比例投影方向法及其在透射层析成像中的应用
Scaled Projected-Directions Methods with Application to Transmission Tomography
摘要：
X射线计算机断层扫描中的统计图像重建产生具有非负边界的大规模正则化线性最小二乘问题，其中操作员的内存占用是一个问题。在柱面坐标中离散化图像导致显著的存储器节省，并且允许并行算子-向量乘积而无需算子的即时计算，而不必降低图像质量。然而，它恶化了操作员的条件。我们通过块循环尺度算子改进了Hessian条件，并提出了一种在有界约束问题的投影方向方法中处理非对角尺度的策略。我们描述了我们的实施的缩放策略，使用两种算法：TRON，一个信赖域方法与精确的二阶导数，和L-BFGS-B，一个有限的内存拟牛顿海森近似的线性插值方法。我们比较我们的方法与一个变量变换的问题。在两个重建问题上，我们的方法比变量变换方法收敛得更快，并且在最优性残差方面比一阶方法实现了更严格的精度。
英文摘要：
Statistical image reconstruction in X-Ray computed tomography yields large-scale regularized linear least-squares problems with nonnegativity bounds, where the memory footprint of the operator is a concern. Discretizing images in cylindrical coordinates results in significant memory savings, and allows parallel operator-vector products without on-the-fly computation of the operator, without necessarily decreasing image quality. However, it deteriorates the conditioning of the operator. We improve the Hessian conditioning by way of a block-circulant scaling operator and we propose a strategy to handle nondiagonal scaling in the context of projected-directions methods for bound-constrained problems. We describe our implementation of the scaling strategy using two algorithms: TRON, a trust-region method with exact second derivatives, and L-BFGS-B, a linesearch method with a limited-memory quasi-Newton Hessian approximation. We compare our approach with one where a change of variable is made in the problem. On two reconstruction problems, our approach converges faster than the change of variable approach, and achieves much tighter accuracy in terms of optimality residual than a first-order method.
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