evenwhen angles cannot be independently measured, the utilizationof range-only in the E-MDS is still advantageous.

However, the problem of finding an appropriate estimatefor ΘE, or equivalently VE, still remains. One method to

14It has been shown [31] that the variables Y and 1/Y are independent,which in turn implies that the ratio Z = X/Y of two independent variates Xand Y is such that Z is independent of both X and Y .

compute ΘE on the basis of the set of anchor-to-target distancemeasurements only is described in the following.15

Let {d1,j, . . . , dNA,j} be the set of distance measurementsbetween the j-th target and the anchor-to-target measurements,then it is well known that [XC ]j can be found jointly solvinga set of i = {1, . . . , BA} constraints [33].

In particular, in the case of planar configurations (η = 2),the extension to the case of 3-D scenarios is straightforward,and denoting with xj = [xj , yj ] the j-th target’s coordinates,then the relation with respect to the i-th anchor can beexpressed by

‖ xi − xj ‖= di,j ⇒ (xi − xj)2 + (yi − yj)

2 = di,j . (39)

Under noisy range measurements equation (39) does nothold. In particular, in presence of NLOS conditions, then theequality in equation (39) modifies into the following inequality[33]

‖ xi − xj ‖� di,j ⇒ (xi − xj)2 + (yi − yj)

2 � di,j . (40)

It is also known that each one of the i-th circular constraintsabove can be relaxed and expressed by a set of η linearconstraints [33] as

xi − x � di,1 and −xi + x � di,1, (41)

yi − y � di,1 and −yi + y � di,1. (42)

In particular, let φ(i) be the area defined by the i-thconstraint relative to the j-th target, than the intersection theintersection of the BA constraints defines a feasibility regionin the space where the target is ensured to be, namely

xj ∈ Φ =

NA⋂i=1

φ(i). (43)

A representation of the feasibility spaces obtained from theBA inequalities given in equation (39) or the linear constraintsdefined in equation (41) is provided in Figure 3 from whichit is evident that the region obtained by the union of all linearconstraints is only an upper bound to one obtained using thecircular constraints.16

In absence of any additional information about the rangemeasurements, e.g statistical models, then each point in thefeasibility region Φ can be assumed as equally likely, implyingthat the most conservative estimate of xj can be obtained asthe center of mass of the region itself. Specifically for thecircular constraints such a point can be found by intersectingthe radial lines defined by the pair or inequalities [34]. Differ-ently the feasibility region defined by the linear constrains canbe fully characterized by a lower and an upper point whosecoordinates are [33]

xl =[mini{xi + di,j},min

i{yi + di,j}

], (44)

xu =[max

i{xi − di,j},max

i{yi − di,j}

]. (45)

15That does not preclude the utilization of any other more sophisticated,if more complex method to compute centroids from feasibility regions. Oneexample of an alternative (substantially more computationally demanding, butalso more accurate) method that could be used can be found, for instance, in[32].

16Also notice that while the system of linear constraints can always beapplied, the system of circular constraints can be solved only in presence ofperfect or positively biased range measurements [33].


It follows that the center of mass of for the linear constraintscan be found by simply averaging the aforementioned points.

V. RESULTS

In order to evaluate the performance of the E-MDS al-gorithm described in Section IV, let us consider a scenarioconsisting of 4 anchor nodes placed at the edges of a 2 × 2square in a η = 2 dimensional space which will also define oursurveillance region. The noisy range measurements are mod-eled as a Gaussian-distributed random variables with meangiven by the true distance and a standard deviation σd, i.e.dij ∼ pGauss(r; d, σd), while the angle error is modeled usingthe Tikhonov model described in Section III. NLOS conditionsare simulated by adding to the mean of the aforementionedGaussian random variable a sample from an random variableuniformly distributed between 0− bd, i.e. U (0, bd).

To begin with, a source localization in a range-only scenariois considered in Figure 4 where the E-MDS-based solutionusing the feasibility constraints defined in Section IV-A iscompared under both LOS and NLOS conditions versus state-of-the-art algorithms.

From Figure 4 it can be appreciated that in fact theapproach of obtaining angle estimates from given distancemeasurements, followed by localization via E-MDS, outper-forms not only the classic (range-only) MDS, but also otherwidely used algorithms such as SMACOF and a semi-definiteprogramming (SDP) formulation of the problem [35].

Also Figure 4 compares the proposed solution versus theposition error bound (PEB) for range-only scenario, namelythe CRLB in LOS and its generalization in NLOS case [4].In particular, the apparently unreasonable result in Figure4(a) that sees the E-MDS outperforming the bound can beexplained by recalling that, even in presence of range-onlyscenario the angle kernel ΘE from the centroids providesextrinsic information to improve the estimation of X.17 Tocorroborate on this notice that also the SDP formulationexploiting heterogenous information proposed in [35], namelythe SDP(AR) solution, when fed with angles computed fromthe centroids also beats the bound.

