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COMBINING RANGE AND DIRECTION FOR IMPROVED LOCALIZATION Gilles Baechler*, Frederike Dümbgen*, Golnoosh Elhami*, Miranda Kreković*, Robin Scheibler, Adam Scholefield, and Martin Vetterli School of Communication and Computer Science, Ecole Polytechnique Fédérale de Lausanne (EPFL) Constrained Edge-Kernel, proposed approach Problem Statement Coordinate Difference Matrices (CDMs) + + + + ∆ ℓ ˜ v i v h v i v k x m x p x n ˜ x m ˜ d mn ˜ α mn d mn α mn ˜ y mn > 0 y mn > 0 ˜ x mn < 0 x mn < 0 *The authors have equal contribution to this work and the order is alphabetical This work was supported by the Swiss National Science Foundation grant number 20FP-1 151073, “Inverse Problems regularized by Sparsity” and by InnoSuisse, grant number 25579.1 PFES-ES. [1] I. Dokmanić, R. Parhizkar, J. Ranieri, and M. Vetterli, “Euclidean Distance Matrices: Essential theory, algorithms, and applications,” IEEE Signal Processing Magazine, vol. 32, no. 6, pp. 12-30, 2015. [2] R. Peng and M. L. Sichitiu, “Angle of arrival localization for wireless sensor networks,” 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and Networks, vol. 1, pp. 374–382, 2006. [3] D. Macagnano, G. Thadeu, F. D. Abreu, and S. Member, “Algebraic Approach for Robust Localization with Heterogeneous Information,” IEEE Transactions on Wireless Communications, vol. 12, no. 10, pp. 5334–5345, 2013. Contact: name.surname@epfl.ch for all authors Background • Goal: multimodal self-localization of nodes in a sensor network • Approach: recover the locations of the nodes given distance and angle measurements • Contributions: two algorithms that outperform the state-of-the art Distance-based approaches • How to measure distances: time-of-arrival, received signal strength • Methods: Euclidean distance matrices, MDS [1] Angle-based approaches • How to measure angles: antenna arrays, leveraging phase differences • Methods: angle-of-arrival [2] Hybrid approaches • Combination of distances and angles • Method: edge-multidimensional scaling [3] Edge-Kernel, prior work Comparison of the two methods Conclusion Comparison with state-of-the-art 0.01 (0.6 ◦ ) 0.5 (28.6 ◦ ) σ α 0.01 0.5 σ d -6.6e-05 0.011 0.021 0.03 0.041 0.05 0.063 0.07 0.083 0.094 0.1 -0.0068 -0.00086 0.0075 0.02 0.032 0.042 0.05 0.063 0.078 0.083 0.095 -0.012 -0.0093 -0.0019 0.0076 0.018 0.028 0.041 0.052 0.064 0.072 0.092 -0.02 -0.017 -0.011 -0.0044 0.004 0.017 0.031 0.039 0.054 0.064 0.075 -0.03 -0.027 -0.025 -0.017 -0.0089 0.0019 0.013 0.024 0.037 0.05 0.063 -0.038 -0.037 -0.033 -0.028 -0.019 -0.0094 -0.0057 0.0084 0.022 0.038 0.047 -0.05 -0.051 -0.043 -0.041 -0.033 -0.027 -0.016 -0.01 0.002 0.012 0.027 -0.066 -0.063 -0.059 -0.058 -0.051 -0.045 -0.029 -0.023 -0.011 -0.0074 0.015 -0.082 -0.077 -0.074 -0.074 -0.07 -0.058 -0.052 -0.041 -0.027 -0.021 -0.0094 -0.098 -0.11 -0.1 -0.094 -0.085 -0.077 -0.071 -0.07 -0.058 -0.041 -0.03 difference CDM - constrained E-MDS −0.08 −0.04 0.00 0.04 0.08 0.0 0.1 0.2 0.3 0.4 0.5 σ d [-] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RMSE ”low” angle noise: σ α =0.11 (6.2 ◦ ) constrained E-MDS MDS CDM E-MDS 0.0 0.1 0.2 0.3 0.4 0.5 σ α [rad] 0.0 0.1 0.2 0.3 0.4 RMSE constrained E-MDS MDS CDM E-MDS 0.0 11.5 22.9 σ α [ ◦ ] ”low” distance noise: σ d =0.06 0.0 0.1 0.2 0.3 0.4 0.5 σ d [-] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RMSE ”high” angle noise: σ α =0.50 (28.6 ◦ ) 0.0 0.1 0.2 0.3 0.4 0.5 σ α [rad] 0.0 0.1 0.2 0.3 0.4 RMSE 0.0 11.5 22.9 σ α [ ◦ ] ”high” distance noise: σ d =0.17 • Experiment: measure the RMSE of the reconstructed locations for different noise on distances ( ) or angles ( ) • Outcome: CDM better for high distance noise, constrained E-MDS better with high angular noise Definitions Input Incomplete noisy measurements ˜ S =(S + Z ) ◦ W Cost function CDM Noise matrix Mask matrix Degree matrix S = s1 > - 1s > Z W Algorithm min s W ◦ (s1 > - 1s > - e S ) 2 F Decompose the edge vectors V into 1D coordinate differences s x and s y . Compute W 0 , ˜ S 0 and ⇤ 0 by removing the first row and column For each dimension d 0 = ⇤ 0 ( ˜ S 0 ◦ W 0 )1 A 0 = I - ⇤ 0 W 0 ˆ s =(A 0 ) -1 d 0 where (A 0 ) -1 has a closed form. Return ⇥ ⇤ Output The 2D points ⇥ ˆ s x ˆ s y ⇤ σ d σ ↵ Constrained E-MDS • Our contribution: enforce additional constraints on • Constraints: (1) is PSD and of rank 2 (2) the entries of satisfy the triangle equality • Solution: lift-and-project, alternative projections onto the sets satisfying constraints (1) and (2) Edge-multidimensional scaling (E-MDS) [3] • Edge kernel matrix • Solution: recover via eigenvalue decomposition of K E K E K E K E • We proposed two algorithms for multimodal sensor localization: • Constrained Edge-Kernel improves on the existing Edge-Kernel method by adding geometric constraints on triplets of points. • Coordinate Difference Matrices allow us to estimate the sensors’ coordinates independently for each dimension. • Numerical simulations demonstrate that both proposed methods significantly outperform existing distance-based and multimodal localization algorithms. State-of-the-art • Distance-only: MDS [1] • Distance + angles: E-MDS [3] Experiment • 6 points chosen uniformly in the unit square • Compute RMSE on reconstructed points RMSE vs. angular noise • RMSE vs. angular noise for two representative distance noise levels RMSE vs. distance noise • RMSE vs. distance noise for two representative angle noise levels W , ˜ S and ⇤. ff pos

