提出可微分SVD解决图像重建中的数值不稳问题。
Differentiable SVD based on Moore-Penrose Pseudoinverse for Inverse Imaging Problems
- 用Moore-Penrose伪逆替代传统SVD，实现可微分计算。
- 在彩色图像压缩感知与动态MRI中验证了稳定性与精度。
- 首次系统分析传统SVD不可微根源，适合图像重建研究者。
基于低秩正则化的深度展开网络在各类逆成像问题（IIPs）中表现优异。然而，当存在重复奇异值时，奇异值分解（SVD）不可微，导致训练过程严重数值不稳定。本文提出一种基于Moore-Penrose伪逆的可微分SVD方法以解决此问题。据我们所知，这是首个对平凡SVD可微性进行全面分析的工作。具体而言，我们揭示了SVD不可微的本质原因在于推导过程中出现的欠定线性方程组。通过使用Moore-Penrose伪逆求解该方程组，我们提出了可微分SVD。同时提供了在逆成像问题背景下的数值稳定性分析。在彩色图像压缩感知和动态磁共振成像（dynamic MRI）重建任务上的实验结果表明，所提方法能有效解决数值不稳定性问题，并保证计算精度。代码已公开于https://github.com/yhao-z/SVD-inv。
原文摘要 · Abstract (English)
Low-rank regularization-based deep unrolling networks have achieved remarkable success in various inverse imaging problems (IIPs). However, the singular value decomposition (SVD) is non-differentiable when duplicated singular values occur, leading to severe numerical instability during training. In this paper, we propose a differentiable SVD based on the Moore-Penrose pseudoinverse to address this issue. To the best of our knowledge, this is the first work to provide a comprehensive analysis of the differentiability of the trivial SVD. Specifically, we show that the non-differentiability of SVD is essentially due to an underdetermined system of linear equations arising in the derivation process. We utilize the Moore-Penrose pseudoinverse to solve the system, thereby proposing a differentiable SVD. A numerical stability analysis in the context of IIPs is provided. Experimental results in color image compressed sensing and dynamic MRI reconstruction show that our proposed differentiable SVD can effectively address the numerical instability issue while ensuring computational precision. Code is available at https://github.com/yhao-z/SVD-inv.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。