On an array localization technique with Euclidean distance geometry Simon Bouley, Charles Vanwynsberghe, Thibaut Le Magueresse, Jérôme Antoni

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Simon Bouley, Charles Vanwynsberghe, Thibaut Le Magueresse, Jérôme Antoni. On an array local- ization technique with Euclidean distance geometry. Euronoise 2018, May 2018, Hersonissos, Greece. ￿hal-01820078￿

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Simon Bouley Univ Lyon, INSA-Lyon, Laboratoire Vibrations Acoustique, F-69621 Villeurbanne, France Charles Vanwynsberghe Univ Lyon, INSA-Lyon, Laboratoire Vibrations Acoustique, F-69621 Villeurbanne, France Thibaut Le Magueresse MicrodB, 28 Chemin du Petit Bois, Écully, France Jérôme Antoni Univ Lyon, INSA-Lyon, Laboratoire Vibrations Acoustique, F-69621 Villeurbanne, France

Summary The localization of sources by an acoustic array of microphones depends to a great extent on an accurate knowledge of the antenna position in its environment. From the geometric data of the array and the object of study, the present work details a methodology to determine the location of the microphones in relation to the object and reproduce the experimental conguration. Reference sources are placed on the object in order to measure the times of ight (ToF) and distances between them and the microphones, connecting the array and the object together. The overall geometric conguration is thus dened by an Euclidean Distance Matrix (EDM), which is basically the matrix of squared distances between points. First, Multidimensional Scaling (MDS) technique is used to reconstruct the point set from distances. Second, this point set is then aligned with the sources whose coordinates are known. This orthogonal Procustes problem is solved by the Kabsch algorithm to obtain the rotation and translation matrices between the coordinate system of the array and the object of study. In addition, a low rank property of Euclidean distance matrices is exploited to evaluate in situ the speed of sound. The main theoretical and algorithmic elements are exposed and a numerical simulation of a geometric conguration, representative of a typical experimental set-up, is carried out. The robustness of the method is nally discussed.

PACS no. 43.60.Fg

1. Introduction

The localization and quantication of acoustic sources 0.5 radiated by a device depend on numerous physical pa- rameters as well as the choice of measurement array 0 of microphones or the back-propagating method. As highlighted recently by Gilquin et al. [1] by means of sentitivity analysis, deviations of the antenna position -0.5 0.4 and orientation in its experimental environment in- 0.2 0.4 0 0.2 0 uence greatly the sound source reconstruction, both -0.2 -0.2 with a classical beamforming technique or a Bayesian -0.4 -0.4 formulation. The antenna positioning problem is illustrated in Figure 1. Position of microphones (red dots) in the array Figs. (1) and (2) : one array of microphones is fac- coordinate system. ing towards a device (here a basic mock-up of an

engine) to obtain a sound source map. The posi- (c) European Acoustics Association tion and orientation of the antenna and the device in their own coordinate system are assumed to be com- Figure 2. Position of the antenna (red dots) in the de- Figure 3. Ground truth of the antenna positioning prob- vice coordinate system (ground truth). The array of mi- lem. Microphones are represented by red dots while blue crophones faces towards a radiating surface to reconstruct. dots depict acoustic sources.

n×n are the squared distances between points x and pletely known. A mesh of the surface of interest is R i achieved to retro-propagate the acoustic eld mea- xj : sured by the array on the device. Several methodolo- gies have been proposed to determine the true posi- 2 dij = kxi − xjk , (1) tion of each sensor in a network, especially through 2 microphone position self-calibration [2]. The aim of where k · k2 is the Euclidean norm. An EDM fullls this study is rather to determine the antenna posi- the following properties : tion in relation to the device during an experimen- tal campaign and collect each microphone coordinates • Non-negativity (dij ≥ 0, i 6= j) in the mesh coordinate system to perform the back- • Hollow matrix (dij = 0 ⇔ i = j) propagation. This application-oriented paper gathers • Symmetry : dij = dji dierent techniques based on Euclidean distance ge- Furthermore, as shown by Gower [3], the rank of ometry to develop a practical tool. The theoretical an EDM D related to the set of points X satises the and algorithmic elements of these methods are de- inegality : tailed and illustrated all along the study with the same numerical simulation, representative of a typi- rank(D) ≥ d + 2. (2) cal experimental set-up. Section (2) rst details how to dene the overall geometric conguration with the Euclidean distance matrix (EDM) and complete it The Euclidean distance matrix is also invariant un- with the help of a set of acoustic sources placed on the der orthogonal and rigid transformations (such as ro- device. The point set is then reconstructed from the tation, reection and translation). As a consequence, distances with a multidimensional scaling technique, the absolute position and orientation of a point set presented in Section (3). The set is then aligned with cannot be reconstructed from the associated EDM. the collection of sources, dened as anchors to pro- Each result is then a rigid transformation of another vide the rigid transformation between the coordinate one. For more details, Parhizkar [4] provides a com- system of the array and the device, with the help of plete description of the EDM algebra. the Kabsch algorithm (Section (4)). Finally, Section (5) proposes a low-rank based criterion to experimen- 2.2. Constructing the EDM tally evaluate the speed of sound, whose knowledge is Figure (3) illustrates the ground truth of the antenna needed in the proposed method. positioning problem, as met in an experimental cam- paign. The coordinates of each microphone (red dots) 2. Euclidean Distance Matrix in the array system are known, as the distances be- tween them. Acoustic sources (blue dots) are placed at some prominent locations of the device. They are 2.1. Properties gathered in a group of four by a structural support Consider a d-dimensional Euclidean space, where n (Fig. (4)), which is moved to each location. This sup- points are set and described by the columns of the port allows increasing the number of sources (and matrix X d×n X d. hence the EDM size) while limiting the number of ∈ R , = [x1, x2, ··· , xn], xi ∈ R The terms dij of an Euclidean distance matrix D ∈ measurement points. The position of these reference 0.1 5

