Nonlinear wavelet thresholding: A recursive method to determine the optimal denoising threshold

Nonlinear thresholding of wavelet coefficients is an efficient method for denoising signals with isolated singularities. The quasi-optimal value of the threshold depends on the sample size and on the variance of the noise, which is in many situations unknown. We present a recursive algorithm to estimate the variance of the noise, prove its convergence and investigate its mathematical properties. We show that the limit threshold depends on the probability density function (PDF) of the noisy signal and that it is equal to the theoretical threshold provided that the wavelet representation of the signal is sufficiently sparse. Numerical tests confirm these results and show the competitiveness of the algorithm compared to the median absolute deviation method (MAD) in terms of computational cost for strongly noised signals.