Analysis of time-frequency scattering transforms (Q2424627)

The authors introduce a new class of frames, referred to as uniform covering frames, which are a certain generalization of Gabor frames. These uniform covering frames are then combined with neural networks to define the Fourier scattering transform $\mathcal{S}_\mathcal{F}$ and the truncated Fourier scattering transform $\mathcal{S}_\mathcal{F}[M,K]$. The parameters $K$ and $M$ are used to control the depth and width of the neural network, respectively. Properties of both transforms are investigated. In order to compute $\mathcal{S}_\mathcal{F}[M,K]$ the fast Fourier scattering transform is introduced and the ensuing algorithm is applied to a model image. There it is found that the first-oder coefficients identify the edges in the image, whereas the second-order coefficients extract global and oscillatory features. A comparison of the theoretical results obtained by the authors with existing ones in the literature is also undertaken.