The multiple-sets split feasibility problem and its applications for inverse problems
The multiple-sets split feasibility problem and its applications for inverse problems
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DOI：
10.1088/0266-5611/21/6/017
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发表时间：
2005-12-01
期刊：
影响因子：
2.1
通讯作者：
Bortfeld, T
中科院分区：
文献类型：
--
作者：
Censor, Y;Elfving, T;Bortfeld, T
多集合分裂可行性问题要求在一个空间中找到一个最接近一族闭凸集的点，使得它在线性变换下的像在像空间中最接近另一族闭凸集。它可以作为许多反问题的模型，其中在线性算子的定义域以及算子的值域中对解都施加了约束。它推广了凸可行性问题以及两集合分裂可行性问题。我们提出一种投影算法，该算法最小化一个衡量一个点到所有集合距离的邻近函数。该公式以及算法推广了早期关于分裂可行性问题的工作。我们还对具有布雷格曼距离的邻近函数进行了推广。该方法在调强放射治疗计划的反问题中的应用在另一篇配套论文中进行了研究，在此仅作简要描述。
The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.