Gilles Baechler*, Frederike Dümbgen*, Golnoosh Elhami ... · mn α˜mn dmn αmn ˜y mn > 0 y mn > 0 x˜mn

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Page 1: Gilles Baechler*, Frederike Dümbgen*, Golnoosh Elhami ... · mn α˜mn dmn αmn ˜y mn > 0 y mn > 0 x˜mn

COMBINING RANGE AND DIRECTION FOR IMPROVED LOCALIZATIONGilles Baechler*, Frederike Dümbgen*, Golnoosh Elhami*, Miranda Kreković*, Robin Scheibler, Adam Scholefield, and Martin Vetterli

School of Communication and Computer Science, Ecole Polytechnique Fédérale de Lausanne (EPFL)

Constrained Edge-Kernel, proposed approachProblem Statement

Coordinate Difference Matrices (CDMs)

+

+

+

+

∆ℓvi

vh

vi

vkxm

xp

xn

xm

dmn

αmndmn

αmn

y mn>

0

y mn>

0

xmn < 0

xmn < 0

Fig. 1. Localization setup in 2D. The measured distance dmn and thedirected angle αmn can be be mutually expressed with the vectorvmn or, equivalently, with the signed projections on the x- and y-axis: xmn and ymn. Triangle ∆ℓ is such that vi − vh = vk.

We define the edge vector vmn corresponding to the pair of points(xn,xm) by

vmn = xm − xn. (1)

The length of the edge vector is the distance between the points,denoted by

dmn = ∥xn − xm∥2, (2)

while dmn refers to its noisy measurement. The orientation of thevector is defined with a directed angle, meaning that it lies between0 and 2π with respect to some common reference direction (in thiscase the x-axis). This assumption is realistic because devices areoften equipped with a compass, providing their absolute orientation.Therefore, we can use any off-the-shelf method [17] to measure thedirections of arrival and then convert them into directed angles. Wedenote the noiseless and noisy measured directed angles by αmn

and αmn, respectively, as depicted in Fig. 1. Then, the coordinatesof edge vectors relate to distances and angles as

(vmn)x = dmn cosαmn and (vmn)y = dmn sinαmn. (3)

Our goal is to estimate the points X , given noisy measurementsof distances dmn and angles αmn.

