A unified treatment of some iterative algorithms in signal processing and image reconstruction
A unified treatment of some iterative algorithms in signal processing and image reconstruction
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DOI：
10.1088/0266-5611/20/1/006
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发表时间：
2004-02-01
期刊：
影响因子：
2.1
通讯作者：
Byrne, C
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文献类型：
--
作者：
Byrne, C
设 T 为希尔伯特空间 R 上的（可能是非线性的）连续算子。如果对于某个起始向量 x，轨道序列 {T(k)x, k = 0, 1....} 收敛，则极限 z 是 T 的不动点；即，Tz = z。希尔伯特空间 H 上的算子 N 是非扩展的 (ne)，如果对于 H 中的每个 x 和 y，\\Nx-Ny\\ 小于或等于 \\x-y\\。即使当 N 具有不动点时，轨道序列 {N(k)x} 也不需要收敛；考虑示例 N = -I，其中 I 表示恒等运算符。然而，对于任何 E (0, 1)，只要存在这样的点，由 x(k+l) = (1-alpha)x(k) + alphaNx(k) 定义的迭代过程就会（弱）收敛到 N 的固定点。这是寻找 ne 算子不动点的 Krasnoselskii-Mann (KM) 方法。信号处理和图像重建以及其他地方使用的各种迭代过程都是 KM 迭代过程的特例，针对 ne 算子 N 的特定选择。其中包括用于带限外推的 Gerchberg-Papoulis 方法、Anderson 和 Kak 的 SART 算法、Landweber 和投影算法 Landweber 算法、用于求解凸可行性问题的联立和顺序方法、用于求解线性方程组的 ART 和 Cimmino 方法、用于求解分裂可行性问题的 CQ 算法以及用于单调算子变分不等式问题的 Dolidze 过程。
Let T be a (possibly nonlinear) continuous operator on Hilbert space R. If, for some starting vector x, the orbit sequence {T(k)x, k = 0, 1....} converges, then the limit z is a fixed point of T; that is, Tz = z. An operator N on a Hilbert space H is nonexpansive (ne) if, for each x and y in H,\\Nx-Ny\\ less than or equal to \\x-y\\.Even when N has fixed points the orbit sequence {N(k)x} need not converge; consider the example N = -I, where I denotes the identity operator. However, for any a E (0, 1) the iterative procedure defined byx(k+l) = (1-alpha)x(k) + alphaNx(k)converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii-Mann (KM) approach to finding fixed points of ne operators.A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator N. These include the Gerchberg-Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.