Sound source localization with data and model uncertainties using the EM and Evidential EM algorithms

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Sound source localization with data and model uncertainties using the EM and Evidential EM

algorithms

Xun Wang

To cite this version:

Xun Wang. Sound source localization with data and model uncertainties using the EM and Evi-

dential EM algorithms. Other. Université de Technologie de Compiègne, 2014. English. �NNT :

2014COMP2164�. �tel-01208211�

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Par Xun WANG

Thèse présentée

pour l’obtention du grade de Docteur de l’UTC

Sound source localization with data and model uncertainties using the EM and Evidential EM algorithms

Soutenue le 09 décembre 2014

Spécialité : Information Technologies and Systems

D2164

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Université de Technologie de Compiègne Thesis

Submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Area of specialization: Information Technologies and Systems by

Xun WANG

Sound Source Localization with Data and Model Uncertainties Using the EM and

Evidential EM Algorithms

Heudiasyc Laboratory, UMR CNRS 7253 Roberval Laboratory, UMR CNRS 7337 Defended on December 9th, 2014.

Thesis Committee:

President: Thierry Denœux Université de Technologie de Compiègne Reviewers: Alain Appriou ONERA

Jean-Marc Girault Université François Rabelais de Tours Examiners: Charles Pézerat Université du Maine

Olivier Strauss Université Montpellier II Supervisors: Jérôme Antoni Université de Lyon

Benjamin Quost Université de Technologie de Compiègne

Jean-Daniel Chazot Université de Technologie de Compiègne

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Abstract

This work addresses the problem of multiple sound source localization for both deterministic and random signals measured by an array of microphones. The problem is solved in a statistical framework via maximum likelihood. The pressure measured by a microphone is interpreted as a mixture of latent signals emitted by the sources; then, both the sound source locations and strengths can be estimated using an expectation-maximization (EM) algorithm. In this thesis, two kinds of uncertainties are also considered: on the microphone locations and on the wavenumber. These uncertainties are transposed to the data in the belief functions framework. Then, the source locations and strengths can be estimated using a variant of the EM algorithm, known as Evidential EM (E2M) algorithm.

The f rst part of this work begins with the deterministic signal model without consider- ation of uncertainty. The EM algorithm is then used to estimate the source locations and strengths: the update equations for the model parameters are provided. Furthermore, experimental results are presented and compared with the beamforming and the statistically optimized near-f eld holography (SONAH), which demonstrate the advantage of the EM algorithm.

The second part raises the issue of model uncertainty and shows how the uncertainties on microphone locations and wavenumber can be taken into account at the data level. In this case, the notion of the likelihood is extended to the uncertain data. Then, the E2M algorithm is used to solve the sound source estimation problem. In both the simulation and real experiment, the E2M algorithm proves to be more robust in the presence of model and data uncertainty.

The third part of this work considers the case of random signals, in which the amplitude is modeled by a Gaussian random variable. Both the certain and uncertain cases are investigated. In the former case, the EM algorithm is employed to estimate the sound sources. In the latter case, microphone location and wavenumber uncertainties are quantif ed similarly to the second part of the thesis. Finally, the source locations and the variance of the random amplitudes are estimated using the E2M algorithm.

Key words: sound source localization, holography, beamforming, acoustic imaging,

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ii

statistical inference from imprecise data, belief functions, Evidential EM algorithm.

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Résumé

Ce travail de thèse se penche sur le problème de la localisation de sources acoustiques à partir de signaux déterministes et aléatoires mesurés par un réseau de microphones. Le problème est résolu dans un cadre statistique, par estimation via la méthode du maximum de vraisemblance. La pression mesurée par un microphone est interprétée comme étant un mélange de signaux latents émis par les sources. Les positions et les amplitudes des sources acoustiques sont estimées en utilisant l’algorithme espérance-maximisation (EM).

Dans cette thèse, deux types d’incertitude sont également pris en compte: les positions des microphones et le nombre d’onde sont supposés mal connus. Ces incertitudes sont trans- posées aux données dans le cadre théorique des fonctions de croyance. Ensuite, les positions et les amplitudes des sources acoustiques peuvent être estimées en utilisant l’algorithme E2M, qui est une variante de l’algorithme EM pour les données incertaines.

La première partie des travaux considère le modèle de signal déterministe sans prise en compte de l’incertitude. L’algorithme EM est utilisé pour estimer les positions et les amplitudes des sources. En outre, les résultats expérimentaux sont présentés et comparés avec le beamforming et la holographie optimisée statistiquement en champ proche (SONAH), ce qui démontre l’avantage de l’algorithme EM.

