Block-iterative interior point optimization methods for image reconstruction from limited data
Block-iterative interior point optimization methods for image reconstruction from limited data
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DOI：
10.1088/0266-5611/16/5/316
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发表时间：
2000-10-01
期刊：
影响因子：
2.1
通讯作者：
Byrne, C
中科院分区：
文献类型：
--
作者：
Byrne, C
图像重建的迭代算法通常涉及最小化一些成本函数h(x)，该函数测量测量数据与理论参数化模型之间的一致程度。此外，人们可能希望x满足某些约束。通常情况下，代价函数是简单函数的和：[GRAPHICS]划分集合（i = 1，…）， 1)为不相交集合B-n， n = 1，…的并集， N，我们让[图形]这里提出的方法是块迭代，在每一步只使用单个h(N)(x)的梯度。与单块（N = 1）方法相比，通过使用适当选择的缩放因子，可以显著加快收敛速度。该算法是一种内点法，即迭代每一步得到的图像x（k+1）满足期望的约束条件。这里的约束是通过使下一个迭代x（k+1）满足梯度方程delF(x(k+1)) = delF(x(k)) - t(n) delh(n) （x(k)）来施加的，对于适当的标量t(n)，其中凸函数F是定义的，并且只能在满足约束的向量上可微。提出了适用于层析图像重建的算法的特殊情况，并允许在单个像素上包含上界和下界。这里的重点是该算法的基本收敛理论的发展。特殊情况的行为已在其他地方得到考虑。
Iterative algorithms for image reconstruction often involve minimizing some cost function h(x) that measures the degree of agreement between the measured data and a theoretical parametrized model. In addition, one may wish to have x satisfy certain constraints. It is usually the case that the cost function is the sum of simpler functions:[GRAPHICS]Partitioning the set (i = 1,..., 1) as the union of the disjoint sets B-n, n = 1,..., N, we let[GRAPHICS]The method presented here is block iterative, in the sense that at each step only the gradient of a single h(n)(x) is employed. Convergence can be significantly accelerated, compared to that of the single-block (N = 1) method, through the use of appropriately chosen scaling factors. The algorithm is an interior point method, in the sense that the images x(k+1) obtained at each step of the iteration satisfy the desired constraints. Here the constraints are imposed by having the next iterate x(k+1) satisfy the gradient equationdelF(x(k+1)) = delF (x(k)) - t(n) delh(n) (x(k)),for appropriate scalars t(n), where the convex function F is defined and differentiable only on vectors satisfying the constraints.Special cases of the algorithm that apply to tomographic image reconstruction, and permit inclusion of upper and lower bounds on individual pixels, are presented. The focus here is on the development of the underlying convergence theory of the algorithm. Behaviour of special cases has been considered elsewhere.