Wavelet Denoising

Wavelet Transform Threshold Denoising Method is a nonlinear signal processing technique initially introduced by Professors Johnstone and Donoho in 1992. This approach achieves near-optimal performance under minimum mean square error criteria while maintaining the simplest implementation structure and lowest computational requirements among denoising methods.

The fundamental principle relies on orthogonal wavelet decomposition's time-frequency localization capability. During signal processing, wavelet components demonstrate significant amplitude variations, contrasting sharply with noise's uniform distribution in high-frequency regions. After wavelet decomposition, larger-magnitude coefficients predominantly represent useful signals, while smaller coefficients generally correspond to noise. Thus, useful signals typically yield larger wavelet coefficients than noise components.

The denoising implementation involves three key steps: First, determining an optimal threshold (commonly using universal threshold rules like VisuShrink or SureShrink). Second, applying thresholding operations where coefficients exceeding the threshold are preserved while smaller coefficients are processed through shrinkage functions (soft thresholding: sign(c)(|c|-λ) for |c|≥λ, hard thresholding: c for |c|≥λ). Finally, reconstructing the denoised signal using the modified coefficients through inverse wavelet transform, typically implemented via MATLAB's wdenoise function or Python's pywt.thresholding methods.