Decay Properties of Riesz Transforms and Steerable Wavelets

On Generalizations of the Nonwindowed Scattering Transform

In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as L q ( ℝ n ) norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of C 2 diffeomorphisms in specific cases. Lastly, we extend our results to formulate an operator invariant to the action of rotations R ∈ SO ( n ) and an operator that is equivariant to the action of rotations of R ∈ SO ( n ) .