This page provides resources and complementary results for the research manuscript:
"Adapted and adaptive linear time-frequency representations: a synthesis point of view"
P. Balazs, M. Dörfler, M. Kowalski and B. Torrésani
accepted for the special issue 'Time-Frequency Analysis and Applications' of the IEEE Signal Processing Magazine.
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 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.
Figure 1: Glockenspiel-Signal. Gabor representations with short window (2.9 ms), resp. long window (92.9 ms).
Figure 2: Glockenspiel-Signal: Nonstationary Gabor representation with time-adaptivity (upper plot) and frequency-adaptivity, corresponding to constant-Q scale (lower plot).
Figure 3: Time-Frequency representations of each denoised layers (Left: transients -- Right: Tonals), for the three methods. From top to bottom: Gabor-GL + Gabor-L, Gabor-GL + Gabor-WGL, Gabor-GL + CQ-Gabor-L
All files are collected in a ZIp-file!
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 social-sparsity operators.
- The NSGT (NonStationary Gabor Transform) toolbox, which provides the implementation, based on the LTFAT toolbox, of the non-stationary Gabor frames, in particular the constant-Q transform.
Acknowledgments: P. Balazs is supported by the Austrian Science Fund (FWF) START-project FLAME ('Frames and Linear Operators for Acoustical Modeling and Parameter Estimation'; Y 551-N13); M. Dörfler is supported by the WWTF project Audiominer (MA09-24); B. Torrésani is supported by the European project UNLocX, grant number 255931, and by the ANR project Metason ANR-10-CORD-010 ; 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.