3. EDGE-KERNEL BASED SOLUTIONS

Before describing our constrained edge-kernel approach, we firstsummarize the original edge-kernel method [16].

3.1. The Edge-Kernel Formulation

In [16], Macagnano et al. propose a new MDS formulation, referredto as edge multidimensional scaling (E-MDS), which allows a uni-fied processing of angle and range information. In the E-MDS-basedformulation, the entities are the vectors pointing from one point toanother, while the measure of dissimilarity is their inner product.

To simplify the notation and allow the construction of a matrixwhose elements depend on two edge vectors i and j, we first re-index

the edges vmn from (1) to vi with i = (m− 1)N − m(m+1)2 + n,

where m = 1, . . . , N − 1, n = m + 1, . . . , N , and i = 1, . . . , E,E = N(N−1)

2 . If we consider a matrix consisting of elements vmn,this corresponds to flattening the elements of its upper triangle in arow-wise order. Then, we define the edge-kernel KE ∈ RE×E of afully connected point set with E edges and the element (i, j) as

(KE)ij = ⟨vi,vj⟩ = ⟨xm − xn,xq − xp⟩= dmndpq cos (φij),

(4)

where dmn is defined in (2), and φij is the inner angle between vec-tors vi and vj . In matrix form, (4) simplifies to

KE = V V ⊤ = dEd⊤E Ω, (5)

where V ∈ RE×2 is the matrix of the edges, dE ∈ RE is the vectorof the edge lengths and represents the Hadamard (entrywise) prod-uct. The elements Ωij of the angle matrix Ω contain the cosines ofthe inner angles between vi and vj . From (5), it is easy to verifythat the rank of the edge kernel is at most 2. Given noisy angle anddistance measurements, the authors in [16] propose to reconstruct adenoised kernel by imposing this low-rank constraint and positive-semidefiniteness:

V = argminV ∈RE×2

∥KE − V V ⊤∥F

= diag!dE

"· [U Ω]1:E,1:2 · [ΛΩ]

121:2,1:2 ,

(6)

where KE = dEd⊤E Ω is the measured edge kernel and U Ω and

ΛΩ are the matrices of eigenvectors and eigenvalues of Ω, respec-tively. Recovering the point set X from V is straightforward oncewe set the translation of X by fixing one point. It consists of solvinga sparse linear system of equations defined in [16].

3.2. Constrained Edge-Kernel

The above solution outperforms previous approaches, which includedistance-only techniques [18] as well as methods combining bothdistances and angles [18, 19]. However, it neglects certain geometricconstraints, which means the recovered vectors are not guaranteedto be geometrically consistent. Indeed, imposing the kernel struc-ture ignores triangle equalities of the edge vectors. Therefore, wepropose a new method, which we call constrained edge-kernel, toaddress this limitation.

To better exploit geometric information for denoising, we intro-duce the novel notion of a triangle constraints matrix M ∈ RC×E .This matrix incorporates the mutual dependencies of edge vectorsforming a triangle, with Mℓi = 1, Mℓk = Mℓj = −1, assumingthat the vectors vi, vj and vk form the triangle ∆ℓ (see Fig. 1).The number of constraints C equals to the number of triangles in thepoint set, C =

#N3

$. The problem given in (6) can then be written as

the rank-constrained optimization,

KE =argminK

∥KE −K∥F

subject to K ∈ C and K ∈ R,(7)

where we introduce the feasibility sets

R =%X ∈ RE×E : X ≽ 0, rank (X) = 2

&, (8)

C =%X ∈ RE×E : MX = 0

&. (9)

*The authors have equal contribution to this work and the order is alphabeticalThis work was supported by the Swiss National Science Foundation grant number 20FP-1 151073, “Inverse Problems regularized by Sparsity” and by InnoSuisse, grant number 25579.1 PFES-ES.

