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 reconstructi...