La deuxième partie considère le problème de l’incertitude du modèle et montre com- ment les incertitudes sur les positions des microphones et le nombre d’onde peuvent être quantif ées sur les données. Dans ce cas, la fonction de vraisemblance est étendue aux don- nées incertaines. Ensuite, l’algorithme E2M est utilisé pour estimer les sources acoustiques.

Finalement, les expériences réalisées sur les données réelles et simulées montrent que les algorithmes EM et E2M donnent des résultats similaires lorsque les données sont certaines, mais que ce dernier est plus robuste en présence d’incertitudes sur les paramètres du modèle.

La troisième partie des travaux présente le cas de signaux aléatoires, dont l’amplitude

est considérée comme une variable aléatoire gaussienne. Dans le modèle sans incertitude,

l’algorithme EM est utilisé pour estimer les sources acoustiques. Dans le modèle incertain,

les incertitudes sur les positions des microphones et le nombre d’onde sont transposées aux

données comme dans la deuxième partie. Enf n, les positions et les variances des amplitudes

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iv

aléatoires des sources acoustiques sont estimées en utilisant l’algorithme E2M. Les résultats montrent ici encore l’avantage d’utiliser un modèle statistique pour estimer les sources en présence, et l’intérêt de prendre en compte l’incertitude sur les paramètres du modèle.

Mots clés: localisation de sources acoustiques, holographie, beamforming, imagerie

acoustique, inférence statistique à partir de données imprécises, fonctions de croyance, al-

gorithme E2M.

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Acknowledgements

Foremost, I would like to express my sincere gratitude to my supervisors Prof. Jérôme Antoni, Dr. Benjamin Quost and Dr. Jean-Daniel Chazot. I would never have been able to f nish my dissertation without their patient guidance, advice, encouragement and careful modif cation of the manuscript they have provided throughout my Ph.D. study and research.

I would like to thank other members of staff at the laboratories Heudiasyc and Roberval, for the assistance they provided at all levels of the research work. In particular, I would like to thank the technicians at Roberval Jean-Marc Gherbezza and Félix Foucart for their helps during the experiments.

I would also like to acknowledge the f nancial support by Collegium UTC/CNRS for providing the funding which allows me to undertake this research and to attend the confer- ences.

Finally, I would like to thank my family. They were always supporting me and encour-

aging me with their best wishes.

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List of Figures

2.1 Spherical waves emitted by a sound source r. . . . 6 3.1 Sound source estimation problem (2D case). The microphone locations are

represented by red crosses, the sound sources by blue crosses. The origin is assumed here to be the center of the array of microphones. . . 16 3.2 Experimental setup and microphone distribution. . . 20 3.3 Comparison between EM, beamforming and SONAH at f=525Hz. Black

crosses represent the loudspeaker center. Subf gure (a): Estimated source position. Subf gures (b-f): Estimated pressure f eld on the source plane. . . 23 3.4 Comparison between EM, beamforming and SONAH at f=1325Hz. Black

crosses represent the loudspeaker center. Subf gure (a): Estimated source position. Subf gures (b-f): Estimated pressure f eld on the source plane. . . 24 3.5 Comparison between EM, beamforming and SONAH at f=5025Hz. Black

crosses represent the loudspeaker center. Subf gure (a): Estimated source position. Subf gures (b-f): Estimated pressure f eld on the source plane. . . 25 3.6 Sound f eld on the microphone plane using (a) measurement (b) EM esti-

mates, S = 2 (c) EM estimates, S = 6 (d) EM estimates, S = 10. f=525Hz. . 26 3.7 Sound f eld on the microphone plane using (a) measurement (b) EM esti-

mates, S = 2 (c) EM estimates, S = 6 (d) EM estimates, S = 10. f=1325Hz. 27 3.8 Sound f eld on the microphone plane using (a) measurement (b) EM esti-

mates, S = 2 (c) EM estimates, S = 6 (d) EM estimates, S = 10. f=5025Hz. 28 5.1 Sound source estimation problem (2D case) with uncertain meta-parameters

(microphone locations and wavenumber). . . 53

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xvi List of Abbreviations and Symbols FIM Fisher Information Matrix

LS least square

ML maximum likelihood

MLE maximum likelihood estimation (or estimate, estimator) NAH near-f eld acoustical holography

PDF probability density function

SONAH statistically optimized near-f eld acoustical holography

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Chapter 1 Introduction

One of the crucial issues in acoustics engineering is to reduce the noise emitted by complex devices. In the automobile industry, for example, due to the requirements expressed by the users (in terms of acoustic comfort) as well as the legislation set by the European community to limit noise pollution, acoustic quality is now accepted as a decisive criterion to judge the performance of a vehicle, as well as its safeness or its fuel consumption. The f rst step of this task is to understand the acoustic behavior of the device in order to focus the efforts on the main sources and the most annoying frequency bands. However, the noise sources emitted by a machine are multiple and structurally complex. Thus, the need for tools allowing visualization, quantif cation and identif cation of noise sources has become signif cant in acoustics engineering.

