Processus alpha-stables pour le traitement du signal

Mathieu Fontaine 1, 2, 3
1 ZENITH - Scientific Data Management
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier, CRISAM - Inria Sophia Antipolis - Méditerranée
2 MULTISPEECH - Speech Modeling for Facilitating Oral-Based Communication
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : The scientific topic of sound source separation (SSS) aims at decomposing audio signals into their constitutive components, e.g. separate singing voice from background music or from background noise. In the case of very old and degraded historical recordings, SSS strongly extends classical denoising methods by being able to account for complex signal or noise patterns and achieve efficient separation where traditional approaches fail. It is classic in signal processing to model the observed signal as the sum of desired signals. If we adopt a probabilistic model, it is preferable that law of the additive processes is stable by summation. The Gaussian process notoriously satisfies this condition. It admits useful statistical operators as the covariance and the mean. The existence of those moments allows to provide a statistical model for SSS. However, Gaussian process has difficulty to deviate from its mean. This drawback limits signal dynamics and may cause unstable inference methods. On the contrary, non-Gaussian α−stable processes are stable under addition, and permit the modeling of signals with considerable dynamics. For the last few decades, α-stable theory have raised mathematical challenges and have already been shown to be effective in filtering applications. This class of processes enjoys outstanding properties, not available in the Gaussian case. A major asset for signal processing is the unique spatial representation of a multivariate α−stable vector, controlled by a so-called spectral measure and a deterministic vector.The spectral measure provides information on the global energy coming from all space directions while the vector localizes the centroid of the probability density function. It shows its usefulness for the socalled independent component analysis (ICA) topic and the SSS. However, those models are only linear instantaneous mixture and do not consider the frequency domain. This thesis introduces several α-stables models, with the aim of extending them in several directions. First, we propose an extension of single-channel α−stable filtering theory to a multichannel one. In particular, a novel spatial representation forα−stable vectors is proposed. Secondly, we develop α−stable models for denoising where each component could admit a different α. This hybrid model provides a rigorous explanation of some heuristic Wiener filters outlined in the 1980s. We also describe how the α−stable theory yields a new method for audio source localization. We use the spectral measure resulting from the spatial representation of α−stable vectors. In practice, it leads to determine whether a source is active at a specific location. Our work consisted in investigating the α-stable theory for signal processing and developing several models for a wide range of applications. The models introduced in this thesis could also be extend to more signal processing tasks. We could use our mutivariate α−stable models to dereverberation or SSS. Moreover, the localization algorithm is implementable for room geometry estimation
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Mathieu Fontaine. Processus alpha-stables pour le traitement du signal. Traitement du signal et de l'image [eess.SP]. Université de Lorraine, 2019. Français. ⟨NNT : 2019LORR0037⟩. ⟨tel-02188304⟩

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