  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. To address these issues, here we show that the long-searched-for
missing link is the convolution framelets for representing a signal by
convolving local and non-local bases. The convolution framelets was originally
developed to generalize the theory of low-rank Hankel matrix approaches for
inverse problems, and this paper further extends the idea so that we can obtain
a deep neural network using multilayer convolution framelets with perfect
reconstruction (PR) under rectilinear linear unit nonlinearity (ReLU). Our
analysis also shows that the popular deep network components such as residual
block, redundant filter channels, and concatenated ReLU (CReLU) do indeed help
to achieve the 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 leads us to propose a novel theory for deep
convolutional framelets neural network. Using numerical experiments with
various inverse problems, we demonstrated that our deep convolution framelets
network shows consistent improvement over existing deep architectures.This
discovery suggests that the success of deep learning is not from a magical
power of a black-box, but rather comes from the power of a novel signal
representation using non-local basis combined with data-driven local basis,
which is indeed a natural extension of classical signal processing theory.
