On a non-local spectrogram for denoising one-dimensional signals
Gonzalo Galiano, Julián Velasco
TL;DR
This work addresses denoising one-dimensional signals by improving their time-frequency representation, the spectrogram, using nonlocal filtering to overcome limitations of local PDE methods. It establishes a mathematical framework where the spectrogram is treated as an image on $\Omega$, connects Neighborhood filtering to Nonlocal Means via the WV distribution with a distance term $F(\mathbf{x},\mathbf{y})=|S(f;\mathbf{x})-S(f;\mathbf{y})|^2$, and provides discrete implementations and comparisons among Neighborhood, Yaroslavsky, and PDE-based filters. Empirical results on synthetic, wolf chorus, and ECG signals show that the fast Neighborhood filter achieves denoising quality similar to Nonlocal Means or diffusion-based methods while offering orders-of-magnitude faster execution (e.g., speedups exceeding 200x). The findings support using nonlocal filtering on spectrograms for reliable instantaneous frequency estimation and critical biomedical signal analysis in practical, time-constrained settings.
Abstract
In previous works, we investigated the use of local filters based on partial differential equations (PDE) to denoise one-dimensional signals through the image processing of time-frequency representations, such as the spectrogram. In this image denoising algorithms, the particularity of the image was hardly taken into account. We turn, in this paper, to study the performance of non-local filters, like Neighborhood or Yaroslavsky filters, in the same problem. We show that, for certain iterative schemes involving the Neighborhood filter, the computational time is drastically reduced with respect to Yaroslavsky or nonlinear PDE based filters, while the outputs of the filtering processes are similar. This is heuristically justified by the connection between the (fast) Neighborhood filter applied to a spectrogram and the corresponding Nonlocal Means filter (accurate) applied to the Wigner-Ville distribution of the signal. This correspondence holds only for time-frequency representations of one-dimensional signals, not to usual images, and in this sense the particularity of the image is exploited. We compare though a series of experiments on synthetic and biomedical signals the performance of local and non-local filters.