Wavelet analysis and deep learning are two popular fields for signal processing. The scattering transform from wavelet analysis is a recently proposed mathematical model for convolution neural networks. Signals with repeated patterns can be analyzed using the statistics from such models. Specifically, signals from certain classes can be recovered from related statistics. We first focus on recovering 1D deterministic dirac signals from multiscale statistics. We prove a dirac signal can be recovered from multiscale statistics up to a translation and reflection. Then we switch to a stochastic version, modeled using Poisson point processes, and prove wavelet statistics at small scales capture the intensity parameter of Poisson point processes. We also design a scattering generative adversarial network (GAN) to generate new Poisson point samples from statistics of multiple given samples. Next we consider texture images. We successfully synthesize new textures given one sample from the texture class through multiscale, multilayer wavelet models. Finally, we analyze and prove why the multiscale multilayer model is essential for signal recovery, especially natural texture images.
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