  This paper examines the possibility of, and the possible advantages to
learning the filters of convolutional neural networks (CNNs) for image analysis
in the wavelet domain. We are stimulated by both Mallat's scattering transform
and the idea of filtering in the Fourier domain. It is important to explore new
spaces in which to learn, as these may provide inherent advantages that are not
available in the pixel space. However, the scattering transform is limited by
its inability to learn in between scattering orders, and any Fourier domain
filtering is limited by the large number of filter parameters needed to get
localized filters. Instead we consider filtering in the wavelet domain with
learnable filters. The wavelet space allows us to have local, smooth filters
with far fewer parameters, and learnability can give us flexibility. We present
a novel layer which takes CNN activations into the wavelet space, learns
parameters and returns to the pixel space. This allows it to be easily dropped
in to any neural network without affecting the structure. As part of this work,
we show how to pass gradients through a multirate system and give preliminary
results.