[1] I. Dokmanić, R. Parhizkar, J. Ranieri, and M. Vetterli, “Euclidean Distance Matrices: Essential theory, algorithms, and applications,” IEEE Signal Processing Magazine, vol. 32, no. 6, pp. 12-30, 2015.[2] R. Peng and M. L. Sichitiu, “Angle of arrival localization for wireless sensor networks,” 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and Networks, vol. 1, pp. 374–382, 2006.[3] D. Macagnano, G. Thadeu, F. D. Abreu, and S. Member, “Algebraic Approach for Robust Localization with Heterogeneous Information,” IEEE Transactions on Wireless Communications, vol. 12, no. 10, pp. 5334–5345, 2013.

Contact: [email protected] for all authors

Background

• Goal: multimodal self-localization of nodes in a sensor network• Approach: recover the locations of the nodes given distance and angle

measurements• Contributions: two algorithms that outperform the state-of-the art

Distance-based approaches• How to measure distances: time-of-arrival, received signal

strength

• Methods: Euclidean distance matrices, MDS [1]

Angle-based approaches• How to measure angles: antenna arrays, leveraging phase

differences

• Methods: angle-of-arrival [2]

Hybrid approaches• Combination of distances and angles

• Method: edge-multidimensional scaling [3]

Edge-Kernel, prior workComparison of the two methods

Conclusion

Comparison with state-of-the-art

0.01 (0.6) 0.5 (28.6)σα

0.01

0.5

σd

-6.6e-05 0.011 0.021 0.03 0.041 0.05 0.063 0.07 0.083 0.094 0.1

-0.0068 -0.00086 0.0075 0.02 0.032 0.042 0.05 0.063 0.078 0.083 0.095

-0.012 -0.0093 -0.0019 0.0076 0.018 0.028 0.041 0.052 0.064 0.072 0.092

-0.02 -0.017 -0.011 -0.0044 0.004 0.017 0.031 0.039 0.054 0.064 0.075

-0.03 -0.027 -0.025 -0.017 -0.0089 0.0019 0.013 0.024 0.037 0.05 0.063

-0.038 -0.037 -0.033 -0.028 -0.019 -0.0094 -0.0057 0.0084 0.022 0.038 0.047

-0.05 -0.051 -0.043 -0.041 -0.033 -0.027 -0.016 -0.01 0.002 0.012 0.027

-0.066 -0.063 -0.059 -0.058 -0.051 -0.045 -0.029 -0.023 -0.011 -0.0074 0.015

-0.082 -0.077 -0.074 -0.074 -0.07 -0.058 -0.052 -0.041 -0.027 -0.021 -0.0094

-0.098 -0.11 -0.1 -0.094 -0.085 -0.077 -0.071 -0.07 -0.058 -0.041 -0.03

difference CDM - constrained E-MDS

−0.08

−0.04

0.00

0.04

0.08

0.0 0.1 0.2 0.3 0.4 0.5σd[-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

RMSE

”low” angle noise: σα = 0.11 (6.2)

constrained E-MDS

MDS

CDM

E-MDS

0.0 0.1 0.2 0.3 0.4 0.5σα[rad]

0.0

0.1

0.2

0.3

0.4

RMSE

constrained E-MDS

MDS

CDM

E-MDS

0.0 11.5 22.9σα[]

”low” distance noise: σd = 0.06

0.0 0.1 0.2 0.3 0.4 0.5σd[-]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

RMSE

”high” angle noise: σα = 0.50 (28.6)

0.0 0.1 0.2 0.3 0.4 0.5σα[rad]

0.0

0.1

0.2

0.3

0.4

RMSE

0.0 11.5 22.9σα[]

”high” distance noise: σd = 0.17

• Experiment: measure the RMSE of the reconstructed locations for different noise on distances ( ) or angles ( )

• Outcome: CDM better for high distance noise, constrained E-MDS better with high angular noise

Definitions

Input Incomplete noisy measurements S = (S +Z) W<latexit sha1_base64="JQV7L+tKM9Oe4yZB9QVVkG0Zh6M=">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</latexit><latexit sha1_base64="JQV7L+tKM9Oe4yZB9QVVkG0Zh6M=">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</latexit><latexit sha1_base64="JQV7L+tKM9Oe4yZB9QVVkG0Zh6M=">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</latexit><latexit sha1_base64="JQV7L+tKM9Oe4yZB9QVVkG0Zh6M=">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</latexit>

Cost function

CDM

Noise matrix

Mask matrix

Degree matrix

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Algorithm

mins

W (s1> 1s> eS)2

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Decompose the edge vectors V into 1D coordinate differences sx

and sy

.

