Qu Gangrong, Wang Caifang, Jiang Ming
IEEE Trans Image Process. 2009 Feb;18(2):435-40. doi: 10.1109/TIP.2008.2008076. Epub 2008 Dec 12.
The Landweber scheme is an algebraic reconstruction method and includes several important algorithms as its special cases. The convergence of the Landweber scheme is of both theoretical and practical importance. Using the singular value decomposition (SVD), we derive an iterative representation formula for the Landweber scheme and consequently establish the necessary and sufficient conditions for its convergence. In addition to verifying the necessity and sufficiency of known convergent conditions, we find new convergence conditions allowing relaxation coefficients in an interval not covered by known results. Moreover, it is found that the Landweber scheme can converge within finite iterations when the relaxation coefficients are chosen to be the inverses of squares of the nonzero singular values. Furthermore, the limits of the Landweber scheme in all convergence cases are shown to be the sum of the minimum norm solution of a weighted least-squares problem and an oblique projection of the initial image onto the null space of the system matrix.
兰德韦伯算法是一种代数重建方法，其特殊情况包含几种重要算法。兰德韦伯算法的收敛性具有理论和实际重要性。利用奇异值分解（SVD），我们推导了兰德韦伯算法的迭代表示公式，从而建立了其收敛的充要条件。除了验证已知收敛条件的必要性和充分性外，我们还发现了新的收敛条件，允许松弛系数在已知结果未涵盖的区间内。此外，发现当松弛系数选为非零奇异值平方的倒数时，兰德韦伯算法可在有限次迭代内收敛。此外，在所有收敛情况下，兰德韦伯算法的极限被证明是加权最小二乘问题的最小范数解与初始图像在系统矩阵零空间上的斜投影之和。
