Grid-based decimation for invertible wavelet transforms in audio processing

The constant center frequency to bandwidth ratio (Q-factor) of wavelet transforms provides a very natural representation for audio data. So far, invertible wavelet transforms have either required non-uniform decimation---leading to irregular data structures that are cumbersome to work with---or require excessively high oversampling with unacceptable computational overhead. Here, we present a novel decimation strategy for wavelet transforms that leads to invertible representations, i.e., representations that allow perfect reconstruction, with minimal oversampling and uniform decimation. Numerical evidence shows that the resulting representation is highly stable in the sense of frame theory. The obtained wavelet coefficients can be stored in a time-frequency matrix with a natural interpretation of columns as time frames and rows as frequency channels. In a special case, the employed decimation strategy corresponds to a time-frequency lattice related to certain one-dimensional low discrepancy sequences. The matrix structure of the wavelet coefficients immediately grants access to a large number of algorithms that have been successfully used in time-frequency audio processing, but could not previously be used jointly with invertible wavelet transforms. We demonstrate the application of our method in processing based on non-negative matrix factorization and phaseless reconstruction.