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15 0.06

20 0.04 25 0.02 30 0 35 0.04 0 40 0.02 -0.02 -0.04 0 45

5 10 15 20 25 30 35 40 45 Figure 4. Structural support of the four acoustic sources (blue dots). At the origin is a fulcrum (small black dot), Figure 5. Incomplete Euclidean distance matrix. The diag- in contact with a prominent location of the device. onal block submatrices represent distances between micro- phones (top-left block) and sources (bottom-right block), while distances between microphones and sources are un- points and the acoustic sources are known in the de- known (blank o-diagonal submatrices). i and j denote vice coordinate system. the index of a particular point in the set-up. At the initializing step of the method, some subma- trices of the EDM are already known (Fig. (5)) : the diagonal block submatrices related to microphones (top-left block) and to sources (bottom-right block). The size of each block is the number of microphones 5 or sources, respectively. The distances between microphones and sources, represented by the blank 10 o-diagonal submatrices are unknown. 15

20 The determination of the distance between a source 25 and microphone is based on time of ight (ToF) mea- surements. This time of propagation between the two 30 points is obtained by cross-correlating the microphone 35 signal with a sound emitted by the synchronized refer- 40 ence source. The benet of the synchronization is the possibility to select the rst peak of cross-correlation 45 related to the straight path, avoiding reection issues 5 10 15 20 25 30 35 40 45 due to the proximity between the array and the de- vice. An evaluation of the speed of sound is nally needed to calculate the distance from the time of Figure 6. Completed Euclidean distance matrix. ight. The sound of reference sources is chosen ac- cording to the quality of the cross-correlation mea- surements. Each source can simultaneously emits un- exceed 3.5 mm. Finally, Fig. (6) illustrates the com- correlated white noise or modulated sweep sines. In pleted EDM. the numerical simulation, the same linear chirp is se- quentially emitted by each source. The frequency lin- 3. Multidimensional Scaling early increases from 0 to 10 kHz and the sampling frequency is 50 kHz. The speed of sound is set at 343 Multidimensional scaling (MDS) refers to a col- m/s. As the peak detection depends on the length lection of techniques for the analysis of similarity of the time sample, the sampling frequency is an es- or dissimilarity in a dataset. Initially developped sential parameter. At a xed sampling frequency fs, in psychometrics [5], MDS allows modeling a wide the maximum error on distance ∆d induced by the range of data as distances and visualizing them as sampling is ∆d = c0/fs. At fs = 50 kHz, ∆d is the- points in a geometric space. The algorithm 1 presents oretically smaller than 7 mm. In the simulation, the the classical MDS, also known as Torgerson-Gower reconstruction error on the distance between sources scaling [6], which nds a coordinate matrix Xb starting and microphones due to this sampling issue does not from an EDM D and the embedded dimension d. set cannot be derived from the multidimensional scal- ing. A secondary step is needed to nd the optimal ro- tation/reection and translation matrices which align the point set in the reference frame of the device.

This is performed with a selection of ns points, de- noted as anchors, whose positions Xs in this particu- 0.2 lar coordinate system are known. This step is usually called orthogonal Procustes analysis [9]. One solution, 0.1 stemmed from crystallography, is the Kabsch algo- rithm [10, 11], which computes the optimal rotation 0 matrix R between two sets of points by minimizing -0.1 the root mean square deviation (least RMSD, Eq.(3)).