Moreover, when distances are measured almost without er-ror, the correspondingly constructed angles from the centroidsbecome almost deterministic, thus providing no additionalstatistical information from which to obtain better results.This is precisely why, in Figure 4(a), it can be seen that theperformance of the proposed S-MDS algorithm reduces to thesame as those of all other methods when σd is small.

Regarding the improved performance of E-MDS in Figure4(b), one reason for such include, as discussed in Section III,the fact that neither the operations of angle calculation fromdistances, nor the construction of the S-MDS kernel, requireall distances (per angle or per kernel element), such that errorpropagation is drastically reduced. The other reason discussedin Section IV-A stems from the deeper implication of the joint

17A more accurate bound bound would need to take into considerationthis information. Also notice that this happens in the LOS case where thecentroids are more informative.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Performance with Range Measurements- LOS Scenario, Source Localization (σd = 0.05) -

σd

C-MDS

SMACOF

SDP(R)

PEB

E-MDS

SDP(AR)

RMSE

(a) Performance in LOS. RMSE as a function of σd .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Performance with Range Measurements- NLOS Scenario, Source Localization (σd = 0.05) -

bd

C-MDS

SMACOF

SDP(R)

SDP(AR)

E-MDS

PEB

RMSE

(b) Performance in NLOS. RMSE as a function of the bias bd.

Fig. 4. Performance of E-MDS algorithm applied to a range only sourcelocalization scenario. Results as function of σd in the LOS case and bd forthe NLOS case.

utilization of angle and distance information onto the FisherInformation Matrix associated with the problem.

Figure 5 offers an additional set of results for the source lo-calization setting in which the E-MDS algorithm is comparedversus the SDP(AR) in a scenario in which both range andangle information are assumed to be measured independentlyfrom each other. Similarly to [35], angles are assumed to bemeasured at the sensors accordingly to their local referencesystem which is assumed to be known only at the anchornodes. Anyhow, by subtracting the relative measurements ateach sensor with respect to other sensors, namely assuming anangle difference of arrival (ADOA) scenario, the dependencefrom the local reference systems is removed from the problem.

Moreover, as explained in [35] the SDP(AR) considers theLaw of Cosines applied to all triplets of points in the scenario,namely a typical ADOA scenario, while the angle kernel ΘE


12 22 32 42 52 62 720

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

0.35

Performance with Range-Angle Measurements- LOS Scenario, Source Localization (σd = 0.05) -

- NLOS Scenario (σd = 0.05, εd = 22◦) -

εθ

bd

SDP(AR)

SDP(AR)

E-MDS

E-MDS

RMSE

RMSE

Fig. 5. Performance of E-MDS algorithm applied to a range-angle sourcelocalization scenario. Results as function of εd for the LOS case and bd forthe NLOS case.

extends it to all pairs of edges in Gη,N (X, �V,D), namely itexploits more information than the SDP(AR) algorithm.18

The second scenario considered consists of a multi targetlocalization scenario in which 5 targets are randomly displacedwithin the surveillance region. As discussed in Section III andIV, the E-MDS framework allows to handle multitarget dy-namic scenarios in which either distance or distance and AOAinformation is available. In what follows the performanceof the E-MDS technique is compared against the SMACOFalgorithm [19] and the C-MDS based method described inSection II-C using both range only and range and anglemeasurements.

The simulations are designed to provide insight on howmuch accuracy can be gained by utilizing angle informationand how robust the edge-kernel formulation is compared tothe other alternatives discussed in Section II.

As discussed in Section IV, although the E-MDS algorithmallows to include range and angle measurements from eachpair of sensors, in practice it can not be expected that everydevice will be able to collect these kind of information. Forthis reason below we will restrict ourselves to a cellularnetwork-like scenario in which only a limited number (NA)of nodes, the anchor nodes, can measure angle and range withall the users in their surroundings.

The results are shown in Figure 6 for a LOS and NLOS con-ditions. The plots reveal that the E-MDS solution implementedas discussed in Section IV has better accuracy than bothSMACOF algorithm19 and the MDS-based method proposed

18More constraints could be added to the SDP formulation, but this wouldresult in a different solution from the one in [35]. This is corroborate by theresults showing a clear gap between the performance of the two algorithms.Similar results stands in the case of minimal angle information, in whichit assumed that only the anchor nodes are able to measure one only angleto the target and this set of measurements is then used to create the set ofADOA use in the SDP(AR) algorithm as well as the edge kernel matrix ΘE.Indeed even in this case, where the same measured information is given toboth algorithms, the E-MDSstill outperforms the SDP approach.

19The algorithm is initialized with the centroids used to compute ΘC .

0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Performance with Range-Angle Measurements- LOS Scenario with 5 Targets (εθ = 15

◦) -

σd

RMSE

SMACOF

C-MDS (Averaged)

E-MDS (Nystrom)

E-MDS (Iterated)

E-MDS (Complete)

Range Only

Range & Angle

(a) Performance in LOS. RMSE as a function of σd .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Performance with Range-Angle Measurements- NLOS Scenario with 5 Targets (εθ = 15

◦, σd = 0.15) -

bd

RMSE

SMACOF

C-MDS (Averaged)

E-MDS (Nystrom)

E-MDS (Iterated)

E-MDS (Complete)

Range Only

Range & Angle

(b) Performance in NLOS. RMSE as a function of the bias bd.

