A New Generalized Projection and Its Application to Acceleration of Audio Declipping
Abstract
:1. Introduction
2. Methods
2.1. Proximal Algorithms
2.2. Proximal Operators
2.3. The New Relation of Projections
3. Discussion on the New Result
4. Experiment
4.1. Problem Formulation
4.2. The Gabor Operators
4.3. Problem Solution
4.4. Condat Algorithm
Algorithm 1: Condat algorithm (CA) adapted to solving Equation (26). |
4.5. Douglas–Rachford Algorithm
Algorithm 2: Douglas–Rachford algorithm (DR) solving Equation (29) |
4.6. Comparison of the Algorithms
- Sparsifying step: one soft thresholding, which is performed elementwise, and thus it is , and one analysis , which is
- Reliable part: one synthesis G and one analysis , both
- Each of the clipped parts: one synthesis, , and one elementwise projection, .
- Sparsifying step: one soft thresholding, which is
- Projection onto K: one synthesis G and one pseudoinverse , which is in the order of , i.e., in our particular setup; projection that is performed elementwise, .
4.7. Redundancy of the Real-Part Operator
4.8. Results
4.9. Other Applications
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
CA | Condat Algorithm |
DCT | Discrete Cosine Transform |
DFT | Discrete Fourier Transform |
DGT | Discrete Gabor Transform |
DR | Douglas–Rachford (algorithm) |
FBB-PD | Forward–Backward-Based Primal–Dual (algorithm) |
FFT | Fast Fourier Transform |
MDCT | Modified Discrete Cosine Transform |
PA | Proximal Algorithm |
SDR | Signal-to-Distortion Ratio |
STFT | Short-Time Fourier Transform |
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Rajmic, P.; Záviška, P.; Veselý, V.; Mokrý, O. A New Generalized Projection and Its Application to Acceleration of Audio Declipping. Axioms 2019, 8, 105. https://doi.org/10.3390/axioms8030105
Rajmic P, Záviška P, Veselý V, Mokrý O. A New Generalized Projection and Its Application to Acceleration of Audio Declipping. Axioms. 2019; 8(3):105. https://doi.org/10.3390/axioms8030105
Chicago/Turabian StyleRajmic, Pavel, Pavel Záviška, Vítězslav Veselý, and Ondřej Mokrý. 2019. "A New Generalized Projection and Its Application to Acceleration of Audio Declipping" Axioms 8, no. 3: 105. https://doi.org/10.3390/axioms8030105