Rotation Equivariant Proximal Operator for Deep Unfolding Methods in
  Image Restoration
Jiahong Fu Qi Xie Deyu Meng Zongben Xu
The deep unfolding approach has attracted significant attention in computer
vision tasks, which well connects conventional image processing modeling
manners with more recent deep learning techniques. Specifically, by
establishing a direct correspondence between algorithm operators at each
implementation step and network modules within each layer, one can rationally
construct an almost ``white box'' network architecture with high
interpretability. In this architecture, only the predefined component of the
proximal operator, known as a proximal network, needs manual configuration,
enabling the network to automatically extract intrinsic image priors in a
data-driven manner. In current deep unfolding methods, such a proximal network
is generally designed as a CNN architecture, whose necessity has been proven by
a recent theory. That is, CNN structure substantially delivers the
translational invariant image prior, which is the most universally possessed
structural prior across various types of images. However, standard CNN-based
proximal networks have essential limitations in capturing the rotation symmetry
prior, another universal structural prior underlying general images. This
leaves a large room for further performance improvement in deep unfolding
approaches. To address this issue, this study makes efforts to suggest a
high-accuracy rotation equivariant proximal network that effectively embeds
rotation symmetry priors into the deep unfolding framework. Especially, we
deduce, for the first time, the theoretical equivariant error for such a
designed proximal network with arbitrary layers under arbitrary rotation
degrees. This analysis should be the most refined theoretical conclusion for
such error evaluation to date and is also indispensable for supporting the
rationale behind such networks with intrinsic interpretability requirements.