Optimized Wavelet Denoising for Self-Similar $\alpha $ -Stable Processes
We investigate the performance of wavelet shrinkage methods for the denoising of symmetric-<inline-formula> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula>-stable (<inline-formula> <tex-math notation="LaTeX">$\text{S} {\alpha }\text{S}$ </tex-math></inline-formula>) self-similar stochastic processes corrupted by additive white Gaussian noise (AWGN), where <inline-formula> <tex-math notation="LaTeX">$ {\alpha }$ </tex-math></inline-formula> is tied to the sparsity of the process. The wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean-square approximation error (MMSE estimator). We derive the corresponding formula for the expected value of the averaged estimation error. We show that the predicted MMSE is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process. Using the calculus of variations, we then optimize this criterion to find the best performing wavelet within the extended family of Meyer wavelets, which are bandlimited. These are compared with the Daubechies wavelets, which are compactly supported in time. We find that the wavelets that are shorter in time (in particular, the Haar basis) are better suited to denoise the sparser processes (say, <inline-formula> <tex-math notation="LaTeX">$ {\alpha <1.2}$ </tex-math></inline-formula>), while the bandlimited ones (including the Held and Shannon wavelets) offer the best performance for <inline-formula> <tex-math notation="LaTeX">$ {\alpha >1.6}$ </tex-math></inline-formula>, the limit corresponding to the Gaussian case (fBm) with <inline-formula> <tex-math notation="LaTeX">$ {\alpha =2}$ </tex-math></inline-formula>.