Deep Convolutional Framelets: A General Deep Learning Framework for Inverse Problems
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Recently, deep learning approaches have achieved significant performance improvement in various imaging problems. However, it is still unclear why these deep learning architectures work. Moreover, the link between the deep learning and the classical signal processing approaches such as wavelet, non-local processing, compressed sensing, etc, is still not well understood. To address these issues, here we show that the long-searched-for missing link is the convolutional framelets for representing a signal by convolving local and non-local bases. The convolutional framelets was originally developed to generalize the recent theory of low-rank Hankel matrix approaches, and this paper significantly extends the idea to derive a deep neural network using multi-layer convolutional framelets with perfect reconstruction (PR) under rectified linear unit (ReLU). Our analysis also shows that the popular deep network components such as residual block, redundant filter channels, and concatenated ReLU (CReLU) indeed help to achieve the PR, while the pooling and unpooling layers should be augmented with multi-resolution convolutional framelets to achieve PR condition. This discovery reveals the limitations of many existing deep learning architectures for inverse problems, and leads us to propose a novel deep convolutional framelets neural network. Using numerical experiments with sparse view x-ray computed tomography (CT), we demonstrated that our deep convolution framelets network shows consistent improvement. 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.
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