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.

通过使用多层卷积框架来实现完美重建（PR）并符合矩形线性单元非线性（ReLU），本文提出了一种新的深度卷积框架神经网络的理论，并证明了这一发现表明深度学习的成功并不是来自于黑箱的魔力，而是来自于非局部基底的新型信号表示方法的力量，这种方法与数据驱动的局部基底的结合，实际上是传统信号处理理论的自然扩展。

深度卷积框架：逆问题的通用深度学习框架

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

We propose an image representation scheme combining the local and nonlocal
characterization of patches in an image. Our representation scheme can be shown
to be equivalent to a tight frame constructed from convolving local bases (e.g.
wavelet frames, discrete cosine transforms, etc.) with nonlocal bases (e.g.
spectral basis induced by nonlinear dimension reduction on patches), and we
call the resulting frame elements {\it convolution framelets}. Insight gained
from analyzing the proposed representation leads to a novel interpretation of a
recent high-performance patch-based image inpainting algorithm using Point
Integral Method (PIM) and Low Dimension Manifold Model (LDMM) [Osher, Shi and
Zhu, 2016]. In particular, we show that LDMM is a weighted
$\ell_2$-regularization on the coefficients obtained by decomposing images into
linear combinations of convolution framelets; based on this understanding, we
extend the original LDMM to a reweighted version that yields further improved
inpainting results. In addition, we establish the energy concentration property
of convolution framelet coefficients for the setting where the local basis is
constructed from a given nonlocal basis via a linear reconstruction framework;
a generalization of this framework to unions of local embeddings can provide a
natural setting for interpreting BM3D, one of the state-of-the-art image
denoising algorithms.

该研究论文提出了一种图像表示方案，其中等价于使用局部和非局部 patches 刻画构建紧密的帧图像，并称其为卷积 framelets。分析该表示方案得出了高性能的 patch-based image inpainting 算法的新解释，并提出了对其的改进。此外，还建立了卷积 framelet 系数的能量集中性质，为一些成像处理算法提供了自然的框架。