Robust wavelet denoising

For extracting a signal from noisy data, waveshrink and basis pursuit are powerful tools both from an empirical and asymptotic point of view. They are especially efficient at estimating spatially inhomogeneous signals when the noise is Gaussian. Their performance is altered when the noise has a long tail distribution, for instance, when outliers are present. We propose a robust wavelet-based estimator using a robust loss function. This entails solving a nontrivial optimization problem and appropriately choosing the smoothing and robustness parameters. We illustrate the advantage of the robust wavelet denoising procedure on simulated and real data.