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An Iterative Wavelet Threshold for Signal Denoising
F. M. Bayer, A. J. Kozakevicius, R. J. Cintra
TL;DR
Threshold selection is critical in wavelet-shrinkage denoising. SpcShrink introduces a wavelet-thresholding scheme, and the authors report it could outperform four competing methods at various noise levels.
Problem
Selecting the threshold is a critical issue in wavelet-shrinkage denoising.
Source: §1 · p. 2Method
SpcShrink is a new wavelet-transform denoising scheme that introduces an approach for obtaining wavelet threshold values.
Source: §1 · p. 2, §5 · p. 16Results
SpcShrink could outperform VisuShrink, SureShrink, BayesShrink, and S-median at various noise levels.
Source: §5 · p. 16Takeaways & Limitations
α1 = 1.5% is suggested for maximizing SNR gain, while α1 = 1.0% is suggested for better visual quality.
Source: §5 · p. 16The method's thresholding rationale assumes transformed noise coefficients are white Gaussian under the null hypothesis.
Source: §2.3 · p. 5
Abstract
from arXiv · showhide
This paper introduces an adaptive filtering process based on shrinking wavelet coefficients from the corresponding signal wavelet representation. The filtering procedure considers a threshold method determined by an iterative algorithm inspired by the control charts application, which is a tool of the statistical process control (SPC). The proposed method, called SpcShrink, is able to discriminate wavelet coefficients that significantly represent the signal of interest. The SpcShrink is algorithmically presented and numerically evaluated according to Monte Carlo simulations. Two empirical applications to real biomedical data filtering are also included and discussed. The SpcShrink shows superior performance when compared with competing algorithms.
1 Introduction
The introduction frames wavelet coefficient threshold selection as critical to denoising and proposes SpcShrink, an SPC-inspired iterative thresholding scheme. It reports competitive simulation performance and feature preservation in biomedical signals.
1 Introduction
- The introduction identifies threshold selection as critical in wavelet shrinkage, whose central operation is coefficient thresholding in the transform domain.Source: §1 · p. 2
- SpcShrink uses successive control limits to iteratively discard wavelet coefficients until all lie within a specified control range.The threshold follows successive limit estimations.Source: §1 · p. 2
- Its threshold estimation explores wavelet coefficients' exponential decay across scales rather than assuming constant variance across coefficients.The paper presents this as part of its threshold-value computation.Source: §1 · p. 2
- SpcShrink is reported to outperform popular thresholding methods on computational quality measures, with competitive simulation performance.The introduction also describes comparisons with classical and state-of-the-art wavelet thresholding schemes.Source: §1 · p. 2
- For real biomedical noisy signals, the denoised output preserves significant features while efficiently discarding noise-related information.The evaluation included two real biomedical datasets and Monte Carlo simulations based on synthetic signals.Source: §1 · p. 2
2 SPC-based shrinkage
This section develops SpcShrink, an adaptive wavelet-thresholding method grounded in statistical process control, reviewing wavelet thresholding and control-chart logic before presenting its iterative, level-dependent procedure. It describes recalculating estimates as detected signal samples are removed and notes that the method is guaranteed to converge.
2.1 Review of wavelet shrinkage
- Wavelet shrinkage thresholds coefficients, eliminating or smoothing those below λ; the authors seek an adaptive, level-dependent λ estimation method robust to hard or soft thresholding.The estimated signal is reconstructed from the modified wavelet coefficients.Source: §2.1 · p. 3
2.2 Review of control charts
- The authors propose using control-chart theory to estimate wavelet threshold values, drawing on charts’ role in identifying sources of process variability.Charts use control limits to distinguish in-control observations from out-of-control ones.Source: §2.2 · p. 4
2.3 Proposed SpcShrink
- SpcShrink iteratively removes wavelet coefficients outside control limits and recalculates the limits until the remaining coefficients determine the threshold λj = dj·sj.Under the null hypothesis, coefficients represent white Gaussian noise; the procedure stops when all coefficients are within the limits.Source: §2.3 · pp. 5–6
- As scale level j increases, significance level αj may increase; increasing type I error probability αj across scales improves hypothesis-test detection power, while the corresponding control-limit distance decreases.The authors adopt a simple linear increase of αj along the scales.Source: §2.3 · p. 6
2.4 Adaptivity and convergence
- SpcShrink adapts thresholds for each wavelet decomposition level by removing detected signal samples and recalculating estimates to match signal characteristics.Source: §2.4 · p. 7
- Numerical convergence is guaranteed, and each level’s bounded, monotonic sequences converge as the number of iterations tends to infinity.Source: §2.4 · p. 7
- The method uses O(N) forward and inverse wavelet transforms and O(N ·log N) find/sort operations, enabling efficient implementation.Per iteration, the algorithm computes standard deviations, evaluates control limits, and removes array elements; in the worst case it discards one coefficient per iteration, with at most N − J0 iterations.Source: §2.4 · p. 7
3 Optimum specification of the control limit distances
The section selects control-limit distances through an optimization of denoising performance and examines how α1 affects SNR and visual smoothness. The reported comparison favors SpcShrink on SNR, while lower α1 can favor smoother-looking output.
