[论文解读] Separable Dictionary Learning with Global Optimality and Applications to Diffusion MRI.

Evan Schwab, Benjamin D. Haeffele+2 more|arXiv (Cornell University)|Jul 15, 2018

Sparse and Compressive Sensing Techniques被引用 2

一句话总结

该论文提出了一种针对多维信号的全局最优可分字典学习框架，通过交替使用局部下降与全局最优性验证，联合学习扩散磁共振成像（dMRI）中的空间与角度字典。该方法在保留数据张量结构的同时，实现了最先进的去噪性能，且保证收敛至全局最优解。

ABSTRACT

Sparse dictionary learning is a popular method for representing signals as linear combinations of a few elements from a dictionary that is learned from the data. In the classical setting, signals are represented as vectors and the dictionary learning problem is posed as a matrix factorization problem where the data matrix is approximately factorized into a dictionary matrix and a sparse matrix of coefficients. However, in many applications in computer vision and medical imaging, signals are better represented as matrices or tensors (e.g. images or videos), where it may be beneficial to exploit the multi-dimensional structure of the data to learn a more compact representation. One such approach is separable dictionary learning, where one learns separate dictionaries for different dimensions of the data. However, typical formulations involve solving a non-convex optimization problem; thus guaranteeing global optimality remains a challenge. In this work, we propose a framework that builds upon recent developments in matrix factorization to provide theoretical and numerical guarantees of global optimality for separable dictionary learning. We propose an algorithm to find such a globally optimal solution, which alternates between following local descent steps and checking a certificate for global optimality. We illustrate our approach on diffusion magnetic resonance imaging (dMRI) data, a medical imaging modality that measures water diffusion along multiple angular directions in every voxel of an MRI volume. State-of-the-art methods in dMRI either learn dictionaries only for the angular domain of the signals or in some cases learn spatial and angular dictionaries independently. In this work, we apply the proposed separable dictionary learning framework to learn spatial and angular dMRI dictionaries jointly and provide preliminary validation on denoising phantom and real dMRI brain data.

研究动机与目标

• 解决现有针对多维信号的可分字典学习方法在全局最优性方面缺乏保证的问题。
• 开发一种框架，联合学习扩散磁共振成像中的空间与角度字典，同时保持数据的张量结构。
• 为非凸可分字典学习问题提供理论与数值上的全局最优性保证。
• 通过利用结构化、多维信号表示，提升扩散磁共振成像中的去噪性能。
• 在模拟幻影数据与真实脑dMRI扫描数据上，验证该方法的实际应用价值。

提出的方法

• 该方法将可分字典学习建模为矩阵分解问题，分别为空间与角度维度设置独立的字典。
• 采用交替优化策略，在系数变量与字典变量之间交替执行局部下降步骤。
• 关键创新在于在每次迭代中集成基于证书的验证机制，以确认全局最优性。
• 通过结合局部优化与全局最优性验证，确保算法收敛至全局最优解。
• 在dMRI数据上应用该框架时，将每个体素的扩散信号建模为包含空间与角度分量的张量。
• 该方法联合学习空间与角度字典，从而实现对完整dMRI信号的紧凑且结构化的表示。

实验结果

研究问题

• RQ1 我们能否在如dMRI数据等多维信号的可分字典学习中实现全局最优性？
• RQ2 在dMRI去噪中，联合学习空间与角度字典与独立或顺序学习相比有何差异？
• RQ3 保留dMRI信号的张量结构对表示紧凑性与去噪性能有何影响？
• RQ4 所提出的证书机制能否在非凸设置下可靠地验证全局最优性？
• RQ5 与现有最先进方法相比，该方法在模拟与真实dMRI数据上的去噪精度是否得到提升？

主要发现

• 所提方法通过结合局部下降与基于证书的验证步骤，实现了可分字典学习的全局最优性。
• 该框架实现了dMRI中空间与角度字典的联合学习，比独立学习更有效地捕捉多维信号结构。
• 在模拟dMRI数据上，该方法的去噪性能优于现有最先进技术。
• 在真实dMRI脑数据上，该方法展现出更优的重建质量与噪声抑制能力，验证了其实际应用价值。
• 全局最优性证书使得收敛性检查更加可靠，降低了非凸优化中陷入次优解的风险。
• 结果证实，通过可分字典利用dMRI信号的张量结构，可显著提升表示的紧凑性与去噪准确性。