Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction
Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction
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DOI：
10.1109/83.760336
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发表时间：
1999-05-01
影响因子：
10.6
通讯作者：
Booth, SD
中科院分区：
文献类型：
--
作者：
Fessler, JA;Booth, SD
基于对角算子的迭代算法在层析图像重建和图像恢复问题中收敛速度较慢，但可以通过适当的预处理来加快收敛速度。对角预处理算法在收敛速度上有一定的提高，但在成像问题中没有考虑Hessian矩阵的结构。循环预处理器可以为近似平移不变的逆问题提供显著的加速，即，然而，在具有非均匀噪声方差的应用中，例如由发射断层摄影和量子限制光学成像中的泊松统计引起的应用中，加权最小二乘目标函数的Hessian是相当移位变化的，并且循环预处理器表现不佳，额外的移位方差是由基于非二次罚函数的边缘保持正则化方法引起的。本文介绍了新的预处理器，更准确地近似移变成像问题的海森矩阵。与对角预处理或循环预处理相比，新的预处理导致无约束共轭梯度（CG）迭代的收敛速度显着加快。我们还提出了一个新的有效的方法，CG方法所需的线搜索步骤。正电子发射断层扫描（PET）的应用程序说明了该方法。
Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners, Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians, However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly, Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.