National Digital Switching System Engineering and Technological Research Centre, Zhengzhou, Henan, China.
J Xray Sci Technol. 2019;27(2):371-388. doi: 10.3233/XST-180455.
Total variation (TV) regularization-based iterative reconstruction algorithms have an impressive potential to solve limited-angle computed tomography with insufficient sampling projections. The analysis of exact reconstruction sampling conditions for a TV-minimization reconstruction model can determine the minimum number of scanning angle and minimize the scanning range. However, the large-scale matrix operations caused by increased testing phantom size are the computation bottleneck in determining the exact reconstruction sampling conditions in practice. When the size of the testing phantom increases to a certain scale, it is very difficult to analyze quantitatively the exact reconstruction sampling condition using existing methods. In this paper, we propose a fast and efficient algorithm to determine the exact reconstruction sampling condition for large phantoms. Specifically, the sampling condition of a TV minimization model is modeled as a convex optimization problem, which is derived from the sufficient and necessary condition of solution uniqueness for the L1 minimization model. An effective alternating direction minimization algorithm is developed to optimize the objective function by alternatively solving two sub-problems split from the convex problem. The Cholesky decomposition method is used in solving the first sub-problem to reduce computational complexity. Experimental results show that the proposed method can efficiently solve the verification problem of the accurate reconstruction sampling condition. Furthermore, we obtain the lower bounds of scanning angle range for the exact reconstruction of a specific phantom with the larger size.
基于全变差（Total Variation，TV）正则化的迭代重建算法在解决有限角度计算机断层扫描（Computed Tomography，CT）中采样投影不足的问题方面具有很大的潜力。分析 TV 最小化重建模型的精确重建采样条件可以确定最小扫描角度数量，并最小化扫描范围。然而，随着测试体模尺寸的增加，大规模矩阵运算成为确定精确重建采样条件的计算瓶颈。当测试体模的尺寸增加到一定规模时，使用现有的方法很难对精确重建采样条件进行定量分析。在本文中，我们提出了一种用于确定大尺寸体模精确重建采样条件的快速有效的算法。具体来说，将 TV 最小化模型的采样条件建模为凸优化问题，该问题是从 L1 最小化模型的解唯一性的充分必要条件推导出来的。我们开发了一种有效的交替方向最小化算法，通过交替求解两个从凸问题中分离出来的子问题来优化目标函数。使用 Cholesky 分解方法求解第一个子问题，以降低计算复杂度。实验结果表明，该方法可以有效地解决精确重建采样条件的验证问题。此外，我们还得到了具有更大尺寸的特定体模精确重建的扫描角度范围的下限。