-0.2 s -0.2 1 X 2 -0.1 lRMSD = argmin |R x − x | . (3) -0.2 bs,i s,i -0.1 0 R∈ d×d ns 0 0.1 R i∈ns 0.1 0.2 First of all, the two set of anchors in both reference

frames (MDS, Xb s and device, Xs) must be translated Figure 7. Multidimensional scaling of the numerical set- to align their centroid with the origin of the coordi- up. The blue and red dots represent the sources and the nate system. The Kabsch algorithm is then based on microphones, respectively. a singular-value decomposition (SVD) of the cross- covariance matrix X XT : b s s The mathematical developments are detailed by U S VH X XT (4) Dokmani¢ et al. [7] and Borg and Groenen [6]. d c = b s s

where (•)H denotes the Hermitian transpose opera- tor and d•c a diagonal matrix. The optimal rotation Algorithm 1 ClassicalMDS(D, d), [7] matrix reads : Require:D , d   1: J(n) = I − 1 11T . Geometric centering matrix 1 0 0 n R V UH (5) 2: G = −(1/2) JDJ . Gram matrix =  0 1 0  , dc ∈ {−1, 1}. 3: U, Λ ← EVD(G) . Eigendecomposition 0 0 dc √ √ T 4: return Xb = [diag( λ1, ··· λd, 0d×(n−d)]U If dc = 1, the two matrices Xb sR + T and Xs are identical whereas one of the matrices is the reection

of the other if dc = −1. All those steps are summa- The method is based on an eigendecomposition rized in Algorithm 2. of the Gram matrix (G = XT X, where (•)T de- notes the transpose operator). According to Gower [8], this matrix can be computed with a double Algorithm 2 Kabsch(Xb s, Xs, d, ns) centering, using the geometric centering matrix J, Require: X , X , , where 1 is a column vector lled with ones. The b s s d ns 1: X = J(n )X . align centroid with origin eigenvalues λ are sorted in order of decreasing am- b s s b s i 2: X = J(n )X . align centroid with origin plitude and only the d rst values are selected. Thus, s s s 3: C = X XT . Cross-covariance matrix the point set Xb is embedded in a d-dimensional space. b s s 4: UdScVH = C . SVD return R VIUH Rotation matrix Figure (7) shows a multidimensional scaling of the 5: = . return T X RX Translation vector numerical set-up. As it can be seen, the position and 6: = s,c − b s,c . return lRMSD least root mean square orientation of the overall geometric conguration are 7: . deviation completely arbitrary, and more, the MDS result is ac- tually a reection of the true set-up. The Kabsch algorithm is then applied to the 4. Orthogonal Procustes Analysis numerical set-up. More specically, this method is carried on to nd the optimal rotation matrix As explained in Section (2.1), the EDM is invariant between the position of sources obtained by the MDS under orthogonal and rigid transformations. There- and their exact position in the device coordinate sys- fore, the absolute orientation and position of the point tem. Figure (8) shows the nal result of the complete 10-3 5

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Figure 8. Final result of the positioning algorithm. The Figure 9. Source positioning error ∆s according the source black dots, the blue and red circles represent the ground index. truth position and the estimated position of the sources and microphones, respectively.

10-3 methodology detailed in this paper. It roughly shows 5 that the array is aligned with the ground truth (black 4.5 dots). The red and blue dots represent the estimated 4 positions of microphones and sources, respectively. If 3.5 required, the algorithm can be performed a second 3 time, taking the position of the microphones in the array and device coordinate systems as inputs. This 2.5 new step provides directly the rigid transformation 2 between the coordinate system of the array and the 1.5 device. 1

0.5 A positioning error is calculated, based on the ∆ 0 distances between the true positions of microphones 0 5 10 15 20 25 30 35 and sources (ground truth, X), and those estimated by the positioning method (Xb ). For a microphone of the array (a) or a source (s), the residue reads : Figure 10. Microphone positioning error ∆a according the microphone index.

X X 2 (6) ∆a,s = k b a,s − a,sk2. sound. Thus, the latter is involved in the o-diagonal As shown by Figs (9) and (10), the maximum errors submatrices of the EDM. The low-rank property im- of reconstruction are 1.46 and 3.12 mm for the sources plies the rst d + 2 singular values si of the Euclidean and microphones, respectively. Those errors come to a distance matrix overwhelm the others. The criterion (Eq.(7)) estimates the information brought by the great extent from the sampling issue detailed in Sec- Λc0 tion (2.2). smaller singular values in the total amount of energy contained in the matrix. Minimizing with respect Λc0 to c0 ensures the Euclidean nature of the geometry, 5. In Situ Evaluation of Sound Speed avoiding any curvature eect. As shown by Eq.(2), the rank of an EDM is at least P s equal to d + 2, where d is the spatial dimension i>d+2 i S (7) Λc0 = P , si , (i, i). of the geometric conguration. This section intro- si i∈N duces a simple criterion built on this specic prop- erty to estimate the experimental value of the speed Figure (11) illustrates the estimation of the speed of sound. As seen in Section (2.2), evaluating the of sound. c0 is set at 343 m/s in the simulation source-microphone distances requires both an accu- and the sampling frequency of the propagating sig- rate estimation of the times of ight and the speed of nals is 50 kHz. A range of possible values is swept truth, with a maximum positioning error around 3 mm. Finally, a low-rank property of the EDM is exploited to evaluate the experimental speed of sound. 0.024 0.023 This method allows one to easily get the position of 0.022 an array in regards to the object of study during an 0.021 experimental campaign. Its advantage is that it can 0.02 benet from extensions provided by the EDM litera- 0.019 ture to tackle many issues faced in domains of source

