` THESE Pour obtenir le grade de ´ DOCTEUR DE LA COMMUNAUTE ´ UNIVERSITE GRENOBLE ALPES Sp´ecialit´e : SIGNAL IMAGE PAROLE TELECOMS Arrˆet´e minist´eriel : 25 mai 2016 Pr´esent´ee par Francesca Elisa Diletta RAIMONDI Th`ese dirig´ee par Pierre COMON, CNRS pr´epar´ee au sein du Laboratoire Grenoble Images Parole Signal Automatique dans l’E´cole Doctorale E´lectronique, E´lectrotechnique, Automatique, Traitement du Signal (EEATS) Traitement d’antenne tensoriel Th`ese soutenue publiquement le 22 septembre 2017, devant le jury compos´e de: M. Olivier J. Michel Professeur des universit´es, GIPSA-lab, Pr´esident du jury M. Pascal Chevalier ´ Professeur du CNAM, Chaire d’Electronique (Conservatoire National des Arts et M´etiers), Rapporteur M. Martin Haardt Professeur des universit´es, Ilmenau University of Technology, Rapporteur M. David Brie Professeur des universit´es, CRAN (Centre de Recherche en Automatique de Nancy), Examinateur M. Pierre COMON Directeur de recherche, GIPSA-lab, CNRS, Directeur de these Tensor Array Processing Francesca Elisa Diletta RAIMONDI A thesis submitted for the degree of Doctor of Philosophy Signal Processing in Supervised by Pierre Comon GRENOBLE ALPS UNIVERSITY GIPSA-lab, CNRS September 22, 2017 Thesis Committee: Chair : Olivier J. MICHEL - GIPSA-lab Reviewers : Pascal CHEVALIER - CNAM Martin HAARDT - Ilmenau University of Technology Examiner : David BRIE - CRAN Thesis advisor : Pierre COMON - GIPSA-lab Acknowledgments A PhD is the outcome of several contributions (beside the endeavour and the efforts of the bewildered student), and I like to interpret it as a Bildungsroman. I am now three years older, hopefully wiser, and a grown up researcher. First, a PhD work, as solitary as it may seem, is founded on a learning process alongside a guide who introduces the student into the world of scientific discovery. This guide, the supervisor, is then a mentor into a new world, science, with its own language, its own codes, its own culture. I can say that I owe that to a wonderful researcher, Pierre Comon: he taught me how to tell a good quality article from a bad one, how to communicate my ideas in a precise and synthetic fashion, and how to understand the requirements of the scientific community. I learned from him that every result should be justified and easily reproducible by other researchers. Whenever I needed his advice, he was there to provide me with a very thoughtful answer. I thank all the invited members of my thesis committee: Martin Haardt who really helped me improving the quality of my work through his meticulous reading of this manuscript; Pascal Chevalier for his attentive reading and inspiring suggestions; David Brie and Olivier Michel for their interesting interpretation of my work. A PhD work also benefits from the exchanges with the natural environment where it is conducted: and for me this was GIPSA-lab in Grenoble, in the department of signal and image processing (DIS). Several were the colleagues with whom I exchanged ideas during these three years. I am thankful for the interesting talks I had the pleasures to have with Olivier Michel, whose insight and intuition are of rare and great value. I also really appreciated being part of a greater project, the ERC DECODA: this allowed me to have very meaningful conversations, in particular with Rodrigo Cabral Farias, who is also a very good friend, Konstantin Usevich, Chengfang Ren, and Souleyman Sahnoun. I also thank my MSc mentor, Umberto Spagnolini, with whom I collaborated for the last chapter of this thesis. I learned a lot from him during the last few years, via his intuitions and pragmatism. I also thank Barbara Nicolas, who was my supervisor during my MSc project. It was also very helpful to be part of the CICS team with weekly meetings and pre- sentations around “complex systems”. I enjoyed then the very positive and interactive atmosphere thanks to my fellow colleagues, with whom it was a pleasure to share this experience. In particular, I want to thank Emmanuelle, for the beautiful friendship that we have, and our sacred gossip time together; my officemates Irina, Saloua and Karina, for their affection: they had to deal daily with the stressed version of me in the end, thank you for understanding my introversion in the cutest possible way; sweet Dana for our talkative daily strolls around the building; the GIPSA-doc team for their cheerful presence and debates: Marc, Alexandre, Quyen, Louis, Florent, Rapha¨el, Marion, Pierre N., Pierre M., Yasmin, Wei, Zhongyang and Guanghan, and all the others; Lucia for her helpful kindness; I want to thank all my friends in Grenoble and scattered in every corner of the world that have enriched my life so far and in particular in the last few years. A warm thankful vi thought to Perrine for our silliness together, and lots of laughs; to Ang´elique and Rapha¨el for our hikes and talks; Letizia for our intellectual connection and Jungian interests; Nina for our lunches and her colourful personality; Raluca and Vincent for our discussions