Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA.
IEEE Trans Image Process. 1999;8(5):688-99. doi: 10.1109/83.760336.
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.
基于梯度的迭代方法通常在层析图像重建和图像恢复问题上收敛缓慢，但可以通过合适的预条件器加速。对角预条件器可以提高收敛速度，但不能将成像问题中的 Hessian 矩阵的结构纳入其中。循环预条件器可以为近似移位不变的逆问题（即具有近似块-Toeplitz 或块循环 Hessian 的问题）提供显著的加速。然而，在具有非均匀噪声方差的应用中，例如发射断层成像和量子限制光学成像中的泊松统计，加权最小二乘目标函数的 Hessian 非常移位变化，循环预条件器的性能很差。基于非二次惩罚函数的保持边缘的正则化方法会导致额外的移位变化。本文描述了新的预条件器，这些预条件器可以更准确地逼近移位变化成像问题的 Hessian 矩阵。与对角或循环预处理相比，新的预条件器可以显著提高无约束共轭梯度 (CG) 迭代的收敛速度。我们还提出了 CG 方法所需的线搜索步骤的一种新的有效方法。正电子发射断层成像 (PET) 的应用说明了这种方法。
