Grid-Based Decimation for Wavelet Transforms with Stably Invertible Implementation

TL;DR

This paper introduces a new decimation method for wavelet transforms that achieves stable, invertible representations with near-unity oversampling and uniform decimation, enabling efficient time-frequency audio processing.

Contribution

A novel decimation strategy for wavelet transforms that maintains stability, invertibility, and uniform data structure, facilitating advanced audio processing applications.

Findings

Energy-preserving finite implementations based on frame theory

Wavelet coefficients stored as a natural time-frequency matrix

Effective application in nonnegative matrix factorization, onset detection, and phase reconstruction

Abstract

The constant center frequency to bandwidth ratio (Q-factor) of wavelet transforms provides a very natural representation for audio data. However, 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 stable representations with oversampling rates close to one and uniform decimation. Specifically, we show that finite implementations of the resulting representation are energy-preserving in the sense of frame theory. The obtained wavelet coefficients can be stored in a timefrequency matrix with a natural interpretation of columns as time frames and rows as frequency channels. This matrix structure immediately grants access to a large number of algorithms that are successfully used in time-frequency audio processing, but could not previously be used jointly with wavelet transforms. We demonstrate the application of our method in processing based on nonnegative matrix factorization, in onset detection, and in phaseless reconstruction.