Wavelet Denoising

Table of contents

  1. Introduction
  2. How to use
  3. Behavior
  4. Method
  5. References

Introduction

Wavelet Denoising reduces spectral noise by decomposing a spectrum into wavelet coefficients, shrinking high-frequency detail coefficients, and reconstructing the signal.

How to use

  1. Upload data, then open Processing Page.
  2. Enable Smoothing.
  3. In Select smoothing function, choose Wavelet Denoising.
  4. Choose Wavelet denoising method, Wavelet family, and Decomposition level.
  5. For Sardy Block Coordinate Relaxation(BCR), set BCR iterations and Robust loss function.
  6. For Standard Universal Thresholding, set Thresholding mode.
  7. Apply the processing workflow to update the spectra.

Behavior

Wavelet denoising decomposes each spectrum into wavelet coefficients, shrinks high-frequency detail coefficients, and reconstructs a smoother spectrum. Standard Universal Thresholding is a single-pass denoising method. Sardy BCR is an iterative robust method that reduces the effect of large residuals while refining detail coefficients.

Method

Standard mode estimates noise from the finest detail coefficients and applies a universal threshold:

\[\lambda = \hat{\sigma}\sqrt{2\log(n)}\]

Sardy BCR scales the same threshold and iteratively updates residual weights:

\[\lambda_{\text{BCR}} = s_{\lambda}\hat{\sigma}\sqrt{2\log(n)}\]
Parameter Tunable or fixed Implementation
Wavelet denoising method Tunable Sardy Block Coordinate Relaxation(BCR) or Standard Universal Thresholding; default Sardy BCR
Wavelet family Tunable sym4, sym8, db4, db8, coif1, coif3, haar; default sym4
Decomposition level Tunable 1-6; UI default 4; internally clipped to the maximum valid level
BCR iterations Tunable 1-15; default 10
Robust loss function Tunable huber or l1; default huber
Thresholding mode Tunable soft or hard; default soft
Huber delta Fixed 1.5 in the Sardy BCR helper
Lambda scale Fixed 1.0 in the Sardy BCR helper

References

  1. Sardy, S., Tseng, P., & Bruce, A. (2001). Robust wavelet denoising. https://www.unige.ch/~sardy/Papers/robustIEEE.pdf
  2. Donoho, D. L., & Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425-455. https://doi.org/10.1093/biomet/81.3.425
  3. PyWavelets Developers. PyWavelets documentation. https://pywavelets.readthedocs.io/