Determination of singular value truncation threshold for regularization in ill-posed problems
Determination of singular value truncation threshold for regularization in ill-posed problems
复制标题
病态问题正则化奇异值截断阈值的确定
DOI：
10.1080/17415977.2020.1832090
复制
发表时间：
2021-08
影响因子：
1.3
通讯作者：
Guirong Liu
中科院分区：
文献类型：
--
作者：
Shuyong Duan;Botao Yang;Fang Wang;Guirong Liu
在应用正则化方法求解不适定反问题时，恰当的正则化参数的选取是解决问题的关键。提出了一种新的截断奇异值分解（TSVD）方法中截断奇异值的确定方法。给出了计算奇异值指数的简单公式，超过该指数，所有较小的奇异值和相应的向量被截断。首先从理论上推导了最佳截断阈值的确定方法。并以二维反问题处理Radon变换为例进行了说明。针对图像分辨率和投影角数不足的问题，推导了相应的计算公式。结果表明，该方法的精度与TSVD方法相当，但效率要高出上级。另一方面，输入数据不足会影响反解的输出精度，可以采用最小二乘法建立截断阈值的计算公式。对于不充足的输入数据集，逆重构信号与TSVD重构信号之间的百分比差约为3%。当TSVD用于求解已知系统特性的反问题时，现有的公式提供了可靠和更有效的方法来计算截断阈值。
Appropriate regularization parameter specification is the linchpin for solving ill-posed inverse problems when regularization method is applied. This paper presents a novel technique to determine cut off singular values in the truncated singular value decomposition (TSVD) methods. Simple formulae are presented to calculate the index number of the singular value, beyond which all the smaller singular values and the corresponding vectors are truncated. The determination method of optimal truncation threshold is firstly theoretically inferred. Two-dimensional inverse problems processing Radon transform are then exemplified. Formulae to solve the problem with insufficient image resolution and projection angle number are derived by the currently proposed method. The results show that accuracy of the current method is similar to that of TSVD but with much superior efficiency. On the other hand, insufficiency in input data affects the output accuracy of the inverse solution, a least square method can be engaged to establish formulae calculating the truncation threshold. For an insufficient set of input data, the percentage difference between inversely reconstructed signal and TSVD reconstructed signal is about 3%. The current formulae offer reliable and more efficient approach to calculate the truncation threshold when TSVD is applied to solve inverse problems with known system characteristics.
登录
查看更多内容
DOI：
10.1007/978-981-10-3090-1
发表时间：
2020-01
期刊：
--
影响因子：
--
作者：
Jie Liu;Xuan Han
通讯作者：
Jie Liu;Xuan Han
影响因子：
2
作者：
A. Lanza;S. Morigi;F. Sgallari
通讯作者：
A. Lanza;S. Morigi;F. Sgallari
影响因子：
10.2
作者：
HANSEN, PC
通讯作者：
HANSEN, PC
DOI：
10.1201/9780203494486
发表时间：
2003-06
期刊：
--
影响因子：
--
作者：
Gui-rong Liu;Xu Han
通讯作者：
Gui-rong Liu;Xu Han
影响因子：
2.8
作者：
Peiliang Xu
通讯作者：
Peiliang Xu