Wavelets

The wavelet transform is a convenient way to represent and manipulate signals featuring sharp transients. It splits the signal between its "low resolution" part and a series of details at different resolutions. The details, which are described by the wavelet coefficients, represent the difference between the representation of the signal at a given resolution and its representation at a coarser resolution. You can find on presentation of wavelets here.

Applications range from denoising to compression.

In control, two kinds of processing can be considered: online and offline. I have mainly studied online applications, i.e. applications where processing delays should be avoided.
Mainstream wavelet processing induces large delays because it uses mirror filters: if a filter is causal, its mirror is anti-causal. To avoid this, I have studied causal wavelet processing, notably for denoising applications. The corresponding applications have been successfully implemented in several real world systems, providing notably denoised derivatives of signals with sharp transients.

On the more theoretical side, I have studied wavelet like decompositions which provide a simple formula that relates the decomposition of a signal and its decomposition after it has been transformed by a polynomial operator.