15.3 Wavelet-based denoising methods

Wavelet transforms are powerful tools for analyzing biosignals. They break down signals into different frequency bands, allowing for multi-resolution analysis. This makes them great for noise reduction, as they can separate signal and noise components while preserving important signal features.

Discrete Wavelet Transform (DWT) and Stationary Wavelet Transform (SWT) are two key techniques for decomposing biosignals. Wavelet thresholding then removes noise by zeroing out small coefficients. These methods effectively increase signal-to-noise ratio while keeping crucial signal information intact.

Wavelet Transforms and Denoising

Fundamentals of wavelet transforms

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• Wavelet transforms decompose signals into different frequency bands using wavelets as basis functions which are localized in both time and frequency domains, enabling multi-resolution analysis of signals
• Suitable for noise reduction in biosignals due to their ability to capture transient and non-stationary features, effectively separate signal and noise components, and preserve signal morphology while removing noise
• Provide a sparse representation of signals by concentrating signal energy into a few large coefficients, while noise is typically distributed across many small coefficients

DWT and SWT for biosignal decomposition

• Discrete Wavelet Transform (DWT) decomposes signal into approximation (low-frequency) and detail (high-frequency) coefficients by applying a series of low-pass and high-pass filters followed by downsampling, with recursive decomposition of approximation coefficients at each level, resulting in coefficients that represent signal information at different frequency bands and time scales
• Stationary Wavelet Transform (SWT), also known as undecimated or shift-invariant wavelet transform, is similar to DWT but without downsampling, applying low-pass and high-pass filters at each level while maintaining the same number of coefficients to ensure shift-invariance and provide a redundant representation of the signal

Wavelet Thresholding and Denoising Performance

Wavelet thresholding techniques

• Wavelet thresholding removes noise coefficients in the wavelet domain
• Hard thresholding sets all coefficients below a threshold to zero and keeps coefficients above the threshold unchanged, defined as: $\hat{x} = \begin{cases} x, & \text{if } |x| > \lambda \\ 0, & \text{if } |x| \leq \lambda \end{cases}$
• Soft thresholding shrinks coefficients towards zero by the threshold value and sets coefficients below the threshold to zero, defined as: $\hat{x} = \begin{cases} \text{sign}(x)(|x| - \lambda), & \text{if } |x| > \lambda \\ 0, & \text{if } |x| \leq \lambda \end{cases}$
• Threshold selection methods include universal threshold $\lambda = \sigma \sqrt{2 \log N}$, where $\sigma$ is the noise standard deviation and $N$ is the signal length, and adaptive thresholds based on signal characteristics or noise estimates

Effectiveness of wavelet-based denoising

• Wavelet denoising aims to increase the Signal-to-Noise Ratio (SNR) of the signal, defined as the ratio of signal power to noise power: $\text{SNR} = 10 \log_{10} \frac{P_\text{signal}}{P_\text{noise}}$, with higher SNR indicating better noise reduction and signal quality
• Denoising should remove noise while preserving important signal features, assessed using metrics such as cross-correlation, mean squared error, or visual inspection
• Factors affecting denoising performance include the choice of wavelet family and decomposition level, thresholding method and threshold selection, and characteristics of the signal and noise (signal bandwidth, noise type, and noise level)
• Performance of wavelet denoising can be compared against other methods, such as linear filtering or adaptive filtering, considering trade-offs between noise reduction and signal preservation