3.1 Optimization problem
- The control-limit distances d_j depend on test significance levels, so the paper chooses α1 by minimizing negative classical SNR across signal instances.Source: §3 · p. 7, §3.1 · pp. 7–8
3.2 Computational search
- Across injected-noise levels, the SNR-minimizing α1 varied; averaging those minima yielded α1 = 1.5% as a compromise for all considered scenarios.Source: §3.2 · p. 9
- For α1 = 1.5%, the selected control-limit distances are d1 = 2.432, d2 = 2.170, d3 = 2.005, d4 = 1.881, and d5 = 1.780.Source: §3.2 · p. 9
3.3 Qualitative analysis
- At 20 dB, α1 = 1.5% generally yielded higher SNR, whereas α1 = 1.0% gave better visual quality; decreasing α1 also makes outputs smoother.Source: §3.3 · p. 9
- Unlike traditional wavelet shrinkage methods, SpcShrink allows a trade-off between quantitative measures such as SNR and smoothness.Source: §3.3 · p. 9
- Figure 5’s SNR comparison reports that SpcShrink outperforms competing thresholding methods.Source: Figure 5 · p. 11
4 Numerical comparisons
The section benchmarks SpcShrink against competing shrinkage schemes on synthetic signals and biomedical data. It reports strong synthetic denoising results alongside visual trade-offs on real signals.
4 Numerical comparisons
- The evaluation combined Monte Carlo tests on synthetic signals with visual-quality assessment on IPD and ECG data.The synthetic experiments varied noise levels and wavelet filter length.Source: §4 · p. 11
4.1 Synthetic data denosing
- Across RMSE comparisons, SpcShrink beat VisuShrink and SureShrink at all input SNRs, while BayesShrink was under 3.3% better for ‘blocks’ and ‘Doppler’ below 10 dB.The RMSE figure compares thresholding methods across noise levels.Source: Figure 7 · p. 12, §4.1 · p. 12
- SpcShrink outperformed competitors in SNR in almost every scenario except ‘Doppler’ below 10 dB; α1=1.5% was best for ‘blocks’ and ‘bumps’, and 1.0% for ‘Doppler’.For input SNR below 10 dB, SNR gains could exceed 12 dB.Source: §4.1 · p. 12
- Table 2 shows superior performance of SpcShrink, especially with α0 = 1.5%, and the method is robust when operating at different scales (J0) of wavelet decomposition analysis.Source: §4.1 · pp. 12, 14, Table 2 · p. 13
4.2 Biomedical signal denoising
- On filtered IPD, the proposed algorithm removed noise while preserving the signal’s general shape and peaks; SpcShrink(1.5%) demanded only 40 and 56 iterations for convergence, respectively.Source: §4.2 · p. 14, Figure 9 · p. 15
- For ECG, SpcShrink and VisuShrink comparably balanced noise reduction and signal shape; SureShrink flattened QRS peaks, while S-median and BayesShrink retained much of the noise.SureShrink performed well visually on IPD, although its ECG result was poor.Source: §4.2 · p. 14, Figure 10 · p. 15
5 Conclusions
SpcShrink combines wavelet shrinkage with an iterative threshold search inspired by statistical process control. Its free parameter trades quantitative error measures against visual quality, and the reported comparisons favor it over several competing methods.
5 Conclusions
- SpcShrink applies wavelet shrinkage with iterative threshold selection inspired by statistical process control control charts.The iterative search establishes a statistical control state to guide threshold selection.Source: §5 · p. 16
- The free parameter α1 trades quantitative error measures against visual quality; the authors suggest 1.5% for SNR-gain maximization and 1.0% for better visual quality.These recommendations draw on an optimization problem and Monte Carlo simulation results.Source: §5 · p. 16
- Across various noise levels, quantitative results favor SpcShrink over VisuShrink, SureShrink, BayesShrink, and S-median.The method was assessed using Monte Carlo simulation and compared across test signals and input-noise levels.Source: §5 · p. 16