Inversion of the scalogram

The wavelet transform modulus, also called scalogram, is a widely-used representation of audio signals, similar to the spectrogram. To better understand its theoretical properties, we study the associated inverse problem: to what extent is it possible to reconstruct a signal from its scalogram? In a first part, we describe this problem and show that, at least for a specific choice of wavelets, it is possible to almost exactly describe which signals have almost identical scalograms. In a second part, we describe the implications of this result for the understanding of a deep representation: the scattering transform, introduced by Mallat.