Filtered wavelet thresholding methods

When working with nonlinear filtering algorithms for image denoising problems, there are two crucial aspects, namely, the choice of the thresholding parameter λ and the use of a proper filter function. Both greatly influence the quality of the resulting denoised image. In this paper we propose two new filters, which are a piecewise quadratic and an exponential function of λ, respectively, arid we show how they can be successfully used instead of the classical Donoho and Johnstone's Soft thresholding filter. We exploit the increased regularity and flexibility of the new filters to improve the quality of the final results. Moreover, we prove that our filtered approximation is a near-minimizer of the functional which has to be minimized to solve the denoising problem. We also show that the quadratic filter, due to its shape, yields good results if we choose λ as the Donoho and Johnstone universal threshold, while the exponential one is more suitable if we use the recently proposed H-curve criterion. Encouraging results in extensive numerical experiments on several test images confirm the effectiveness of our proposal.