基于图和有限元的漫射光学层析成像反问题的全变分模型
Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography
摘要：
全变分（TV）是一种功能强大的正则化方法，已被广泛应用于不同的成像应用中，但由于复杂和非结构化的几何结构、数据拟合和正则化项的非线性以及正则化项的不可微性，难以应用于漫射光学层析成像（DOT）图像重建（逆问题）。我们开发了几种方法来克服这些困难：i）定义离散微分算子的非结构化几何使用有限元和图形表示; ii）开发一个优化算法的基础上交替方向法的乘数（ADMM）的不可微和非线性最小化问题; iii）研究TV正则化的各向同性和各向异性变体，并比较它们的基于有限元和基于图形的实现。这些方法进行评估的模拟数据和实际数据从组织幻影上的实验。我们的研究结果表明，有限元和基于图形的TV正则化是能够准确地重建稀疏和非稀疏分布没有过度平滑效果的Tikhonov正则化和过度稀疏效果的L $_1$正则化。对于低分辨率网格，图形表示法的性能优于有限元法，而对于高分辨率网格，有限元法的精度更高。
英文摘要：
Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to complex and unstructured geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for unstructured geometries using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L$_1$ regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.
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