Wavelet Threshold Denoising: Implementation and Comparative Analysis of Soft vs. Hard Thresholding Techniques

In this article, we explore a crucial technique in digital signal processing: wavelet threshold denoising. This method effectively removes noise from signals, making them clearer and more suitable for analysis. We implement this technique through a three-step computational process: first performing wavelet decomposition using functions like wavedec() to obtain coefficients across different frequency bands, then applying thresholding operations to the detail coefficients, and finally reconstructing the signal using waverec(). The core algorithm involves calculating appropriate thresholds (often using universal threshold formulas like σ√(2logN)) and applying them to wavelet coefficients.

We conduct a detailed comparison between two primary thresholding approaches: soft thresholding and hard thresholding. Soft thresholding implements a continuous shrinkage function that reduces all coefficients by the threshold value (implemented as sign(c).*max(abs(c)-threshold,0)), while hard thresholding uses a binary keep-or-discard approach (c.*(abs(c)>threshold)). Through MATLAB code examples and performance metrics, we demonstrate how soft thresholding provides smoother results but may oversmooth signals, whereas hard thresholding preserves sharp features but can introduce artifacts.

This comparative analysis helps understand the trade-offs between different methods, enabling selection of the most appropriate approach for specific applications. We examine implementation considerations including threshold selection strategies (VisuShrink, SureShrink), wavelet basis choices, and level-dependent thresholding. By diving deep into wavelet threshold denoising, we uncover its significant applications and potential in digital signal processing scenarios such as biomedical signal cleaning, image denoising, and communication signal enhancement.