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Recently, deep learning approaches with various network architectures haveachieved significant performance improvement over existing iterativereconstruction methods in various imaging problems. However, it is stillunclear why these deep learning architectures work for specific inverseproblems. To address these issues, here we show that the long-searched-formissing link is the convolution framelets for representing a signal byconvolving local and non-local bases. The convolution framelets was originallydeveloped to generalize the theory of low-rank Hankel matrix approaches forinverse problems, and this paper further extends the idea so that we can obtaina deep neural network using multilayer convolution framelets with perfectreconstruction (PR) under rectilinear linear unit nonlinearity (ReLU). Ouranalysis also shows that the popular deep network components such as residualblock, redundant filter channels, and concatenated ReLU (CReLU) do indeed helpto achieve the PR, while the pooling and unpooling layers should be augmentedwith high-pass branches to meet the PR condition. Moreover, by changing thenumber of filter channels and bias, we can control the shrinkage behaviors ofthe neural network. This discovery leads us to propose a novel theory for deepconvolutional framelets neural network. Using numerical experiments withvarious inverse problems, we demonstrated that our deep convolution frameletsnetwork shows consistent improvement over existing deep architectures.Thisdiscovery suggests that the success of deep learning is not from a magicalpower of a black-box, but rather comes from the power of a novel signalrepresentation using non-local basis combined with data-driven local basis,which is indeed a natural extension of classical signal processing theory.

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