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Compute W 0, S0

and 0by removing the first row and column W , S and .

<latexit sha1_base64="6oG5NvAOUdXMcuTYQ4//XhUsKGA=">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</latexit><latexit sha1_base64="6oG5NvAOUdXMcuTYQ4//XhUsKGA=">AAAFQnicdVPNbhMxEN60AUr5a+HIxWpSgRCNskEIuFWqVHFAopSGVsqGyOudTaz6Z2V7UyJ3n4ILL8RD8AycEFeEsHdT2k1bS7saf/ONPfPNOM4Y1abb/dFYWm7euHlr5fbqnbv37j9YW3/4SctcEegTyaQ6irEGRgX0DTUMjjIFmMcMDuPjHe8/nILSVIoDM8tgyPFY0JQSbBw0Wvu2I3mWG0DtaArE2MPiSfv52SYylCVgPxYORFgk//F37oIEezSeIQVcTqkYIzMBlFKlDVLypOS7/HIuzs++6ugrT253RmutbqdbLnTZCOdGK5ivvdH68tcokSTnIAxhWOtB2M3M0GJlKGFQrEa5hgyTYzyGgTMF5qCHtpSwQJsOSVAqlfuEQSV6McJirvWMx47JsZnoRZ8Hr/INcpO+HloqvMiCVBelOUNGIt8PlFDlCmczZ2CiqMsVkQlWmBjXtdotMa/VYBPtU63XpQlmkAztBNgUvE+BgBMiOXciP4tSzCmbJZDinDmxdTo3XVqb6L1wWdC0bKMfqUoJWQFnyiKqfeoxII2FRhoUTWsZHIRD6wN9sXWR2Fi6+ib8Gpgu8L9UnVmNXI5uuMud5TTZSrA63horPCts9e92Xrx0vPNCbTlMxcClEsWSJb5xktlWWLhCvcv39iIdx7pkn7bCU0dBbu8m172KKWY51LlCKl4d7dhRSffQZdLBBdbI9oozJupd5u7WuLvn3N06l2YVkUFqIobFmIELUXQ8MZEqtz6SCgEKZUomOTH1A/aLmiSRH9g4tvtFsaAgFfvVVVRcE/C5FLQeJQXooiIl2jq3e8bh4qO9bPR7nTed8EOvtb0xf88rweNgI3gahMGrYDt4G+wF/YAEfxvtxlaj0/ze/Nn81fxdUZca85hHQW01//wDeCHSPA==</latexit><latexit sha1_base64="6oG5NvAOUdXMcuTYQ4//XhUsKGA=">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</latexit><latexit sha1_base64="6oG5NvAOUdXMcuTYQ4//XhUsKGA=">AAAFQnicdVPNbhMxEN60AUr5a+HIxWpSgRCNskEIuFWqVHFAopSGVsqGyOudTaz6Z2V7UyJ3n4ILL8RD8AycEFeEsHdT2k1bS7saf/ONPfPNOM4Y1abb/dFYWm7euHlr5fbqnbv37j9YW3/4SctcEegTyaQ6irEGRgX0DTUMjjIFmMcMDuPjHe8/nILSVIoDM8tgyPFY0JQSbBw0Wvu2I3mWG0DtaArE2MPiSfv52SYylCVgPxYORFgk//F37oIEezSeIQVcTqkYIzMBlFKlDVLypOS7/HIuzs++6ugrT253RmutbqdbLnTZCOdGK5ivvdH68tcokSTnIAxhWOtB2M3M0GJlKGFQrEa5hgyTYzyGgTMF5qCHtpSwQJsOSVAqlfuEQSV6McJirvWMx47JsZnoRZ8Hr/INcpO+HloqvMiCVBelOUNGIt8PlFDlCmczZ2CiqMsVkQlWmBjXtdotMa/VYBPtU63XpQlmkAztBNgUvE+BgBMiOXciP4tSzCmbJZDinDmxdTo3XVqb6L1wWdC0bKMfqUoJWQFnyiKqfeoxII2FRhoUTWsZHIRD6wN9sXWR2Fi6+ib8Gpgu8L9UnVmNXI5uuMud5TTZSrA63horPCts9e92Xrx0vPNCbTlMxcClEsWSJb5xktlWWLhCvcv39iIdx7pkn7bCU0dBbu8m172KKWY51LlCKl4d7dhRSffQZdLBBdbI9oozJupd5u7WuLvn3N06l2YVkUFqIobFmIELUXQ8MZEqtz6SCgEKZUomOTH1A/aLmiSRH9g4tvtFsaAgFfvVVVRcE/C5FLQeJQXooiIl2jq3e8bh4qO9bPR7nTed8EOvtb0xf88rweNgI3gahMGrYDt4G+wF/YAEfxvtxlaj0/ze/Nn81fxdUZca85hHQW01//wDeCHSPA==</latexit>

