Rotationally invariant time-frequency scattering transforms

In this paper we construct directionally sensitive functions that can be viewed as directional time-frequency representations. We call such a sequence a rotational uniform covering frame and by studying rotations of the frame, we derive the rotational Fourier scattering transform and the truncated rotational Fourier scattering transform. We prove that both operators are rotationally invariant, are bounded above and below, are non-expansive, and contract small translations and additive diffeomorphisms. We also construct finite uniform covering frames.