In this thesis, we investigate the problem of multiple sound source localization. The sound signals are subdivided into two classes: deterministic and random signals. The former assumes that the amplitude is a constant to be estimated, while in the latter case the amplitude is modeled as a random variable. The task of this work is to estimate the sound source locations and amplitudes for deterministic signals, or the sound source locations and variances of amplitudes for random signals, using sound pressures measured by an array of microphones.

Part I of this thesis studies the sound source localization problem for deterministic signals in a maximum likelihood framework. The sound pressure measured by a microphone is interpreted as a mixture of signals emitted by the sources, and the contributions of this measurement from various sources are assumed as latent variables. In this case, the EM algorithm is used to solve the estimation problem.

However, contrary to the ideal experimental setting, different kinds of uncertainties may

pervade the sound propagation and measurement process. For example, sometimes the mi-

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2 Introduction crophone array has to be placed on a vibrating object so that the microphone locations are never precisely measured. Moreover, the sound propagation medium can be uncertain as well. For instance, the sound speed, thus the wavenumber, may vary signif cantly due to changes of temperature. Therefore, in Part II, we take into account these two kinds of uncertainties in the estimation process. The uncertainties are f rst transposed to the data, via f rst-order approximations. The resulting uncertain pressures are then represented using contour functions, which are mathematical tools proposed within the framework of belief functions. Then, parameter estimation can be carried out using the Evidential EM (E2M) algorithm, which is a variant of the EM algorithm to perform maximum likelihood estimation from uncertain data.

Finally, in Part III, the source localization problem for random signals is investigated.

First, under the assumption of certain measurements, the source locations and variances of random amplitudes are estimated using the EM algorithm, as in Part I. Then, the uncertainties of the microphone locations and the wavenumber are considered. The parameters are estimated using the E2M algorithm as in Part II.

The simulated and real experiments are presented for both deterministic and random

signals. In the certain case, the EM algorithm is compared with beamforming and SONAH,

which are popular methods for identifying the sound sources. The results show that the EM

algorithm can clearly localize the sources while beamforming and SONAH are restricted to

the studied frequency. Moreover, when the data are pervaded with uncertainties, the EM

and E2M algorithm are compared to each other. The experimental results show that E2M

algorithm performs better than the EM in the sense of estimation error.

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Part I

Sound Sources Localization From

Certain Measurements

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14 State of Art considered, but the propagation operator was taken without attenuation, i.e., the propagation operator from m-th microphone to s-th source was assumed to be G(r

_s

| r

^′_m

) = e

^jk|r^s^−r^′^m^|

.

Some researchers have investigated different kinds of superimposed signals using the EM algorithm. Cirpan and Cekli [11, 12] and Kabaoglu et al. [42, 43] used a similar EM approach to study the localization of near-f eld sources. Each of these four contributions addresses a specif c case, namely: deterministic signals in 2D space and 3D space, and Gaussian random signals in 2D and 3D space. However, note that the near-f eld was just taken into account with the Fresnel approximation of the source-microphone distance in the time delay (without the 1 /r spatial attenuation in Eq. (2.1)), and that even in the deterministic case [11, 42] the signal strength depends on the snapshots (unlike in Eq. (2.1), where the strength A is a constant parameter). Lu et al. [46] and Yan et al. [73] studied the source localization of wideband signals, in which the basic assumption of the propagation operator is more or less same as in [11, 12, 42, 43]. Moreover, [46] solved the problem with a diagonal but non-identity noise covariance matrix, and [73] analyzed the computational complexity of the algorithm. Sheng and Hu [61] and Meng and Xiao [47] investigated the multiple source localization problem via acoustic energy measurement using the EM algorithm. In these two papers, only the strength and the attenuation of the sound (A and 1/r) are considered, but without regard to the phase (the term e

^jkr

in Eq. (2.1)). Frenkel and Feder [31] applied the EM algorithm to the multiple target tracking problem. At each time interval, the sonar sends a signal that is ref ected from the moving targets and received by the sensors. The purpose of the algorithm is to estimate the locations and velocities of the multiple moving targets.

In the next chapter, we propose a solution of the multiple sound sources localization

problem presented in Section 1.1, using the EM algorithm. The methodology is very similar

to that described in Ref. [28], but the assumptions underlying our model differ from all the

works mentioned above.

(35)

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Sound source localization with data and model uncertainties using the EM and Evidential EM algorithms

HAL Id: tel-01208211

https://tel.archives-ouvertes.fr/tel-01208211

Submitted on 2 Oct 2015

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