Generalized Scattering Transforms: Stationarity, Inversion, and Applications

Abstract

This dissertation investigates theoretical and computational aspects of scattering transforms, as well as applications to the problem of hyperspectral reconstruction. Scattering transforms are operators from harmonic analysis with a similar structure to convolutional neural networks but with predefined convolutional filters with known mathematical structure.

The first focus of this dissertation is on generalizing theoretical results, which have previously been proved for scattering transforms with directional wavelet-based convolutional filters, to scattering transforms with more general kinds of filters. First, we prove results on the limiting translation invariance of sequences of scattering transforms with general families of filters. Then, we extend the theory of scattering transforms of strictly stationary random processes from wavelet-based filters to more general filters. In particular, we study stationary scattering transforms whose filters extract time-frequency features of strictly stationary processes, called stationary Fourier scattering transforms.

The second focus of this dissertation is to applications of scattering transforms to the hyperspectral reconstruction problem, in which hyperspectral images are generated from smaller and easier-to-obtain images. We develop a pipeline in which wavelet scattering transform features are mapped between these two modalities by a shallow convolutional neural network, followed by a different network which inverts the scattering embedding. We optimize our pipeline hyperparameters over many experiments, and we report our results on the Hyper-Skin and ARAD datasets.

The final focus of this dissertation is dedicated to theoretical and computational aspects of inverting Fourier scattering transforms. For the former, we construct generalized Fourier scattering transforms which are invertible in some cases, and we prove energy decay and bi-Lipschitz results for these transforms. For the latter, we study two different attempts at numerically inverting such transforms.