Denoising by multiscale product coefficient semi-soft thresholding

In multiscale product coefficient hard thresholding, how to determine the optimal threshold is the main problem due to the discontinuity of MSE. Here a semi-soft thresholding function is constructed in the product form of shrinkage coefficient function and wavelet coefficients. This function is infinite-order differentiable with respect to wavelet coefficient, and can adaptively shrink wavelet coefficient in the neighborhood of the threshold. Through minimizing the Stein Unbiased Risk Estimate (SURE) based on the function, the optimal threshold, varying with the signal and noise, is obtained in the Mean Square Error (MSE) sense. In simulations to denoise multiple classic noisy signals, the multiscale product coefficient thresholding is improved through our semi-soft thresholding function.