Titolo della tesi: Wavelet Scattering Transform for Signal Approximation and Classification
The Wavelet Scattering Transform (WST) is a recently introduced time-frequency transform that generates a representation invariant to data translation and stable to small deformations, which allows to overcome some of the limits of classical time-frequency transforms – like wavelets of Fourier. WST can be seen as a scattering convolution network with fixed wavelet filters which involves no learning and in which robust feature representations can be extracted across various scales without the need of training data, leading to compact representation of the significant structures of the signal.
However, scattering coefficients are typically highly redundant, especially when the chosen wavelet filters have a significant frequency overlap. It is in consideration of this problem that we developed an innovative and automatic procedure for optimizing the selection of the scattering features in the frequency domain. The procedure is based on the Minimum Description Length (MDL) principle, a well-known and powerful tool to estimate the best data model among a class of candidates; in this case, the considered class is the one of uniform sampling models. The proposed MDL-based selection strategy has been applied to different datasets of sound signals, and the results have been evaluated in terms of classification accuracy and compared with other methods, giving promising results.
Another problem that we tried to address is the approximation of a signal starting from its WST. Indeed, due to the energy dispersion along the layers it is not possible to analytically define an exact inverse transform, while the high complexity and the multi-layered structure of the transform make the numerical inversion challenging. Nonetheless, obtaining an approximate inverse for the WST opens up a range of possibilities for using the scattering transform in real-world applications and various processing tasks.
The strategy developed in this work relies on atomic approximation, a model for compacting and approximating wavelet coefficients information, and works as follows: through a deconvolution step, a coarse estimate of the signal is obtained; then the wavelet transform (WT) of this deconvolved signal is computed, and approximated with atoms; it follows an iterative procedure whose output is a reproduction of the WT of the original signal, obtained by trying to minimize the error on the basis of the comparison with the scattering; finally, the WT is inverted to find the desired approximation.
A first implementation of the model and its application to some study cases are presented.