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Time-frequency and time-scale transforms

Bruno Torresani
Centre de Physique Theorique, Marseille

Current multimedia technologies call for efficient ways of representing signals. Efficiency may be understood from several points of view, in particular the following three: efficiency from the point of view of signal compression, i.e. for representing maximal amount of information within minimal amount of data. Or efficiency from the point of view of signal analysis, i.e. ability of providing a representation which is simple enough so that information may be easily retrieved. Or efficiency from algorithmic points of view.
We shall review several methods for signal representation, which may be termed ``efficient'' from the above mentioned points of view. We shall especially focus on some representations based upon the notion of localization in time-frequency or time-scale spaces.
We shall pay special attention to the representations which are adapted to ``non-stationary'' features of signals, in particular the classes of bilinear representations (in particular the Wigner representations, and their generalizations), and their approximations using time-frequency atoms (mainly wavelet transforms and Gabor transforms).
Among the applications of such methods, we shall more particularly emphasize concrete (and potential) applications to signal compression and signal de-noising, as well as applications to detection and parameter estimation problems.

B. Torresani was born in Marseille, France, on November 23th, 1961. He received his PhD degree in theoretical physics (July 1986) from the Universite de Provence (Marseille), and his habilitation degree (October 1993) from the Universite de la Mediterranee (Marseille). He has been appointed Alexander von Homboldt fellow in Bochum (FRG), and associate professor in St Louis (Missouri), Irvine (California), Louvain-la-Neuve (Belgium) and La Plata (Argentina). He currently holds a research position at CNRS, and a joint appointment at the Centre de Physique Theorique and the Laboratoire d'Analyse, Topologie et Probabilites.
B. Torresani's research interests are mainly in signal analysis and time-frequency methods, numerical electromagnetic scattering, mathematical physics and computational biology.

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