and friendship; Sophie for her kindness; Pauline and Henri for their congeniality; my virtual Italian sense8s Alberto, Elide, Roberto, Andrea, Mirko, Sabrina, and Xavier in particular; my “girls only” group with Tiphaine, Ang´elique, Olha, and Audrey; my ASP friends Alessandra, Cristina, Boris, Michele; Ilaria, Valeria R., Elisa, Daniel, Francesca, Valeria Miranda, Valeria D. M., Giada, Cecio, Viola, Valerio, Ilaria P. from Italy that I would like to see more often; Tommaso for his friendship and advice. I want to thank my Karate senseis from my past, for their training in this magnificent martial art, and all their life teachings, as well as my training mates at the dojo of sensei Marchini. From the last two years, I want to thank my Iyengar Yoga teacher Juliette who introduced me to the endlessly profound world of yoga and pranayama, that changed my life forever. I could not find a more resourceful and gifted teacher. I want to thank my family, and in particular my two wonderful as much as different sisters: Laura for being the best cheerleader and singing teacher ever and for laughing together at each other and at the world, and Valeria for our deep exchange about psy- chology and existence, nothing is ever prosaic with you two at my side; my mother for her support and anxious affection; my grandfather, who was a bright mathematician and who was so affectionate and who would be so proud of me; my father for his levity and Italian sense of humour; all my grandparents for their affection and their teachings; Valentina for her fondness; Alessandro for his relentless optimism and wittiness. I want to thank above all Josselin for loving me, for being there for me, for being the best lover, life companion, yoga mate, hiking pal, intellectual partner, French teacher and French cook (especially when I was hungry and composing this manuscript) that I could wish for. You really supported me with endless empathy, infinite curiosity for the complex human being that I am, and always believed in me when I had ceased to. Last, but not least, I want to thank our marvellous Russian Blue, Artu`, for loving me in his feline way, purring me out of anguish, meowing out of love and hunger, attention seeking, and curling up on my lap even and especially when I was working at this thesis. Namaste. Abstract English Abstract Source estimation and localization are a central problem in array signal processing, and in particular in telecommunications, seismology, acoustics, biomedical engineering, and astronomy. Sensor arrays, i.e. acquisition systems composed of multiple sensors that receive source signals from different directions, sample the impinging wavefields in space and time. Hence, high resolution techniques such as MUSIC make use of these two ele- ments of diversities: space and time, in order to estimate the signal subspace generated by impinging sources, as well as their directions of arrival. This is generally done through the estimation of second or higher orders statistics, such as the array spatial covariance matrix, thus requiring sufficiently large data samples. Only recently, tensor analysis has been applied to array processing using as a third mode (or diversity), the space shift translation of a reference subarray, with no need for the estimation of statistical quanti- ties. Tensor decompositions consist in the analysis of multidimensional data cubes of at least three dimensions through their decomposition into a sum of simpler constituents, thanks to the multilinearity and low rank structure of the underlying model. Thus, ten- sor methods provide us with an estimate of source signatures, together with directions of arrival, in a deterministic way. This can be achieved by virtue of the separable and low rank model followed by narrowband sources in the far field. This thesis deals with source estimation and localization of multiple sources via these tensor methods for ar- ray processing. Chapter 1 presents the physical model of narrowband elastic sources in the far field, as well as the main definitions and assumptions. Chapter 2 reviews the state of the art on direction of arrival estimation, with a particular emphasis on high- resolution signal subspace methods. Chapter 3 introduces the tensor formalism, namely the definition of multi-way arrays of coordinates, the main operations and multilinear decompositions. Chapter 4 presents the subject of tensor array processing via rotational invariance. Chapter 5 introduces a general tensor model to deal with multiple physical diversities, such as space, time, space shift, polarization, and gain patterns of narrow- band elastic waves. Subsequently, Chapter 6 and Chapter 8 establish a tensor model for wideband coherent array processing. We propose a separable coherent focusing operation through bilinear transform and through a spatial resampling, respectively, in order to ensure the multilinearity of the interpolated data. We show via computer simulations that the proposed estimation of signal parameters considerably improves, compared