This page provides resources and complementary results for the research manuscript:
"Adapted and adaptive linear timefrequency representations: a synthesis point of view"
P. Balazs, M. Dörfler, M. Kowalski and B. Torrésani
accepted for the special issue 'TimeFrequency Analysis and Applications' of the IEEE Signal Processing Magazine.
Abstract:
In this paper, we give an overview of linear timefrequency representations, focusing mainly on two fundamental aspects. The first one is the introduction of flexibility, more precisely the construction of timefrequency 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 thir 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.
Figures:

Figure 1: GlockenspielSignal. Gabor representations with short window (2.9 ms), resp. long window (92.9 ms).

Figure 2: GlockenspielSignal: Nonstationary Gabor representation with timeadaptivity (upper plot) and frequencyadaptivity, corresponding to constantQ scale (lower plot).

Figure 3: TimeFrequency representations of each denoised layers (Left: transients  Right: Tonals), for the three methods. From top to bottom: GaborGL + GaborL, GaborGL + GaborWGL, GaborGL + CQGaborL
Codes:
All files are collected in a ZIpfile!
For this code several other packages are needed:
 The LTFAT toolbox, which provides an implementation of the Gabor analysis and synthesis operations, with a C backend for efficiency.
 The StrucAudioToolbox, which provides an implementation of various thresholding operator, and in particular the socialsparsity operators.
 The NSGT (NonStationary Gabor Transform) toolbox, which provides the implementation, based on the LTFAT toolbox, of the nonstationary Gabor frames, in particular the constantQ transform.
Acknowledgments: P. Balazs is supported by the Austrian Science Fund (FWF) STARTproject FLAME ('Frames and Linear Operators for Acoustical Modeling and Parameter Estimation'; Y 551N13); M. Dörfler is supported by the WWTF project Audiominer (MA0924); B. Torrésani is supported by the European project UNLocX, grant number 255931, and by the ANR project Metason ANR10CORD010 ; M. Kowalski, benefited from the support of the "FMJH Program Gaspard Monge in optimization and operation research", and from the support to this program from EDF.