[论文解读] Deep Convolutional Framelets: A General Deep Learning Framework for Inverse Problems

Jong Chul Ye, Yoseob Han+1 more|arXiv (Cornell University)|Jul 3, 2017

Sparse and Compressive Sensing Techniques被引用 4

一句话总结

本文提出深度卷积框架（Deep Convolutional Framelets），一种新颖的深度学习框架，用于反问题求解。该框架通过使用带有ReLU激活的多层卷积框架，将经典信号处理理论与深度学习相统一，实现了完美重建。该框架揭示了残差块、冗余滤波通道和CReLU层可实现完美重建，而池化层则需配备高通分支以保持此特性，为反问题网络提供了理论基础明确的设计方法。

ABSTRACT

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.

研究动机与目标

• 识别深度学习与经典信号处理在反问题中理论联系的机制。
• 解释为何特定深度学习组件（如残差块、CReLU）可在反问题中实现完美重建。
• 开发一种通用的深度学习框架，将框架理论与深度网络统一，以提升反问题求解性能。
• 揭示现有架构的局限性，并基于信号表示理论提出设计原则。

提出的方法

• 提出一种使用ReLU激活的多层卷积框架网络，在特定条件下实现完美重建（PR）。
• 通过ReLU和CReLU操作将特征图分解为正负部分，推导出完美重建的条件。
• 引入对称的编码-解码结构，其中滤波器组和偏置被设计为能精确重建原始信号。
• 以Hankel矩阵为基础的框架，将低秩矩阵方法推广至深度学习框架。
• 证明残差块和冗余滤波通道对实现完美重建至关重要。
• 揭示池化层必须附加高通分支才能保持完美重建特性。

实验结果

研究问题

• RQ1 为何某些深度学习架构在缺乏理论基础的情况下，仍能在反问题中表现优越？
• RQ2 是什么数学机制使得深度卷积网络能够实现完美重建？
• RQ3 标准组件如残差块和CReLU如何在反问题中促进完美重建？
• RQ4 为何池化和反池化层通常无法在未经修改的情况下实现完美重建？
• RQ5 如何将经典信号处理理论（如框架和压缩感知）与深度学习统一于反问题求解中？

主要发现

• 所提出的深度卷积框架网络在滤波器组和偏置满足特定对称条件时，可利用ReLU激活实现完美重建。
• 残差块、冗余滤波通道和CReLU层在数学上是实现完美重建的必要条件，解释了其在实践中取得成功的原因。
• 池化层若不附加高通分支，则无法保证完美重建，这是关键的设计洞见。
• 该框架揭示，深度学习在反问题中成功的原因在于一种新型信号表示，结合了非局部与数据驱动的局部基。
• 数值实验表明，该方法在低剂量CT和压缩感知MRI等多样化反问题中，持续优于现有深度架构。
• 该理论提供了原则性的设计框架，表明深度学习的威力并非源于黑箱学习，而是经典信号处理的自然延伸。