For each dimension

<latexit sha1_base64="fCJdShX3aFfPlTp1bepFdLU7FkE=">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</latexit><latexit sha1_base64="fCJdShX3aFfPlTp1bepFdLU7FkE=">AAAEvHicdZPNbhMxEMfdNkApX205crEaVUJIjbJBCDhRCSniRqkaWikbKq93NmvFHyvbGxq5+xTcucIr8TbYu6naTcCH1XjmN/Z/ZtZJwZmx/f6fjc2tzr37D7Yf7jx6/OTps929/a9GlZrCiCqu9EVCDHAmYWSZ5XBRaCAi4XCezD6G+PkctGFKntlFARNBppJljBLrXZe7+0OlMRCa45QJkKZ2dvu9fr3wuhEtjS5arpPLva0fcapo6fMt5cSYcdQv7MQRbRnlUO3EpYGC0BmZwtibkggwE1eLr/Ch96Q48zIyJS2uvXczHBHGLETiSUFsblZjwfmv2Li02buJY7IoLUjaXJSVHFuFQyd8wRqo5QtvEKqZ14ppTjSh1verdUsiWjW41ASp7boMJRzSicuBzyHENEj4TpUQRKav4owIxhcpZKTk1sUmW5pe1iH+LL0KlmGbAw7DbDqhGsdNZzEzQXoC2BBpsAHNspaCs2jiQmIott0kPlW+vlz8x81W+KtmMjux1+h/q3rnBEuPUqJnR1NNFpVrvv3e6zeeuy3UxXPf02rspcSJ4mkYnOKuG1W+0BAKs72Lk8TU9HU3uvYI9nvF/cDwnPAS2qxUWjRHezqu8eBah87uUJduUN2QeLDODlvs8JYdtllWNCCHzMacyCkHn6LZNLexrrchk0kJGhdapSW17QNOq1ZL4vDDJok7raqVDjJ52lzF5H8SvtUNbWcpCaZqoNQ4H/bPOFp9tOvGaNB734u+DLrHB8v3vI1eoAP0EkXoLTpGn9AJGiGKrtBP9Av97nzoQGfWEQ26ubHMeY5aqzP/Cwd9oKs=</latexit><latexit sha1_base64="fCJdShX3aFfPlTp1bepFdLU7FkE=">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</latexit><latexit sha1_base64="fCJdShX3aFfPlTp1bepFdLU7FkE=">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</latexit>

d0 = 0(S0 W 0)1<latexit sha1_base64="w6uRMpjGE4Oe/iAduyZaB95Mwok=">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</latexit><latexit sha1_base64="w6uRMpjGE4Oe/iAduyZaB95Mwok=">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</latexit><latexit sha1_base64="w6uRMpjGE4Oe/iAduyZaB95Mwok=">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</latexit><latexit sha1_base64="w6uRMpjGE4Oe/iAduyZaB95Mwok=">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</latexit>

A0 = I 0W 0<latexit sha1_base64="LxTXFA84q/AHH+6nim+PpFCVVT4=">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</latexit><latexit sha1_base64="LxTXFA84q/AHH+6nim+PpFCVVT4=">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</latexit><latexit sha1_base64="LxTXFA84q/AHH+6nim+PpFCVVT4=">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</latexit><latexit sha1_base64="LxTXFA84q/AHH+6nim+PpFCVVT4=">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</latexit>

s = (A0)1d0where (A0)1

has a closed form.