Fig. 6. Performance of E-MDS algorithm applied to a multitarget scenario.Results as function of σd and bd, for range only and range-bearing measure-ments.

in Section II-C averaged over all NA anchor nodes. As Figure6(b) shows, the advantage of the E-MDS solution becomesparticularly evident in presence of NLOS measurements.Moreover, similarly to iterated Bayesian solutions, e.g. iteratedKalman filter [29], in the case of range only observations anadditional improvement in performance can be achieved byupdating ΘC on the basis of the solution obtained by theE-MDS and running the algorithm once more time over theobserved range measurements.

VI. CONCLUSIONS

We proposed a comprehensive algorithm to handle theproblem of simultaneously localize a large number of targetsin LOS-NLOS with no a priori measurement models. Thesolution allows for both distance and angle information tobe processed algebraically, without iteration, simultaneously


and at a computational cost of the algorithm is of a fewmatrix multiplications only, solving a well known bottleneckof the C-MDS solution. We compared the proposed E-MDSalgorithm to a constrained optimization technique specificallydesigned to handle localization in heterogeneous conditionsshowing the superiority of our solution. More importantly ithas been shown that the E-MDS framework allows to exploitthe eventual angle information to improve the localizationestimate over biased range information. This has been shownto be true even in presence of range only information wherethe E-MDS still outperforms well established solutions.

APPENDIX APROCRUSTES TRANSFORMATION

The procrustes transformation is an affine transformation,namely a rigid transformation20 and a scaling factor .

Given the coordinates of at least NA > η anchor nodesXNA ∈ R

NA×η, then the estimated X can be recoveredfrom the output Y ∈ RN×η of a scaling algorithm, e.g. theC-MDS, using the following isotropic dilation and the rigidtransformations

X = α ·Y ·FR + cD ⊗ 1N , (46)

where the scalar α, the matrix FR ∈ Rη×η and the vectorcD ∈ Rη are the coefficients of the Procrustes transformation,calculated as

α = tr([HT

NA·HNA ]

12

)· ‖XNA‖F

‖YNA‖F, (47)

FR = UTL ·UR, (48)

cD =1TNA

NA·XNA −

α · 1TNA

NA· [Y]1:NA,1:η · FR. (49)

In the above, UL - UR are the left and right matrices ofsingular vectors of HNA = XT

NA· YNA = UL · Σ ·UR, and

XNA and YNA are the centralized true and rotated coordinatematrices of the anchor nodes, given by

XNA = XNA −1TNA

NA·XNA ⊗ 1NA , (50)

YNA = YNA −1TNA

NA· [Y]1:NA,1:η ⊗ 1NA . (51)

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Davide Macagnano (S’06-M’12) obtained hisM.Sc. degree from the Politecnico di Milano, Italyin 2005 and the Ph.D. degree from the University ofOulu, Finland in 2012. Since January 2006 he is a re-searcher at the Centre for Wireless Communications,University of Oulu, Finland. His research interestsare Bayesian algorithms, convex optimization andgraph theory for statistical signal processing withfocus on localization and tracking algorithms. In2012 he was co-recipient of the Best Paper Awardat the Workshop on Positioning Navigation and

Communications.

Giuseppe T. F. de Abreu (S’99-M’04-SM’09) isan Associate Professor of Electrical Engineering atJacobs University, Bremen. He received the B.Eng.degree in Electrical Engineering and a specialization(Latu Sensu) degree in Telecommunications Engi-neering from the Universidade Federal da Bahia(UFBa), Salvador, Bahia, Brazil, in 1996 and 1997,respectively; and the M.Eng. and Ph.D degrees inPhysics, Electrical and Computer Engineering fromthe Yokohama National University, Japan, in March2001, and March 2004, respectively, having been the

recipient of the 2000 Uenohara Award by Tokyo University for his Master’sThesis. Prior to joining Jacobs University he was an Adjunct Professor(Docent) on Statistical Signal Processing and Communications Theory at theDepartment of Electrical and Information Engineering, University of Oulu,Finland. Dr. Abreu is a member of the Technical Program Committee ofvarious IEEE conferences, an author/co-author of over 130 published technicalarticles, patents and a few book chapters. He co-authored articles selected asthe best paper of the Signal Processing and Communications Track at theAsilomar Conference on Signals Systems and Computers in 2009 and 2012,respectively, and was co-recipient of the Best Paper Award at the Workshopon Positioning Navigation and Communications in 2012. Prof. Abreu is activein various topics ranging from communications theory, signal processing,statistical modeling, localization and tracking algorithms, cognitive radio,wireless security and others.

algebraic approach for robust localization with heterogeneous information

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