Rotationally Invariant Time-Frequency Scattering Transforms

The focus of this paper is on the mathematical construction of a transform that is invariant to a finite rotation group and is stable to small perturbations. A key step in our theory lies in the construction of directionally sensitive functions that are partially generated by rotations. We call such a family a rotational uniform covering frame and by studying rotations of the frame, we derive the desired operator, which we call the rotational Fourier scattering transform. We prove that the transformation is rotationally invariant to a finite rotation group, is bounded above and below, is non-expansive, and contracts small translations and additive diffeomorphisms. To address the numerical aspects of this theory, we also construct digital versions of the frame and show how to faithfully truncate the transform. We also discuss connections between this new family of directional representations with previously constructed ones.