Adapted and adaptive linear time-frequency representations: a synthesis point of view

International audienceTo display the time and frequency content of a given signal a large variety of techniques exist. In this paper, we give an overview of linear time-frequency representations, focusing mainly on two fundamental aspects. The first one is the introduction of flexibility, more precisely the construction of time-frequency waveform systems that can be adapted to specific signals, or specific signal processing problems. To do this, we base the constructions on frame theory, which allows a lot of options, while still ensuring perfect reconstruction. The second aspect is the choice of the synthesis framework rather than the usual analysis framework. Instead of the correlation of the signal with the chosen waveforms, i.e. the inner product with them, we look at how the signals can be constructed using those waveforms, i.e. find the coefficient in their linear combination. We show how this point of view allows the easy introduction of prior information into the representation. We give an overview over methods for transform domain modeling, in particular those based on sparsity and structured sparsity. Finally we present an illustrative application for these concepts: a denoising scheme