  Both classical Fourier transform-based methods and neural network methods are
widely used in image processing tasks. The former has better interpretability,
whereas the latter often achieves better performance in practice. This paper
introduces ButterflyNet2D, a regular CNN with sparse cross-channel connections.
A Fourier initialization strategy for ButterflyNet2D is proposed to approximate
Fourier transforms. Numerical experiments validate the accuracy of
ButterflyNet2D approximating both the Fourier and the inverse Fourier
transforms. Moreover, through four image processing tasks and image datasets,
we show that training ButterflyNet2D from Fourier initialization does achieve
better performance than random initialized neural networks.