to existing narrowband tensor processing and wideband MUSIC. Throughout the chapters we also compare the performance of tensor estimation to the Cram´er-Rao bounds of the multilinear model, which we derive in its general formulation in Chapter 7. Moreover, in Chapter 9 we propose a tensor model via the diversity of propagation speed for seismic waves and illustrate an application to real seismic data from an Alpine glacier. Finally, the last part of this thesis in Chapter 10 moves to the parallel subject of multidimensional spectral factorization of seismic ways, and illustrates an application to the estimation of the impulse response of the Sun for helioseismology. viii Keywords Tensor, Array, Wideband, Source Separation, Localization, Direction of Arrival, Seismic Resum´e en Franc¸ais L’estimation et la localisation de sources sont des probl`emes centraux en traite- ment d’antenne, en particulier en t´el´ecommunication, sismologie, acoustique, ing´enierie m´edicale ou astronomie. Une antenne de capteurs est un syst`eme d’acquisition compos´e par de multiples capteurs qui rec¸oivent des ondes en provenance de sources de direc- tions diff´erentes: elle ´echantillonne les champs incidents en espace et en temps. Pour cette raison, des techniques haute r´esolution comme MUSIC utilisent ces deux ´el´ements de diversit´e, l’espace et le temps, afin d’estimer l’espace signal engendr´e par les sources incidentes, ainsi que leur direction d’arriv´ee. Ceci est g´en´eralement atteint par une es- timation pr´ealable de statistiques de deuxi`eme ordre ou d’ordre sup´erieur, comme la covariance spatiale de l’antenne, qui n´ecessitent donc de temps d’observation suffisam- ment longs. Seulement r´ecemment, l’analyse tensorielle a ´et´e appliqu´ee au traitement d’antenne, graˆce `a l’introduction, comme troisi`eme modalit´e (ou diversit´e), de la transla- tion en espace d’une sous-antenne de r´ef´erence, sans faire appel a` l’estimation pr´ealable de quantit´es statistiques. Les d´ecompositions tensorielles consistent en l’analyse de cubes de donn´ees multidimensionnelles, au travers de leur d´ecomposition en somme d’´el´ements constitutifs plus simples, graˆce a` la multilin´earit´e et a` la structure de rang faible du mod`ele sous-jacent. Ainsi, les mˆemes techniques tensorielles nous fournissent une estim´ee des signaux eux-mˆemes, ainsi que de leur direction d’arriv´ee, de fac¸on d´eterministe. Ceci peut se faire en vertu du mod`ele s´eparable et de rang faible v´erifi´e par des sources en bande ´etroite et en champs lointain. Cette th`ese ´etudie l’estimation et la localisation de sources par des m´ethodes tensorielles de traitement d’antenne. Le premier chapitre pr´esente le mod`ele physique de source en bande ´etroite et en champs lointain, ainsi que les d´efinitions et hypoth`eses fondamentales. Le deuxi`eme chapitre passe en revue l’´etat de l’art sur l’estimation des directions d’arriv´ee, en mettant l’accent sur les m´ethodes haute r´esolution a` sous-espace. Le troisi`eme chapitre introduit la notation tensorielle, a` savoir la d´efinition des tableaux de coordonn´ees multidimensionnels, les op´erations et d´ecompositions principales. Le quatri`eme chapitre pr´esente le sujet du traitement ten- soriel d’antenne au moyen de l’invariance par translation. Le cinqui`eme chapitre introduit un mod`ele tensoriel g´en´eral pour traiter de multiples diversit´es a` la fois, comme l’espace, le temps, la translation en espace, les profils de gain spatial et la polarisation des ondes ´elastiques en bande ´etroite. Par la suite, les sixi`eme et huiti`eme chapitres ´etablissent un mod`ele tensoriel pour un traitement d’antenne bande large coh´erent. Nous proposons une op´eration de focalisation coh´erente et s´eparable par une transform´ee bilin´eaire et par un r´e-´echantillonnage spatial, respectivement, afin d’assurer la multilin´earit´e des donn´ees interpol´ees. Nous montrons par des simulations num´eriques que l’estimation propos´ee des param`etres des signaux s’am´eliore consid´erablement, par rapport au traitement tensoriel classique en bande ´etroite, ainsi qu’a` MUSIC coh´erent bande large. Egalement, tout au long de la th`ese, nous comparons les performances de l’estimation tensorielle avec la borne de Cram´er-Rao du mod`ele multilin´eaire associ´e, que nous d´eveloppons, dans sa forme la plus g´en´erale, dans le septi`eme chapitre. En outre, dans le neuvi`eme chapitre nous illus- trons une application a` des donn´ees sismiques r´eelles issues d’une campagne de mesure sur un glacier alpin, graˆce `a la diversit´e de vitesse de propagation. Enfin, le dixi`eme et dernier chapitre de cette th`ese traite le sujet parall`ele de la factorisation spectrale multi- dimensionnelle d’ondes sismiques, et pr´esente une application `a l’estimation de la r´eponse ix impulsionnelle du soleil pour l’h´eliosismologie. Keywords Tenseur, Antenne, Bande Large, S´eparation de Sources, Localization, Direction d’Arriv´ee, Sismique This work was supported by the ERC Grant AdG-2013-320594 “DECODA”.