SEMINAIRE DU 4 DÉCEMBRE 2014 – 16H @ MAP5 – SALLE DU CONSEIL
Phase retrieval for the wavelet transform
The wavelet transform of a signal f is obtained by convolving f with a set of complex-valued functions, called wavelets. In many cases, the wavelet transform provides a representation of signals which is better suited to their automated processing than their raw « natural » representation. In audio processing, one generally uses only the modulus of the wavelet transform of the signals; the phase is discarded. Indeed, in practice, it seems that the modulus alone contains all the information we may need about the considered audio signals. To understand and theoretically justify this property, we study the corresponding inverse problem: to what extent is it possible to reconstruct a signal from the modulus of its wavelet transform?
This problem is a particular case of a wider class of inverse problems, the so-called « phase retrieval problems ». As we will see, it is in general difficult, in this kind of problems, to prove the uniqueness of the reconstruction and, even more, its stability to noise. However, in the wavelet transform case, for a specific choice of the wavelet family, it is possible to prove a unicity result and a partial stability result. After describing these results, we will present the algorithms which, in practice, can be used for reconstruction.
Irène Waldspurger est doctorante dans l’équipe DATA à l’Ecole Normale Supérieure.