We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues.We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction.
Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.
We studied the ability of deep neural networks (DNNs) to restore missing audio content based on its context, a process usually referred to as audio inpainting. We focused on gaps in the range of tens of milliseconds, a condition which has not received much attention yet. The proposed DNN structure was trained on audio signals containing music and musical instruments, separately, with 64-ms long gaps. The input to the DNN was the context, i.e., the signal surrounding the gap, transformed into time-frequency (TF) coefficients. Two networks were analyzed, a DNN with complex-valued TF coefficient output and another one producing magnitude TF coefficient output, both based on the same network architecture. We found significant differences in the inpainting results between the two DNNs. In particular, we discuss the observation that the complexvalued DNN fails to produce reliable results outside the low frequency range. Further, our results were compared to those obtained from a reference method based on linear predictive coding (LPC). For instruments, our DNNs were not able to match the performance of reference method, although the magnitude network provided good results as well. For music, however, our magnitude DNN significantly outperformed the reference method, demonstrating a generally good usability of the proposed DNN structure for inpainting complex audio signals like music. This paves the road towards future, more sophisticated audio inpainting approaches based on DNNs.
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