BACKGROUND:The limited-angle reconstruction problem is of both theoretical and practical importance. Due to the severe ill-posedness of the problem, it is very challenging to get a valid reconstructed result from the known small limited-angle projection data. The theoretical ill-posedness leads thenormal equation A T Ax = A T b of the linear system derived by discretizing the Radon transform to be severely ill-posed, which is quantified as the large condition number of A T A. OBJECTIVE:To develop and test a new valid algorithm for improving the limited-angle image reconstruction with the known appropriately small angle range from [0,π3]∼[0,π2] . METHODS:We propose a reweighted method of improving the condition number of A T Ax = A T b and the corresponding preconditioned Landweber iteration scheme. The weight means multiplying A T Ax = A T b by a matrix related to A T A, and the weighting process is repeated multiple times. In the experiment, the condition number of the coefficient matrix in the reweighted linear system decreases monotonically to 1 as the weighting times approaches infinity. RESULTS:The numerical experiments showed that the proposed algorithm is significantly superior to other iterative algorithms (Landweber, Cimmino, NWL-a and AEDS) and can reconstruct a valid image from the known appropriately small angle range. CONCLUSIONS:The proposed algorithm is effective for the limited-angle reconstruction problem with the known appropriately small angle range.

中文翻译：

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有限角度图像重建的预处理landweber迭代方案

背景：有限角度重建问题具有重要的理论意义和实际意义。由于该问题具有严重的不适定性，因此从已知的小角度有限角度投影数据中获得有效的重建结果非常具有挑战性。理论上的不适定性导致将Radon变换离散化得到的线性系统的正规方程AT Ax = AT b严重不适定，量化为AT A的大条件数。改进有限角度图像重建的有效算法，已知适当的小角度范围为 [0,π3]∼[0,π2] 。方法：我们提出了一种改进A T Ax = A T b 条件数的重新加权方法和相应的预条件Landweber迭代方案。权重是指将 A T Ax = A T b 乘以与 A TA 相关的矩阵，加权过程重复多次。实验中，重加权线性系统中系数矩阵的条件数随着加权次数接近无穷大而单调递减至1。结果：数值实验表明，该算法明显优于其他迭代算法（Landweber、Cimmino、NWL-a和AEDS），可以从已知的适当小角度范围内重构出有效图像。结论：该算法对于已知适当小角度范围的有限角度重建问题是有效的。随着加权时间接近无穷大，重新加权线性系统中系数矩阵的条件数单调减少到 1。结果：数值实验表明，该算法明显优于其他迭代算法（Landweber、Cimmino、NWL-a和AEDS），可以从已知的适当小角度范围内重构出有效图像。结论：该算法对于已知适当小角度范围的有限角度重建问题是有效的。随着加权时间接近无穷大，重新加权线性系统中系数矩阵的条件数单调减少到 1。结果：数值实验表明，该算法明显优于其他迭代算法（Landweber、Cimmino、NWL-a和AEDS），可以从已知的适当小角度范围内重构出有效图像。结论：该算法对于已知适当小角度范围的有限角度重建问题是有效的。