0.018 localization and acoustic array processing.

0.017 Acknowledgement 0.016 This study has been produced in the framework of 0.015 LUG2 supported by Région Auvergne Rhône-Alpes 325 330 335 340 345 350 355 360 365 and BPIFrance (FUI22). It was performed within the framework of the Labex CeLyA of Université de Lyon, operated by the French National Research Agency Figure 11. Evaluation of the experimental speed of sound (ANR-10-LABX-0060/ANR-11-IDEX-0007). . cb0 References (c ∈ [325, 365] m/s) and the criterion Λ reaches a 0 c0 [1] Laurent Gilquin, Simon Bouley, Jérôme Antoni, Clé- unique and well-dened minimum. The optimal speed ment Marteau, and Thibaut Le Magueresse. Sensi- of sound is then equal to m/s. cb0 343.017 tivity analysis of acoustic inverse problems. In Pro- ceedings of NOVEM 2018, Ibiza, Spain, 2018. [2] Axel Plinge, Florian Jacob, Reinhold Haeb-Umbach, 6. Conclusions and Gernot A. Fink. Acoustic microphone geometry calibration: An overview and experimental evaluation An array localization technique has been detailed of state-of-the-art algorithms. IEEE Signal Process- in this paper. It gathers a collection of methods ing Magazine, 33(4):1429, 2016. based on Euclidean distance geometry. The main [3] John C. Gower. Properties of euclidean and non- theoretical element is the Euclidean distance matrix, euclidean distance matrices. Linear Algebra and its which reports on the geometric conguration of a Applications, 67, 1985. set-up. Acoustic sources, placed on several locations [4] Reza Parhizkar. Euclidean distance matrices: Proper- of the device, act as anchors to connect the antenna ties, algorithms and applications. PhD thesis, École Polytechnique Fédérale de Lausanne (EPFL), 2013. of microphones to it. As the EDM deals with pairwise distances between points, these sources have to be [5] Warren S. Torgerson. Multidimensional scaling: I. theory and method. Psychometrika, 17(4):401419, visible from the microphones. In addition, as the 1952. EDM is invariant under orthogonal transformation, [6] Ingwer Borg and Patrick J. Groenen. Modern multidi- especially reection, the anchors positions must form mensional scaling: Theory and applications. Springer- an asymmetrical conguration in a three-dimensional Verlag New York, 2 edition, 2005. space. Once the anchors are set and their positions [7] Ivan Dokmani¢, Reza Parhizkar, Juri Ranieri, and known in the device coordinate system, the source- Martin Vetterli. Euclidean distance matrices: essen- microphone distances are calculated to complete the tial theory, algorithms, and applications. IEEE Signal EDM. The time of propagation between the two Processing Magazine, 32(6):1230, 2015. points is obtained by cross-correlating the micro- [8] John C. Gower. Euclidean distance geometry. Math. phone signal with a calibration sound emitted by Sci, 7(1):114, 1982. the synchronized source. The sampling of the signal [9] John C. Gower. Procrustes methods. Wiley involves a limit on the resolution of the rst peak, Interdisciplinary Reviews: Computational Statistics, representing the straight path between the source and 2(4):503508, 2010. the microphone. The sampling frequency is therefore [10] Wolfgang Kabsch. A solution for the best rotation an inuent parameter on the distance estimation to relate two sets of vectors. Acta Crystallographica Section A: Crystal Physics, Diraction, Theoretical and the global reconstruction error. Once the EDM and General Crystallography, 32(5):922923, 1976. is complete, the positioning problem is solved using [11] Wolfgang Kabsch. A discussion of the solution for the a multidimensional scaling technique and a Kabsch best rotation to relate two sets of vectors. Acta Crys- algorithm to nd the optimal rotation and translation tallographica Section A: Crystal Physics, Diraction, matrices to align the array with its ground truth Theoretical and General Crystallography, 34(5):827 position. The results show a good agreement between 828, 1978. the estimated position of the array and the ground