Deep Convolutional Framelets: A General Deep Learning Framework for Inverse Problems

Recently, deep learning approaches with various network architectures have achieved significant performance improvement over existing iterative reconstruction methods in various imaging problems. However, it is still unclear why these deep learning architectures work for specific inverse problems. Moreover, in contrast to the usual evolution of signal processing theory around the classical theories, the link between deep learning and the classical signal processing approaches, such as wavelets, nonlocal processing, and compressed sensing, are not yet well understood. To address these issues, here we show that the long-sought missing link is the convolution framelets for representing a signal by convolving local and nonlocal bases. The convolution framelets were originally developed to generalize the theory of low-rank Hankel matrix approaches for inverse problems, and this paper further extends this idea so as to obtain a deep neural network using multilayer convolution framelets with perfect reconstruction (PR) under rectified linear unit (ReLU) nonlinearity. Our analysis also shows that the popular deep network components such as residual blocks, redundant filter channels, and concatenated ReLU (CReLU) do indeed help to achieve PR, while the pooling and unpooling layers should be augmented with high-pass branches to meet the PR condition. Moreover, by changing the number of filter channels and bias, we can control the shrinkage behaviors of the neural network. This discovery reveals the limitations of many existing deep learning architectures for inverse problems, and leads us to propose a novel theory for a deep convolutional framelet neural network. Using numerical experiments with various inverse problems, we demonstrate that our deep convolutional framelets network shows consistent improvement over existing deep architectures. This discovery suggests that the success of deep learning stems not from a magical black box, but rather from the power of a novel signal representation using a nonlocal basis combined with a data-driven local basis, which is indeed a natural extension of classical signal processing theory.