Denoising

Wavelets are suited to the denoising of signals with sharp transients. A threshold is used to delete the wavelet coefficients where the signal is smooth (thus leaving the denoising to the low pass cascade) and preserve these coefficients when they are large.

Using wavelets designed for the whole real axis result in structural delay which may be large. For instance, in the case of orthogonal wavelets, if a filter used by the algorithm is causal, then its mirror, which is also used in the algorithm, is anticausal.

To avoid this, we used a transform which is adapted to the half axis (in the real time context, this means past values). It is based on a combination of averaging and polynomial interpolation (see, for instance, Donoho). The interpolation scheme is very flexible and can be adapted to the half axis. Without much effort, a redundant version can be derived which significantly improves the denoising.

This scheme has been successfully implemented and tested on a satellite model where a feedback control was performed using an acceleration term whereas only a velocity measure was available (with P. Tsiotras et D. Jung). The Matlab source code is available here, and the corresponding journal paper is here.
More recently, the same algorithm has been successfully used to detect shocks from velocity measurements. Detection speed was important here for safety reasons.