 Wavelet coefficient thresholding

Hard thresholding keeps detail coefficients whose magnitude exceeds a prescribed threshold and sets the rest to zero, while retaining the coarse approximation coefficients. Soft thresholding instead uses $\mathrm{sgn}(\hat{b})(|\hat{b}|-\tau {)}_{+}$. The simultaneous wavelet coefficient noise bound motivates thresholds of order $\sigma \sqrt{\log n/n}$ in a normalized Gaussian regression model with at most order $n$ details. This is a nonlinear selection rule: coefficients can be removed individually rather than discarding an entire finest resolution.