<latexit sha1_base64="8EmOcZJyO5/MfVA5lqvbgxrqaVs=">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</latexit><latexit sha1_base64="8EmOcZJyO5/MfVA5lqvbgxrqaVs=">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</latexit><latexit sha1_base64="8EmOcZJyO5/MfVA5lqvbgxrqaVs=">AAAFA3icdZPNbhMxEMfdNkApH23hyMVqUrUgNcoGIeCAVIRUcaNUDS3KppHXO5u16rVXtjclcvfKC/AanBBXxHPwAFzhFfDupmo3oZYSjWd+s575exyknGnT6fxaWFxq3Lh5a/n2yp279+6vrq0/+KBlpij0qORSHQdEA2cCeoYZDsepApIEHI6C0zdF/GgMSjMpDs0khUFCRoJFjBLjXMO1jy1/DNRYPybG6jzHr/B25Xmdbz0+sTtejqt9mG+18FkMCnBrFmnhmGhMMOVSQ4gjqZL2cK3ZaXfKhecNb2o00XTtD9eXvvihpFkCwlBOtO57ndQMLFGGUQ75ip9pSAk9JSPoO1OQBPTAlhrkeDObnux+wuDSezXDkkTrSRI4MiEm1rOxwvm/WD8z0YuBZSLNDAhaHRRlHBuJC0FxyJTTgk+cQahirlZMY6IINU722ilBUuvBhrootd6XpoRDOLAx8DEUMQUCzqhMEiLCJ35EEsYnIUQk4+7SdDQ1XVmb+J1wVbAImxhwMROVErJyXCiLmS5KDwBrIjTWoFhUq+DQG9gisWi2LhIfSddfnFzjZjP8p+pmVnxXo5vOcmcTFu6ERJ3ujBSZ5Lb677SfPnPcZaO2nK+870rxA8nD4uIkt03PDehmESru9ipOAl3S503v3CHY7SV3F4bHhGdQZ4WbzurTjvZLvHDNQ4dXqKHt5hck7s6zezV275Ldq7MsrUAOkfE5ESMOLkWxUWx8VW6LTCYEKJwqGWbU1D9wkNck8YuBDQJ7kOczCjJxUB3FxDUJJ6Wg9SwpQOcVFGrrwu4Ze7OPdt7oddsv2977bnN3Y/qel9EjtIG2kYeeo130Fu2jHqLoJ/qN/qC/jc+Nr41vje8VurgwzXmIaqvx4x/PcLs1</latexit><latexit sha1_base64="8EmOcZJyO5/MfVA5lqvbgxrqaVs=">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</latexit>

Return s =0 s

><latexit sha1_base64="feXxLtIV6U2Z1es1oPuSHn147To=">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</latexit><latexit sha1_base64="feXxLtIV6U2Z1es1oPuSHn147To=">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</latexit><latexit sha1_base64="feXxLtIV6U2Z1es1oPuSHn147To=">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</latexit><latexit sha1_base64="feXxLtIV6U2Z1es1oPuSHn147To=">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</latexit>

OutputThe 2D points

sx

sy

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Constrained E-MDS• Our contribution: enforce additional constraints on

• Constraints: (1) is PSD and of rank 2

(2) the entries of satisfy the triangle equality

• Solution: lift-and-project, alternative projections onto the sets satisfying constraints (1) and (2)

Edge-multidimensional scaling (E-MDS) [3]• Edge kernel matrix

• Solution: recover via eigenvalue decomposition of

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• We proposed two algorithms for multimodal sensor localization:• Constrained Edge-Kernel improves on the existing Edge-Kernel

method by adding geometric constraints on triplets of points.• Coordinate Difference Matrices allow us to estimate the sensors’

coordinates independently for each dimension.• Numerical simulations demonstrate that both proposed methods significantly

outperform existing distance-based and multimodal localization algorithms.

State-of-the-art• Distance-only: MDS [1]

• Distance + angles: E-MDS [3]

Experiment• 6 points chosen uniformly in the unit square

• Compute RMSE on reconstructed points

RMSE vs. angular noise• RMSE vs. angular noise for two representative distance noise levels

RMSE vs. distance noise• RMSE vs. distance noise for two representative angle noise levels

Compute W 0, S0

and 0by removing the first row and column W , S and .

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Decompose the edge vectors V into 1D coordinate differences sx

and sy

.

<latexit sha1_base64="UTwQwtgs9XqVuR/F7imQcRiANfY=">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</latexit><latexit sha1_base64="UTwQwtgs9XqVuR/F7imQcRiANfY=">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</latexit><latexit sha1_base64="UTwQwtgs9XqVuR/F7imQcRiANfY=">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</latexit><latexit sha1_base64="UTwQwtgs9XqVuR/F7imQcRiANfY=">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</latexit>

Decompose the edge vectors V into 1D coordinate differences sx

and sy

.

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